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USC Computer Science Technical Reports, no. 555 (1993)

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USC Computer Science Technical Reports, no. 555 (1993)

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Content Imaging Hyp erplanes and the Decomp osition of the
P olar Set T ransformation
ChaoKuei Hung
ckhungusc edu
Doug Ierardi
ierardiu sc edu
No v em b er Abstract
Motiv ated b y the con v ex h ull nding problem for curv ed ob jects w e lo ok at the clas
sical p olar set tr ansformation whic his in timately related to con v ex sets By partitioning
the set of h yp erplanes in the space in to three subsets with resp ect to a giv en set Aw e
conclude that
inversion IH bs
where denotes the p olar set transformation IH denotes a simple sc heme whic h images
ah yp erplane as a p oin t and bs
denotes the set of b ounding and supp orting h yp erplanes
of A The decomp osition theorem rev eals more of the nature of dualit y In a subsequen t
rep ort w e will showho w the h yp erplane imaging sc heme ma y b e generalized to giv e
a framew ork for con v ex h ull nding of curv ed p olyhedra in E
d
When comp osed with
p ersp ectiv e pro jection this sc heme also nds application in the tolerance problem arising
from computer aided man ufacturing
In tro duction
This tec hnical rep ort furnishes the mathematical results of our in v estigations on the classical
p olar set transformation W e nd it helpful to decomp ose the transformation in to three more
primitiv e transformations including a simple sc heme that images a h yp erplane as a p oin t
The results will serv e as a reference and theoretical foundation for our ongoing researc hon
three seemingly indep enden t problems whic h arise from geometric mo deling and computer
vision
Use of dualit y is classical in transforming the problem of nding the con v ex h ull of a
nite set of p oin ts tofrom the in tersection of h yp erplanes Rag And Dos W eha v e
ho w ev er b een unable to lo cate an y literature on the generalization of this idea to curv ed
ob jects or gures In studying the problem of nding the con v ex h ull of curv ed p olyhedra
in E
w e come to in v estigate the reasons that dualit y helps in the con v ex h ull problem The
ideas turn out to b e applicable to the tolerance problem in computer aided man ufacturing
and the construction of asp ect graphs in computer vision as w ell

Section reviews basic prop erties of con v ex sets found in La y and giv es additional
ones whichw e need in later sections In a similar fashion w e review and extend results on
p olar sets in Section Motiv ated b y the results in the previous sections w e then pro ceed to
dene a simple represen tation for h yp erplanes in Section A simple observ ation leads us to
review inversion another w ell studied transformation in E
d
The ma jor claim that the p olar
set transformation is the comp osition of three more primitiv e transformations is then pro v ed
Finally in Section w e conclude with applications to t w o geometric mo deling problems w e
are curren tly w orking on
Prop erties of Con v ex Sets and Con v ex Hulls
In this section w e dev elop sev eral prop erties of con v ex sets whichw e shall mak e use of in later
sections Occasionally w e will b e concerned only with compact sets They are nice enough
for the statemen ts and pro ofs to b e concise and y et general enough to include practically
ev erything in geometric mo deling to w ards whic h the applications are geared
Let us recall a few denitions and lemmas
Lemma If S is a c omp act set then conv S is c omp act L ay The or em Lemma Supp ose A and B ar e nonempty c omp act sets Then ther e exists a hyp erplane
H which strictly sep ar ates A and B if and only if conv A conv B L ay The or em
Denition The hyp erplane
fx E
d
f x g
dene d by the line ar functional f is said to b ound the set S E
d
if either f S or
f S It strictly b ounds S if the e quality never holds is said to cut S if it do es not
b ound S L ay Denition augmente d
Denition Ahyp erplane is said to supp ort aset S at a p oint x S if x and b ounds S L ay Denition Lemma If x isab oundary p oint of a closedc onvex subset S of E
d
then ther e exists at
le ast one hyp erplane supp orting S at x L ay The or em Lemma L et f b e a line ar functional and S ac omp act c onvex set Then ther e exist
extr eme p oints x and y of S such that
f x max
x S
f x and
f y min
x S
f x L ay The or em
Figure Three t yp es of planes with resp ect to a set Sho wn is a disconnected compact set
c
s
b
Next wedev elop the additional prop erties of con v ex sets w e need
Denition L et A E
d
Dene the fol lowing sets of hyp erplanes
b
A f strictly b ounds Ag
s
A f supp orts Ag
c
A f cuts Ag
bs
A b
A s
A Note b
A s
A and c
A form a partition of the set of all h yp erplanes in E
d
That is
ah yp erplane b elongs to exactly one of the sets Figure sho ws a disconnected compact set
and one straigh t line from eac h set
The follo wing lemma though simple to pro v e is the k ey to the cen tral idea of our
presen tation Stated informally successful classication of all planes in space according as
b
A s
A or c
Aisprett ym uc h the same as nding its con v ex h ull
Lemma L et S E
d
Then
b
S b
conv S
s
S s
conv S c
S c
conv S pr o of
If cuts S meaning that there are p oin ts of S on b oth sides of then a fortiori cuts
conv S since S conv S If do es not cut S meaning that one side of con tains
no p ointof S then no p oin tof conv S can b e on that side either since all p oin ts of
conv S are necessarily con v ex com binations of p oin ts of STh us c
S c
conv S If strictly b ounds conv S then it necessarily b ounds S since S conv S If strictly b ounds S then again it b ounds all con v ex com binations of S since is linear
Th us b
S
b
conv S Finally b
S b
CH S b y the observ ation immediately follo wing the denition

