Perception of 3-D shape from 2-D image of contours

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Content PER C E PTIO N OF 3-D SHAPE FROM 2-D IM AGE O F CONTOURS
by
Fatih Ulupm ar
A D issertation Presented to the
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In P artial Fulfillment of the
Requirem ents for the Degree
D O C TO R O F PH ILO SO PH Y
(Com puter Engineering)
December 1991
Copyright 1991 Fatih U lupm ar
UMI Number: D P22839
All rights reserved
INFORMATION TO ALL USERS
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a note will indicate the deletion.
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UNIVERSITY OF SOUTHERN CALIFORNIA
THE GRADUATE SCHOOL
UNIVERSITY PARK
LOS ANGELES, CALIFORNIA 90089-4015
This dissertation, written by
Fatih Ulupinar
under the direction of /u s Dissertation
Committee, and approved by all its members,
has been presented to and accepted by The
Graduate School, in partial fulfillment of re
quirements for the degree of
P L D .
CpS
" 9 )
0 * 7
Hs7 B /'^°
DOCTOR OF PHILOSOPHY
Dean of Graduate Studies
Date
A ugust 16* 1991
DISSERTATION COMMITTEE
Chairperson
Dedication
This thesis is dedicated to my wife who supported me throughout my Ph.D . and
to my son Doga Can Ulupinar.
ii
Acknowledgments
I would like to thank Prof. Ram akant Nevatia, my thesis advisor, who was
always there when I needed him and m ade this thesis possible. He formed an
environm ent th a t supported creative thinking and hard but com fortable working
w ithout any pressure. I would like to thank Dr. G erard Medioni who continuously
provided me w ith very valuable feedbacks. I would like to thank Dr. K eith
Price and Andreas H uertas for providing and m aintaining the com putational
environm ent for this thesis. I would like to thank my com m ittee m em bers, Prof.
Ram akant Nevatia, Dr. G erard Medioni and Dr. Francis Bonahon, for their
valuable participation in my thesis. I would like to thank all IRIS faculty, staff
and students for providing a friendly environm ent to work in.
Finally, I would like to thank my wife who supported me, throughout my
Ph.D . studies, emotionally, psychologically, and even financially and w ithout
whom my studies here would not be as meaningful.
iii
Contents
D ed ica tio n ii
A ck n o w led g m en ts iii
L ist O f F igu res v iii
A b stra ct xii
1 In tro d u ctio n 1
1.1 Sum m ary of the Previous W o r k ................................................................ 4
1.2 Sum m ary of the Proposed M ethod ........................................................... 5
1.2.1 Original C o n trib u tio n s .................................................................... 7
1.3 O rganization of the T h e s is .......................................................................... 9
2 P r ev io u s W ork 10
2.1 G radient S p a c e ............................................................................................... 11
2.2 E xtrem al M ethods ........................................................................................ 12
2.2.1 Smoothness M easures...................................................................... 12
2.2.1.1 Barrow and T e n e n b a u m .................... 12
2.2.1.2 Weiss ................................................................................. 13
2.2.2 Com pactness Measures (Brady and Y u ille ) ............................ 13
2.2.3 O ther E xtrem al M e a s u r e s ............................................................ 14
2.3 Constraint A p p ro a c h e s .................. 15
2.3.1 K anade ............................................................................................... 15
2.3.2 Using Lines of C u r v a tu r e .............................................................. 17
2.3.2.1 S t e v e n s .............................................................................. 17
2.3.2.2 Xu and T s u j i ................................................................... 18
2.3.3 Horaud and B r a d y .......................................................................... 19
2.3.4 O t h e r s ................................................................................................... 20
3 S y m m e tr ie s, Q u a lita tiv e In feren ces and C o n stra in ts 21
3.1 S y m m e trie s ...................................................................................................... 21
iv
3.1.1 Parallel S y m m e tr y ........................................................................... 22
3.1.2 Line-Convergent S y m m e tr y ......................................................... 23
3.1.3 Skew S y m m e try .................................................................................. 24
3.2 Q ualitative Shape In feren ces.................................................................. . 25
3.2.1 Case I ................................................................................................... 26
3.2.2 Case I I ................................................................................................... 28
3.2.3 C a s e U I ................................................................................................ 28
3.3 C o n s tr a in ts ................................................................................................. 30
3.3.1 Curved Shared Boundary C onstraint (CSBC) ...................... 31
3.3.2 Inner Surface C onstraint (ISC) .................................................. 32
3.3.3 O rthogonality C onstraint ( O C ) .................................................. 35
4 Z e ro G a u ssia n C u r v a tu r e S u rfa c e s 37
4.1 Symmetries and ZGC S u rfaces............................................................... 38
4.2 Q uantitative Shape Recovery of ZGCs w ith Parallel Sym m etry . . 43
4.2.1 Recovering R u lin g s ........................................................................... 44
4.2.2 Application of CSBC and I S C ...................................................... 44
4.2.3 Combining Three C o n s tr a in ts ......................... 45
4.2.3.1 Cylindrical S u r f a c e s .................. 46
4.2.3.2 Circular C o n e s ................................................................ 48
4.2.3.3 General ZGC S u rfa c e s .................................................. 48
4.2.4 E stim ating (pc, qc) 49
4.2.4.1 Com puting (pc?9c) given a line I : ............................. 50
4.2.5 Com putational R e s u lts .................................................................... 54
4.2.5.1 Synthetic I m a g e s ............................................................. 56
4.2.5.2 A Real Image E x a m p le .................................................. 56
4.3 Q uantitative Shape Recovery of ZGCs Cut by Non Parallel Planes 59
4.3.1 Shape R e c o v e ry .................................................................................. 60
4.3.2 R e s u lts ................................................................................................... 62
5 M u ltip le Z G C S u rfa c e s 65
5.1 Representing S urfaces................................................................................ 65
5.1.1 Planar S u rfa c e s .................................................................................. 66
5.1.2 ZGC S u rfa c e s ..................................................................................... 66
5.2 Combined Shape R e c o v e ry ...................................................................... 69
5.2.1 Internal Constraints ................................................................ 70
5.2.2 External C o n s tra in ts ....................................................................... 72
5.2.3 Solving Constraint E q u a tio n s ...................................................... 74
5.3 R e s u lts ............................................................................................................ 74
6 S tra ig h t H o m o g e n e o u s G e n e ra liz e d C y lin d e rs 78
6.1 Surfaces and Their Limb E d g e s ........................................................... 79
v
6.2 Properties of S H G C s ................................... ? ............................................. 82
6.2.1 Recovering the Cross S e c tio n s .................................................... 84
6.2.2 Observing S H G C s............................................................................. 86
6.3 Q uantitative Shape Recovery of S H G C s.................................................. 86
6.3.1 C S B C ................................................................................................. 87
6.3.2 I S C ........................................................................................................ 87
6.3.3 P lanarity of M e rid ia n s ................................................................... 88
6.3.4 O rth o g o n a lity ................................................................................... 89
6.3.5 R e su lts.............................................................................................. . 90
7 P la n a r R ig h t C on stan t C ross S ectio n G en era lized C y lin d ers 92
7.1 Properties of C G C s ........................................................................................ 93
7.1.1 Observing P R C G C s........................................................................ 96
7.2 Q uantitative Shape Recovery of PRCG Cs ........................................... 96
7.2.1 C S B C ................................................................................................. 96
7.2.2 I S C ........................................................................................................ 97
7.2.3 O rth o g o n a lity ................................................................................... 97
7.3 Recovering Cross Section Curves ............................................................ 98
7.3.1 Com puting (pa>qa) ............................................................................ 101
7.3.2 R e su lts.....................................................................................................103
8 C on clu sion 104
8.1 Future R esea rc h ................................................................................................... 105
R eferen ce L ist 106
A p p en d ix A
P l'O o fs .............................................................................................................................. I l l
A .l Proofs of Theorems for ZGC S u rfa c e s....................................................... I l l
A .1.1 Proof of Theorem 5 112
A .1.2 Curves of M axim um Curvature for ZGC S u r f a c e s .................113
A. 1.3 Proof of the Inner Surface Constraint for Z G C s .................... 114
A p p en d ix B
Experim ents on Perception of qc by H u m a n s...................................................... 115
A p p en d ix C
Prespective P r o je c tio n ...............................................................................................123
C .l C onstraints Under Perspective P ro je c tio n ...................................................125
C.1.1 Choosing a Representation for Surface O rien ta tio n ................. 126
C .l.2 Shared Boundary C o n s tra in t...........................................................126
C .l.3 Parallelity T h e o re m ............................................................................ 127
vi
C .l.4 O rthogonality C o n stra in t.................................................................. 128
C .l.5 Convergent S y m m e tr y ......................................................................130
C .l.5.1 Com puting O rientation Using Convergent Sym
m etry .................................................................................... 132
C.2 Usage of the Constraints for Polyhedral O b j e c t s ..................................136
C.3 Extensions to Curved S u rfaces..................................................................... 138
C.3.1 Parallel S y m m e tr y ............................................................................ 138
C.3.2 Analysis of a Conic Surface ........................................................... 140
C.4 C onclusion.......................... 142
vii
List Of Figures
1.1 Some objects th a t we can readily perceive a 3-D shape from the
contours alone.................................................................................................... 2
1.2 Shape perception; (a) A smooth curve (b) two sym m etric curves
(c) two non sym m etric c u r v e s ................................................................... 5
1.3 The shaded images of the of the com puted 3-D shapes for the
contours given in figure 1 .1 ......................................................................... 8
2.1 Each point on the ellipse on th e image plane may be the projection
of any point on the line of sight.................................................................. 11
2.2 (a) A typical rectangle and its skew symmetry, (b) corresponding
constraint hyperbola in p — q space........................................................... 16
2.3 Two cylinders resting at different slant angles......................................... 18
3.1 Examples to (a) and (c) parallel sym m etry w ith curved contours,
(b) parallel sym m etry w ith straight contours. T he dotted curves
are axes of sym m etry and the dashed lines are lines of symm etry. 22
3.2 Two line-convergent sym m etric curves....................................................... 23
3.3 Examples to (a) skew sym m etry w ith curved contours, (b) and
skew sym m etry w ith straight contours. The dotted curves are axis
of sym m etry and the dashed lines are lines of sym m etry................... 24
3.4 Some exam ples of case I.............................................. 27
3.5 (a) A figure w ith two skew sym m etries, (b) addition of an extra
curve clarifies the perceived s h a p e ............................................................ 29
3.6 (a) Face of a cylinder w ith a clipped corner, (b) the parallel sym
m etry cover only part of the surface, (c) the top curve is extended
for the parallel sym m etry to cover the whole face, (d) the top of
th e surface is also in c lu d e d .......................................................................... 30
3.7 Two curved surfaces meeting along a curve T ............................................ 32
3.8 Two cylinders (a) is cut along the curves of maximal curvature, and
(b) is cut in an arbitrary direction while preserving parallel symmetry,
now we have the perception of an elliptical cylinder................................... 35
3.9 Orthogonality c o n stra in t.................................................................................. 36
viii
4.1 Exam ples to ZGC surfaces.............................................................................. 38
4.2 A ZGC surface cut along th e “ruling”........................................................ 41
4.3 Form ation of the line-convergent sym m etry w ith a ZGC surface an
two non parallel planes................................................................................... 42
4.4 O bjects w ith cross sections having (a) only one skew sym m etry,
(b) two skew s y m m e trie s............................................................................. 43
4.5 The three degrees of freedom present, pc,qc,d, in a ZGC surface after
applying the constraints ISC and CSBC........................................................ 45
4.6 (a) A cylindrical surface w ith axis of sym m etry and th e rulings
m arked, (b) the constraints ISC, CSBC and the orthogonality for
the cylindrical surface ................................................................................. 47
4.7 (a) A cylindrical object and th e ellipse fitted to the cross section,
(b) th e orientation (pe, < 7 e ) th a t would make th e ellipse a circle and
its projection on the q axis gives qe, first approxim ation to qc. . . 51
4.8 O bjects and ellipses fit for their cross sections. The cross sections
of the objects are segmented based on their concavities (or inflec
tion points) and the whole cross section has the same slan t............. 53
4.9 Sample contours, the needle images com puted and their images
after shading the object w ith the com puted orientation at every
point on the surface......................................................................................... 57
4.10 The processing of a real image; the cone image, edges, com puted
surface norm als and the shaded image w ith the com puted surface
norm als................................................................................................................. 58
4.11 A conic surface w ith line-convergent sym m etry....................................... 60
4.12 Three ZGCs cut by non parallel planes...................................................... 61
4.13 Constraints on the orientation of the cutting planes of a ZGC surface. 63
4.14 Recovering the surface shape of a ZGC cut by non parallel planes. 64
5.1 Some objects consisting of m ultiple planar and curved surfaces. . 66
5.2 The param eters of a ZGC surface and the constraints in the gra
dient (p, q) of the surface along the ruling r. . . . . ......................... 68
5.3 The segmented surfaces, and the sym m etries com puted for each
surface. T he skew sym m etry of planar surfaces are shown by
crosses, the long line is the axis of sym m etry and the short one
is th e direction of the lines of sym m etries. Parallel and line-
convergent sym m etries are shown by their axis..................................... 75
5.4 T he needle and the shaded images obtained from the com puted
surface norm als for the objects in 5 . 1 ..................................................... 77
6.1 An SHGC along the z coordinate axis w ith both m eridians and
cross sections m arked...................................................................................... 79
6.2 Sample Straight Homogeneous Generalized Cylinders.......................... 80
IX
6.3 Tangent line, L v, of a surface S at point P in direction V ................ 80
6.4 Tangent plane, Tp, of a surface, S, containing all th e tangent lines
at point P ......................................................................................................... 81
6.5 (a) An SHGC, and its tangent lines, in the direction of the axis
em itting from a single cross section, intersecting at a single point
on the axis, (b) The tangent lines, X }, of limb edges are not the
sam e as the tangents lines, Tm, of th e m eridians in 3-D..................... 83
6.6 Image of an SHGC cut along its cross sections. Image of the top
cross section curve is the bottom one is Cy and the limb bound
aries are on the left Ci and on the right Cr............................................. 85
6.7 Images of th e cross sections and axes, the dashed lines, recovered
for the SHGCs in figure 6.2 ...................................................................... 86
6.8 T he needle images and the shaded images generated w ith the com
puted gradients at each point of the SHGCs in figure 6 . 2 .............. 91
7.1 Sam ple PR C G C s............................................................................................... 93
7.2 A PR C G C w ith both meridians and cross sections m arked.............. 94
7.3 A PRC G C (half of a torus) (a) from a general view and (b) semi
transparent top view with th e limb edges of the previous view and
th e m eridians passing from the points Pi and P 2 m arked along
w ith their tangent lines................................................................................... 95
7.4 A PR C G C w ith a non-rotationaliy sym m etric cross section............. 100
7.5 A PR C G C w ith, (a) none, (b) one, and (c) both end cross sections
available............................................................................................................... 100
7.6 The recovered cross sections for th e PRCG Cs in figure 7.1. . . . 101
7.7 T he recovered orientations shown by b o th needle image and by
shading the objects for th e PRCG Cs in figure 7.1................................ 102
B .l The test figures used in th e experim ent................................................... 116
B.2 The reference cylinders w ith cross section plane slants, qc, ranging
from 20% th e top left one, to 75% the bottom right one, w ith 5°
increm ents........................................................................................................... 117
B.3 Com parison of hum an responses and th e algorithm discussed in
section 4.2.4. In the graph the x axis is th e object num ber and
th e y axis is the slant angle in degrees. For each object the shaded
bars show the interval of uncertainty ( range of the slant angles
containing the 90% of th e hum an responses), w hite circles shows
the m ean of the slant angles given by hum an subjects, and the
crosses show the slant angle com puted by the algorithm ................... 121
X
B.4 The cum ulative distribution of the errors. T he x axis indicates
the error in degrees (the difference between th e response of the
algorithm and the m ean of the hum ans), and th e y axis indicate
the num ber of objects. T he graph shows the num ber of objects
having error which is greater than or equal to a given error, for
each error value..................................................................................................... 122
C .l (a) An arrow like planar object w ith its axis of sym m etry, solid
vertical line, and lines of sym m etry, dashed horizontal lines, (b)
Projection of th e arrow like object and its convergent sym m etry
lines; dashed lines are th e lines of sym m etry m eeting at the point
(wc, vc), Li is one of the lines of sym m etry m eeting the boundary
at points E and F. T he vertical solid line is th e axis of sym m etry,
L s, having N IG P of A = (a, 6, c)..................................................................... 131
C.2 A cube under perspective projection (a), and com puted orienta
tions for the faces shown as points on th e p — q space w ith the
shared boundary constraints overlayed, dashed lines, (b )......................... 137
C.3 Contours of a conic surface under perspective projection, w ith the
point of convergence for the rulings P, and th e line L($) ......................139
xi
Abstract
In to d ay ’s technology, the capabilities of machine vision system s are far be
hind those of the hum an vision system . To be able to design and m anufacture
vision system s th a t rival th e hum ans, we have to study and understand th e prin
ciples and tools th a t the hum an visual system utilizes. One of th e m ost im portant
features of these systems is the variety of sources of inform ation used. Hum ans
can perceive definite shapes for a large class of objects from their sim plest ab
straction, nam ely their contours. The most common exam ple is our ability to
“see” and understand cartoons.
Shape com putation from a single 2-D image of object contours is an in trin
sically am biguous problem , i.e., given a contour in the image plane there are
infinitely m any objects th a t can produce th e given contour. Various preference
criteria, in addition to th e constraints available, m ust be used to resolve the
am biguity in the perception process. Previous work on shape from contour con
centrated on objects w ith planar faces. In this thesis the problem of shape from
contour for curved surfaces is addressed.
