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USC Computer Science Technical Reports, no. 719 (1999)
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USC Computer Science Technical Reports, no. 719 (1999)
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Content
Image Retriev al By Shap e A Comparativ e Study
Ma ytham Safar Cyrus Shahabi and Xiaoming Sun
In tegrated Media Systems Cen ter
Departmen t of Computer Science
Univ ersit y of Southern California
Los Angeles California saf ar shahabi xi aomins usc edu
No v em b er
Abstract
Besides traditional applications eg CADCAM and T rademark registry new m ultimedia applicatio ns suc h
as structured video animation and MPEG standard require the storage and managemen tof w elldened ob jects
This study compared four shap ebased ob ject retriev al tec hniques FD GB DTand TPV AS The simi
larit y retriev al accuracy of our metho d TV PASw as comparable to the other metho ds while it had the lo w est
computation cost to generate the shap e signatures of the ob jects Moreo v er it has lo w storage requiremen t and
a comparable computation cost to compute the similaritybet w een t w o shap e signatures In addition TPV AS
requires no normalization of the ob jects and is the only metho d that has direct supp ort to RS T query t yp es W e
also in tro duced a new shap e description taxonom y In tro duction
Man y applications in the areas of CADCAM and computer graphics require to store and access large databases
These ob ject databases are then queried and searc hed for dieren t purp oses A sample query migh tbe
nd al l the sc enes that c ontain a c ertain obje ct Therefore giv en a query ob ject w ew ould lik e to obtain a list of
ob jects from the database whic h are most similar in some asp ects to the query ob ject
Shap e of an ob ject is an imp ortan t feature for image and m ultim edia similarit y retriev als Therefore this study
fo cuses on shap ebased ob ject retriev al tec hniques There is a v arietyoftec hniques that has b een prop osed in the
literature for shap e represen tation In shap e represen tation tec hniques are divided in to t w o categories
b oundarybased and regionbased Boundary based metho ds use only the con tour or b order of the ob ject shap e and
completely ignore its in terior On the other hand the region based tec hniques takein to accountin ternal details
eg holes b esides the b oundary details The purp ose of this pap er is to ev aluate and compare the p erformance of
four b oundary based metho ds for shap e represen tation and retriev al F ourier descriptors metho ds FD based on
ob jects shap e radii gridbased metho d GB based on c hain co des Delauna y triangulation metho d DT based on corner p oin ts and T ouc hp oin tv ertexanglesequence TPV AS based on minim um b ounding circles and
angle sequences TPV AS approac h to shap e represen tation and similarit y measure w as prop osed b y the authors
in W e compare its retriev al p erformance with that of the other more established metho ds ie FDand GB
metho ds that w ere used in some commercial systems and as a basis for dieren t comparison studies in In addition it w as compared to a new metho d based on a new indexing tec hnique ie DT
Although TPV AS metho d uses simple attributes suc h as minim um b ounding circles and angle sequences simple
to extract these attributes w ere sho wn to b e translation rotation and scale in v ariantin F urthermore TPV AS
has lo w storage requiremen t and has the lo w est computation cost to generate the shap e signatures of the ob jects
Moreo v er it has a comparable computation cost to compute the similarit ybet w een t w o shap e signatures In addition
the similarityretriev al accuracy of TV PAS comparable to the other metho ds
This researchw as supp orted in part b y NASAJPL con tract nr and unrestricted cashequipme n t gifts from In tel NCR
and NSF gran ts EEC IMSC ER C and MRI
In some application domains the supp ort of RS T query t yp es in whic h the matc hing ob jects could b e within
a sp ecied rotation angle scaling factor translation v ector or an ycom bination of the three is imp ortan t see for further details An example is searc hing for similar tumor shap es in a medical image database T umors
are presen ted b y a set of D images that represen t slices through it A metho d for retrieving similar tumor shap es
w ould help to disco v er correlations b et w een tumor shap e and certain diseases Besides the shap e the size and the
lo cation of the tumor w ould help in the iden tication of patien ts with similar health history With TV PAS metho d
a subset of the MBC features eg r adius rc enter C and start p oint SP could b e utilized to supp ort RS T query
t yp es On the other hand the other three metho ds lac k the supp ort of RS T queries This is due to either the use
of normalization or lac k of features that could supp ort suc h queries In addition this study in tro duces a new shap e