Though the next t w o lemmas will not b e used directly in dev eloping the algorithm their
inclusion here should motiv ate some useful denitions to come
Lemma L et A beac omp act set in E
d
and b e the set of al l halfsp ac es c ontaining A e ach dene dby a supp orting hyp erplane of A Then
conv A
pr o of
Ob viously conv A T
Supp ose p T
conv A The singleton fpg is a
con v ex set whose con v ex h ull is itself Since fpg conv A the latter b eing com
pact b y Theorem w ecan ndah yp erplane H strictly separating fpg and conv A b y Theorem Let H b e dened b y the linear functional f x Using Theo
rem it is straigh tforw ard to construct a halfspace that do es not include pa
con tradiction
Lemma L et A bea c omp act set in E
d
Then
conv A
s A conv A pr o of
Let p conv A By Theorem conv A is compact By Theorem there exists
a p
s
A supp orting conv Aat p By denition p conv A conv Aisa
con v ex com bination p k
i i
x
i
of a nite n um b er of elemen ts x
x
k
of A with
i
for all i k It is straigh tforw ard to v erify that i
for all x p
using the fact that p
b ounds AHence p S
s A conv A Con v ersely supp ose q conv q
A for some q
supp orting A It is straigh tforw ard to
v erify that p conv A Tw o commen ts are in order First w e note that the con v ex h ull nding op eration on the righ t
hand side is one in a lower dimension Secondly in the case where A is a con v ex smo oth
h yp ersurface suc hasah yp ersphere b oth lemmas just rewrite conv Ain a w a y more
reminiscen t of its iden tit y as the en v elop e of its tangenth yp erplanes The latter observ ation
is crucial to the design of the transformation that will facilitate con v ex h ull nding
P olar Set
Duality isanin teresting phenomenon in mathematics Besides its pure mathematical b eaut y it also has applications in areas suc h as linear programming problems whic h arise in man y
practical elds ranging from engineering to economics Dos And The p olar set of a
giv en set is an example of dualit y when the giv en set is a con v ex p olytop e
Figure The cub e and its dual
Denition L et K b e any nonempty subset of E
d
Then the p olar set K
of K is dene d
by
K
f y E
d
x y for al l x K g L ay Denition Note that the p olar set is a transformation applied to the original set as a whole In particular
it is not a function from E
d
to E
d
Tw o examples of p olar sets are
If K is a singleton other than the origin denoted O from this p oin t on then K
is the
closed halfspace con taining O and b ounded byah yp erplane
If K is a closed ball of radius r cen tered at O then K
is the closed ball of radius
r cen tered at O In particular the p olar set of the unit closed ball is itself
Figure sho ws an example of the p olar set of a con v ex set The dualit ybet w een v ertices in
one gure and faces in the other has b een observ ed in the past Shortly w e shall p oin t out
exactly ho w this dualit y arises
Lemma L et K K
K
and K
A b e nonempty sets Then
A
K
A
K
K
is a closedc onvex set c ontaining O the origin and
If K
K
then K
K
L ay The or em
Lemma L et K b e a closedc onvex set c ontaining the origin Then K
K L ay The or em As an aside w e started our in v estigation b y designing the transformation to b e giv en in
Section and pro ceeded to comp ose it with the in v ersion map Not un til w e had conjectures
for the follo wing lemmas did w e b egin to susp ect that the result of our comp osition is exactly
the p olar set transformation
Lemma L et A E
d
Then A
conv A pr o of
By Lemma conv A A
since A conv A Supp ose no w that y A
Let
x conv A b e arbitrary Then there exist k Z x
x
k
X
k
suc h that x P
k
i i
x
i
Th us
y x k
X
i
i
y x
i
k
X
i
i
Hence y conv A Lemma L et B E
d
beac omp act c onvex set which c ontains O in its interior Then
ther e exists a unique c omp act c onvex set A such that A
B pr o of
Existence Consider A B
whic h is closed and con v ex b y Lemma It is b ounded
b ecause O
A
By Lemma A
B Uniqueness Supp ose no w that A
is a compact con v ex set whose in terior con tains
the origin and suchthat A
B By a straigh tforw ard generalization of Lemma A
A
B
A Corollary L et A E
d