By studying the geom etric and differential properties of various types of sur
faces, the constraints im posed on the object surfaces by images of its contours
are form ulated. It is shown th a t three classes of sym m etries, form ed by object
contours, convey im portant shape inform ation. These sym m etries, called par
allel sym m etry line-convergent sym m etry and skew sym m etry, have enabled us
to analyze a wide class of objects, such as objects w ith zero Gaussian curvature
surfaces, classes of generalized cylinders w ith straight axis (SHGC) and with
non-straight axis (CGC).
xii
Image contours containing the above type of sym m etries impose two essential
constraints on th e shape of the surface; th e “Curved Shared Boundary Con
strain t” and th e Inner Surface C onstraint. A pplication of these constraints to
surface shape leaves three or four degrees of freedom depending on th e type of
surface. These degrees of freedom are fixed by using assum ptions of regularity
inspired from hum an visual system , like preference of orthogonality. Various al
gorithm s are devised to reconstruct th e surfaces from their contours for th e object
classes m entioned above.
xiii
Chapter 1
Introduction
One of th e basic goals of com puter vision is to be able to “perceive” the environ
m ent from 2-D pictures of it w ithout having specific models of the com ponents of
th e environm ent. Recovering 3-D shape of the objects constituting the environ
m ent from their 2-D images is an essential com ponent of perception. 3-D shape
is needed to achieve m ost of the higher level goals of com puter vision. For object
recognition, recovering 3-D shape enables us to obtain a richer description of the
scene, thus, enhancing the ability to discrim inate between wider range of model
objects. And for m any other applications th at requires the “understanding” of
th e environm ent, 3-D shape recovery is essential.
Hence, one of the essential goals of the mid-level vision is to recover the local
orientations of the surfaces of the objects in a scene. T he basic difficulty, of
course, is th a t an im age is a 2-D projection of th e 3-D scene, therefore, the 3-D
structure can not be recovered w ithout some assum ptions. In spite of the inherent
am biguities in a single view, hum ans are able to perceive 3-D surfaces in single
images.
Much effort has been devoted in the past few years to understanding ability
to perceive 3-D shape and has led to development of techniques such as shape
from shading, shape from tex tu re and shape from contour (sometimes also known
as shape from shape).
1
Figure 1.1: Some objects th a t we can readily perceive a 3-D shape from the
contours alone.
We believe th a t of all th e m onocular cues, shape of the 2-D contour itself is the
m ost im portant cue for th e shape of the 3-D surfaces. Strictly speaking, not only
is such interpretation infinitely ambiguous but the contours can only give shape
inform ation near the contours; shape of the surface in between can vary sm oothly
w ithout producing other contours. Nonetheless, hum ans when presented with
contours of various, not necessarily fam iliar, objects perceive com plete surfaces
(and even solids). Some exam ples for such contours th a t we readily perceive a
shape are given in figure 1.1.
Barrow and Tenenbaum [BT81] show, by some exam ples, th a t in case of
conflict betw een contour and shading inform ation hum ans use th e contour in
form ation. Biederm an [Bie87] claims th a t in the experim ents w ith hum ans the
2
recognition of a full colored image of an object is not faster th an the recogni
tion of th e line draw ing of the object. He also shows th a t we do not necessarily
need to have any fam iliarity w ith the object in order to perceive a shape from its
boundary. We conjecture th a t th e reason for preferring shape from contour over
other cues, such as shading, m ay be th a t even though shape from contour m eth
ods need to m ake some assum ptions, other m ethods need to m ake even stronger
assum ptions. Shape from shading m ethods, for exam ple, need to assum e th a t the
reflectance properties of the surface are known, th a t th e albedo is constant, and
th a t the light sources are known precisely. On the other hand, contour is m ostly
invariant under large variations of the lighting and surface reflectance.
These observations are not to argue th a t only shape from contour is needed,
b u t th a t it is an essential elem ent in m onocular perception th a t can not be ig
nored. We believe th a t such an ability will also be essential for m achine vision
system s, if they are to work w ith m onocular images in the absence of highly
specific models. Even stereo vision systems ignoring all the m onocular cues and
relying solely on stereo inform ation, as they are done today, handicaps these sys
tem s severely. For such systems to rival or exceed th e hum an visual system , they
should use the m onocular shape cues available in th e images too.
In this thesis, we study the perception of object shape from 2-D images of the
contours of the surfaces surrounding the object. Specifically, we recover the local
surface norm als of th e object surfaces. W ithout knowing th e com plexity of the
processes involved, hum ans can easily perceive a 3-D shape from 2-D contours
for a large variety of objects. Cartoons and engineering drawings or any drawing
in millions of the books are some exam ples. However this does not m ean th at
hum ans can perceive a 3-D shape for any given random line drawing. Only the
line drawings w ith certain properties, carry inform ation for 3-D shape. In this
thesis we a ttem p t to identify some of the properties of the line drawings required
for 3-D shape recovery. Zero Gaussian curvature surfaces, straight homogeneous
generalized cylinders, and constant cross section generalized cylinders cut by pla
nar planes are studied in this thesis, because, such surfaces produce contours th a t
3
contain the properties required to recover 3-D shape. A lthough we do not claim
th a t all th e classes of contour th a t hum ans can perceive a 3-D shape are identi
fied and studied, we believe th a t these classes of surfaces constitute an im portant
subclass and a line drawing carries a little (if not none) shape inform ation, if it
does not fit into one of th e categories, studied in this thesis, or some derivation
of these categories.
1.1 Summary of the Previous Work
The early work on inferring 3-D structure from a 2-D shape was focused on anal
ysis of line drawings of polyhedra [Huf71], [Clo71], [Mac73], [Kan81], [KK83]. In
polyhedral scenes, the problem is th a t of segm entation and estim ating orienta
tions of faces. In early 70’s [Huf71], [Clo71], [Mac73] th e work on contours has
concentrated on line drawing interpretation and edge labelling. K anade [Kan81]
has pioneered th e work on shape recovery from contours for polyhedral scenes us
ing skew symm etry. Techniques for non-polyhedral scenes have been proposed in
[BT81], [Wei88], [BY84], [Ste81], [XT87], [HB88]. Barrow and Tenenbaum [BT81]
and Stevens [Ste81] pioneered the work on shape recovery for curved surfaces from
2-D contours. Barrow and Tenenbaum proposed a m ethod th a t minimizes the
smoothness of the curve in 3-D and Stevens has proposed a m ethod th a t is only
applicable to cylindrical surfaces. Xu and Tsuji .[XT87] extended Stevens work
to other surfaces than cylindrical surfaces. Weiss [Wei88] has proposed another
smoothness based m ethod w ith a different definition of smoothness. B rady and
Yuille [BY84] proposed a com pactness based m ethod for planar surfaces th a t are
not necessarily enclosed by straight lines. Later Horaud and Brady extended this
work to apply to conic surfaces. We discuss these techniques in more detail in
section 2 and contrast w ith our work later. It is interesting to note th at, m ost of
these techniques exam ine a single surface in the scene whereas our perception of
a surface can be strongly influenced by our perception of the entire object.
4
(a) (b) (c)
Figure 1.2: Shape perception; (a) A sm ooth curve (b) two sym m etric curves (c)
two non sym m etric curves
Recently some papers (e.g. [PCM89, Nal89]), have been published th a t study
th e projection and geom etry of certain classes of curved surfaces, however the
analysis is not sufficient to recover the 3-D surfaces.
1.2 Summary of the Proposed Method
In this thesis we present m ethods for recovering the 3-D shape of th e surfaces from
th eir contours. We follow a constraint based approach, where constraints on shape
are separated from th e assum ptions of regularity. T he two essential constraints on
shape im posed by the contours are Curved Shared Boundary C onstraint (CSBC)
and the Inner Surface C onstraint (ISC). CSBC restricts th e orientation of the
neighboring surfaces. This constraint was first used in the study of th e planar
surfaces and is extended to apply to curved surfaces here. ISC restricts the
orientation of th e inner points w ithin a curved surface along certain selected
curves.
T he perception of the shape of a contour is sharply influenced by its rela
tionship w ith other contours. One of the m ost im portant relationships is th at
of sym m etry. For exam ple in figure 1.2(a) the single curve does not give any
strong 3-D shape im pression, but in figure 1.2(b) another curve which is per
fectly sym m etric to th e first one is added and a definite shape is perceived. In
figure 1.2(c) two non sym m etric curve are displayed and again there is no definite
shape perception.
5
Sym m etries play an im portant role in our analysis in two ways. F irst, sym m e
tries carry im portant inform ation about the surface type: by ju st looking at the
sym m etries of the contours of a surface, in m any cases, we can identify th e type of
the surface w ithout even perform ing any q uantitative analysis, and in some cases
we show th a t th e surface m ust be of the identified type under the assum ption
of general viewpoint. Second, sym m etries provides vital inform ation needed for
quantitative analysis. For exam ple, one of th e two essential constraints, th e inner
surface constraint described in section 3.3.2 is based m ostly on a sym m etry th a t
we call parallel sym m etry, and orthogonality assum ption is based on sym m etry
properties of the contours.
We also conjecture th a t little shape inform ation is available in figures lacking
sym m etries for hum ans as well. Our work m ay be viewed as being based on gener
alizations of concepts th a t have been used previously such as by K anade [KanSl]
for polyhedral scenes and by Stevens [Ste81] and Xu and Tsuji [XT87] for curved
surfaces. We believe th a t our m ethod is of wide applicability. In particular,
we provide a detailed analysis for the case of Zero-Gaussian C urvature (ZGC)
surfaces, Straight Homogeneous Generalized Cylinders (SHGCs), P lanar Right
C onstant cross section Generalized Cylinders (PRCG C s) and objects form ed by
com bination of planar and ZGC surfaces.
T he types of surfaces, analyzed in this thesis, are commonly found in ev
eryday life, especially in m an-m ade objects. For each class of surface, the con
straints available are identified. Com bination of constraints reduces, but does
not uniquely determ ine the degrees of freedom of surface orientations. For ZGCs,
after the CSBC and ISC are applied; only three degrees of freedom rem ain to
recover the local surface gradient at each point on the surface. For SHGCs and
CGCs four degrees of freedom are left. T he assum ption driven constraints are
sym m etry based and they are applied to obtain a unique or a few m ost likely
interpretations. The m ain assum ption used is the 3-D orthogonality of 2-D sym
m etries. W hat exactly is m ade orthogonal depends on the type of surface studied.
For planar surfaces the skew sym m etry is m ade orthogonal in 3-D. For curved
6
surfaces th e intersection of the lines and curves of parallel sym m etry are m ade
orthogonal. In all of th e surfaces, we have studied, application of all the con
straints, including th e one based on orthogonality assum ption, consistently left
one degree of freedom undeterm ined. The shape of th e cross section curve is used
to fix this last degree of freedom.
For objects form ed by m any surfaces the constraints and assum ptions for each
surface are identified. T he constraints are essentially th e curved shared boundary
constraints between surfaces. In com puting a consistent global shape for the
object, these constraints are satisfied exactly while the error of not satisfying the
orthogonality constraints are minimized. T he set of param eters of th e surfaces
th a t minimizes the constraint m inim ization define the object shape.
In figure 1.3 th e synthetically shaded images of the objects, given in figure
1.1, are shown after the local surface orientations are recovered using the m ethods
described in this thesis. These are typical objects th a t we are able to handle by
the m ethods described here.
O ur m ethod assumes th a t clean, closed boundaries are given (or can be ex
tracted from th e real image). We do not address th e issue of separating object
boundaries from surface m arkings, or other perceptual grouping operations here.
We believe th a t th e specific conditions needed for an object surface to be recon
structed by our m ethod will provide further constraints for the perceptual group
ing operations when surface m arkings and other noisy boundaries are present.
1.2.1 O rigin al C o n trib u tio n s
T he following item s are individual im portant contributions introduced in this
thesis.
• Definition of sym m etries, especially parallel and line-convergent symm etry,
found in various classes of curved surfaces.
•! -N W 'W ''5
Figure 1.3: T he shaded images of the of th e com puted 3-D shapes for th e contours
given in figure 1.1
8
• Explicit analysis of contour based constraints on surface shape for curved
surfaces.
• Introduction of Inner Surface C onstraint for curved surfaces.
• A detailed analysis of th e following classes of surfaces and objects;
— Zero Gaussian curvature surfaces,
— Straight homogeneous generalized cylinders,
— P lan ar Right Constant cross section generalized cylinders.
• Com bining constraints from intersecting surfaces to obtain a consistent re
construction for an object consisting of planar and ZGC surfaces.
• Extension of all the constraints and the analysis of ZGC surfaces to P er
spective Projection.
1.3 Organization of the Thesis
In chapter 2, we discuss previous related m ethods. In chapter 3, we define two
kinds of sym m etries and discuss the constraints we obtain from contours and the
sym m etries. In chapter 4 we describe our technique for qualitative and quan
titativ e shape recovery of Zero Gaussian C urvature (ZGC) surfaces. Straight
Homogeneous G eneralized Cylinders (SHGCs) are studied in chapter 6 and Con
stan t Cross section generalized Cylinders (CGCs) are studied in chapter 7. In
chapter 5 we discuss objects composed of planar and ZGC surfaces. Finally we
state our conclusion in chapter 8.
9
Chapter 2
Previous Work
Here we present an overview of the im portant classes of previously m ethods. We
also give our view of their strengths and weaknesses. As our m ethod builds on
some of th e previous work, this section will also help provide some of th e relevant
background for describing our work.
3-D in terp retatio n of a 2-D line drawing is, of course, inherently ambiguous.
Given a curve in th e image, we can find an infinite num ber of 3-D curves th a t
the 2-D curve could be a projection of. T he construction is simple. Consider
figure 2.1. Given a point in the image, we can associate a ray in 3-D w ith it on
which th e corresponding point of th e 3-D curve m ust lie. Any choice of points on
these rays will project in the same 2-D curve. Note th at inferring the 3-D curve
gives th e surface orientation only if the curve is planar (and closed). Otherwise,
another process of estim ating surface orientations from the bounding curve m ust
be applied.
Two classes of m ethods have been used to reduce this am biguity. One is to
choose a shape th a t satisfies some preference criteria over the others. The second
approach is to define some constraints on the 3-D interpretation- some of the
constraints come from purely geom etric considerations, others require assum ption
of some regularity in the 3-D shape which m ay be signaled by some regularity in
the 2-D shape. In the following, we survey th e two classes of m ethods. F irst we
10
Image plane
Figure 2.1: Each point on the ellipse on the im age plane m ay be the projection
of any point on the line of sight.
give a definition of th e gradient space th at has been used by m any researchers
before and is extensively used in this thesis.
2.1 Gradient Space
W e assume orthographic projection throughout the paper unless specifically m en
tioned otherwise. In appendix C, we have shown how th e constraints for ortho
graphic projection can be transform ed to the case of perspective projection.
In this paper we will use Gradient Space to represent the orientation of surfaces
(given by their norm als). To review, th e normal, N , of a plane ax + by + cz + d — 0
is given by the vector N = (a,6, c). This can be rew ritten as (p, q, 1), where
p = a/c and q = b/c. N ote th a t this excludes cases where c = 0, however, such
planes are parallel to the line of sight and are not im aged under orthographic
projection from a general view-point anyway, (p, q) can be thought of as defining
a two dim ensional space, called the gradient space, such th a t every point in this
space corresponds to the norm al of a plane in 3-D.
11
2.2 Extremal Methods
In this class of m ethods, each interpretation (of a curve or figure) has, a m easure
of some desirable property, such as sm oothness or com pactness, associated with
it. In th e following we briefly describe some of th e m ore influential m ethods.
2 .2 .1 S m o o th n e ss M ea su res
Sm oothness is the first regularity m easure tried by researchers like [BT81, Wei88].
In th e following these m ethods are described briefly.
2 .2 .1 .1 B arrow and T en en b au m
Barrow and Tenenbaum [BT81] proposed to use the sm oothness of th e curve as
th e preference criterion. Their m ethods is not restricted to planar curves. The
sm oothness function they propose is an integral function of curvature and torsion
of th e 3-D curve. Over all possible 3-D curves th a t can generate a given 2-D image
curve, they choose the one th a t minimizes th e integral functional
where k is the curvature, t is the torsion and s is the arc length of th e 3-D
curve. They im plem ented an iterative optim ization procedure to com pute the
m inim um of the integral in equation 2.1. B ut they noted th at the convergence
of th e algorithm was slow. If planar surfaces are assum ed, only the first term in
th e integrand above need be considered.
Given an ellipse in the image, this m ethod will find a 3-D circle as th e curve
th a t m inim izes th e above function; an inference generally in agreem ent w ith
hum an perception. (In fact, for a circle the m easure in equation 2.1 is identical
to zero everywhere.)
This m ethod has th e advantage of having a well-defined com putational model
though th e com putation m ay require a search over an n dim ensional space where
(2.1)
1 2
n is th e num ber of points on the curve. It is also applicable to non-planar curves.
However, the answers are not always in agreem ent w ith th e hum an perception.
This m ethod is likely to be highly sensitive to noise as it uses the derivatives of the
curvature which are th ird order derivative operations. As an exam ple consider
a circle w ith a small notch. T he contribution of th e notch to th e smoothness
integral m ay be m uch larger th an the contribution of all other parts of the circle,
in this case, th e resulting 3-D curve m ay not resem ble a circle. Strictly speaking,
this m ethod is applicable to sm ooth image curves only and not to polygons, for
exam ple.
2 .2 .1 .2 W eiss
Weiss [Wei88] has proposed a modified m easure th a t uses curvature rather th an
its derivative and also handles polygons in a cleaner way. For planar curves
he proposes to m inim ize the integral of the square of the curvature (norm alized
by the length of the curve). This m ethod is slightly less sensitive, compared
to th e previous m ethod, to small changes on a curve (like a notch on a circle)
but it is still quite sensitive. Although, this m ethod interprets an ellipse on
the im age plane as a circle in 3-D, partial contours result in a consistent bias
due to norm alization, therefore a section of a ellipse will not be interpreted as
a section of a circle. He also proposes using square of th e angles of corners for
polygonal scenes as a m inim ization criteria. And he shows th a t a parallelogram is
interpreted as a slanted rectangle which may be in agreem ent w ith hum an visual
system . However, if a corner of th e same parallelogram is clipped th e m ethod
interprets it as a pentagon rath er than a clipped rectangle.
2 .2 .2 C o m p a c tn e ss M easu res (B ra d y and Y u ille)
B rady and Yuille in [BY84] used the “com pactness” of a figure as th eir preference
criterion. M easure of com pactness is chosen to be (area) / (perimeter)2. (This
m easure im plicitly assumes th at the curve is planar.) This m easure is com puted
13
over all 3-D orientations and th e m axim um chosen. This requires a search though
th e search can be speeded up for an approxim ate answer.
