Clustered Signatures for Modeling and Recognizing 3D Rigid Objects

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1 World Academy of Scece, Egeerg ad Techology Iteratoal Joural of Computer, Electrcal, Automato, Cotrol ad Iformato Egeerg Vol:, No:5, 008 Clustered Sgatures for Modelg ad Recogzg 3D Rgd Obects H. B. Darbad, M. R. Ito, ad J. Lttle Iteratoal Scece Idex, Computer ad Iformato Egeerg Vol:, No:5, 008 waset.org/publcato/9067 Abstract Ths paper descrbes a probablstc method for three-dmesoal obect recogto usg a shared pool of surface sgatures. Ths techque uses flatess, oretato, ad covexty sgatures that ecode the surface of a free-form obect to three dscrmatve vectors, ad the creates a shared pool of data by clusterg the sgatures usg a dstace fucto. Ths method apples the Bayes s rule for recogto process, ad t s extesble to a large collecto of three-dmesoal obects. Keywords Obect recogto, modelg, classfcato, computer vso. I. INTRODUCTION HREE-DIMENSIONAL model-based obect T represetato [ uses the geometry of a obect for modelg. I ths method, a obect s geometrc propertes ad relatos are extracted ad stored as a model of that partcular obect. Durg the matchg process, the same procedure s appled to the test obect, ad ts geometrc propertes ad relatos are compared agast the models for detfcato purposes. The ma goal of all modelg techques s to extract suffcet obect features to eable relable obect recogto durg the matchg process. Recetly, due to the decreasg cost of 3D scaers, more complex methods that use patches created from dese mages have bee troduced the lterature. These methods clude varety of dfferet techques such as sp mages [, surface pot sgatures [3, surface splash [4, harmoc shape cotexts [5, flatess ad oretato sgatures [6, ad the tesor method, whch models ad recogzes 3D obects cluttered evromets [7. A good survey of dfferet techques ca be foud [8 ad [9. The sze of the model created by all the above metoed modelg techques creases learly as more obects are added to the lst of the model lbrary. As a result, the effcecy stll remas as oe of the ma problems wth these modelg techques. Ths problem creases the sze of the lbrary model, ad makes the recogto process too costly. I ths paper t wll be show that the flatess, oretato, ad covexty sgatures (FOC sgatures) [0 ca be shared by dfferet obects a pool of dscrmatve sgatures. A H. B. Darbad ad M. R. Ito are wth Electrcal ad Computer Egeerg, Uversty of Brtsh Columba (e-mal: hosseb@ece.ubc.ca, mto@ece.ubc.ca). J. Lttle s wth Computer Scece, Uversty of Brtsh Columba (e-mal: lttle@cs.ubc.ca). probablstc method the ca be used to accurately recogze ad match the obects o the scee wth the lbrary of the models. The sze of the model created by clustered sgatures s substatally smaller tha the orgal models used [0, ad the recogto results are better. The rest of ths paper s orgazed as follows: I Secto II, the FOC sgatures are explaed, ad set of models used ths paper are descrbed. Secto III descrbes detals of the proposed modelg techque. I Secto IV, the results ad aalyss of the expermets are preseted ad compared wth the orgal method ad sp mages, ad secto V the cocluso ad suggesto for further vestgato are dscussed. II. BACKGROUND Ths paper uses FOC sgatures for modelg threedmesoal rgd obects by ecodg the fluctuato of the surface ad the varato of ts ormal aroud a oreted surface pot, as the surface expads. I ths method, the surface of a obect s ecoded to three dscrmatg vectors o each oreted pot o the surface of the obect termed the flatess, oretato, ad covexty sgatures. The collecto of these three sgatures s the used to model ad recogze the obect. A. Revew The basc elemet used to model ad recogze a obect [0 s referred to as oreted pot. A oreted pot s a pot (P) o the surface of a obect alog wth ts ormal (N) at pot P. Cosder a oreted pot, P, o the surface of a obect (please refer to the left-had colum of Fg. ). Now, assume a sphere wth radus R, cetered o P. If S s the total area of the obect crcumscrbed sde the sphere, ad A s the proecto of S o a plae Π, ormal to N, the A F = 0 F () S F specfes the flatess of the area aroud a oreted pot, P, o the surface. For a flat surface, F s equal to ; for a curved surface, F s less tha oe. The more curved the surface s, the lower the value of F would be. As R creases, the rato of A/S chages, creatg a graph that s the flatess sgature of the surface aroud pot P. The flatess sgature of a flat area s a horzotal le. Smlarly, the agle of the ormal, θ, whch s equal to the average of the ormal of the patches eclosed the sphere, also Iteratoal Scholarly ad Scetfc Research & Iovato (5)