description taxonom yF urthermore w e classify tec hniques that w ere not describ ed in previous studies eg turning angle collinearit y etc
The remainder of this pap er is organized as follo ws Sec pro vides some bac kground material on shap e descrip
tion tec hniques while Sec describ es dieren tt yp es shap e retriev al metho ds considered in this study In Sec w e
pro vide cost analysis for the dieren t metho ds and compares their eciency in terms of computation and storage
requiremen ts Sec lists some of the dra wbac ks of the alternativ e shap e description tec hniques Sec rep orts on
the p erformance results obtained from a set of exp erimen ts Finally Sec concludes the pap er and pro vides an
o v erview on our future plans
Shap e Description T ec hniques
Shap e description is an imp ortan t issue in ob ject recognition and its ob jectiv e is to measure geometric attributes of
an ob ject that can b e used for classifying matc hing and recognizing ob jects There are v arious metho ds for shap e
represen tation
An o v erview of shap e description tec hniques is providedin It categorizes the tec hniques in to b oundary
based and region based metho ds Boundary based metho ds use only the con tour of the ob jects shap e while on
the other hand the region based metho ds use the in ternal details eg holes in addition to the con tour The
regionbased metho ds are further brok en in to spatial and transform domain subcategories dep ending on whether
direct measuremen ts of the shap e are used or a transformation is applied A dra wbac k of this categorization is that
it do es not further subcategorizes b oundary based metho ds in to spatial domain and transform domain metho ds F or
example F ourier descriptors metho d can b e considered as a transform domain tec hnique while c hain co des can b e
considered as spatial domain tec hnique Another dra wbac k of this categorization is that it assumes that structural
tec hniques eg Dstrings are a subcategory of regionbased spatial domain tec hniques Ho w ev er w e b eliev e
that structural tec hniques as DStrings are spatial similarit y based tec hniques in whic h the retriev al of ob jects is
p erformed based on the spatial relationships among ob jects and not shap e retriev al tec hniques see Fig
In another study shap e description tec hniques w ere brok en in to t w o dieren t categorizes transformation
based and measuremen t based categories They further breakdo wn the transformation based category in to t wosub categories functional and structural categories ho w ev er it is not clear what criteria is used to this end Some
dra wbac ks of this categorization are that it do es not distinguish b et w een b oundary or region based tec hniques and
sometimes it miss categorizes some tec hniques F or example c hain co de tec hnique is categorized as a transformation
based tec hnique while it is a measuremen tbased tec hnique Another example is that silhouette moments could b e
used as a regionbased tec hnique but not as a b oundarybased tec hnique Therefore w ein tro duce a new shap e
description taxonom y Our complete taxonom y of shap e description tec hniques is illustrated in Fig W e also added
further tec hniques that w ere not describ ed in eg turning angle collinearit y etc
Ob ject Shap e Descriptions
This study compares four NI P b oundary based metho ds for ob ject shap e descriptions F ourier descriptors metho d
FD gridbased metho d GB Delauna y triangulation metho d DT and T ouc hp oin tv ertexangle
sequence TPV AS
Boundary Based Methods Region Based Methods
Spatial Domain (Geomtric)
Transform Domain
Partial
(Occlusion)
Complete
(No Occlusion)
Corner Points
Chain Codes
Shape Number
Perimeter
Area
Elongation
Compactness
Significant Points
Boundary Coordinates
Turning Angle
Collinearity
Co-Curvilinearity
Parallelism of Lines or Curves
Convergence of Lines or Arcs
Physical Proximity
Length
Radius of Shape Boundary
Symmetry
Fitting line
Touch Points
Touch Points Angle Sequence
V ertex Angle Sequence
Curvature
Boundary Moment Invariants
Circularity
Aspect ratio
Discontinuity Angle
Right-Angelness
Length Irregularity
Complexity
Sharpness
Directedness
Grid Cover
Edge Direction
Hough Transform
Walsh Transform
Wavelet Transform
Contour Segments
Breakpoints
Area
Holes
Euler Number
Moment Invariants
Silhouette Moments
Zernike Moments
Pseudo-Zernike Moments
Max-Min Radii
Compactness
Elongation
Symmetry
Rectangle Cover
Grid Cover
Matrix Decomposition
Polygon Cover
Circularity
Aspect ratio
Spatial Domain (Geometric) Transform Domain
Complete
(No Occlusion)
Partial
(Occlusion)
Contour Segments
Curvature
Discontinuity Angle
Right-Angelness
Length Irregularity
Complexity
Sharpness
Directedness
Content Retrieval
Structural
Primitives
Rules
2D Strings
2D B -Strings
2D C -Strings
2D C+ -Strings
Extended 2D Strings
θR String
Spatial Orientation Graph
Quad-tree Spatial Arrangement
Texture Description Color Description Spatial Location Shape Description
Discrete Fourier
UNL Fourier
NFD Fourier