T he area can be com puted using th e Stoke’s theorem , which is:
tion efficient. This m ethod should be insensitive to sm all noise as the area of a
surface is alm ost unaffected by noise on th e boundary, and the perim eter is at
th e orientation. Also, this m ethod processes sm ooth curves and polygons in a
unified way.
A lthough, this m ethod has m any nice features from a com putational point of
view, the real question is: how often does the m ethod give the desired answers.
The m ethod works perfectly when th e the input is an ellipse and when the input is
a projection of a square. However, it fails on m any very sim ple shapes. For exam
ple it interprets a rectangle in th e picture as a slanted square; not in conform ity
w ith th e hum an perception. Also when the boundary is not com plete th e behav
ior of the m ethod is different th an w ith a com plete boundary which is not totally
desirable. In a recent paper H oraud and Brady [HB88] presents a m ethod for
interpreting generalized cylinders w ith the help of (area)/(perimeter)2 m ethod.
T heir m ethod is discussed in the next subsection.
2 .2 .3 O th er E x tr e m a l M ea su res
We have studied other preference m easures ourselves, for exam ple th e regularity
of a figure, which for a polygonal figure can be defined to be the equality of
th e angles of th e polygon. Again, this m easure works well on simple exam ples,
including a rectangle not handled by the Brady-Yuille m ethod. However, this
(a re a )n
where (area) is a scalar, n is a vector norm al to th e boundary curve and r is a
vector coordinate system in th e plane of the figure. This allows both area and
perim eter to be com puted by an integral on the boundary m aking the com puta-
m ost affected by a linear factor and this does not change rapidly when we change
14
m easure is difficult to apply to continuous curves, is sensitive to sm all changes
in th e boundary and also does not always produce results th a t agree w ith our
perception. For exam ple, a rectangle with a clipped corner is interpreted as a
slightly distorted pentagon.
It is our conclusion th a t the m ethods based on m axim izing some simple prop
erty of a 3-D figure viewed in isolation are not very effective even for planar
figures and generally do not apply to curved surfaces at all.
However, we believe th a t the m ajor deficiency of these m ethods rem ains,
nam ely th a t they use only the inform ation of a single curve and ignore all other
context.
2.3 Constraint Approaches
In this class of m ethods, constraints on 3-D surface orientations are obtained
by a variety of observations, w ith the expectation of getting unique (or a few)
solutions when the various constraints are combined. Som etim es, the constraints
are based on an assumption th at an observed regularity in th e image corresponds
to a regularity in the 3-D scene. Almost all of these techniques rely on making
some lines (or norm als) orthogonal in 3-D. If it is possible to find two lines (or
vectors) in an image, th a t we have reasons to believe are orthogonal in 3-D, we can
derive some constraints on the plane containing th e two lines (vectors). K anade
[Kan81] used the observation of skew symmetry to signal such orthogonality,
others [Ste81] [XT87] have used the assum ptions th a t some of th e curves in image
are lines of curvature. We briefly survey these im portant techniques below.
2.3.1 K a n a d e
A sym m etric figure in 3-D, w ith a straight axis and lines of sym m etry th a t are
orthogonal to it, projects in a figure which is skew symmetric (under orthographic
projection), i.e. th e lines of sym m etry are no longer orthogonal to the axis but
15
(b) (a)
Figure 2.2: (a) A typical rectangle and its skew sym m etry, (b) corresponding
constraint hyperbola in p — q space.
are at a constant angle to it. K anade [Kan81] showed th a t if we assum e th a t the
inverse also holds, i.e. th a t an observed skew sym m etry in the image is in fact
due to orthogonal sym m etry in 3-D, some useful constraints can be obtained.
Consider Figure 2.2(a) showing a parallelogram w ith its axis and skewed line of
sym m etry m aking angles a and fi w ith the horizontal. Imposing the requirem ent
th a t these lines are orthogonal in 3-D constraints the possible orientations of
the figure in 3-D. Expressing the orientation of th e surface in gradient space
(p,q ), the constraint on th e surface orientation is given by a hyperbola. Figure
2.2(b) shows th e constraint hyperbola for th e parallelogram in figure 2.2(a). Note
th at the skew sym m etry observation by itself does not give a unique orientation.
K anade suggests th a t in the absence of any other constraint one m ay choose
th e orientation th a t gives m inim um slant; this is the point on the hyperbola
th a t is nearest to th e origin (the two possible answers are equivalent to a “Necker
reversal”). However, this simple selection m ethod does not always give th e desired
answers.
For polyhedral scenes, K anade combined the skew sym m etry constraints with
w hat we have called the shared boundary constraints to obtain unique (or a small
set of consistent orientations). T he answers given by this m ethod appear to be
consistent w ith hum an interpretation. Of course, this m ethod applies only when
skew sym m etric objects are present.
16
2 .3 .2 U sin g L in es o f C u rvatu re
Some researchers [Ste81, XT87] have proposed using the lines of curvature to
recover th e shape assum ing th e object is cut along th e lines of curvature.
2 .3 .2 .1 S tev en s
Stevens [Ste81] studied cylindrical surfaces using orthogonality property. A cylin
drical surface is one where one of the principal curvatures is zero and th e lines
of zero curvature (the rulings of the surface) are parallel to each other. For such
a surface th e lines of m axim um curvature are planar and parallel to each other.
Stevens assumes th a t th e lines of m axim um curvature are given. T he rulings can
be obtained from these by connecting points w ith the same tangent. T he surface
is thus covered by a grid of curves, w ith the property th a t on the actual surface,
th e curves are orthogonal at th e points of intersection. Thus, constraints sim ilar
to those of th e skew sym m etry analysis can be applied. Stevens chooses to use
the slant1 and tilt representation instead of (p, q) representation.
As before, th e constraint is not enough to give unique orientations. However,
Stevens observes th a t slant and tilt param eters can be bounded and th a t the
bound depends on the angle between the two intersecting curves, w ith error
in tilt approaching zero as fi approaches 7 r. This happens near th e occlusion
boundaries of a cylindrical surface. S tarting from these points where tilt can be
fixed accurately, Stevens gives a m ethod of propagating th e estim ates along the
lines of m axim um curvature by th e following form ula (given w ithout proof here):
tan Ti ta n /A = ta n r 2 tan /?2 (2-3)
where Ti are th e tilt angles (3{ are the angles between th e lines of m axim um
and m inim um curvature, at two points along a line of m axim um curvature.
1Orientation of a surface, having gradient (p,q), can be alternately described in terms of
its tilt and slant which can be viewed as the polar coordinates of a point in gradient space,
specifically tilt = arctan slant = arctan \/p 2 + < 7 2
17
(b) (a)
Figure 2.3: Two cylinders resting at different slant angles.
This m ethod, however, does not always give correct results even when a cir
cular cylinder is given to it, as can be shown by a sim ple exam ple. Consider the
two cylinders in figure 2.3. T he points on the limb edges for both cases have /3j
equal to it which give unam biguous values of zero tilt. Using Stevens m ethod
to extrapolate, along th e cross-section, we will get the same orientations for the
m id-points of th e two cylinders where /?2 is ?r/2, in clear contradiction to our per
ception (which indicates th a t the top surface of (b) appears much m ore slanted
to us th an th a t of (a)).
2 .3 .2 .2 X u and T su ji
Xu and Tsuji [XT87] have described an extension of this m ethod to apply to
m ore general, curved surfaces. Their m ethod does not require th a t all lines of
curvature be given but th a t th e surface is cut along these lines of curvature. Given
a figure w ith four sides, such th at two of the opposite sides are lines of m axim um
curvature and th e other two are lines of m inim um curvature, they show how a
net (or a grid) over th e figure can be constructed w ith th e expected property
th a t th e corresponding net on the 3-D surface is orthogonal (this construction is
strictly valid for a restricted class of surfaces only). Surface orientations are first
com puted at special points on th e net where error in tilt is sm all (as in Stevens’
m ethod) and then propagated to other points on the net.
18
This m ethod seems to work well in some cases but has several draw backs. The
propagation scheme for non-cylindrical surfaces is only approxim ate and errors
can add up. It will always find an answer for a four-sided figure, even when
hum ans perceive no specific 3-D shape in it. Like Stevens’ m ethod, this m ethod
also would not differentiate betw een the two cylinders in the figure 2.3.
2 .3 .3 H orau d an d B ra d y
H oraud and B rady [HB88] present a m ethod for interpreting linear straight ho
mogeneous generalized cylinders (LSHGCs). Their m ethod makes the following
assum ptions: a) th e axis of the LSHGC projects as th e axis of th e ribbon form ed
by th e two limb contours in th e im age plane, b) th e cross section of th e LSHGC
is planar, and c) th e cross section is orthogonal to the axis in 3-D. Satisfying
assum ption (b) above gives a constraint th a t the orientation of th e cross section
m ust be along a certain curve in th e orientation space (the curve is shaped like
th e character “s” and hence called “s-curve”). They also require th a t th e back
projected cross section satisfy the Brady-Yuille com pactness m easure. If an ori
entation satisfies both constraints then th at orientation is chosen. They do not
specify w hat should be done if this is not the case, though a n atu ral extension
would be to take the m ost com pact shape constrained by the s-curve. Finally they
suggest sweeping the reconstructed cross section along the 3-D axis to reconstruct
the surface of the LSHGC.
This m ethod has the attractiv e property th a t it attem p ts to com bine the con
straints from two surfaces. However it has several deficiencies. T he com pactness
m easure can only be applied to com plete cross sections. M ore seriously, the as
sum ption (a) above is incorrect. Given the image of an LSHGC, we can choose
any axis th a t passes through th e apex of the LSHGC in th e im age plane and
reconstruct an orthogonal LSHGC in 3-D, including th e ends, th a t have th e same
projection in th e image. To see this, take any backprojection of the chosen axis,
backproject th e two cross sections (the top and the bottom ) on any two planes
19
orthogonal to th e backprojected axis; th e orthogonal LSHGC can be com pleted
by joining th e points on the cross sections, such th a t th e lines joining these points
pass through th e backprojected apex. Finally even if an axis is chosen in the im
age plane, th e point through which the 3-D axis pierces the reconstructed 3-D
cross section m ust be chosen in order to reconstruct the LSHGC. This point is
not addressed in th e H oraud and B rady paper.
2 .3 .4 O th ers
In other work, Nalwa [Nal87] has derived a sym m etry condition th a t m ust be
satisfied by the lim b boundaries of a solid of revolution (sufficiency of these
conditions is also claim ed under certain general viewing conditions). However,
this paper does not show how to actually reconstruct the surface. Ponce et al.
[PCM89] have derived properties th a t m ust be satisfied by a broader class of
surfaces known as straight homogeneous generalized cylinders (SHGCs). Again,
these properties by them selves are not sufficient to reconstruct the 3-D surface
of SHGCs.
Recently Gross and Boult [GB90] have provided a m ethod for recovering
SHGCs from contour and shading data. Their analysis of the contour is quite
sim ilar to ours, however they use shading inform ation to further constrain the
possible orientations. They claim th at, using shading inform ation elim inates the
need to m ake assum ptions about the shape of the object, as needed for shape
from contour m ethods. However, using shading inform ation itself requires drastic
assum ptions about the surface reflectance properties of th e objects.
20
Chapter 3
Symmetries, Qualitative Inferences and
Constraints
O ur proposed technique is based on observations of sym m etries in figures. We
believe th a t sym m etries have an im portant role in shape perception, this also
has been noted and used by m any researchers [NB77b, Nal87, Kas88, Kan81,
Ste81, MN89]. We define three types of sym m etries, th a t we call parallel sym m e
try, line-convergent sym m etry and skew sym m etry. Parallel and line-convergent
sym m etries are m ainly found in curved surfaces, skew sym m etry is usually an
indicator of planar surfaces. O ther types of sym m etry relations betw een curves
specific to various classes of surfaces will also be discussed as the surfaces gen
erating such curves are discussed. Later, we discuss qualitative shape inference
from th e sym m etries. Finally, constraints th a t constitute th e basis of surface
reconstruction for all types of surfaces, studied in this thesis, are presented.
3.1 Symmetries
We define two new sym m etries called parallel sym m etry line-convergent sym m etry
and redefine skew sym m etry, previously defined and used by K anade [Kan81].
For curves to be sym m etric (parallel or skew) certain point-wise correspondences
betw een two curves m ust exist. We will call the lines joining th e corresponding
points on th e curves as the lines of symmetry, the locus of th e m id points of these
21
'\
\ \
\
\
(a) (b) (c)
Figure 3.1: Exam ples to (a) and (c) parallel sym m etry w ith curved contours,
(b) parallel sym m etry w ith straight contours. T he dotted curves are axes of
sym m etry and th e dashed lines are lines of sym m etry.
lines as the axis of symmetry, and the curves form ing th e sym m etry as the curves
of symmetry.
3.1.1 P a ra llel S y m m e tr y
Consider two curves Xi(s) = (®,-(s), yi(s)), for i = 1,2, param eterized by arc
length s. Let T)(s) = (:r'(s), t/8 '(s)) be th e unit tangent of th e curves. Then,
X i(s) and A T 2(.s) are said to be parallel sym m etric if there exists a correspondence
function / ( s ) between them such th a t,
T .M = r 2( /( s ) ) (3.1)
for all values of s for which X \ and X 2 are defined and f(s) is a continuous
m onotonic function. Note th a t com puting sym m etry between two curves using
this definition requires estim ating the function f(s) as well. A useful special case
is when f(s) is restricted to be a linear function. In th a t case, the sym m etry
condition becomes:
Ti(s) = T2(as + b) (3.2)
where a and b are constant (a m ay be thought of as a scale param eter). Some
exam ples of parallel sym m etry are given in figure 3.1; th e correspondence function
22
Figure 3.2: Two line-convergent sym m etric curves.
f(s) is linear for (a) and (b) but not for (c). Note th a t th e above definition of
parallel sym m etry also holds for curves consisting of straight lines and corners
as in figure 3.1 (b). Presence of parallel sym m etry w ith non straight curves of
sym m etry, is a very strong indication of curved surfaces. We can extend the
definition of parallel sym m etry to 3-D curves. T he parallel sym m etry in 3-D is
still defined by equation 3.1 where Ti is the unit tangent in 3-D.
3.1.2 L in e-C o n v erg en t S y m m e tr y
Two image curves C\ and Ci are line-convergent sym m etric if the tangents of C\
and C > 2 j &t the corresponding points, intersect along a line, say I, on th e image
plane. This is shown in figure 3.2. This sym m etry is found in curves obtained
by cutting ZGC surfaces w ith two non parallel planes. Also this sym m etry is
present between th e projection of limb edges of straight homogeneous generalized
cylinders (SHGCs) described in chapter 6.
23
2
(a)
Figure 3.3: Exam ples to (a) skew sym m etry w ith curved contours, (b) and skew
sym m etry w ith straight contours. T he dotted curves are axis of sym m etry and
th e dashed lines are lines of symm etry.
3 .1 .3 S k ew S y m m e tr y
In this sym m etry, the point-wise correspondence should be such th a t the axis
of the sym m etry is straight, and th e lines of sym m etry are at a constant angle
(not necessarily orthogonal) to the axis of sym m etry. Skew sym m etry was first
proposed by K anade [Kan81] and used in the analysis of scenes of polyhedral
objects.
Ponce [Pon88] has given point-w ise conditions for two curves to be skew sym
m etric. We state these here w ithout proof. For the case when th e lines of sym
m etry are orthogonal to the axis of sym m etry th e criterion for the two curves,
X l(s) and 2f2(s), to be skew sym m etric is:
/c i(s ) = — /c2 (s + b) (3 -3 )
where k ( s ) is th e curvature, and b is the offset. In general, let a,-(s) be
the angle betw een the line of sym m etry and th e tangent to th e curve i at the
corresponding points, then the necessary condition for the two curves to be skew
sym m etric is:
Ki(s) sin3(o:2(s)) = — k 2( s + b) sin3(o:2(s + b)) (3.4)
24
An exam ple is given in figure 3.3 (a). The above conditions are only valid
for curves, not for lines, th a t is curvature should be non zero. For lines the
first definition of skew sym m etry can simply be applied as: two lines are skew
sym m etric if another set of two lines th a t joins th e end points of th e given lines
are parallel to each other. In this case the new two lines are th e lines of sym m etry
and the lines joining the end points of these lines is the axis of sym m etry. An
exam ple to skew sym m etry for straight lines is given in figure 3.3 (b).
3.2 Qualitative Shape Inferences
We now describe some qualitative inferences about th e shape of surfaces from their
sym m etries. We prove some of the inferences th a t we m ake w ith th e assum ption
of general viewpoint defined as:
D e fin itio n 1 G e n e ra l V ie w p o in t : A scene is said to be imaged from a gen
eral viewpoint} if perceptual properties of the image are preserved under slight
variations of the viewing direction.
Specifically, the properties we are interested in are: straightness and paral-
lelity of lines and sym m etry of curves
Here, we discuss th e interpretation of individual surfaces independently. In an
object, of course, several surfaces may be visible and th eir interpretations m ust be
m utually consistent. This can provide a m echanism for either reinforcing individ
ual surface interpretations or choosing among possible m ultiple interpretations of
individual surfaces.
It will be useful to consider figures as belonging to one of th e following three
classes:
25
3.2.1 C a se I
Here, one skew sym m etry covers the entire boundary of the surface. We allow
m ore th an one alternative descriptions for a figure (figure 3.4 shows two exam
ples). For exam ple, th e ellipse in figure 3.4 (a) can be described as being skew
sym m etric about any axis th a t passes through its center, and th a t all sym m etries
include all the points on th e ellipse boundary.
Surfaces belonging to case I are generally perceived to be planar. We prove
th a t if a contour belongs to case I bounded by non-lim b edges, then the contour
has to be planar under the assum ption of general viewpoint and if the correspon
dence is static w ith respect to chcnging viewpoint. Limb edges (or limbs) of a
surface are generated by points on the surface whose norm al is orthogonal to the
viewing direction. Such edges changes its position on the surface as the viewpoint
changes. Non-limb edges, on the other hand, do not change their position on the
surface as th e view point moves, they include creases and wireframes.