beac omp act set c ontaining O Then conv A is the unique
c omp act c onvex set whose p olar set c oincides with that of A pr o of
The p olar set of conv A is the same as that of A b y Lemma conv Aiscon v ex b y
denition and compact b y Lemma The statemen t then follo ws from Lemma In a sense the p olar set of A con tains exactly the information no more and no less w e need
for conv A Please note a similar commen t to Lemma Applying transitivit y to these
t w o informal statemen ts weem bark on seeking a direct explicit connection b et w een A
and
the three sets s
A b
A c
A
Figure Tw ostraigh t lines in E
and their images
O
IH IH Imaging Hyp erplanes
With Lemma in mind w e switc h our atten tion from the giv en set A E
d
itself to the three
sets of h yp erplanes s
A b
A and c
A Sp ecically w ew ould lik e to b e able to refer to
anyh yp erplane in space b y something simpler than its dening equations F or the momen t
lets consider the set of h yp erplanes not con taining the origin A natural represen tation of
ah yp erplane is
Denition L et b e the hyp erplane
fx E
d
n x g
wher e and jnj Dene IH n Figure sho ws the simple sc heme for imaging h yp erplanes in the case of E
Geometrically w e iden tify a h yp erplane b y the co ordinates of the p erp endicular fo ot on it dropp ed from
the origin IH isaw ell dened onetoone corresp ondence b et w een and E
d
fOg The follo wing lemma stipulates that the top ology of the images of the h yp erplanes classied
according to a closed con v ex set A reect that of A in the original space
Lemma L et A E
d
b e closedand c onvex c ontaining O in its interior Then IH c
A is op en with b oundary IH s
A and exterior IH b
A in the r elative top olo gy
of E
d
fOg sketch of pr o of
W e resort to the picture Figure for the pro of Let c
Aand p q A suchthat
p and q fall on dieren t sides of Since A is con v ex the straigh t line segmen t pq A If at ev ery p oin t r pq w e image the h yp erplanes passing r and cutting Aw e get the
ab o v e picture It is straigh tforw ard to v erify that IH is indeed in the in terior of the
shaded area comprising of the images of the aforemen tioned cutting h yp erplanes of A The pro of for the b ounding h yp erplanes b eing mapp ed to the exterior is similar
Finally the remaining h yp erplanes the supp orting ones can only b e mapp ed exactly
to the b oundary
Figure Lo ci of the p erp endicular fo ots of the straigh t lines that necessarily cuts A O
p
q
IH If one go es in to the computational details in the pro of ab o veone w ould see the motiv ation
of what wew ould lik e to review next namely in v ersion Consider imaging the set of supp orting
planes of a xed p oin t p E
An elemen tary geometric argumen t immediately rev eals that
as the supp orting plane hinges on p the p erp endicular fo ot dropp ed from O to traces
the circle C with diameter pO That is C IH s
fpg Since a singleton fpg is so nice
a subset of E
w e naturally tend to exp ect s
p to b e mapp ed to something nice p erhaps
at the cost of an additional transformation to mak e it something ev en simpler than a circle
passing the origin
Denition Denition PS In v ersion in E
d
isap ointtop oint tr ansformation
which maps a ve ctor v applie d to the origin to the ve ctor e v v jvj
applie d to the origin
The inversion f e x x Ag of a set A wil l b e denotedby
e
A Wema y think of in v ersion as a bijectiv e function inv E
d
fOg E
d
fOg W e list a few
noticeable prop erties of in v ersion see also Figure Hyp erspheres are mapp ed to h yp erspheres
The in terior of an op en ball B is mapp ed to the in terior resp ectiv ely exterior of an
op en ball if and only if B do es not resp ectiv ely do es con tain O Hyp erspheres passing O are mapp ed to h yp erplanes and vice v ersa
The p oin ts on the unit h yp ersphere radius with cen ter at O are the xed p oin ts of
in v ersion
Incidence of p oin t sets in E
d
fOg not in P
d
is preserv ed under in v ersion
Prop ert y is exactly what w e are after W ealso men tion the ob vious top ological prop ert y
that in v ersion is a homeomorphism from E
d
fOg to itself