Fourier Descriptors
Figure T axonom y of Shap e Description T ec hniques
F ourier Descriptors Metho d
The FD metho d obtains the ob ject represen tation in the frequency domain as complex
co ecien ts of the F ourier series expansion of the ob jects shap e signature The metho d starts b y obtaining a feature
function of the ob ject called shap e signature f k whic h could b e curv ature based radius based or b oundary
co ordinates based f k is also called the F ourier descriptors FDs of the b oundary In the next step a discrete
F ourier transform of the shap e signature is obtained The F ourier co ecien ts obtained are then used for shap e
represen tation as index and for shap e similarit y calculation The discrete F ourier transformation DFT of a shap e
signature f k is giv en b y
F u
N
N X
k f k exp j uk N u N Where N is the n umberof samplesof f k Direct represen tation eg radii captures eac h individual detail of a shap e ho w ev er it is v ery sensitiv e to small
c hanges and noise As a consequence a small c hange in the co ordinates of the ob jects b oundary p oin ts ma y lead to
av ery dieren t shap e signature hence v ery p o or retriev al p erformance On the other hand F ourier transformation
captures the general feature of a shap e bycon v erting the sensitiv e direct represen tation measures in to frequency
domain As a result the data is more robust to small c hanges and noise Therefore F ourier transformation is used
as shap e represen tation instead of the direct represen tation see for further details A p opular shap e signature
is the shap e radii or cen troidal distance whic h computes the distance b et w een p oin ts uniformly sampled along
the ob ject b oundary and its cen troid its cen ter of mass When shap e radii are used only the distances b et w een
p oin ts are recorded and not the exact co ordinates of the p oin ts Th us the shap e signatures of the ob jects are
translation in v arian t Since the magnitude information of F u ie jF u jis in v arian t to rotation Hence rotation
normalization is ac hiev ed taking only the magnitude information from F u and ignoring its phase information
Moreo v er jF j reects the energy of the shap e radii th us j F u j j F j will b e scale in v arian t Therefore in order
to scale normalize the ob jects FDs the magnitude of all F us are divided b y the magnitude of F As a result of
A sampling point
Centroid
f(k) = [r1,r2,r3,r4,r5,r6,r7,r8]
r1
r2
r3
r4
r5
r6
r7
r8
a An ob ject with sampling p oin ts b F ourier descriptor of the ob ject
Figure F ourier Descriptors Metho d
normalization a new feature v ector whic hisin v arian t to translation rotation and scale can b e generated as follo ws
FN jF j jF jjF j jF j jF N j jF j T
The dierence b et w een t w o ob jects is dened as the Euclidean distance b et w een their corresp onding feature
v ectors Hence the similarit y measure of FD metho d is that t w o ob jects are similar in shap e if and only if the
Euclidean distance b et w een their feature v ectors is less than a presp ecied threshold In other w ords they ha v e the
same set of distances b et w een their cen troids and their sampled b oundary p oin ts
An example is illustrated in Fig where Figa sho ws an ob ject with its sampling p oin ts and its cen troid
and Figb sho ws the radii distances and the F ourier descriptors f k of the b oundary Grid Based Metho d
An alternativ e approac h for shap e represen tation is GB metho d With this metho d an ob ject is
rst normalized for rotation and scale Then the ob ject is mapp ed on a grid of xed cell size Subsequen tly the grid
is scanned and a or is assigned to the cells dep ending on whether the n um b er of pixels in the cell whic h are inside
the ob ject are greater than or less than a predetermined threshold Finally a unique binary n um b er is obtained as
the shap e represen tation b y reading the s and s assigned to the cells from left to righ t and top to b ottom T o
impro v e the eciency of this metho d another shap e feature called eccen tricity w as used Eccen tricit y of shap e
is the ratio of the n um b er of cells used in xdirection to the n um b er of cells used in the ydirection to represen t
a shap e Therefore for t w o ob jects to b e similar their shap e signatures and their eccen tricities v alues should b e
similar
The dierence b et w een t w o ob jects is the n um b er of cells in the grids whichare co v ered b y one shap e and not
the other whic h is the same as the sum of s in the result of the exclusiv e OR of their binary n um b ers Hence the
similarit y measure of GB metho d is that t w o ob jects are similar in shap e if and only if the dierence b et w een their
binary represen tations is less than a presp ecied threshold and they ha v e similar eccen tricities
An example is illustrated in Fig where the ob jects are mapp ed on to a grid of xed cell size in a manner suc h
that the ob jects are justied to the top left corner ie assuming that the ob jects do not need to b e normalized