L e m m a 1 A 3-D skew symmetric figure projects as a skew symmetric figure
under orthographic projection.
P r o o f It is a direct result of the property of the orthographic projection th at
parallel lines project as parallel lines and th a t m id points of lines project as mid
points of th e projected lines. Therefore, the 3-D lines of sym m etry project as the
lines of sym m etry on the image plane and the projection of th e 3-D axis is the
line joining the m id points of the lines of sym m etry on th e image plane.
T h e o re m 1 If a 3-D contour, formed by non-limb edges, produces a skew sym
metric line drawing in the image plane such that the 3-D correspondence is in
variant under small perturbations of the viewpoint then the 3-D contour must be
planar (under the assumption of general viewpoint).
P ro o f: Since the 3-D correspondence is invariant w ith respect to variations
of the view point (th at is the projection of the same set of 3-D points correspond
26
(a) < b )
Figure 3.4: Some exam ples of case I.
from different view points) then, the assum ption of general viewpoint implies th a t
parallel lines in th e image plane m ust be th e projection of parallel 3-D lines,
otherw ise they would not project parallel from nearby view points. Therefore we
conclude th a t th e 3-D lines, say U, th a t project as the lines of skew sym m etry
on the im age plane, m ust be parallel to each other in 3-D, because lines of skew
sym m etry are parallel to each other in the im age plane. T he axis of sym m etry
in 3-D, which can be obtained by joining the m idpoints of the 3-D lines /j, m ust
be straight because its projection on the image plane, which is the axis of skew
sym m etry is straight. Therefore, the lines have to lie on a plane, because they
are parallel to each other and a single line, th e 3-D axis of sym m etry, intersects
them . Hence the 3-D contour, which encloses the lines k is planar. □
Lem m a 1 shows th a t planar skew sym m etric figures projects as skew sym m et
ric figures on th e im age plane. Theorem 1 shows th e reverse is also tru e under
the stated conditions. We conjecture th a t invariance of th e correspondence is not
necessary, b u t have not proved it.
Note th a t, if the 3-D contour form ing th e skew sym m etry on the image plane is
limb edge, th en th e 3-D contour could be non-planar. For exam ple, lim b edges on
surfaces of revolution produce an orthogonal skew sym m etry, [Nal89]. Generally,
such surfaces also produce a parallel sym m etry together w ith skew sym m etry and
they belong to case II, defined below.
27
3 .2 .2 C a se II
Here, th e boundary of th e figure is covered by exactly two sym m etries, identified
a th e first and the second sym m etry. The first sym m etry is required to be a
parallel sym m etry (or could also be a line-convergent sym m etry in the case of
ZGC surfaces). We will argue th a t case II figures are the ones th a t give us the
m ost inform ation about the surface shape and th a t such cases are com m on in
scenes of everyday experience. T he first four objects in figure 1.1 shows some
exam ples of this case. In this thesis we concentrate on such surfaces. Depending
on the second sym m etry the type of the perceived surface changes. In this thesis
we identify and study three types of surfaces:
• If th e second sym m etry is a skew sym m etry w ith straight curves of sym m e
try then the surface is hypothesized to be a zero Gaussian curvature (ZGC)
surface. Also if th e first sym m etry is an line-convergent sym m etry term i
nated by line segments, the surface is hypothesized to be a ZGC surface.
In chapter 4, ZGC surfaces are studied under th e assum ption of general
view point.
• If the second sym m etry is a line convergent sym m etry th a t satisfies the limb
boundary condition given in section 6.2, then the surface is hypothesized
to belong to Straight Homogeneous Generalized Cylinders (SHGCs). Such
surfaces are studied in chapter 6.
• If the second sym m etry satisfies the term inator condition given in section
7.3 then th e surface is hypothesized to belong to planar right constant cross
section generalized cylinders (PR C G C s). This class of objects are studied
in chapter 7 in detail.
3 .2 .3 C ase III
This class includes all rem aining cases. Two interesting sub-classes occur here.
28
(b)
(a)
Figure 3.5: (a) A figure w ith two skew sym m etries, (b) addition of an ex tra curve
clarifies th e perceived shape
a) We hypothesize th e presence of some boundaries not present in the im age to
convert the figure into case I or II. For exam ple consider figure 3.5 (a) w ith
two skew sym m etries. Note th a t we have no strong feel for the 3-D shape of
this surface. However, if we assum e th a t there is one missing boundary th a t
would introduce a parallel sym m etry (and an additional skew sym m etry)
as shown in figure 3.5 (b), th e surface shape becomes very distinct. Of
course, m ore th an one such construction m ay be possible, each giving an
alternative interpretation; some constructions m ay be preferable according
to some heuristic criteria.
A nother interesting case is where two sym m etries cover m ost of the bound
ary b u t not all of it; an exam ple is given in figure 3.6 (a) Here, two choices
are available. E ither we can inscribe a sm aller figure inside the larger one,
or ex trapolate some of the boundaries to m eet the requirem ents of case II.
T he two choices are shown in figure 3.6 (b) and (c). N ote th a t the ex
trapo latio n is preferred if the “to p ” surface is also shown as in figure 3.6
(d).
b) All other cases. We have not studied such surfaces, b u t we conjecture th a t
it is difficult to perceive a specific shape in such cases in general.
29
~ r A - "
(a)
(c)
Figure 3.6: (a) Face of a cylinder w ith a clipped corner, (b) the parallel sym m etry
cover only p art of the surface, (c) the top curve is extended for th e parallel
sym m etry to cover the whole face, (d) the top of th e surface is also included
3.3 Constraints
We now give th e constraints th a t derive from observations of th e sym m etries and
other boundaries in the image. T he constraints are:
• Curved shared boundary constraint (CSBC) : relates the orientation of two
intersecting surfaces along the curve of intersection.
• Inner surface constraint (ISC) : restricts th e orientation of the neighboring
points on a surface.
• Orthogonality constraint (OC) : uses the assum ption th a t some observed
regularity is a result of some 3-D orthogonality.
Usage of these constraints will be discussed when individual types of surfaces
are studied. Here we give th e definitions and proofs, when needed, of these
constraints.
30
3.3.1 C u rv ed S h ared B o u n d a ry C o n stra in t (C S B C )
T his constraint relates the orientations of th e two surfaces on opposite sides of
an edge. T he planar version has been used since early days in polyhedral scene
analysis [Mac73]. Shafer et al. [SKK83] extended it to th e case of intersection of
curved surfaces.
Consider two surfaces X i(u ,u ) and X 2(u,v) m eeting along a curve
T (s) = (a;(s), y(s ), 2 (5 )) as in figure 3.7. Let N i(u , u) and N 2(u, v ) be th e norm als
of X i and X 2 respectively. Along the curve T(s) we can represent the norm als
Ni and N 2 as N{(s) = Ni(ui(s),Vi(s)). Since T'(.s) is on the tangent planes of
bo th X i and X 2, T '(s) is orthogonal to both N i(s ) and N 2(s) in 3-D. T h a t is
Ni(s) ■ T'{s) = 0 N 2( s ) ■ r ( s ) = 0 (3.5)
We can rew rite it as r 7 (s) • (N 2(s) — iVi(s)) = 0.
Say the norm als Ni(s) are represented in p —q space as Ni(s) = (Pt(s), 1).
S ubstituting these in the above equation gives:
(x \s),y '(s):z \s )) ■ ((p2(s),q2(s),l) - (p1(s),q1(s),l)) = 0
^'(s)(p2(s) - p d s )) + y/(s)(92(3) - qi(s)) = 0 (3.6)
This is the Curved Shared Boundary C onstraint (CSBC) which states th a t
along th e curve T(s) th e orientation of th e surfaces X i and X 2 are constrained
by th e tangent, (x'(s),y'(s)) of th e im age of th e curve T(s) under orthographic
projection.
A stronger constraint can be obtained if we can assum e th a t the intersection
curve, T, is planar. Say, T lies in a plane w ith orientation (pc,qc). W ith the
assum ption of p lanarity th e constraint equation becomes:
- p(s)) + y(s)(qc - q(s)) = 0 (3.7)
For ZGC, SHGC and CGC surfaces, we will assum e th a t th e parallel sym m et
ric curves are planar either due to some regularity in the figure or by definition
of th e individual surface model.
31
Figure 3.7: Two curved surfaces meeting along a curve F
3 .3 .2 In n er S u rface C o n stra in t (IS C )
T he inner surface constraint restricts th e relative orientations of th e neighboring
points, w ithin a surface. Consider a curve C{t) = y(t ), z{t )) on a C 2 surface
S. For each point P £ C associate a vector R £ Tp such th a t
gn
■ dNp = 0 (3.8)
w here Tp is th e tangent plane of th e surface S at th e point P and (INp is the
derivative of the norm al N of th e surface S in the direction R.
T h e o r e m 2 Inner Surface Constraint: Under orthographic projection, if an im
age curve Ci is the projection of the curve C on the surface S and R i = (rx,ry)
is the projection of the vector R satisfying equation 3.8, then the change of the
orientation, (p,q), of the surface S, along the curve C, in the p — q space is
restricted by the image vector Rj, as:
d(p,q)c > ■ R i = 0 (3.9)
P r o o f Let X (n ,u ) be the local param eterization of the surface S around the
point P £ C(t) such th a t for P = X (n 0,uo), th e curve X(u,t>o) is th e curve C
and th e curve X { u q , v ) is in th e direction R. T h at is, u param eter curve is along
th e curve C and v param eter curve is in th e direction R at th e point P. Here
we have to show th a t ■ R j = 0 where (p, q) is the norm al of th e surface in
32
the gradient space, du is in the direction of C', R j is th e im age, (xv,yv), of th e
vector R = X v = (x v, yv,z v) under orthographic projection.
N orm al, N , of this surface at any point is given by:
N = i y ' v y \ (310)
\ S \ U X u \ y J
T hen, th e functions dC}dt and dNn are:
dC d X v 1A T d N A T
—- = — — = X v, dNR = -j,— = N u (3-11)
dt ov ou
By equation 3.8 we have X v ■ N u = 0. Let the norm al N of th e surface around
point P is represented in th e (p,q) space as N — c(p,q, 1). W here c is the scale
coefficient and equal to (p 2 T q2 + l ) -1/ 2. D ifferentiation of N w ith respect to the
p aram eter u gives:
Nu = cu(p,q, 1) + c(pw ,? u,0) = — iV + c(pu,? tt,0) (3.12)
c
If we set X v ■ N u = 0 where X v — (x v,yv, zv) and N u is given in equation 3.12
we get:
X v • N u = — X v • N + c(xv, yv, zv) ■ (pu, qu, 0) = 0 (3.13)
c
We also have N • X v = 0 from 3.10. Therefore
x vpu + yvqu = ■ R i = 0 (3-14)
□
To apply this constraint, we need to identify a curve C in the im age plane
for which th e orientation R can be determ ined. In appendix A. 1.3 we show
th a t for zero G aussian curvature surfaces any curve on th e surface can be the
C curve if th e direction R is chosen to be th e direction of th e rulings of the
surface. Following theorem shows how we can use parallel sym m etric curves for
this purpose for other classes of surfaces.
T h e o r e m 3 Let the family of curves,{Ci}, be on a surface S such that the curves,
Ci, are parallel symmetric in 3-D. If the curves Ci are used as the C curves of
33
equation 3.8 then, the tangent of the curves obtained by joining the symmetric
points of the curves Ci gives the direction R of the ISC. Conversely, if the curves
obtained by joining the parallel symmetric points of curves, Ci, are used as C
curves of equation 3.8 then the tangents of the curves Ci give the direction R.
P r o o f Consider th e param etric representation S(u, u) of th e surface S' such th a t
th e u param eter curves are parallel sym m etric to each other (th e {Cf} family
of curves) and v param eter curves join the parallel sym m etric points of th e u
param eter curves.
For th e first p art of the theorem we have to show th a t equation 3.8 holds or
w ith th e current param eterization
Su -N v = 0 (3.15)
is tru e, w here N = is the unit norm al of the surface. N ote th a t N ■ Su —
P u X |
N • Sv = 0 by definition. We can substitute — Suv ■ N for Su ■ N v since:
0 = — Sg v N ^ = S UV- N + SU- N V ^ S u - N v = - S uv - N (3.16)
S u is th e tangent of th e u param eter curves, and since th e v param eter curves
join th e parallel sym m etric points of u param eter curves th e direction of S u(u, v)
is independent of the v param eter, th a t is 5 u(u, v) = c(v)Su(u) where c is a scalar
function. And;
S U v = = c » s » (3.17)
By substituting this in equation 3.16 we get
N v ■ Su = - N • Suv = —c'(v)(N • Su(u)) = 0 (3.18)
For the second p art of the theorem we have to show th a t Sv • N u — 0. Using
equation 3.18 we get:
0 = N v ■ Su = - N ■ Suv = - N ■ Svu = Sv ■ N u (3.19)
□
34
«
Figure 3.8: Two cylinders (a) is cut along the curves of maximal curvature, and (b)
is cut in an arbitrary direction while preserving parallel symmetry, now we have the
perception of an elliptical cylinder.
3 .3 .3 O rth o g o n a lity C o n stra in t (O C )
C ertain properties or sym m etries invoke the assum ption of orthogonality in 3-D.
T he assum ption of orthogonality was first studied for skew sym m etric contours
by K anade [Kan81]. We will assum e orthogonality between th e axis of parallel
sym m etry and the lines of parallel sym m etry. For a ZGC surface, this is equivalent
to slicing th e surface along rulings to obtain th in skew sym m etric planar strips
and assum ing th a t these strips are orthogonally sym m etric in 3-D. This preference
is illu strated in 3.8 where in (a) we see a circular cylinder, b u t in (b) we prefer
to see an orthogonal elliptic cylinder rath er th an a slanted cylinder. For other
surfaces, SHGCs and CGCs, the curves of parallel sym m etry will be assum ed to
be orthogonal to th e curves obtained by joining points of correspondence.
On th e surface analyzed, say th e tangent of the parallel sym m etry curve makes
an angle a w ith th e horizontal and the tangent of the other curve (the curve th a t
parallel sym m etry curve will be m ade orthogonal w ith) makes an angle at some
point on the surface, as in figure 3.9. Let the norm al of the surface be N = (p, q, 1)
at th a t point. Since th e 3-D tangent vectors A and B lie on th e tangent plane of
the surface they can be represented as:
A = (cos(a), sin (a ),p c o s(a ) + q sin(a))
B — (cos(/?),sin(/3),pcos(/3) + qsin(P)) (3.20)
35
Figure 3.9: Orthogonality constraint
and from th e orthogonality of th e 3-D vectors A and B we get: A ■ B — 0,
this is the equation of a hyperbola as:
(3.21) cos(a — /?) + (p cos a -f q sin a)(p cos j3 -f q sin /3) = 0
This is th e equation of a hyperbola in the p — q space as shown in figure 2.2,
constraining the possible orientations for th e surface norm al N.
Chapter 4
Zero Gaussian Curvature Surfaces
A Zero Gaussian C urvature (ZGC) surface is one where th e th e G aussian cur
vature (the product of th e m axim um and m inim um principal curvatures) of the
surface is zero everywhere. Cylinders and cones are exam ples of a ZGC surface.
Some exam ples are given in figure 4.1. These surfaces are also called developable
surfaces since they can be generated from a piece of paper by rolling an d /o r
bending w ithout cutting. We feel th a t ZGC surfaces com prise a large and useful
class and th a t they represent a n atu ral step up in com plexity from the study of
planar surfaces th a t have dom inated previous work in th e field. Lines of m inim um
curvature for a ZGC surface, also called rulings, are straight, i.e. it is possible to
em bed straight lines on a ZGC surface along these rulings.
In section 3.2.2 we argue th a t a surface bounded by parallel sym m etric curves
term in ated by straight lines are perceived as Zero G aussian C urvature (ZGC)
surfaces. In this chapter we actually prove th a t such a surface m ust be ZGC,
along its contours, w ith th e assum ption of general viewpoint. L ater we apply the
constraints discussed in section 3.3 to recover the shape of th e ZGC surfaces. At
th e end of th e chapter we extend th e reconstruction algorithm s to ZGC surfaces
th a t are cut by non parallel planes.
37
Figure 4.1: Exam ples to ZGC surfaces.
4.1 Symmetries and ZGC Surfaces
In this section th e types of sym m etries found in ZGC surfaces cut by parallel
planes are discussed.
T he following theorem asserts th a t case II figures satisfying specific properties
m ust have Zero G aussian curvature along its skew sym m etry contours.
T h eo r e m 4 If a surface patch generates one parallel sym m etry and one skew
sym m etry, with straight curves of skew sym m etry on the image plane, and the
straight curves of skew sym m etry are also the lines of sym m etry for the parallel
sym m etry, then the Gaussian curvature of the surface must he zero along the
curves of skew symm etry.
P ro o f: There are two sub-cases, depending on w hether th e curves of skew
sym m etry are limb edges or not.
a) T h e straight curves of skew sym m etry are produced by lim b edges: In this
case ju st th e straightness of th e limb is sufficient for the surface to have zero
38
G aussian curvature along the limb. This can be inferred as a special case
of a theorem given by K oenderink [Koe84]. We give an alternative proof
here th a t does not need the additional assum ptions used in K oenderink’s
proof. Let th e surface X (u ,v ) be param eterized such th a t a u param eter
curve is along the limb boundary for v = va. Since th e curve X { u ,v 0) is
along th e lim b boundary and the projection of this curve is straight the
surface norm al JV * along this curve is constant, th a t is Afu(u,v0) = 0. This
condition is a sufficient condition for th e G aussian curvature of th e surface
along the X ( u ,v 0) to be zero. T he G aussian curvature, k , of a surface is
given by;
L N - M 2
* E G - F 2 ^ ^
where L , M, N are the coefficients of th e second fundam ental form of the
surface and E , F, G are th e coefficients of the first fundam ental form. The
equations of these coefficients are given in equation A .3. Particularly, the
coefficients L and M can be w ritten as;
L = - X u -Mu M = - X v ■ Afu (4.2)
Since N (u ,v 0) = 0, th e G aussian curvature, /c, m ust be zero along th e limb.