Figure Examples of in v ersion The dotted circle is S
the unit circle
O
O
Let b e a set of h yp erplanes The in v ersion of their represen tation will b e denoted b y
f
IH The pro of of the follo wing lemma is just straigh tforw ard computation and hence
omitted
Lemma L et b e the hyp erplane fx R
d
n x g Then
f
IH n W e are no w ready to pro v e the simple y et imp ortan t
Theorem Decomp osition of the P olar Set T ransformation
L et A E
d
and O
A
Then A
fOg f
IH
bs
A pr o of
Let p E
d
fOg Then
p A
p x for all x A denition of p olar set
The h yp erplane fx p x g bounds A since A
p f
IH f
IH bs
A Lemma A t this p ointitbecomesob vious that instead of excluding h yp erplanes not passing the
origin from consideration w ema y include all h yp erplane at dieren t directions at innit y
in to the ab o v e denitions and lemmas First observethat E
d
fOg the range of IHis
homeomorphic to S
d I
where S
d is the unit d sphere and
I
the op en unit in terv al
Wema y patcht w o copies of S
d at the ends of this nite cylinder similar to what one do es
in patc hing one p oin t to compactify a space Gem The in v ersion map ma y b e extended
naturally to map p oin ts on one of the patc hed spheres to those on the other Without going