The binary n um b ers obtained for the ob jects in Figa and b are and resp ectiv ely Hence
the dierence b et w een the ob jects is Binary n um b er obtained for the same ob ject with a dieren t orien tation in space or with a dieren t scale will b e
dieren t Therefore the normalization of the ob ject b oundaries prior to indexing is crucial to meet the uniqueness
criteria of the binary n um ber Rotation Normalization
Rotation c hanges the spatial relationships b et w een the grid cells and the ob jects b oundaries whic h leads to dieren t
binary n um b er represen tations for the same ob ject Therefore ob jects m ust b e normalized for rotation This is
ac hiev ed b y obtaining the major axis of the shap e whic h is the line joining the t wopoin ts on the shap e b oundary
farthest a w a y from eac h other Subsequen tly the shap e is rotated to mak e the ma jor axis parallel to the xaxis
An example of rotation normalization of an ob ject is sho wn in Fig where in Figa an ob ject is sho wn prior
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a Ob ject b Ob ject
Figure Grid Based Metho d Ob ject Mapping and Represen tation
x
y
Major Axis
x
y
Major Axis
Minor Axis
a Original ob ject b After rotation normalization
Figure Grid Based Metho d Rotation Normalization
to rotation normalization and Figb sho ws the same ob ject after b eing rotated so that the ma jor axis b ecomes
parallel to the xaxis
Scale Normalization
Both the grid and the ob ject sizes aect the binary n um b er deriv ed for an ob ject Therefore ob jects m ust b e
normalized for scale This is ac hiev ed byc ho osing a xed length of the ma jor axis called the standar dize d major
axis Subsequen tly scaling normalization is ac hiev ed b y scaling along the ma jor axis of the ob ject suc h that its
ma jor axis b ecomes equal to the length of the standardized ma jor axis T o main tain the p erceptual similarit y of the
ob ject it is also scaled along the minor axis prop ortionally The minor axis is p erp endicular to the ma jor axis and
of suc h length that a rectangle with sides of ma jor axis and minor axis denes the minim um b ounding rectangle of
the ob ject An example of scale normalization of an ob ject is sho wn in Fig where in Figa and b t w o similar
ob jects are sho wn prior to scale normalization while Figc sho ws the same ob jects after b eing scaled along the
xaxis and yaxis suc h that the length of the ma jor axis b ecomes equal to the standardized ma jor axis
DelaunayT riangulation Metho d
This approac h of shap e represen tation is histogrambased Giv en an ob ject cor ner points are used as the
feature p oin ts of the ob ject Corner p oin ts are generally highcurv ature p oin ts lo cated along the crossings of an
ob jects edges or b oundaries Then a Delauna y triangulation of these feature p oin ts is constructed Consequen tly a feature p oin t histogram is obtained b y discretizing the angles pro duced b y this triangulation in to a set of bins
x
y
Major Axis
x
y
Major Axis
x
y
Standardised
Major Axis
a Original ob ject b A similar ob ject c After scale normalization
Figure Grid Based Metho d Scale Normalization
Feature Point Histogram
Bin number
Number of angles
123456789 101112131415161718
1
2
3
4
5
a An ob ject with p oin ts b Delauna y triangulation c F eature p oin t histogram
Figure Delauna yT riangulation Metho d
and coun ting the n um b er of times eac h discrete angle o ccurs in the triangulation In building the feature p oin t
histogram a selection criteria of whic h angles will con tribute to the nal feature p oin t histogram is c hosen F or
example the selection criteria could b e coun ting the t w o largest angles the t w o smallest angles or all three angles
of eac h individual Delauna y triangle
The dierence b et w een t w o ob jects is the Euclidean distance b et w een the corresp onding bins of the ob jects
feature p oin t histograms Hence the similarit y measure of DT metho d is that t w o ob jects are similar in shap e if
and only if the euclidean distance b et w een their feature p oin t histograms is less than a presp ecied threshold In
other w ords they ha v e the same set of feature p oin ts Th us eac h pair of the corresp onding Delauna y triangles in the
t w o resulting Delauna y triangulations m ust b e similar to eac h other The angles of the resulting Delauna y triangles
of a set of p oin ts is in v arian t under uniform translations scalings and rotations Therefore the similarit y measure
is indep enden t of ob jects p osition scale and rotation ie ob jects do not ha v e to b e normalized This approac h
also supp orts an incremen tal approac h to image ob ject matc hing from coarse to ne b yv arying the bin sizes
An example is illustrated in Fig where Figa sho ws an ob ject that consist of feature p oin ts corner p oin ts
Figb sho ws the resulting Delauna y triangulation While Fig sho ws the resulting feature p oin t histogram built