T he above proof does not require the assum ption of general view point,
hence, it only shows th a t along the curve of th e limb boundary th e surface
has zei'o G aussian curvature. W ith the assum ption of general view point,
we conclude th a t an open region surrounding the limb boundary also has
zero G aussian curvature.
b) T he straight curves of skew sym m etry are cut edges. Consider figure 4.2,
w here a pair of parallel sym m etric curves on a ZGC surface cut along a
ruling is shown. Since, in the im age plane, th e tangents t\ and t2 of the
top and bottom curves are parallel, by the assum ption of general viewpoint
they m ust be parallel in 3-D. Also, since the skew sym m etry curves (one of
which is th e ruling in figure 4.2) are straight on th e im age plane, th e 3-D
39
corresponding curves m ust also be straight. T h a t is, the surface em beds
straight lines. Therefore th e surface can locally be represented as a ruled
surface having equation:
X(v,v) = f ( v ) + ug(v) (4.3)
w here f(v) and g(v) are arbitrary vector functions of th e param eter v only.
T he vector function g(v ) indicates th e direction of th e ruling which are also
th e u param eter curves. T he norm al of this surface is:
M (u< v ) = ( / ' ( » ) + x g ' W ) X g(v)
|(/'( v ) + ug’(v)) X g(v) I
For figure 4.2, let the dotted line (ruling) be th e cut boundary for v = v0,
and J\fi(ui,v0) and Af2(u2, v 0) be the norm als of th e surface at points where
th e ruling intersects th e parallel sym m etry curves. Since the tangents t\
and t2 are the sam e and of course the tangent of th e ruling is constant along
it, the surface norm als and A/2 , which are th e cross products of C and
t2 w ith th e tangent of th e ruling, m ust be th e same. T h a t is:
M (u i,v 0) = J\f(u2,v0)
( f ( vo) + uxg'(v0)) x g{va) = ( f ( v 0) + u2g'(v0)) x g(v0) (4.5)
w here = indicates parallelism of th e vectors. T hen, we have th a t either
= u2 or th e three vectors f , g \ g are dependent. Clearly «i u2, so
/ ' , 1 q\ g are dependent and hence the surface norm al J\f is independent of the
the u p aram eter curve, th a t is J\f(u,v0) = 0. As in th e case (a) above, this
is a sufficient condition th a t the Gaussian curvature of the surface along
th e X (u,v0) curve is zero. □
G e n e r a liz a tio n If we assum e th a t the type of th e surface does not change
w ithout producing a visible edge, th en we conclude th a t the whole surface m ust be
a ZGC surface if it satisfies th e property given in theorem 4. This generalization
40
,ruling
Figure 4.2: A ZGC surface cut along th e “ruling”.
m ay appear to be a rath er sweeping one. However, it is no m ore so th an the
com m on assum ption th a t a polygonal line drawing corresponds to polyhedral
objects.
It follows th a t if the parallel sym m etry has a linear correspondence function
then th e surface is conic, and if th e correspondence function is an identity then
th e surface is cylindrical. We now show how we can infer th e rulings and th e cross
sections of the ZGC surface. Rulings are the lines along which th e curvature of
th e surface is zero. Cross sections are th e transverse (not necessarily orthogonal)
curves; specifically th e curves th a t project into parallel sym m etric curves. We
first give two theorem s th a t is key in inferring properties of cross sections and
rulings.
T h eo r e m 5 Curves obtained by intersecting a Z G C surface with two parallel
planes are parallel sym m etric such that the lines of sym m etry are the rulings of
the surface.
T he proof of this theorem is given in appendix A. 1.1.
T h e o r e m 6 Curves, C\ and C%, obtained by cutting a Z G C surface, S, by two
non parallel planes, IR and H2 project as line-convergent sym m etric curves such
that the lines joining the corresponding points of the image curves are the projec
tions of the rulings of S and the line I form ed on the image plane by joining the
intersection points of the tangent lines of the line-convergent sym m etric curves
is the projection of the 3-D intersection line of the planes and II2 .
41
Figure 4.3: Form ation of the line-convergent sym m etry w ith a ZGC surface an
two non parallel planes.
P r o o f T he above theorem is visualized in figure 4.3. T he key to the proof of
this theorem is th a t th e tangent plane, plane T , in figure 4.3 of th e ZGC surface
S is sam e along th e rulings of S. Therefore, both the tangent lines, t\ and £2 , of
th e curves C\ and C2 from points P\ and P2 are on plane T. Also the tangent line
tx is on plane III and 12 is on n 2. Therefore intersection of G and t 2 is necessarily
at th e intersection point of the three planes III, II2 and T. For other rulings the
same things repeat for a different T plane, and all the tangent line intersections
tak e place along th e line I, th e intersection line for planes Ifi and 1I2. Hence, on
th e im age plane too th e intersection of th e tangents takes place on the projection
of the line I.
N ote th a t th e reverse of these theorem s, th a t parallel sym m etry or line-
convergent sym m etry curves must come from parallel or non parallel planar cuts,
is not valid; In appendix A. 1.2 we show th a t lines of m axim um curvature which
are not necessarily planar, also project as parallel sym m etric curves. However, we
believe th a t it is reasonable to infer th a t parallel sym m etry (or line-convergent)
curves are planar, unless we have evidence to th e contrary. Lines of curvature
42
(a)
Figure 4.4: O bjects w ith cross sections having (a) only one skew sym m etry, (b)
two skew sym m etries
of a ZGC, in general, can be very complex and it is unlikely th a t an observed
surface would be cut in this way. If the curves are neither planar nor along the
curves of m axim um curvature, it is quite difficult to obtain parallel symm etry.
For exam ple, in order to obtain parallel sym m etry for a conic surface, a sub
case of ZGC surfaces, by cutting w ith non-planar cross sections the cuts m ust be
tran slated along th e axis of the cone and scaled exactly w ith th e scaling function
of the cone.
O ur interpretation does allow for piecewise planar cross sections as indicated
by m ultiple skew sym m etries. Figure 4.4 shows an exam ple, th e cross section of
th e object in figure 4.4 (a) has a single skew sym m etry and is perceived planar,
whereas the cross section of th e object in figure 4.4 (b) has two skew sym m etries
and the perception is th a t th e cross section has two planar parts. T h at is if the
cross section has m ultiple skew sym m etries then it will be piecewise planar such
th a t each planar section has one skew symmetry.
4.2 Quantitative Shape Recovery of ZGCs with
Parallel Symmetry
We now describe our technique of quantitative shape recovery for ZGC surfaces.
R em em ber th a t presence of ZGC surfaces is indicated by observing the properties
given in theorem 4. T he constraints described in section 3.3 will be applied to
43
ZGC surfaces to recover the local surface orientations. In order to apply these
constraints we need to recover the rulings of ZGC surfaces.
4.2.1 R ec o v erin g R u lin g s
We can infer the rulings of the surface by joining the corresponding points on the
two curves forming the parallel sym m etry by straight lines, as shown in figure 3.1
(c) (the corresponding points on th e two curves have the same tangent). Note
th a t the orientation of a ZGC surface does not change along a ruling (this is also
proved as a byproduct of th e above proofs in th e appendix). Therefore, if we find
th e orientation of th e surface at a single point on a ruling we can extend it along
th e ruling.
4 .2 .2 A p p lic a tio n o f C S B C an d ISC
T he curved shared boundary constraint (CSBC) given in section 3.3.1 is applica
ble along the parallel sym m etry curve which is generated by th e intersection of
the ZGC surface and the cross section plane (the top plane in figure 4.5). In dis
crete dom ain we need to quantize (p(s),q(s)) as (pt -, 3). Thus
in general, it m ay not be possible to find an in terp retatio n for th e contours such
th a t th e surface obeys all the given constraints exactly. This is directly related
to th e conflict introduced in section 3.3.3, th a t strict orthogonality forces parallel
sym m etry curves to be curves of m axim um curvature which are not planar. How
ever for special b u t im p o rtan t cases, these set of constraints are dependent and
m ay give a unique answer or even leave one degree of freedom unconstrained. In
the following, special cases of cylindrical surfaces and circular cone are analyzed
th en a general algorithm is given for all types of ZGC surfaces.
4 .2 .3 .1 C y lin d rica l S u rfaces
A cylindrical surface is a ZGC surface for which rulings are parallel to each other,
in 3-D. An exam ple is given in figure 4.6(a). Let this surface be param eterized
by W (it,u) = (x (u ,u ), j/(u, u),;z(u,d)) such th a t u is along th e axis of sym m etry
and v is along th e rulings. As we move along th e axis of sym m etry let th e angle
betw een th e tangent of the axis of sym m etry and th e horizontal be a(u) note th a t
a is a function of u only, and let the angle betw een th e ruling and th e horizontal
be /? as in figure 4.6. Note th a t 0 is constant since all rulings are parallel. We
can always ro tate the coordinate system to m ake /3 equal to II/2 , as in the figure.
W ith these angles we have
X u = (xu, yu, zu) = (cos or, sin a, zu)
X v — (xv,yv,z v) — (cos/?, sin /?, zv) = (0, l , z v) (4.6)
O ur purpose is to com pute th e surface orientation (p(u),q(u)) along th e axis
of sym m etry. A pplying inner surface constraint in equation 3.8 gives
pux v + quyv — 0 = $ ■ q(u) = q0(constant) (4.7)
46
CSBC
ISC
( a )
Figure 4.6: (a) A cylindrical surface w ith axis of sym m etry and th e rulings
m arked, (b) th e constraints ISC, CSBC and the orthogonality for th e cylindrical
surface
T h a t is th e ISC curve is a horizontal line on the p — q plane as shown by
d o tted line in figure 4.6(b).
Say th e orientation of th e cross section plane, is ( <7c)• Then the curved
shared boundary constraint gives:
Xu(pc - p(u)) + yu(qc - qo) = 0
co s(a(u ))(p c — p(u)) + sin(o:(n))(^c — qo) = 0
- » > - + » . < « )
cos(a(tt))
T h a t is, if we fix pc,qc and qo then the surface orientation (p(u), qo) for all
values of u is fixed. T he last constraint is given by orthogonality as given in
equation 3.21. Since ( 3 = 7r/2 we have:
sin (a(n )) + qop(u) cos(a(u)) + q$ sin (a(u )) = 0 (4-9)
su b stitu tin g p(u) by 4.8 in above equation gives
sin (a (u ))(l + q0qc) + Pc cos(« (« )) = 0 (4.10)
Since th e above equation is equal to zero for all values of u then we both have
pc = 0 and
l + ?o9c = 0 =4 q0 = - l f q c (4-11)
47
W ith th e orthogonality constraint we have pc = 0 and qo = — 1 / qc leaving
qc as a variable. T h at is th e three constraints CSBC, ISC and OC, are satisfied
for a cylindrical surface and still one degree of freedom , nam ely qc, rem ains. In
section 4.2.4 we describe a m ethod to estim ate qc. T he m ethod uses th e shape of
th e parallel sym m etry curves.
4 .2 .3 .2 C ircu lar C o n es
A circular cone is a Linear Straight Homogeneous G eneralized Cylinder LSHGC
whose cross section is a circle. The im portance of circular cones is th a t these are
the only ZGC surface th a t have a unique (2 corresponding to Neckers reversal)
solution to th e three constraints (ISC, CSBC and OC) given before. In [UN88] we
have analyzed the im age of a cone under these constraints and a unique solution
is found which is also in agreem ent w ith th e assum ption th a t the ellipse of the
cross section in the im age plane is th e projection of a circle in 3-D. The details
are o m itted from th e thesis.
4 .2 .3 .3 G en era l Z G C S u rfaces
For surfaces other th an cylindrical surfaces and th e circular cone, the three con
straints can not be satisfied exactly. We believe th a t in m ost cases the planarity
assum ption is stronger th a n th e orthogonality assum ption. Therefore, the fol
lowing process tries to m axim ize the orthogonality while keeping the constraints
ISC and CSBC satisfied exactly.
As discussed in section 4.2.2 there are 3 degrees of freedom left for recon
structing a ZGC surface. T he free variables are (pc,? c) and d. We choose the
values for these free variables th a t m inim izes th e orthogonality error:
E = £ c o s 0 t - (4-12)
»=l
48
W here 0, is the angle betw een the two 3-D vectors (A and B in figure 3.9 whose
projection on th e im age plane m ake angles a ; and w ith th e horizontal, cos 0t
is given by
(cos(aj — 1 3 j) -f (p i cos c tj + q j sin a,•)(/?,• cos ( 3 j -f q % sin A ))2
(1 -f (p i cos a i + q i s in a i)2)(l + (p i cos # + q i sin # )2)
H ere (p.;, q, ) are dependent on (pc, qc) and d as given by constraints ISC and CSBC.
We w ant to m axim ize th e orthogonality by m inim izing the above function E for
(pc, qc) and d. We can convert this problem into a 2-D m inim ization problem by
associating a d value to each choice of (pc, qc) th a t m inim izes E.
U nfortunately, for a general ZGC surface the global m inim um for E occurs
when (p c, qc) = (0, 0) and d = oo; this is an infeasible in terpretation. However, the
function E, in term s of (po^c) has a “valley” of local m inim a (passing through
the origin of the p — q space) and the valley is typically a straight line. Any
choice of ( p c , # c ) along this valley is essentially equally acceptable, i.e., we have
one degree of freedom to fix. In section 4.2.4 we discuss how to choose a specific
value of (pc,qc) on this line using th e shape of th e cross section.
4 .2 .4 E stim a tin g (pc, qc)
As discussed in section 4.2.3 the previous three constraints (ISC, CSBC, OC)
leave one degree of freedom, such th a t constraining the orientation of th e cross
section plane, ( p c , # c ) , is constrained to be along the m inim um line of th e function
E. It is expensive to com pute this m inim um line. Instead we use the following
gradient descent algorithm to com pute (pc,g c).
1. Choosing a starting line, l0, passing through th e origin, in th e p — q space,
in th e direction of th e skew sym m etry axis. Set the current line I = Iq-
2. C om pute th e (pc,q c) for the line I using the m ethod described below.
3. C om pute th e value of E for (pc,gc), check if (pc,qc) is along the m inim um
line of E by repeating the above process for lines ±60 degrees off the line
and by com paring th e E values for these lines.
49
4. If (pc,qc) is along the m inim um line of E stop. O therw ise choose another
line by rotating the line I by 66 degrees in the direction of descending E,
and go to step 2.
For th e above algorithm we use the im age axis, the line joining the m id points
of th e lines joining end points of parallel sym m etry curves, as the startin g line,
lo-
4 .2 .4 .1 C o m p u tin g (pcQc) g iv en a lin e /:
We ro ta te the coordinate system such th a t th e line I is aligned w ith th e q axis of
th e p — q plane then we have pc = 0 and qc is the unknow n quantity.
To fix qc, we use the shape of th e cross section. If the cross section is skew
sym m etric, th e lines and the axis of sym m etry are m ade orthogonal in 3-D. Ap
plication of this orthogonality constraint, given in equation 3.21, restricts (pc,9c)
to be on a hyperbola in the p — q space as given in figure 2.2 (b). T he intersection
of th e line I and this hyperbola uniquely determ ines (pc, qc). If the cross section is
not skew sym m etric, we propose another m ethod based on perceptual properties
rath er th an on m athem atical constraints. Specifically, our inform al studies of
hum an perception indicate th a t we prefer com pact shapes (as also observed in
[BY84]) and th a t we prefer interpretations w here the slant of th e surface is not
very high or very low.
O ur m ethod is based on th e following observations of hum an perception th a t
• We prefer com pact shapes,
• We prefer m edium slant to very high or very low slant and
• We have a large range of uncertainty for the perceived slants.
Based on these observations we propose a two stage m ethod for determ ining
qc. F irst we estim ate a value for qc then we u p d ate it w ith a bias tow ards 45°.
50
Fiist approximation to q « q
(a) (b)
Figure 4.7: (a) A cylindrical object and th e ellipse fitted to th e cross section, (b)
th e orientation (pe,qe) th a t would m ake th e ellipse a circle and its projection on
th e q axis gives qe, first approxim ation to qc.
For th e first estim ation, an ellipse is fit to th e cross section and back projected to
an orientation th a t makes it a circle (apart from being m uch faster, this has the
advantage over (area)/ (perimeter)2 m easure used by B rady and Yuille [BY84]
in th a t it does not require th a t closed contours be given ). T he two steps are
described in detail below.
F irst E stim a tio n o f qc: An ellipse fitting process is utilized as a first approx
im ation for qc. An ellipse is fit to th e cross section contour, then the orientation
of the circle (pe,qe), th a t would project as th e fitted ellipse is projected on the q
axis, on th e p — q plane to obtain th e first approxim ation of qc, call it qe. Figure
4.7 shows an exam ple. N ote th a t there are two values of (pe,qe) th a t m ake a
circle project as the ellipse in the im age plane. These correspond to a N ecker’s
reversal and we choose th e one th a t gives a solid shape in terp retatio n to the one
th a t gives th e in terp retatio n of a hole.
51
T he behavior of the m ethod is dependent on th e choice of th e ellipse fitting
algorithm used. We have experim ented w ith two different ellipse fitting algo
rithm s. F irst one is based on the scattering of the boundary points. Covariance
m atrix of the equally spaced contour points is com puted by:
c = i n * * - * ) 2 l T . { x i - x ) { y i - y )
^ £ 0 * - - x)(y{ - y) ^ J 2 ( y i - y ) 2
W here (Xi,yi) are th e equally spaced boundary points and (x,y) is th e mean.
T he scattering of these contour points is given by th e eigenvalues ei and e2 of
th e covariance m atrix C. Say th e un it vectors v\ and v2 are eigenvectors of C
corresponding to ei and e2 respectively, then we can approxim ate the cluster
of points w ith an ellipse whose m ajor and m inor axes are in the direc
tions v\ and v2 w ith m agnitudes yjei/2 and yJe2/2. This m ethod is quite robust
when the contour is closed, however for open contours, th e m ethod consistently
underestim ates th e eccentricity of the ellipse.
The second m ethod is a regular least squares fit of the param eters of the
q uadratic representation of the ellipse to th e boundary points. This m ethod is
robust when th e contour is sim ilar to an ellipse w hether it is closed or not, but
m ay give a bad fit if the contour is not sim ilar to an ellipse. We apply both
m ethods to a contour and choose the one having the sm aller fit error (the e in
equation 4.15).