2 World Academy of Scece, Egeerg ad Techology Iteratoal Joural of Computer, Electrcal, Automato, Cotrol ad Iformato Egeerg Vol:, No:5, 008 Fg. Modelg. Left: Accumulated method. Mddle: Dfferetated method. Rght: Sample sgatures Iteratoal Scece Idex, Computer ad Iformato Egeerg Vol:, No:5, 008 waset.org/publcato/9067 fluctuates from ts orgal posto, N, as R creases, creatg aother curve, the oretato sgature. For a symmetrc surface, the oretato sgature s a horzotal le f P s set o the symmetrcal pot o the surface. The combato of oretato sgature ad flatess sgature models the obect o pot P o the surface of a obect. The collecto of the sgatures s used to model the etre obect. Each sgature s a vector of d elemets, whch d s the umber of tervals used to geerate the sgature. For each terval, the radus of the sphere s set to R = R + ΔR.. To fd the sgature for each oreted pot o a vertex, the ormal of the vertex, N, s frst calculated by averagg the ormal of the surfaces aroud the vertex. To decrease ose effect, the ormal of the patches s averaged sde a sphere wth a specfc radus called the base radus. By fdg the ormal, N, ad Π, the plae ormal to N, the two vectors S = [ s 0 s,..., s d A = [ a 0 a,..., a d () ca be calculated. The flatess sgature, F = [ f 0 f..., f d, of the oreted pot P s the calculated from F = A /( S + ε ). Here, ε s added to each elemet of S to avod dvso by zero. Cocurretly, the oreted sgature O = [ o 0 o,..., o d s foud. The collecto of F ad O sgatures models the surface of the obect at the selected vertex. Ths method s called accumulatve method [6. The ma problem wth ths method s that as R creases, the sgatures reach a relatvely steady state. Cosequetly, the dsparty of the sgatures decreases as R creases. To overcome ths problem, stead of usg the surface of the obect crcumscrbed sde a sphere as R creases, two cocetered spheres are used, as show the mddle colum of Fg.. The area crcumscrbed betwee two spheres wth radus R -R s the used for sgature creato. Ths method s called dfferetated method [0. The flatess sgature ths method s calculated by F = A /( S + ε ) where S = Δs = s s Δs 0 Δs,..., Δs [, d [ Δa = a a Δa 0 Δa Δa A =,...,, (3) d Fg. Covexty ad cocavty of a sample surface Expermets show that the sgatures created by dfferetated method are sestve to ose. Whe Δs ad Δa are too small, a small perturbato o the surface of the obect caused by ose affects the flatess sgature. To remedy ths problem, the flatess sgatures are smoothed wth a hgh-pass half a bell-shape flter wth parameters μ = mea(s) ad δ = std ( S for all Δs < μ) for all the values of S<µ. B. Covex ad Cocave Surfaces The method troduced so far creates detcal sgatures for both cocave ad covex surfaces. To dstgush betwee cocave ad covex surfaces, the surface s dvded to postve ad egatve patches (covex ad cocave surfaces respectvely) based o the drecto of ther ormal relatve to a oreted pot. Cosder the cross-secto of surface S Fg.. O s a pot o the surface, ad N s ts ormal. Let us assume we are terested fdg the sgatures of the surface relatve to oreted pot O. To fd the covexty ad cocavty of the surface o pots O ad O 3 relatve to O, coect O to O ad O to O 3 to create two vectors, O O ad O O 3. The fd the proecto of N ad N 3 o O O ad O O 3, T ad T 3, respectvely. If the drecto of O O x ad T x are the same, the the surface at pot O x s covex relatve to pot O. If the drecto of O O x ad T x are opposte, the the surface at pot O x s cocave relatve to O. For example, relatve to the oreted pot O, the surface at O ad O 3 are covex ad cocave, respectvely. I addto to flatess ad oretato sgatures, the total covex ad cocave area at each step of sgature creato are Iteratoal Scholarly ad Scetfc Research & Iovato (5)