b y coun ting all three angles of eac h individual Delauna y triangle with bin size of degrees
T ouchP ointV ertex Angle Sequence Metho d
This approac h TPV AS for shap e represen tation w as prop osed b y the authors in another study In that study w e presen ted a m ultistep query pro cessing metho d that ecien tly supp orts the retriev al of D ob jects b yshape W e
prop osed three index structures I
TP AS
I
V AS
and I
TPV AS
based on features that are extracted from the ob jects
MBC Those features are the cen ter co ordinates of the MBC the radius of the MBC the set of touc h p oin ts on
MBC TP s touc hpoin ts angle sequence TPAS v ertex angle sequence VAS and the start p oin t of the angle
sequence SP Fig pro vides an example of an ob ject with its six iden tied features In Fig b the MBC of an
ob ject is computed and its touchpoin ts are iden tied as the v ertices that la ys on its MBC denoted b y X Figs c
and d sho w b oth the touc h p oin ts and v ertex angle sequences resp ectiv elyF or the algorithms to extract the MBC
features as w ell as their formal denitions see A ma jor observ ation is that those features are unique p er ob ject In addition a subset of the MBC features eg
TPAS VASand TP s are translation scaling and rotation indep enden t This subset of features could b e utilized
C r
X
X
X
ϕ1
ϕ4
ϕ3
ϕ2
X
X
X
ϕ1 ϕ2
ϕ3
ϕ4
ϕ9
ϕ8
ϕ7
ϕ6 ϕ5
a Original ob ject
b MBC of the original ob
ject with its cen ter C radius
r TP denoted byX and SP
denoted b y bullet
c TP AS
d V AS
Figure MBC F eatures
to supp ort query t yp es in whic h the matc hing ob jects could b e in v arian t with resp ect to translation rotation and
scaling It could also b e utilized to supp ort query t yp es in whic h the matc hing ob jects could b e within a sp ecied
rotation angle scaling factor translation v ector or an ycom bination of the three RS T queries W e use ob jects
SP to c hec k if it is rotated use ob jects r to c hec k if it is scaled and the v ertex co ordinates of the ob jects C to
c hec k if it is translated see for further details
With TPV AS metho d the shap e descriptor of an ob ject dep ends on a subset of the six MBC features The
metho d starts b y obtaining a feature function of the ob ject called shap e signature f k T o obtain f k four steps
are required The minim um b ounding circles of the ob jects are rst found Second MBC features suc h as angle
sequences VASand n um ber of touc h p oin ts TP are iden tied Third the ob jects shap e signature is iden tied
as the sequence com bination of TP and VAS Finally as with the FD metho d a unique ob ject represen tation is
obtained as the discrete F ourier series expansion of the ob jects shap e signature The F ourier co ecien ts obtained
are used for shap e represen tation as index and for shap e similarit y calculation The discrete F ourier transformation
DF T of a shap e signature f k is giv en b y
F u
p
N
N X
k
f k exp j uk N u N Where N is the n um b er of p oin ts in the p olygon represen tation of an ob ject
Although TPV AS metho d uses simple attributes suchas minim um b ounding circles and angle sequences these
attributes w ere sho wn to b e translation rotation and scale in v ariantin When angle sequences are computed
only the angle b et w een t wov ectors connecting the cen ter of the MBC and the ob jects p oin ts w ere recorded Th us
it do es dep end on the exact co ordinates of the p oin ts Therefore the shap e signatures of the ob jects are translation
in v arian t Rotating t woin tersecting v ectors in space do es not c hange the angle b et w een them In addition b y
taking the magnitude information of F u ie j F u j w e are insuring that the angle sequence is indep endentof
the start p ointc hoice Hence TPV AS is also in v arian t to rotation When applying uniform scaling to an ob ject
assuming that the origin of the co ordinate system is the cen ter of the MBC w e are only c hanging its size and
not its shap e Changing the size means that only the prop ortion of the distances b et w een the p oin ts of the ob ject
and the cen ter of its MBC c hanges Therefore the new ob jects AS is equiv alen t to the old ob jects AS Hence
TPV AS is also scaling in v arian t F or further details refer to Therefore the similarit y measure is indep enden t
of ob jects p osition scale and rotation ie ob jects do es not ha v e to b e normalized
The dierence b et w een t w o ob jects is the Euclidean distance b et w een their corresp onding feature v ectors Hence
the similarit y measure of TPV AS is that t w o ob jects are similar in shap e if and only if the Euclidean distance
bet w een their feature v ectors is less than a presp ecied threshold In other w ords they ha v e similar v ertex angle
sequences and n um b er of touc h p oin ts
An example is illustrated in Fig Figa sho ws an ob ject with v ertices with its minim um b ounding circle
and touc h p oin ts Figb sho ws the v ertex angle sequence of the ob ject and its shap e descriptors f k
Cr
X
X
X
TP
TP
TP
TP and SP
MBC
ϕ1
ϕ2
ϕ3
ϕ4
ϕ9
ϕ8
ϕ7
ϕ6
ϕ5