If th e cross section has repetitive parts, as in figure 4.8, then th e slant percep
tion is governed by th e shape of individual parts rath er th an th e overall figure.
For closed cross sections we segment them by finding the concavities on the two
sides and then m atching them . For open cross sections (as in figure 4.8 (e)) we
sim ply segm ent it at inflection points. An ellipse is fit to each p art and corre
sponding qe s are com puted for each part. T he qe for the whole cross section
is given by weighted average of th e qe values, where the weight is given by the
length of th e curve to which the ellipse was fit. Note th a t this is different than
th e segm entation of th e cross section described in section 3.2.2 which is based on
52
(a)
Figure 4.8: O bjects and ellipses fit for th eir cross sections. T he cross sections of
the objects are segm ented based on th eir concavities (or inflection points) and
the whole cross section has th e sam e slant.
broken skew sym m etry axis and results in a different slant com putation for each
segm ented p art as in figure 4.4.
Figure 4.8 shows ellipses fit to th e cross sections of various objects. The
objects in (b) and (c) are segm ented by th e above m ethod and an ellipse is fit to
each part.
U p d a tin g qc: T he purpose of this updating process is to sim ulate th e behavior
th a t hum ans have in preferring m edium slant to very high and very low slant.
We u p d ate qe to obtain th e final qc as follows (after converting qc into degrees):
qc = 45° + X(qe - 45°) (4.14)
W here A is a confidence factor in th e range [0,1] and is a function of how
well th e ellipse approxim ates th e cross section curve. Intuition suggests th a t the
b e tte r the approxim ation of th e ellipse the higher the value of A should be and
th e closer the qc is to the 45° th e less th e correction should be. T he A we are
using :
A(£) = ( l - £ 2) (4.15)
W here e is the ellipse fit error given by d /yfab, where d is the average distance
of the contour points from th e fitted ellipse and a and b are th e half-lengths of
53
th e m ajor and m inor axes of the ellipse. We use an approxim ation (not described
here) to com pute d. Note th a t e is in the range [0,1].
We believe th a t th e exact form of the function is not critical. Small changes
in qc do not radically affect th e perceived surface shape and hum ans too estim ate
qc rath er imprecisely.
V a lid a tio n We have conducted a psychological experim ent w ith hum an sub
jects on th e perception of qc for cylindrical and conic objects. Results of the
experim ent show th at; th e standard deviation of th e perceived angle for th e top
plane is quite high, w ith an average standard deviation, a, of 8°. T he interval of
uncertainty for the slant of each object, which is th e angle interval th a t contains
90% of th e responses given for th a t object, is 24°. T he com parison shows th a t
th e algorithm perform s quite well for a variety of shapes. T he average of th e dif
ferences betw een th e m ean of the hum an response and com puted slants is only 6°
(sm aller th an th e average standard deviation of hum an responses). In appendix
B we provide th e details and results of this analysis.
4 .2 .5 C o m p u ta tio n a l R e s u lts
For the results shown in this section the following im plem entation is used. The
input to the program are the segm ented curves th a t define th e contour of each
object. These segm ented curves are grouped into closed regions using continuity.
Each closed region is taken to correspond to an object surface. N ext, we find
sym m etries am ong segments of a surface. Every segm ent in a surface is checked
for parallel sym m etry against every other segment in the surface. Two segm ents
are considered to be parallel sym m etric, if they retu rn a low parallel sym m etry
error defined as;
j^ J o \ / ( 4 0 ) - x i(p a ))2 + (y'2(s) - y[(ps))2ds (4.16)
54
w here th e segm ents Ci(.s) = (ari(s), y-i(s)) and C 2{s) = ( ^ ( s ) , V2{s)) are para
m etrized in term s of their arclength s, p = I1/I 2 is a scaling p aram eter where h
and l2 are th e lengths of the segm ents C\ and C2.
T he above error m easure is effective only if the entire lengths of two segments
are parallel sym m etric to each other. Also, this m easure is lim ited to linear
parallel sym m etry (found in cylindrical and conic surfaces).
Segm ents are also checked for having the same curvature sign at th e corre
sponding points. This m easure is especially useful when th e segm ents are alm ost
straight, in which case th e error m easure given in equation 4.16 m ay be low even
if th e segm ents are not parallel sym m etric.
T he surfaces containing parallel sym m etric segment pair are tre ated as curved
and others are treated as planar. For curved surfaces th e curves joining parallel
sym m etric curves are checked if they are straight to confirm th a t the surface is a
ZGC. T he curved surfaces are associated w ith their planar neighbors, which will
be treated as th e cross section having th e norm al (pc> < 7 o 1)-
For each object th e orientation of the planar cross section, ( p c ><7c), is com
pu ted using the m ethod describe in section 4.2.4. Then th e angle param eter d is
com puted by m inim izing th e orthogonality error E given in equation 4.12. T he
surface orientation, [pll < ? ,- ) at each point th en is com puted by using constraints
ISC and CSBC as illustrated in figure 4.5.
In th e following, we show results on some synthetic exam ples as well as a
real image. Evaluation of shape from contour results is difficult as there is no
real “ground tru th ” . Even when contours are derived from a real object, or from
projection of a synthetic object, th e sam e boundaries could have been derived
from a projection of infinitely m any other real or synthetic objects. Thus, in a
sense, the only good m easure of the perform ance of our algorithm s is a com parison
w ith hum an perform ance. We use two graphical m ethods to display th e com puted
orientations. T he first one shows the surface norm als as oriented needles along
one cross section. T he orientations along other cross sections are th e same, as the
orientations are tran slated along rulings. T he orientation and length of a needle
55
is th e projection of a unit surface norm al at th a t point (for us to perceive 3-D
orientation from this requires solving a shape from contour problem in itself). The
second m ethod is to display th e surface orientations by constructing a synthetic
shaded im age from th e reconstructed surface, by assum ing L am bertian reflection
and a point source of light (for hum ans to perceive this requires solving a shape
from shading problem ). We believe the needle diagram s to be m ore effective th an
shaded images for this purpose. We present graphical results for a reader to make
h is/h e r own judgm ent. U nfortunately, this can only give qualitative rath er than
q u an titativ e evaluation. For th e real im age, we also give a com parison w ith the
real object.
4 .2 .5 .1 S y n th e tic Im a g es
Figure 4.9 shows com puted surface orientations using our m ethod from contours
of objects in figure 4.1. T he input to the algorithm are th e curves (given as a list of
points) defining the contour for each object. In our judgm ent, the reconstruction
is consistent w ith hum an perception of the given figures. It is w orth noting th a t
for all th e objects the com puted orientation at th e lim b boundaries of th e objects
is orthogonal to the boundary, even though this is not an explicit constraint in
our m ethod.
T he cross section of the object in the bottom last row is segm ented into
two planar sections based on the observation of th e skew sym m etry of th e cross
section. Each section is processed individually b u t th e inner surface constraint is
required to apply betw een th e two sections of the object.
4 .2 .5 .2 A R ea l Im a g e E x a m p le
To apply our m ethod to real images, we need to first find th e boundaries of the
objects and then the sym m etries, if any, contained in them . In general, we can
expect object boundaries to be fragm ented, and several intensity boundaries th a t
correspond to surface m arkings, shadows and noise to be present. To separate the
56
Figure 4.9: Sam ple contours, th e needle images com puted and th eir images after
shading th e object w ith the com puted orientation at every point on th e surface.
57
(a) (b) (c) (d)
Figure 4.10: T he processing of a real image; th e cone im age, edges, com puted
surface norm als and th e shaded image w ith the com puted surface norm als.
object boundaries from these other boundaries, and to fill in th e gaps in object
boundaries as appropriate, is a difficult problem in m onocular im age analysis and
this thesis is not about such analysis. P erceptual grouping has been suggested
as one solution to such problem s and in our group we have developed such tech
niques th a t we believe are p art of the solution to these problem s [MN89]. Also,
th e fact th a t we are seeking certain specific relations betw een curves should help
in th e process of perceptual organization. Here, we only show an exam ple where
boundaries can be detected cleanly and no surface m arkings, shadows or high
lights are present. Nonetheless, we still deal with the noise in the location of the
detected edges and the effects of this noise on com puting tangents.
In th e exam ple to be shown, we first detect edges in the im age using a Canny
edge detector [Can86]. Edges are then linked into curves. T he curves are seg
m ented into sm aller curves by detecting corners using a m ultiscale version of the
curvature based corner finder described in [MY87]. T hen th e segm ented curves
are given to th e reconstruction system as before. T he tangent of th e axis of the
parallel sym m etric curves (necessary for th e curved shared boundary constraint)
is com puted by convolving the larger of th e sym m etric curves w ith w ith a first
derivative of G aussian kernel having a large standard deviation (a = 10.0), to
sm ooth out th e noise.
In figure 4.10 we show the results on a real image. T he im age (245 x 300 x 8)
is th a t of a circular cone. Figure 4.10 (b) shows edges, figure 4.10 (c) shows the
58
recovered surface norm als as needles, and figure 4.10 (d) shows a reconstructed
im age assum ing L am bertian reflection and point source of light. We believe th a t
th e results agree well w ith hum an perception of th e original image. T h e average
error of th e surface norm al from the actual cone is about 5° (the average error
in tangent estim ates of the im age curves is also about 5°). M ost of this error is
concentrated near the lim b boundaries where th e im age tangent estim ates also
have higher error. N ote th a t th e im age was obtained by perspective projection
b u t we have processed it as if it were obtained by an orthographic projection.
For this exam ple, th e difference in the two projections apparently does not create
a large error.
This exam ple is intended to dem onstrate th a t our reconstruction algorithm is
robust enough to work w ith real images at least those obtained under controlled
conditions. We do not, however, claim to have solved other problem s of m onocular
im age analysis.
4.3 Quantitative Shape Recovery of ZGCs Cut
by Non Parallel Planes
If a ZGC surface is cut by two non-parallel planes, we get a line-convergent
sym m etry instead of a parallel sym m etry. In order to com pute line-convergent
sym m etry betw een two curves on a general ZGC surface we m ust try all possi
ble m onotonic point correspondences betw een th e curves. This is a very costly
search. However, for th e case of cylindrical and conic surfaces, th e com putation
of line-convergent sym m etry is m uch simpler. M ost of the ZGC surfaces th a t
we encounter in our environm ent are in fact cylindrical or conic surfaces. M ore
over, we can always segment a general ZGC surface into cylindrical and conic
sections (m ostly at the inflection points) and process each section in d iv id u al^
w ith appropriate constraints applied along the lines of segm entation.
59
A p e x
Figure 4.11: A conic surface w ith line-convergent sym m etry.
For conic surfaces correspondence of line-convergent sym m etry is restricted
to be along th e lines th a t pass through a com m on point, th e apex of th e cone.
In figure 4.11 line-convergent sym m etry is shown for a cone. T he com putation of
line-convergent sym m etry for conic surfaces is, therefore, restricted to checking
for the correspondences betw een the curves such th a t the rulings, when extended,
intersect at a single point on th e image plane. T he process is further simplified
when th e end points of the curves are available. In th a t case the apex point
can easily be com puted on th e im age plane and it is only needed to check the
correspondence for th a t apex point. For cylindrical surfaces th e apex point is at
infinity, therefore, the direction of the apex is used rath er th a n th e location of it.
4 .3 .1 S h a p e R e c o v e r y
Process of recovering local surface norm als for ZGCs cut by non parallel planes
is sim ilar to th a t of recovering ZGCs cut by parallel planes. F irst we need to
decide which cross section curve is to be m ade orthogonal to th e rulings. Figure
4.12 shows three possibilities. In (a) the general preference is to m ake the top
cross section curve orthogonal, in (b) the b o tto m one is preferred and in (c)
we prefer th e m iddle curve, which is the axis of line-convergent sym m etry. In
60
I,III' " " 1
l,‘llllllllfllllllllll"',l"“
(a ) (b ) (c)
Figure 4.12: T hree ZGCs cut by non parallel planes.
our im plem entation the curve th a t “looks” orthogonal to th e im age axis, which
is the line joining the m id points of th e lines joining th e end points of the line-
convergent sym m etry curves, is chosen as th e curve th a t will be m ade orthogonal.
How orthogonal a curve looks is determ ined by th e angle betw een th e im age axis
and th e line joining two ends of th e curve.
For a ZGC surface cut by non parallel planes there are two additional un
knowns which are th e gradient param eters of th e second cutting plane. Also,
there are two additional constraints on th e orientation of th e planes cutting the
ZGC surface. T he first one is a shared boundary constraint; for a ZGC surface
S, w ith line-convergent sym m etry, let (pt,qt) be the gradient of th e top plane
and let (pb, qb) be th e gradient of th e bottom plane, and let th e intersection line
have direction (lXJly) on th e im age plane. Since the top and th e bottom planes
actually intersect each other along th e line I in 3-D we have the shared boundary
constraint as;
(P t Pbt qt ~ qb) ' (C> ly ) — 0 (4-17)
61
Consider figure 4.13; let (pr,qr) be th e local surface gradient of th e surface S
along ruling r. This constraint enforces th a t th e 3-D lines tx and t2 be on the
sam e tangent plane having gradient (pr,qr )• T he constraint is:
t\ x r = t2 x r (4.18)
in long form;
(a?! - x, y x - y, ~ ( p b( x i - x) + qb{ y x - y))) x
(;x 2 - 2/2 — 2/1, ~ ( P r ( x 2 - x x) + qr (y2 - t /i) ) ) =
( x 2 - x, 2/2 - y, ~ ( P t ( x 2 - x ) + qt(y 2 - y ) ) ) x
( x 2 - Xi, V2-yi, ~ { p r { x 2 - Xi) + qr (y 2 - y i ))) (4-19)
W ith these two constraints we can com pute th e orientation of th e plane con
taining th e curve th a t is not chosen to be m ade orthogonal. Once the orientation
of the cross section th a t is chosen to be m ade orthogonal is com puted, com puta
tion of the local surface norm als is exactly th e sam e as for ZGCs cu t w ith parallel
planes. T he curved shared boundary constraint is applied betw een th e orientation
of th e chosen cross section and th e ZGC surface (in the case of parallel sym m etry
this constraint is applied betw een any cross section and th e ZGC surface). Inner
surface constraint is applied along th e sam e cross section.
4 .3 .2 R e s u lts
A ctual com putation of local surface norm als for ZGCs cut by non parallel planes
is perform ed by the algorithm for m ultiple ZGC surfaces described in chapter
5. T he m ultiple surface algorithm finds a best fit solution th a t m inim izes the
error of not satisfying each constraint available for these surfaces. In a single non
parallel cut ZGC surface case, the algorithm ignores the constraints obtained
from intersection of m ultiple surfaces (except for the intersection of th e ZGC
surface and th e cross section plane).
62
(x,y)
(x2,y2)
Figure 4.13: C onstraints on th e orientation of the cutting planes of a ZGC surface.
Figure 4.14 shows th e results for ZGC surfaces cut by non parallel planes.
T he figure shows th e local surface norm als as oriented needles and by a shaded
im age obtained by using a Lam bertian shading m odel. For planar surfaces we
use a little coordinate fram e, w ith two direction lines orthogonal to th e surface
norm al are joined by a line, instead of a single line parallel to th e norm al of the
surface, to enhance th e perception of the com puted surface orientation.
63
v S\l/'///M " "
Figure 4.14: Recovering th e surface shape of a ZGC cut by non parallel planes.
64
Chapter 5
Multiple ZGC Surfaces
M any objects of interest consists of several curved surfaces. Here th e recovered
3-D individual surfaces m ust be in agreem ent w ith th e neighboring surfaces, i.e.,
surfaces sharing a common boundary. We describe a technique for such integrated
m ultiple surface recovery for objects consisting of planar and ZGC surfaces. Fig
ure 5.1 shows some sam ple objects. T he problem of finding a consistent shape
for all the surfaces is form ulated as a constraint optim ization problem , where
the curved shared boundary constraints and inner surface constraints are satis
fied exactly and m inim ization is perform ed on assum ption driven constraints of
orthogonality of surfaces.
5.1 Representing Surfaces
In order to be able to include the contributions of the constraints from each sur
face and inter-surface constraints into pool of constraints, appropriate param eter
representation for each surface is very im p o rtan t. We use the following param e
terization for th e two surface types.
65
Figure 5.1: Some objects consisting of m ultiple planar and curved surfaces.
5.1.1 P lan ar Surfaces
For planar surfaces th e gradient space representation (p , q) of th e surface norm al
of the plane is used. This is the n atu ral and m ost versatile (for our purposes)
representation for planar surfaces.
5.1.2 Z G C Surfaces
Due to difficulties of com puting sym m etries for a general ZGC surface, and since
a ZGC surface can be decom posed into cylindrical and conic surfaces, in this
section we study cylindrical and conic surfaces only.
If a ZGC surface is cut by parallel planes, (i.e., it has parallel sym m etry),
then it has three degrees of freedom as discussed in section 4.2.2. In chapter
4 these are represented as (pcQc) and d, where (pc, qc) is th e gradient of th e
cross section planes and d is the angle param eter as stated in section 4.2.2. In
th e m ore general case of ZGC surfaces cut by non parallel planes, studied in
section 4.3, two additional param eters are involved, th e gradient of the second
cross section plane. In to tal a ZGC surface cut by non parallel planes have five
degrees of freedom . Since parallel cut ZGCs are special cases of non-parallel cut
ZGCs, we will always use th e non-parallel cut ZGC case. Of th e five degrees
66
of freedom , th e four param eters are the gradients (pt,qt ) and (pb, qb) of the top
and th e b o tto m planes cutting th e ZGC surface. T he fifth p aram eter is related
to th e angle param eter d introduced in section 4.2.3. However, here we will
use a different param eter th an the angle param eter. If th e angle param eter
is used, in order to com pute a local surface norm al at a given location, the
surface norm als for the whole surface needs to be com puted. This posed no
problem for th e single surface m inim ization problem described in section 4.2.3.3,
because for th a t m inim ization all surface norm als were needed at each step of
th e iteration. However, in m ultiple surfaces case the local surface orientations
need to be com puted at random locations (especially at the ends of the surface)
only. Therefore we use a different param eter in place of th e angle param eter,
th a t enables us to com pute local surface norm als at any point on th e surface.