3 World Academy of Scece, Egeerg ad Techology Iteratoal Joural of Computer, Electrcal, Automato, Cotrol ad Iformato Egeerg Vol:, No:5, 008 Fg. 3 Left: Test obects. Rght: Surface of the test obects removed form 0% up to 50% of ther total surface Iteratoal Scece Idex, Computer ad Iformato Egeerg Vol:, No:5, 008 waset.org/publcato/9067 stored separately. The calculato of covex ad cocave areas has o maor effect o processg tme sce the same procedures are used for calculatg flatess ad oretato sgatures. The covexty sgature s calculated by dvdg the covex area to the total area at each step of sgature creato. The covexty ca hold a value betwee oe ad zero. A value of oe for covexty meas that the area beg cosdered for sgature creato s a complete covex area. A value of zero meas that the area s a complete cocave or a flat area. Covexty sgatures, alog wth flatess ad oretato sgatures, are used to model the surface aroud a oreted pot. A smlar flter used to smooth fatess sgatures s appled to covexty sgatures to reduce the ose effect. Please refer to the rght-had colum of Fg. for sample FOC sgatures o the surface of a obect. C. Models ad Settg The Prceto bechmark [ models (Fg. 0) were chose as expermetal models to test our approach. The models were scaled to have a maxmum dmeso equal to 00 uts for cosstecy. Normal ose wth stadard devato equal to was added to each vertex the ormal drecto of the surface of the models to create osy test models (please refer to the left-had colum of Fg. 3). To demostrate the robustess of the sgatures to clutter, a large porto of the osy test models were removed by selectg a radom patch as seed patch o the surface of the test obect, ad surface patches totalg up to 50 percet of the total surface of the obect were removed. Ths s a vald assumpto for cluttered surfaces, sce subtractg ad addg extra surface have smlar effect o the sgatures created by the proposed method. The rght-had colum of Fg. 3 shows sample test obects that ther surfaces were removed up to 50 percet of ther total surface area. Sgatures created wth the proposed modelg techque deped o two parameters that lmt the modelg area of a surface aroud a oreted pot. The supportg dstace lmts the value of R. Small values of R model the local deformato aroud pot P, ad large values of R model the global deformato of the surface relatve to the oreted pot P. Due to occluso, the etre obect caot be see from a sgle vewpot. It s logcal to assume that f the ormal of a patch makes a agle greater tha a threshold agle wth the oretato of pot P, t caot be see from the same agle that sees pot P. Therefore, those patches caot be used for modelg the obect from that vewpot. These two parameters are called support dstace ad support agle, respectvely [. The drecto of the ormal of the patches s a mportat factor creatg the sgatures. The ormal should pot to outsde of the obect to create cosstet sgatures. We determed the drecto of the ormal of the patches by assumg dfferet vew-pots far from the obect, ad the the drectos were calculated by a method smlar to ray tracg method [3. D. Effect of Samplg Itervals o Sgatures It was show [0 that FOC sgatures are robust to scale, oretato, occluso ad clutter, patch resoluto (samplg rate), ad they tolerate ose. Our expermets dcate that FOC sgatures are also robust to samplg tervals ( Δ R ). Dfferet samplg tervals ( Δ R ) create dfferet flatess ad cocavty sgatures. To deal wth ths problem, stead of storg flatess ad cocavty sgatures, ther compoets were stored as modelg parameters, ad the the sgatures were calculated from these compoets. Flatess ad covexty sgatures are calculated as A Acovex + Acocave Flatess = = S Scovex + Scocave ad Scovex Covexty = Scovex + Scocave where A covex, A cocave, S covex, ad S cocave are A ad S compoets of covex ad cocave surfaces. Fg. 4 shows the four compoets of a sample vertex o the surface of M48. I tur, A ad S compoets of dfferetated sgatures ca be calculated from A ad S compoets of accumulatve sgatures (please refer to equatos ad 3). As a result, Iteratoal Scholarly ad Scetfc Research & Iovato (5)