VAS = [ϕ1,ϕ2,ϕ3,ϕ4,ϕ5,ϕ6,ϕ7,ϕ8,ϕ9]
X
X
X
f(k) = [TP,ϕ1,ϕ2,ϕ3,ϕ4,ϕ5,ϕ6,ϕ7,ϕ8,ϕ9]
a An ob ject with its MBCcen ter C radius r TP s and
SP
b V ertex angle sequence VAS and shap e descriptor f k Figure T ouchP ointV ertex Angle Sequence Metho d
Computation and Storage Cost Analysis
This section pro vides cost analysis for the dieren t metho ds and compares their eciency in terms of computation
and storage requiremen ts
Grid Based Metho d
F or the GB metho d w e assume that the grid cell size gnX g n the standardized ma jor axis size mj total n um ber of
pixels of the grid px mj
n um b er of co ordinates of ob ject N and binary sequence size SS mj g n b ytes
Then the follo wing storage is required for the shap e signatures
b ytes to store the eccen tricit y of eac h shap e one real n um b er
SS mj g n by tes to store one binary sequence
n SS nbytes to store n shap e signatures assuming at least sequences p er ob ject
A temp orary storage of px b ytes p er ob ject
T o compute the shap e signature for the query ob ject and computing the similaritybet w een the query ob ject and
the ob jects in the database the follo wing ma jor op erations are required T o nd ma jor axis O N
T o nd minor axis O N T o rotate and scale O N Toc heck ifapoin t is inside a p olygon O N T o generate a binary n um ber weneed to c hec k all the grid cells pixels to see if they are inside the shap e
O px N Usually
p
px N therefore O px N O N
F or eac h similarit y calculation weneed excl usiv e O R bet w een the query binary sequence and the storedshap e
binary sequence and coun t the n um b er of ones in the result Hence similarit y calculation tak es around SS
bits X OR op erations and SS bits additions
F ourier Descriptors Metho d
F or the FD metho d w e assume that n um b er of radii equals to rn um ber of F ourier co ecien ts is fnn um ber of
co ordinates of an ob ject N Then the follo wing storage is required for the shap e signatures
T o store one shap e signature f n by tes T o store n shap e signatures n f n by tes T o compute the shap e signature for the query ob ject and computing the similaritybet w een the query ob ject and
the ob jects in the database the follo wing ma jor op erations are required T o nd cen troid O NlogNw e compute the Delauna y triangulation to this end
T o nd all radii and to compute FD co ecien ts O r
T o generate shap e signature O r required for normalization step
F or eac h similarit y calculation w e need fn realn um b er subtractions fn realn um ber m ultipli cations fn realn um b er additions
DelaunayT riangulation Metho d
F or the DT metho d w e assume that n um ber of bins nb and n um b er of co ordinates of ob ject N Then the follo wing
storage is required for the shap e signatures
T o store one shap e signature nb by tes T o store n shap e signatures n nb by tes T emp orary storage N p er ob ject
T o compute the shap e signature for the query ob ject and computing the similaritybet w een the query ob ject and
the ob jects in the database the follo wing ma jor op erations are required T o compute the Delauna y triangulation of a shap e O NlogN T o generate shap e signature O N required to nd all angles
F or eac h similarit y calculation weneed nb in tegern um b er subtractions nb in tegern um ber m ultipli cations
nb in tegern um b er additions
TPV AS Metho d
F or the TPV AS metho d w e assume that n um ber of F ourier co ecien ts fn and n um b er of co ordinates of an ob ject
N Then the follo wing storage is required for the shap e signatures
T o store n um b er of touchpointsb yte
T o store one shap e signature f n by tes T o store n ob jects n f n by tes T o compute the shap e signature for the query ob ject and computing the similaritybet w een the query ob ject and
the ob jects in the database the follo wing ma jor op erations are required T o nd MBC O N T o compute angle sequence and nd all DF T co ecien ts of an angle sequence O N
F or eac h similarit y calculation w e need fn realn um b er subtractions fn realn um ber m ultipli cations fn realn um b er additions
Discussion
Supp ort for RST queries F or some application the concept of shap e similarit y is that the matc hing ob jects
are in v arian t to scale size and orien tation On the other hand for other applications it is imp ortan t that the
matc hing ob jects b e within a sp ecied rotation angle scaling factor translation v ector or an ycom bination of the
three RS T query With TV PAS metho d a subset of the MBC features eg r Cand SP could b e utilized to
supp ort RS T query t yp es in whic h the matc hing ob jects could b e within a sp ecied rotation angle scaling factor
translation v ector or an y com bination of the three see for further details On the other hand the GB FD and DT metho ds are unable to supp ort suc h query t yp e With the GB and FD metho ds normalization is used
so that the ob jects t in to a presp ecied mesh or force the b oundaries to ha v e a standard size and orien tation