We can m odel a conic surface by using any'3-D axis th a t goes through th e apex
of th e cone. We use th e 3-D axis th a t projects as the 2-D axis of th e straight edges
of th e cone. If the im age direction of the axis is {a,x,ay) then the 3-D direction of
th e axis in gradient space representation is (uax, uay) w here u is a free variable,
and it is th e fifth param eter of th e ZGC representation. Given (pt,qt)i (pf>,< 7& ),
and u, surface gradient can be com puted at any point on th e surface. Consider
figure 5.2, let (pt, qt) be the gradient of th e cross section plane th a t is chosen to
be m ade orthogonal to th e surface. T he gradient (p, q) of th e surface along the
ruling r is given by com bination of two linear constraints given below. T he first
one is th e curved shared boundary constraint, given in section 3.3.1, betw een
gradients (p, q) and (pt,qt) using th e tangent (x',y ') of th e intersection curve at
th e point th e curve touches the ruling r. T he equation of the constraint is;
(p - p t , q - qt) ■ (x',y') = 0 (5.1)
This constraint is shown in the gradient space by the line labelled T (x(, y')
in figure 5.2.
T he second linear constraint is inherently equivalent to th e inner surface con
strain t given in section 3.3.2. T he constraint is th a t th e 3-D gradient (pr,qr, 1) of
67
Figure 5.2: T he param eters of a ZGC surface and the constraints in th e gradient
(Pi < ? ) o;f th e surface along the ruling r.
th e ruling r m ust m ust be orthogonal to the 3-D gradient, (p, q, 1), of the surface
along ruling r, th a t is;
(p,q, 1) • (Pr,tfr, 1) = 0 (5.2)
In figure 5.2 this constraint is shown by the line labelled _ L (pr,qr), which
is th e orthogonal line of th e gradient (pr,q T ), i.e., th e gradient of th e set of
th e directions th a t are orthogonal to (pr,qr) in 3-D. Note th a t this line is also
orthogonal to th e 2-D direction of th e im age of th e ruling.
T he gradient (pr, q r ) of th e ruling r is obtained by reconstructing the axis line
( a x , a y ), and th e line betw een points (x , y ) and (x p , y p ) in 3-D (i.e., com puting
th e z coordinates of these image points). Since the gradient of the axis line is
(u a x , u a y ), fixing zp — 0 , the z coordinate of the point (Xtip , y u p ), Ztip , is given
by;
Ztip
'Etip 3 C p
(5 .3 )
T he z coordinate of th e point ( x , y ) is com puted using the gradient ( p t , q t ) as;
z — p t ( x - x p ) + q t ( y - y p ) (5.4)
T hen th e gradient (p r , q r ) of th e ruling is given by;
1
( P r , q r ) =
Ztip — Z
(x t ip x , y tip y ) (5.5)
68
5.2 Combined Shape Recovery
The shape of all the surfaces is recovered sim ultaneously by finding appropriate
values for th e param eters of each surface. T he values of the surface param eters
are com puted by solving th e following constraint m inim ization problem;
m in Ei subject toEx = 0 (5-6)
w here Ei stands for error term s resulting from internal constraints of each surface
and E x are th e external term s, th a t is, th e constraints obtained by intersection
of surfaces.
B oth internal and external constraints are based on three simple essential
constraint functions. These constraints have been defined before, however, here
we redefine th em such th a t the error given by these constraints is uniform and
norm alized, i.e., in th e range [0,1.0]. T he gradient space is not uniform , i.e., a
constant shift at th e center of the gradient space corresponds to a larger vector
difference in 3-D th an th e same shift somewhere farther away from the center.
Therefore, th e uniform ity of the error function im plies th a t th e error returned by
th e function depends on the 3-D vector differences rath er th an th e differences in
gradient space. T he draw back of this norm alization is th a t linear error functions
are no longer linear.
T he first constraint function is the redefined shared boundary constraint given
in equation 3.6. This constraint is applied betw een two gradients. (pi,qi) and
(P2 j< ? 2 ), and a 2-D vector (x,y). The constraint is;
o z ^ v 'k ((P2 ~ Pi)x + (q2 ~ qi)yY ft. ^
S B C (p 1,qi,p2,q2,x ,y ) = —---------- — — ---------- • (5.7)
{{P2 - Pi )2 + {q* ~ qi )2 + l)(z 2 + r )
T he second constraint function is equality of two gradients (pi, qi) and (p2, q2).
Since the gradient space is-not uniform , using Euclidean distance betw een vectors
(pi,pEp + Y2 w°Eo + Y2 wcEc (5.10)
W here each ws are weight and Ep is th e error term for th e orthogonality constraint
of th e planes, E 0 is th e error term for the orthogonality constraint of th e ZGC
surfaces and E c is the error term for th e im plicit constraint of th e param eters of
ZGC surfaces. These error term s are described in m ore detail in the following.
E p is th e error term for the orthogonality constraint of planes. If a planar
surface has a skew sym m etry th en this is the orthogonality function of th e lines
and axis of skew sym m etry as given in 5.9, where and (^ 2 , 2/2) are the
im age directions of the lines of sym m etry and the axis of sym m etry and (p, q) is
th e gradient of the plane. wp is th e weight of E p and is proportional to the total
length of th e contour enclosing th e surface. T he form ula used for wv is wp — y/Tc
w here lc is th e to tal length of th e curve enclosing th e surface. If the surface does
not have skew sym m etry E p is zero.
70
E 0 is th e error term for th e orthogonality of ZGC surfaces. This error m easure
is an approxim ation of the one given by £ in equation 4.12. However com puting
£ requires com putation of local norm als at each point on th e surface at each
iteratio n of m inim ization, which is very tim e consuming. Therefore, we use an
approxim ation of the tru e orthogonality error as follows (this error term is exact
for cylindrical surfaces and is an approxim ation for conic surfaces):
E 0 = E a x + E t (5.11)
w here E a x = cos2 (a ), and a is the angle betw een th e gradient (pt, qt) of th e plane
containing th e parallel (or line-convergent) sym m etry curve th a t is decided to be
m ade orthogonal and the direction of the im age axis (ax,ay). E ax forces the 3-D
orthogonal cone axis to be as close as possible to th e image axis. E t = (u in t — u )2
w here u is th e u-param eter of the ZGC surface and Uint is set at the initialization
by m inim izing orthogonality error £ given in equation 4.12. wQ is th e weight of
the orthogonality term and is proportional to th e to tal length of the perim eter
of th e surface, w0 = y/ Tc where lc is th e total length of th e contour enclosing the
surface.
E c is th e error term for im plicit constraints of th e param eters of ZGC surfaces.
Let (pt,qt) and (pb,qb) be th e gradients of the planes containing the two parallel
(or line-convergent) sym m etry curves of the ZGC surface. If the ZGC surface
has a parallel sym m etry then (pt,qt) should be equal to (Pb,qb), therefore, E c =
Eq(Pt,qt,Pb,qb), where E q () is given in equation 5.8. If th e ZGC surface has
an line-convergent sym m etry then E c is th e addition of th e constraints given in
equations 4.17 and 4.19. wc is the weight and is inversely proportional to the
eccentricity of the parallel (or line-convergent) sym m etry curves. If th e parallel
(or line-convergent) sym m etry curves are highly eccentric, i.e., th ey are alm ost
straight, then th e weight of this constraint is low. T he form ula for wc = 1/ecc,
were ecc is the eccentricity of the to tal cross section curve (the eccentricity of a
71
curve is given by y e i / e 2 where e\ and e2 are the first and second eigenvalues of
the covariance m atrix of th e curve given in equation 4.13).
N ote th a t although ISC is an internal constraint of ZGC surfaces, it is not
included in in Ei, because th a t constraint is inherently included in th e param
eterization of the ZGC surfaces and in effect it is always satisfied exactly, not
m inim ized.
5.2.2 E x tern a l C on strain ts
E xternal constraints are the inter-surface restrictions im posed by each surface on
neighboring surfaces. E xtrem al constraints have th e following form:
Ex = w E v p + H wEpz + w E z z (5.12)
w here w is th e weight of each constraint and is equal to y/ Tc where lc is th e length
of the curve produced by intersection of the surfaces. Epp, E pz and E zz are the
error term s for shared boundary constraint between planes, betw een planes and
ZGC surfaces, and betw een ZGC surfaces respectively. In detail, the individual
error term s are:
E pp is th e error of th e shared boundary constraint betw een th e gradients of th e
two intersecting planes as given in equation 5.7.
E pz is th e error term for shared boundary constraint betw een a plane and a
ZGC surface. T here are two possibilities; th e intersection is along a ruling of the
ZGC surface or the intersection is along a parallel (or line-convergent) sym m etry
of the ZGC surface. If the intersection is along th e ruling of the ZGC surface
then Epz is the error from the shared boundary constraint betw een th e plane
and th e local surface norm al of the ZGC surface at th e ruling of intersection. If
th e intersection is along one of th e parallel (or line-convergent) sym m etry curves
then;
Epz = Eq(p,q,pt,qt) (5.13)
72
w here (pt,Qt) is the param eters of th e ZGC surfaces which is the gradient of the
plane containing th e intersection curve and (p, q) is th e gradient of th e planar
surface.
E zz is th e error term for th e shared boundary constraint betw een two ZGC
surfaces. T here are various ways two ZGC surfaces m ay intersect each other. Here
we only handle the intersections th a t produce a planar intersection curve. There
are two types of such intersections: along th e rulings of th e ZGC surfaces or along
th e parallel (or line-convergent) sym m etry of the ZGC surfaces. If th e intersection
is along th e rulings of the ZGC surfaces th en Shared Boundary C onstraint given
in equation 5.7 is applied betw een the local surface norm als of ZGC surfaces
at the ruling of intersection. If the intersection is along th e parallel (or line-
convergent) sym m etry curves, then, let ( p i, 9 i) and (p2,q2) be th e gradients of
th e planes containing the intersection curve in th e representations of the first and
th e second intersecting ZGC surfaces. T he error term is:
E zz = Eq(p1,q1,p2,q2) (5-14)
W hen two ZGC surfaces intersect each other along their parallel sym m etry
(or line-convergent sym m etry) curves, how orthogonal b o th surfaces can be m ade
depends on how parallel their im age axes are. Therefore we form a new orthog
onality error term E on for the intersecting ZGC surfaces to replace their original
orthogonality error term s (E0's). Let a be the angle betw een the im age axis of
these surfaces, let E 01 and E Q2 be th e error term s for the orthogonality of the
intersecting ZGC surfaces. Then the new com bined orthogonality error term is;
E on = cos2 ( a ) ( £ ol + E o2) + sm2(a)(EolE o2) (5.15)
E on em phasizes th e orthogonality of b o th of th e ZGC surfaces when th e image
axis are alm ost parallel to each other, and it em phasizes th e orthogonality of
either of the ZGC surfaces when th e im age axes are alm ost orthogonal to each
other.
73
5.2.3 S olvin g C on strain t E qu ation s
T he to tal error function E is solved using a constraint m inim ization technique,
where E x consists of “m ust-satisfy” constraints and Ei consists of assum ption
driven error term s as defined earlier. To solve this constraint m inim ization (where
th e constraints are non-linear), the problem is converted into a m inim ization form
as follows:
E = Ei + X E x (5.16)
E is m inim ized for successively larger values of A, thus, em phasizing E x m ore
at each m inim ization cycle. At the end E x constraints are satisfied alm ost exactly
and ECs are m inim ized to th e extent possible. T he set of param eters of the
surfaces m inim izing E is taken as th e solution set and used to reconstruct the
local surface gradients.
5.3 Results
Input to our im plem entation are clean segm ented curves. The surfaces and sym
m etries w ithin each surface are com puted by th e m ethod described in section
4.2.5.
Some surfaces are com bination of various curved surfaces and there is no
distinctive boundary between them . This is the case for the curved surfaces of
the left object in figure 5.1. Such surfaces contain m ore th an one parallel (or line-
convergent) sym m etry and they are segm ented into sm aller surfaces containing
only one parallel (or line-convergent) sym m etry. Figure 5.3 shows the segmented
surfaces and the sym m etries (skew, parallel or line-convergent) for each surface.
The constraints for each surface and inter-surface constraints including the
newly form ed intersections are extracted form ing the error function E. Initial
values of th e param eters of E, i.e., th e param eters of all th e surfaces involved in
E, are com puted by an initializer. The initializer starts w ith an arb itrary ZGC
surface, and sets its param eters as if it is an isolated surface. Then, the initializer
74
Figure 5.3: T he segm ented surfaces, and th e sym m etries com puted for each sur
face. T he skew sym m etry of planar surfaces are shown by crosses, th e long line is
th e axis of sym m etry and the short one is th e direction of the lines of sym m etries.
Parallel and line-convergent sym m etries are shown by their axis.
sets the param eters of the neighboring surfaces by keeping th em consistent with
th e first surface, and, th e neighbors of these surfaces are processed progressively
until th e param eters of all th e surfaces are initialized. T hen E is m inim ized by
constraint m inim ization discussed in section 5.2.3.
Figure 5.4 shows th e result of m inim ization for the objects in figure 5.1. The
com puted surface norm als are shown by needle diagram , as needles sticking to
th e surface in the direction of th e local surface norm als. For planar surfaces,
a sm all coordinate fram e is used to b ette r show th e com puted surface norm al.
We also provide th e shaded images of the objects shaded w ith com puted surface
norm als using a Lam bertian shading model.
It is interesting to note th a t, for the object on the bottom , th e m iddle sur
face, w ith a parallel sym m etry, is initially classified as a curved surface based on
th e parallel sym m etry of the surface. However, th e only consistent in terp reta
tion for th e whole object is th a t, th e m iddle surface is planar (like a m ountain
road), because the top surface is planar. A fter the m inim ization the com puted
75
orientation for th e m iddle surface is, in fact, planar (w ithin error bounds of the
m inim ization).
We believe th a t th e com putational results are in agreem ent w ith hum an per
ception.
76
Figure 5.4: T he needle and th e shaded images obtained from th e com puted sur
face norm als for the objects in 5.1
77
Chapter 6
Straight Homogeneous Generalized Cylinders
G eneralized cylinders (GCs) are obtained by sliding a cross section (m ostly pla
nar) along an arb itrary axis while scaling the cross section by an arb itrary scaling
function. GCs are used as a powerful m odeling tool by th e com puter vision com
m unity since early 1970s. T he ability of GCs to m odel a wide variety of objects
a ttra cted m any researchers. GCs were first introduced by Binford [Bin71] and
popularized by Agin and Binford [AB73] and N evatia and Binford [NB77a] then
by M arr [Mar82]. Shafer [SK83] classified generalized cylinders according to the
properties of th e cross section, axis and th e scaling function of th e GCs. He
identified an im p o rtan t subclass of generalized cylinders th a t he called S traight
Homogeneous Generalized Cylinders (SHGCs). This is the class of GCs where
th e axis is straight and th e planar cross section is scaled but kept homogeneous.
SHGCs have becom e the focus of m any researchers topics [Kas8 8 , PCM 89, GB90]
because of their ability to m odel a wide class of objects while keeping th e geom ery
relatively sim ple to deal with.
Straight Homogeneous Generalized Cylinders (SHGCs) are obtained by sliding
a cross section, say C , along a straight axis, say A. T he cross section is also scaled
as it is swept along th e axis by a scaling function, say r. We can param eterize
th e surface, S, of an SHGC, given th e planar cross section C{u) = (x(w), y(u), 0 ),
and th e scaling function r(t), as :
S(u,t) = (r(t)x(u),r(t)y(u),i) (6 .1 )
78
Cross sections
Figure 6.1: An SHGC along the 2 coordinate axis w ith both m eridians and cross
sections m arked.
T he axis of th e SHGC in this case is the z axis of th e coordinate system . An
exam ple is shown in figure 6.1. N ote th a t th e cross section curves are generated by
fixing t and varying u. We will call th e curves generated by fixing u and varying
t as th e meridians of th e surface. N ote th a t the cross sections of an SHGC are
p lanar because th e cross section function C'(w) is planar, and th e m eridians of an
SHGC are planar since th e SHGC has no tw ist in its sweep. Let m eridian edges
of an SHGC be edges th a t are along th e m eridians of the SHGC. Usually images
of SHGCs do not contain m eridian edges, however, such edges m ay be present if
th e cross section has a tangent discontinuity, a corner, (see b o tto m left SHGC
in figure 6.2). Figure 6.2 shows some sam ple SHGCs. N ote th a t, even though
th e bottom rightm ost exam ple does not seem to have a straight axis, it can be
described by an SHGC w ith a vertical axis located outside th e object.
6.1 Surfaces and Their Limb Edges
Limb boundaries are very im portant for describing surfaces they enclose. Here
we prove a theorem for the projection of limb boundaries th a t enables us to
prove im p o rtan t properties of the limb boundaries of SHGC and CGC surfaces.
T he definitions and theorem s in this section are valid for both orthogonal and
perspective projection geometries.
79
Figure 6.2: Sample Straight Homogeneous Generalized Cylinders.
D e fin itio n 2 Tangent line, L v, of a surface, S, at point, P, in a given direction,
V , is the line from the point P in the direction of the tangent of the curve, C ,
obtained by cutting the surface by a plane, n , that passes through P, and contains
the normal, N , of the surface at P and the direction given by the vector V.
Figure 6.3 shows an exam ple. It is a well known property in differential
geom etry [DC76] th a t the tangent lines, L v i, of a surface, S , at point, P , in all
Figure 6.3: Tangent line, L v, of a surface S at point P in direction V.
80
Figure 6.4: Tangent plane, Tp, of a surface, S, containing all the tangent lines at
point P
possible directions, Vi € R 3, are on a plane, Tp, called the tangent plane of the
surface at P . M oreover th e plane Tp is orthogonal to th e norm al, T V , of th e surface
at P. This property is shown graphically in figure 6.4.
N ext, we define lim b edges and their projections for sm ooth surfaces.
D e fin itio n 3 The limb edge of a surface is a viewpoint dependent curve on the
surface such that at each point on the curve the surface normal is orthogonal to
the viewing direction.
T he limb edges project on the im age plane as the bounding curve of th e sur
face. At these edges the surface sm oothly curves around to occlude itself. This
definition of limb edges holds both for orthographic and perspective projection.