in to all the tec hnical details w econ ten t ourselv es with the in tuition that the ab o v e denitions
can b e mo died in the compactied space and the statemen ts of the lemmas will b e prett y
m uc h the same only simpler Th us for example w e will state the decomp osition theorem as
A
f
IH bs
A for A con taining the origin In the ev entthat A is closed and O A the ab o v e set will
be un b ounded It is not di cult to v erify that dually if A is closed and un b ounded then
O A In summary w eha vepro v ed the claim weha v e long promised from the b eginning
of the c hapter namely that the p olar set tr ansformation is just the c omp osition of inversion
with IH bs
Lemma no w b ecomes an immediate corollary of Theorem and Lemma The
geometric meaning of the facts listed in Lemma also b ecomes clear In addition w eha v e
Corollary L et A E
d
bea c onvex close d set c ontaining O Then A
is c onvex and
close d and it c ontains the origin F urthermor e
A
f
IH
b
A A
f
IH
s
A exterior A
f
IH
c
A Corollary L et A E
d
bea c omp act set c ontaining the origin Then
conv A
f
IH bs
f
IH
bs
A pr o of
By Lemma conv A conv A
By Lemma it is equal to A

No w apply
Theorem t wice to obtain the desired conclusion
Conclusion
Weha vesho wn that the p olar set transformation whic h when applied t wice tak es a closed set
to its con v ex h ull is simply the in v erted image of the p erp endicular fo ots of the collection of
the sets b ounding and supp orting h yp erplanes The connection b et w een the p olar set and the
con v ex h ull of a set b ecomes clear under this decomp osition Sev eral classical results follo w
as immediate corollaries In particular the corresp ondence b et w een the v ertices and faces of
the dual p olytop es is made in tuitiv e The dual of a v ertex of a p olytop e A is the in v ersion of
the images of the supp orting planes of A at the v ertex and hence a face and vice v ersa
The decomp osition is called up on and some comp onen ts are rened andor comp osed
with other transformations in the follo wing practical problems that arise in geometrical mo d
elingcomputer vision

T olerance Giv en a nite set of p oin ts A in space usually represen ting the measured co or
dinates of a supp osedly at face of a man ufactured part nd a closest pair of parallel
planes that enclose A Curv ed Con v ex Hull Compute the con v ex h ull of a set in space with piecewise smo oth
curv ed b oundary Asp ect Graph Construct a graph with v ertices corresp onding to top ologically distinct views
at innit y of a giv en ob ject in space and edges corresp onding to p ossible direct tran
sitions from one view to another
The rst problem is solv ed b y comp osing the IH transformation with p ersp ectiv e pro jection
W e showho w to generalize IH to smo oth manifolds other than h yp erplanes and obtain a
framew ork for solving the second problem in an y nite Euclidean spaces in HI W eha v e
just b egun lo oking at the application of the results in this rep ort to the third problem
References
And E J Anderson Line ar Pr o gr amming in InniteDimensional Sp ac es The ory and
Applic ationsWiley In terscience Publication Other authors Edw ard J An
derson and P eter Nash
Dos Nazir G Dossani Duality The ories in Line ar Quadr atic and Convex Pr o gr amming
a Survey Philadelphia Resional Science Researc h Institute Gem Mic hael C Gemignani Elementary T op olo gy Do v er Republication of the
second edition of the w ork originally published in b y AddisonW esley HI ChaoKuei Hung and Doug Ierardi Generalized IH for con v ex h ull of curv ed p oly
hedra ! a general framew ork and an optimal d algorithm T ec hnical Rep ort USC
CS Univ ersit y of Southern California La y Stev en R La y Convex Sets and Their Applic ations John Wiley " Sons Inc original edition Mas William S Massey A Basic Course in A lgebr aic T op olo gy SpringerV erlag PS F ranco P Preparata and Mic hael Ian Shamos Computational Ge ometry A n Intr o
duction SpringerV erlag Rag Prabhak ar Ragha v an Lecture notes on randomized algorithms T ec hnical rep ort
IBM ResearchW atson ResearchCen ter Decem b er

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Description

Chao-Kuei Hung and Doug Ierardi. "Imaging hyperplanes and the decomposition of the polar set transformation." Computer Science Technical Reports (Los Angeles, California, USA: University of Southern California. Department of Computer Science) no. 555 (1993).

Asset Metadata

Creator Hung, Chaeo-Kuei (author), Ierardi, Doug (author)

Core Title USC Computer Science Technical Reports, no. 555 (1993)

Alternative Title Imaging hyperplanes and the decomposition of the polar set transformation (title)

Publisher Department of Computer Science,USC Viterbi School of Engineering, University of Southern California, 3650 McClintock Avenue, Los Angeles, California, 90089, USA (publisher)

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Unique identifier UC16270212

Identifier 93-555 Imaging Hyperplanes and the Decomposition of the Polar Set Transformation (filename)

Legacy Identifier usc-cstr-93-555

Format 11 pages (extent),technical reports (aat)

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Repository Name USC Viterbi School of Engineering Department of Computer Science

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Description Archive of computer science technical reports published by the USC Department of Computer Science from 1991 - 2017.

Coverage Temporal 1991/2017

Repository Email csdept@usc.edu

Repository Name USC Viterbi School of Engineering Department of Computer Science

Repository Location Department of Computer Science. USC Viterbi School of Engineering. Los Angeles\, CA\, 90089

Publisher Department of Computer Science,USC Viterbi School of Engineering, University of Southern California, 3650 McClintock Avenue, Los Angeles, California, 90089, USA (publisher)