resp ectiv ely While DT metho d represen tation pro vides no information ab out the size and orien tation of the original
ob jects
Normalization F or the GB and FD metho ds the shap e represen tations obtained for the same ob ject with
dieren t orien tation in space or with a dieren t scale are dieren t Therefore the normalization of the ob ject
b oundaries prior to indexing is crucial to meet the uniqueness criteria of the shap e descriptor The DT and TPV AS
metho ds use simple shap e attributes suc h as the angles of the resulting Delauna y triangles of a set of p oin ts minim um
b ounding circles and angle sequences Ho w ev er these attributes w ere sho wn to b e translation rotation and scale
in v arian t Therefore their similarit y measures are indep enden t of ob jects p osition scale and rotation ie ob jects
do es not ha v e to b e normalized
Exp erimen ts
In man y applications an imp ortan t criterion for testing the ecacy of the shap e retriev al metho ds is that for eac h
query ob ject the relev an t items similar shap es in the database should b e retriev ed Therefore an exp erimen tw as
designed to measure the eectiv eness of the similarit y retriev al of the four dieren t shap e represen tation metho ds
ie FD GB DTand TPV AS in terms of recall and precision that are commonly used in the information
retriev al literature
W e implemen ted a protot yp e shap e retriev al system in Ja v a on a SUN UltraSparcI I w orkstation The shap e
database of the system consists of sh and to ol shap es pro vided to us b y In order to reduce the
n umberofv ertices used to represen t the shap es while main taining its general c haracteristics w e emplo y ed a three
pass algorithm In eac h pass w e try to iden tify the straigh t lines on the b oundary of the shap e and eliminate the
extra p oin ts represen ting the line Optimally a straigh t line could b e represen ted b y its t w o end p oin ts Ho w ev er
the edge detection algorithm that w as used to generate the shap e les in tro duced more than t wopoin ts to represen t
a straigh t line The extra p oin ts mak es the straigh t line lo oks lik e a jagged line lik esa w edge
Fifteen shap e queries w ere selected randomly from the database where eac h query ob ject has similar relev an t
ob jects in the database The ten similar ob jects v arian ts w ere constructed as three rotation v arian ts three rotation
scaled up v arian ts three rotation scaled do wn v arian ts and one translated v arian t F or our exp erimen ts w e simply
assume that eac h shap e is relev an t only to itself and to its ten v arian ts As advised in for the GB
metho d w e emplo y ed a grid cell size of X pixels and the length of the standardized ma jor axis w as xed at pixels With FD metho d w e used radiusbased signature with uniformly sampled b oundary p oin ts F or b oth FD
and TPV AS metho d w ek ept lo w frequency co ecien ts of the F ourier transform Finally for the DT metho d w e
used t w o largest angle histogram with bins of size degrees
Giv en a query ob ject the system retriev es relev an t shap es from the database in decreasing order of similarit yto
the query shap es First the shap e features of the query ob ject using the four dieren t metho ds are extracted Second
the Euclidean distances b et w een the shap e signature of the query ob ject and all other ob jects shap e signatures are
computed
Third the Euclidean distances are ordered in an ascending order for all the metho ds Finally the
accuracy of the retriev al metho ds w as calculated as precisionrecall T o w ards this end w eiden tify the ranks of the
relev an t ob jects to the query ob ject inside the ordered list according to Euclidean distance
The recall and precision results for query shap es are a v eraged and sho wn in Fig F rom Figa it is observ ed
that the FD metho d p erforms w orse than the other three metho ds This p o or p erformance is explained b ythe fact
that the shap e signature is constructed using shap e radii of p oin ts uniformly sampled along the ob ject b oundary These p oin ts are not the exact v ertices of the ob ject therefore using dieren t sampled p oin ts leads to dieren t shap e
signatures In other w ords the rotation normalization is not truly ac hiev ed b y taking only the magnitude information
from the F ourier co ecien ts F us and ignoring its phase information as explained in Figb sho ws the
The retriev al can b e done more ecien tly using index structures but since this is not the fo cus of this study w e assumed the sequen tial
scan of the database
Precision
0
20
40
60
80
100
120
1 0 20 30 40 50 60 70 80 90 100
TPV AS
GB
FD
DT
Recall
GB
91
92
93
94
95
96
97
10 20 30 40 50 60 70 80 90 100
TPV AS
DT
Recall
Precision
GB
a F our Metho ds b Three metho ds
Figure RecallPrecision Curv es
T able Costs
a Storage cost
GB DT FD TPV AS
nb fn fn
SS n nb n fn n fn
n SS temp N
temp px b Shap e signature generation cost
GB DT FD TPV AS
O N
O NlogN O N log N O N O N O N O r
O N O N O r O N O px N c Similarit y computation cost