Limb edges (also called “occluding contours”) can give some very im p o rtan t in
form ation about th e 3-D surface they come from; K oenderink [Koe84] has given
a nice analysis in previous work. We will show how th e lim b edges help us recover
3-D surface shape later in this paper.
T h e o r e m 7 All the tangent lines of a surface at a point, P , which is on a limb
edge of the surface for a given projection geometry, project as the same line on
the image plane.
P r o o f The proof involves a sim ple com bination of th e definition of lim b edges
and th e property of tangent planes. Since th e norm al of the tangent plane at
81
P (which is also th e norm al of the surface at P ) is orthogonal to the viewing
direction, th e tangent plane projects as a line on th e im age plane. Therefore all
the tangent lines at P , which are included in the tangent plane also project to
th e one line th a t the plane projects into. □
This theorem , though sim ple and rath er obvious, tu rn s out to be highly useful
in proving other im portant properties of limb boundaries.
6.2 Properties of SHGCs
Here we present and prove th e properties of the limb boundaries and the cross
sections of SHGCs. These properties enable us to identify SHGCs from images
of their contours.
T h e o r e m 8 For an SHGC, the tangent lines of the surface in the direction of
the axis from the points of any given cross section intersect at a common point
on the axis of the SHGC.
A proof of this theorem m ay be found in [SK83]. Figure 6.5 (a) graphically
illustrates th e property.
C o ro lla ry The tangents of all m eridian edges at th e points they intersect a
single cross section intersect the axis of the SHGC at a single point. Therefore
th e tangents of the images of the m eridian edges, at th e point they intersect
a single cross section, intersect the im age of th e axis in a single point, under
orthographic or perspective projection.
It has been shown by Shafer[Sha83] th a t th e lim b edges on an SHGC are not
necessarily planar. Therefore the lim b edges of an SHGC are not necessarily along
its m eridians, and the tangents of th e lim b boundaries at th e point they intersect
th e sam e cross section do not intersect the axis in 3-D. Figure 6.5 (b) shows the
lim b edge and its tangent for an SHGC after rotating it, to show th a t in 3-D the
tangent of th e limb edge does not intersect the axis of the SHGC. However, it has
82
Limb
Meridians
Figure 6.5: (a) An SHGC, and its tangent lines, in the direction of th e axis
em itting from a single cross section, intersecting at a single point on th e axis, (b)
T he tangent lines, Ti, of lim b edges are not th e sam e as th e tangents lines, Tm,
of th e m eridians in 3-D.
been shown by Ponce [PCM89] th a t under orthographic projection the tangents
of th e lim b edges, at the point they intersect the same cross section, intersect
th e im age of th e axis at a single point. Here we give a sim pler proof which is
independent of th e projection geometry.
T h e o r e m 9 The tangents of the projections of the limb edges at the points they
intersect the same cross section, when extended, intersect the image of the axis
of the SHGC at the same point.
P r o o f Say th e lim b edge intersects a given cross section at point P (see figure
6.5). Since th e tangent line Tm from point P in the direction of th e axis of the
SHGC (the tangent line of the m eridian passing through th e point P ) intersect
th e axis of the SHGC, by theorem 7, the im age of the tangent line Ti from point
P in th e direction of th e tangent of th e lim b edge project as th e sam e line as the
tangent line Tm and thus im age of th e line T\ intersect th e im age of th e axis at
th e sam e point as th e im age of the line Tm intersects. □
Since theorem 7 holds both under perspective and orthographic projection,
th e above theorem and th e proof hold for b o th of th e projection geom etries.
83
In the following we show th a t th e cross sections of an SHGC are parallel
sym m etric in 3-D w ith th e m eridian curves joining the parallel sym m etric points
of th e cross sections.
T h e o re m 1 0 The cross sections of an SHGC are parallel symmetric in 3-D with
each other such that the meridian curves join the parallel symmetric points of the
cross sections.
P r o o f Given th e param eterization in equation 6 . 1 for an SHGC, we have to
show th a t th e direction of the tangent of th e cross sections is independent of the
t param eter curve. The tangent of the cross sections (u p aram eter curves) is
given by:
= (r(t)x'(u), r{t)y'(u), 0 ) = r(t)(x'(u ), y'(u), 0 ) (6 .2 )
Clearly the direction of Su is independent of th e t param eter. □
C o ro lla ry T he projection of th e cross section curves of an SHGC are also
parallel sym m etric in the image plane. And the correspondence function is linear
because cross sections are obtained by scaling a reference cross section curve
w ithout deform ing it.
6.2.1 R ecoverin g th e C ross S ection s
We next show how to find th e projections of cross sections in the im age of an
SHGC, given th e images of its external contours. O ur m ethod does not require
com plete cross sections, b u t only th e part th a t lies on th e visible face of the SHGC.
However, we require th a t the SHGC be cut along its cross sections, otherw ise we
would not have a parallel sym m etry between th e im age curves of th e two extrem e
cross-sections (C* and Cb in figure 6 .6 ). We conjecture th a t hum ans too do not do
well if this condition is not satisfied. The following algorithm recovers th e im age
curves C % th a t correspond to the projections of th e cross sections of the SHGC.
For each point Pi € Ci do:
84
Figure 6 .6 : Im age of an SHGC cut along its cross sections. Im age of th e top cross
section curve is Ct, the bottom one is Cb and the lim b boundaries are on the left
Ci and on th e right Cr.
1 . Find the point Pci E Ct such th a t C[{Pi) ~ C^(Pci).1
2 . T ranslate th e cross section curve Ct such th a t th e point Pci € Ct coincides
w ith th e point Pi, obtaining th e curve Cit.
3. F ind the point Pcr € Ctt th a t m inim izes th e function f ( P cr) = + d^fdx
which is the am ount of scaling required to be applied on the curve Ctt to
bring the point Pcr to th e point Pr. T he quantities d\ and are th e length
of th e line segm ents from Pi to Pcr and from Pcr to Pr. It can be shown th at
local m inim a of the function /(• ) above gives th e correct point Pcr € Ctt
such th a t the limb boundary condition C'tt(Pcr) = C'T{Pr) is m et.
4. Scale th e curve Ctt by f( P cr) so th a t th e point Pcr m eets w ith th e point Pr ,
obtaining th e curve Ci.
T he curve Ci obtained by this algorithm is precisely the im age of th e cross
section curve between the points Pi and Pr of th e SHGC. Once th e correspondence
of th e points Pi and Pr betw een the limb edges Ci and Cr is obtained, we can
recover th e im age of the axis of th e SHGC by using theorem 9. Figure 6.7 shows
JT he = operator is used for parallelity o f vectors, that is, if Vi = V 2 then V i = AV2 for som e
nonzero scalar A.
85
Figure 6.7: Images of the cross sections and axes, th e dashed lines, recovered for
th e SHGCs in figure 6.2
the com puted images of th e cross section curves and the axes for SHGCs in figure
6.2. If th e parallel sym m etric points of th e cross section curves are joined, by
theorem 1 0 , we obtain the m eridian curves.
6.2.2 O b servin g S H G C s
If there are two parallel sym m etric curves w ith a linear correspondence function
such th a t they are bound by curves th a t has a straight axis when th e axis is
com puted by the above algorithm , then we can hypothesize th a t th e line drawing
results from an SHGC.
6.3 Quantitative Shape Recovery of SHGCs
To com pute the shape of an SHGC along each recovered cross section curve we
can apply th e constraints discussed in section 3.3 as they are applied to a ZGC
surface in section 4. For th e following; say th a t th ere are m cross section curves
and we would like to com pute the orientation of th e surface at n points along
86
a cross section. T hen we have 2nm unknowns, initially, corresponding to the
gradient (p,q) of the surface at nm points. Let (pc,qc) be th e orientation of the
cross section of the SHGC, which is th e sam e for all cross sections.
6.3.1 C S B C
T he curved shared boundary constraint applies betw een th e orientation, (pc,qc),
of the cross section curves Cj and the orientation, (pi,Qi) of each of th e point
on th e surface along a cross section. Note th a t (pc,qc) is th e sam e for all cross
section curves. T he curved shared boundary states th a t th e line in th e p — q
space from th e gradient (/y, qt ) of a point P,, 6 Cj to th e gradient (pc, qc) of the
cross section plane is orthogonal to th e tangent, Cj(Pi), of th e cross section Cj
at point Pi. Then the constraint equation is:
(Pc - P i , q c - q.) ■ C '(P ) = 0 VP; € Cj (6.3)
This provides n constraints along each cross section curve.
6.3.2 ISC
Inner surface constraint is applied along a cross section using th e tangents of
the m eridians at each point. The theorem 3 indicates th a t ISC is applicable
along th e cross section curves because cross section curves are parallel sym m etric
by theorem 1 0 w ith th e m eridian curves joining the parallel points of th e cross
section curves. Inner surface constraint states th a t change of th e orientation
(pi+i — pi, #m)j discussed in th e previous section. We
would like to m inim ize the function E for (pc, qc) and (pm, qm). However from our
experim ents we observe th a t m inim ization of E chooses values th a t are always
consistent w ith the assum ption th a t the 3-D axis of the SHGC is orthogonal to
its cross section.
If we enforce th e cross sections to be orthogonal to the axis of the SHGC, the
orientation (pc?5 c) of the cross section lies along a line in the p — q space th a t
passes through the origin and is in the direction of the im age of the axis of the
SHGC. This constraint also, in effect, enforces th e gradient (pm,qm) of th e plane
of the m eridians to be orthogonal to the gradient (pc,3 a) of the plane IIa containing the axis is com puted by per
form ing a search in th e gradient plane. T he objective of the search is to com pute
(paiQa) th a t gives a valid reconstruction. A valid construction is one th a t makes
the projection of th e cross section points Cp(Px) and Cp(P 2) exactly th e sam e as
th e points Px and P 2 on th e im age plane (see figure 7.4). We form an objective
function which is th e average distance, on th e im age plane, of th e reconstructed
and projected point Cp(P 2) to the point P 2 w hen Cp(Px) and Px is aligned exactly.
T hen this objective function is m inim ized for (pa,qa).
T he search is facilitated by finding a good initial point for (pa> < Z a ) using the
shapes of th e end cross sections. T he analysis in section 7.2 show th a t th e gradient
101
,\'' 1 /
Figure 7.7: T he recovered orientations shown by b o th needle im age and by shad
ing th e objects for th e PR C G C s in figure 7.1.
102
(.Pc, Q c) of th e cross section at one end is constraint to be on a line in th e gradient
space. A particular value on th a t line m ay be chosen by using th e ellipse fit
discussed in section 4. Sim ilar analysis applies to the other end of th e PRC G C
(if available). Say th e orientation of th e plane containing th e other end cross
section Cn is (pn, 9n)- Then the plane of Cn is orthogonal to th e plane IIa. If
(pn, qn) is not equal to (0, qc) we can com pute an initial norm al N a = (pa, qa, 1) of
n a as N a = (pn, gn,l) x ( 0 ,^ c,l). If the other end cross section Cn is not available
th en th e gradient (pa,qa) is constrained to be on a line by its orthogonality to
(0,gc). T he equation of the line containing (pa,qa) is (0,qc, l ) • (pa,qa, l ) = 0.
Any particular value of (pa,qa) m ay be chosen on this line as th e initial (pa,qo)-
Figure 7.5 shows th a t perception is m ore definite when both ends are available,
which confirms th e above observation th a t two ends are m ore inform ative than
only one.
7 .3 .2 R e su lts
We have im plem ented th e cross section recovery m ethod described in section 7.3.
In the im plem entation first the orientations (pc,?c) and (pn,q n) of the end cross
sections are com puted. T hen the norm al N a of IIa is found by searching around
the gradient given by (p c, qc, 1 ) x (pn, qn, 1 ) th a t gives a valid reconstruction. The
3-D position of each cross section is then found by tran slatin g the end cross
section rotating and aligning it w ith the limb boundaries and the plane of the
axis na. Figure 7.6 shows the recovered cross sections and figure 7.7 shows the
recovered orientations by both needle and shaded images for th e PR C G C s given in
figure 7.1. As m entioned in sections 4.2.5 and 6.3.5 there is no real ground tru th
to com pare the com puted surface norm als. Since m any objects could produce
th e shown contours and th e specific m odel used to generate th e im age contours
has no particular im portance, the real com parison is w ith hum an perception. In
our opinion the results are in agreem ent w ith hum an perception. We provide
graphical display for th e reader to judge th e results on their own.
103
Chapter 8
Conclusion
We have presented a theory of how to infer 3-D shape from the contour of curved
surfaces if certain sym m etry properties are present. We have given a detailed
analysis for ZGC surfaces, objects composed of planar and ZGC surfaces, straight
homogeneous generalized cylinders and constant cross section generalized cylin
ders.
O ur theory does m ake certain assum ptions, as m ust all shape from contour
m ethods, to extract shape from contour, b u t we believe th a t our assum ptions are
m inim al and th a t th e results agree w ith th e hum an perception. We have ai’gued
th a t, in a certain sense, this is the only evaluation th a t can be m ade for shape
from contour m ethods, as m any shapes can produce the same contour.
We have presented results on several synthetic exam ples and real images. To
apply our m ethod to complex real images, where surface m arkings, shadows and
highlights m ay be present, will require solution of other m onocular im age analysis
problem s. However, we believe th a t our approach can help provide a m odel for
such analysis.
We have shown th a t contours carry useful, im portant and in m any cases
sufficient inform ation for 3-D surface perception. This is not to say th a t all the
3-D inform ation is in the contours and no study of other sources of inform ation,
like shading, is needed. R ather, we suggest th a t contours should not be om itted
in shape perception and recovery.
104
In this thesis we have provided theorem s and m ethods for effective usage of
contours for shape recovery. We believe th a t for the surfaces and objects studied
here, the m ethods proposed perform ed very well.
8.1 Future Research
Shape from contour m ethod described here has certain lim itations. The algo
rithm s described here work on perfect line drawings. However, the edges ob
tained by state of th e a rt edge detectors are far from perfect. Also n atural effects
like texture, m arkings on th e objects, lighting conditions, and shadows lim it the
ability of local edge detectors to ex tract real edges of th e objects. We believe
th a t the class of algorithm s known as perceptual grouping are quite promising
for inferring real edges of objects. T he fact th a t our algorithm s require certain
sym m etry relations also provides constraints for such algorithm s.
A lthough a wide class of object surfaces is studied in this thesis, there still
rem ain other classes of surfaces for which hum ans can perceive a 3-D shape but
are not handled by our algorithm s. M any of these cases are some extensions
of the classes studied here, like a ZGC surface having planar cut on one side
b u t non-planar cut on th e other side. Such objects do not produce th e types of
sym m etries used in our analysis. However, 3-D shape of such surfaces can be
recovered w ith the m ethods studied here if we know which side is the planar cut.
For a ZGC surface w ith non planar cuts on either side, we conjecture th a t, either
one of th e sides is perceived planar or a specific shape is not perceived. However,
surfaces w ith non planar cuts needs to be studied in detail.
Contours contain the m ost im portant shape inform ation in m onocular im
ages, however other shape recovery techniques contain im p o rtan t inform ation
too. Com bination of other shape-from techniques, like shading, texture, even
stereo, has a p otential to produce b e tte r results. Shading inform ation m ay pro
vide useful qualitative shape inform ation which m ay supplem ent the quantitative
techniques detailed in this thesis. Also, in this thesis th e effects of gravity and
105
support are not discussed; our algorithm s, as they are stated , are invariant to
rotatio n in th e im age plane. Including a gravitational bias m ay enhance the
perform ance of th e algorithm s in some cases.
106
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1 1 0
Appendix A
Proofs
A .l Proofs of Theorems for ZGC Surfaces
In this section we give three proofs; two are related to the existence of parallel
sym m etries on Zero Gaussian C urvature surfaces (theorem s 12 and 5) and the
other proves the Inner Surface C onstraint. All of the proofs uses th e following
surface representation.
Let X ( u ,v ) = {x(u, v), y(u, v), z{u, u)) be a (u,v) param etric representation
of the class C 2 Zero Gaussian C urvature surface X . L et’s assum e th a t the v pa
ram eter curves are along the lines of m inim um curvature (rulings) of the surface.
N orm al, Af, of this surface at any point is given by:
( a - 1 }
where x is the vector product operator, and |V | is the length of th e vector V .
Note here th a t \Af\ — 1 . F irst, I, and second, I I , fundam ental forms of such a
surface are given by:
I ( X udu + X vdv) = E d u 2 + 2Fdudv + Gdv2 (A. 2)
11 (X udu + X vdv) = Ldu2 2M dudv + N d v 2
where
E = X u - X u F = X u • X v G = X v • X v
L = X uu • A f M = X u v -A f N — X vv ■ Af
(A.3)
111
Since th e param eter v is along the ruling (a line) the norm al curvature of the
surface in the direction X v, given by I I ( X V ), should be zero, then we have:
II( X V) — N = 0 (A .4)
G aussian curvature, k, of such a surface is given by [Lip69]
L N - M 2
K =
E G - F 2
(A.5)
Since th e G aussian curvature of the surface is zero setting k = 0 , w ith substi
tu tin g 0 for N by equation A.4 gives:
M = 0 (A .6 )
A .1.1 P r o o f o f T h e o r e m 5
Consider th e surface X , as given above. Also assum e th a t the u param eter curves
on the surface X are planar and parallel to each other. We have to show th a t the
tangent of the u param eter curves, is constant w ith respect to v (i.e. jjj^j is
a function of u only).
Let the planes, th a t th e u param eter curves lie on, have the norm al V (P is
constant). T hen we have:
• V = 0 = ► 0 = d

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Creator Ulupinar, Fatih (author)

Core Title Perception of 3-D shape from 2-D image of contours

Contributor Digitized by ProQuest (provenance)

School Graduate School

Degree Doctor of Philosophy

Degree Program Computer Engineering

Degree Conferral Date 1991-12

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Type texts

Source University of Southern California Dissertations and Theses (collection), University of Southern California (contributing entity)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the author, as the original true and official version of the work, but does not grant the reader permission to use the work if the desired use is covered by copyright. It is the author, as rights holder, who must provide use permission if such use is covered by copyright.

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA

Repository Email uscdl@usc.edu

Linked assets

University of Southern California Dissertations and Theses