GB DT FD TPV AS
S S bit xor nb int sub fn float sub f n f loat sub
S S bit add nb int mult fn float mult f n f loat mult
nb int add fn f loat add fn f loat add
recallprecision curv es of the metho ds excluding FD metho d F rom Figb w e mak e the follo wing observ ations
First that the accuracy of our prop osed metho d TV PAS is comparable to the other metho ds Second unlik e
what w as exp ected the results obtained for the DT metho d w ere b etter than what w as rep orted in This is
due to the fact that in their approac h they reduce the n um ber of v ertices used to representan object b y iden tifying
highcurv ature p oin ts called c orner p oints On the other hand in our exp erimen ts w e used the three pass algorithm
In their exp erimen ts the n um ber of v ertices used to represent anobjectw as appro ximately p oin ts ho w ev er in
our approachitw as from p oin ts Although our approac h used more v ertices to represen t the ob jects ho w ev er
the represen tation is more discriminativ e in shap e signatures W e can conclude that the DT metho d p erformance
highly dep ends on the tec hnique used to nd the represen tativev ertices of the ob jects
Sec pro vides cost computation analysis for the dieren t metho ds and compares their eciency in terms of
computation and storage requiremen ts F rom T able a w e can conclude that b oth TPV AS and FD ha v e the
lo w est storage requiremen t while DT metho d requires the largest storage requiremen t GB requires larger storage
than the DT metho d if the ob jects ha vem ultiple ma jor axes Both GB and DT require some additional temp orary
storage to p erform their algorithms T able b sho ws the computation cost to generate the shap e signatures of the
ob jects W e can conclude that TPV AS has the lo w est computation cost and GB the highest Both DT and FD
metho ds ha v e comparable computation costs due to the use of Delauna y triangulation algorithm Ho w ev er T able c
sho ws that GB requires the lo w est computation cost to compute the similaritybet w een t w o shap e signatures while
the other three metho ds ha v e comparable computation costs
Conclusion and F uture W ork
This study compared four shap ebased ob ject retriev al tec hniques FD GB DTand TPV AS The similarit yre triev al accuracy of our metho d TV PASw as comparable to the other metho ds while it had the lo w est computation
cost to generate the shap e signatures of the ob jects Moreo v er it has lo w storage requiremen t and a comparable
computation cost to compute the similaritybet w een t w o shap e signatures In addition TPV AS requires no nor
malization of the ob jects and is the only metho d that has direct supp ort to RS T query t yp es W e also in tro duced
a new shap e description taxonom y Wein tend to extend this w ork in sev eral directions First w eplan toin v estigate ho w to handle ob jects with
curv ed parts holes and op en curv es or lines Second partial similarit ymatc h is not supp orted b y all the systems
therefore wein tend to iden tify b etter features to b e used to for partial similarit y matc h queries Third in our
exp erimen ts w e assume that eac h shap e is relev an t only to itself and to its ten v arian ts Ho w ev er h uman p erceptions
on shap e similarit y are appreciably dieren t Therefore w ein tend to incorp orate h uman p erception in the future
ev aluation of the eectiv eness of the shap e matc hing metho d F ourth w ein tend to test the metho ds for robustness
to noise b y p erforming queries on a database of noisy shap es Finally w ew an t to extend this w ork to supp ort three
dimensional ob jects Our preliminary in v estigations sho w that analogous to MBC features for D ob jects w ecan
extract features from D ob jects b y using their minim um b ounding spheres
Ac kno wledgmen t
The authors thanks Geo Leac h for pro viding us with the Ja v a co de for implemen ting Delauna y triangulation
algorithm a v ailable at h ttpgoannacs rmiteduau glresearc hcomp geomdelauna ydelauna y h tm l Sp ecial
thanks also to F Mokh tarian X Ding and S Abbasi for pro viding us with the sh shap es data set a v ailable
at ftpftpeesurrey acukpubvisionmiscsh con tourstarZ W ew ould also lik e to thank J Gary for kindly
pro viding us with the to ol shap es data set
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Description
Maytham Safar, Cyrus Shahabi and Xiaoming Sun. "Image retrieval by shape: A comparative study." Computer Science Technical Reports (Los Angeles, California, USA: University of Southern California. Department of Computer Science) no. 719 (1999).
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Shahabi, Cyrus
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Sun, Xiaoming
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USC Computer Science Technical Reports, no. 719 (1999)
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Image retrieval by shape: A comparative study (
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