4 World Academy of Scece, Egeerg ad Techology Iteratoal Joural of Computer, Electrcal, Automato, Cotrol ad Iformato Egeerg Vol:, No:5, 008 Iteratoal Scece Idex, Computer ad Iformato Egeerg Vol:, No:5, 008 waset.org/publcato/9067 Fg. 4 Compoets of flatess ad covexty sgatures of a sample vertex o the surface of M48 stead of storg flatess ad covexty sgatures, four dscrmatve vectors of the accumulatve method, A covex, A cocave, S covex, ad S cocave were stored as compoets of the sgature of the model o each oreted pot. Sce the values of the compoets creases cotuously, they ca be re-sampled accurately wth ay tervals. The flatess ad covexty sgatures of both accumulatve ad dfferetated methods are the calculated from these compoets. The compoets of oretato sgatures caot be stored. However, expermets dcated that oretato sgatures ca be dow-sampled safely. Note that oretato sgatures caot be up-sampled. III. PROPOSED METHOD The authors [0 used ormalzed cross correlato to compare sgature of the models. However, the expermets dcate that the smple proposed dstace measuremet works much faster ad provde better measuremet to compare the sgatures of the models. The dstace of two sgatures (S ad S ) s calculated as ther ormalzed Eucldea dstace ds( s, s ) = S S (4) d where d s the dmeso of the vectors S ad S. The dssmlarty of the oreted pots s calculated from; / dssmlar ty = ( ds( f, f ) + ds( o, o) + ds( c, c) ) (5) where f x, o x, ad c x are the FOC sgatures of the oreted pot x. A. Dscrmatory Power of the Sgatures To study dscrmatory power of the sgatures, radom vertces were selected o the surface of the test obects, ad the ther sgatures were compared wth the sgatures of the orgal model o the same oreted pots usg equato (5). For comparso purposes, radom vertces were also selected o the surface of the lbrary models ad ther sgatures were compared wth the sgatures of other radomly selected vertces. Fg. 5 shows the cremetal hstogram of the dssmlarty of the oreted pot where Nosy Sgatures are the sgatures of the osy-test models compared wth the sgatures of the orgal models, ad Radom sgatures are the sgatures of radomly selected vertces ad compared wth other radomly selected sgatures. As dcated the fgure, the dssmlarty creates a dstctve measuremet for comparso. For example, whle about 80% of the sgatures of the osy models compared wth the sgatures of the orgal models have a dssmlarty less tha a threshold value, oly about 0.7% of the Radom sgatures have a dssmlarty less tha the same threshold value. B. Sharg Sgatures Expermets dcate that oreted pots close to each other o the surface of a obect have smlar FOC sgatures. Furthermore, smlar surfaces have smlar FOC sgatures. As a result, FOC sgatures ca be shared to model a varety of obects. The sgatures were clustered by usg a method smlar to BIRCH [4. Sce we do ot kow the umber of the clusters, we set a threshold for groupg smlar sgatures a cluster. To calculate the threshold value, we referred to our expermets the prevous Secto. The threshold value to cluster the sgatures was set to the meda dstace of the osy sgatures compared wth the orgal sgatures for each set of F, O, ad C sgatures. threshold = meda( ds( osy _ sgaures, orgal _ sgatures)) The threshold was used as the maxmum ter-cluster dstace betwee the sgatures of the clusters [5. Here the commo covarace matrx ( Σ ) was used for the clusters. The shared covarace matrx was calculated by the followg equato σ Σ = ( σ det( σ ) 0) Fg. 5 Dssmlarty of the oreted pots where σ s the covarace matrx of cluster, ad s the umber of the sgatures the cluster. The top-row of Fg. 6 shows the umber of the clusters versus the umber of the sgatures for each set of F, O, ad C sgatures. As dcated the fgure, the umber of clusters does ot crease learly as more sgatures are added to the lst of the sgatures. Iteratoal Scholarly ad Scetfc Research & Iovato (5)

5 World Academy of Scece, Egeerg ad Techology Iteratoal Joural of Computer, Electrcal, Automato, Cotrol ad Iformato Egeerg Vol:, No:5, 008 Iteratoal Scece Idex, Computer ad Iformato Egeerg Vol:, No:5, 008 waset.org/publcato/9067 The mddle-row of the fgure shows the hstogram of the sgatures per clusters. As dcated the fgure, half of the sgatures were grouped clusters wth sgatures ad more, ad aother half of the sgatures were grouped to clusters wth sgatures per cluster or less. I our expermets, sce obects were symmetrcal, clusters wth oe sgature were deleted as outlers. Fally, the bottom-row of the fgure shows the umber of the clusters versus umber of the dstctve models each cluster. For example, as show the fgure, there were 48 clusters that each cluster was shared by sgatures of four dfferet models. C. Obect Recogto The probablty of a obect gve sgature s from a lst of models s calculated by Bayes rule: s O ) O ) (6) O s) = s O ) O ) Sce each sgature may come from ay cluster, the P ( s O ) = s C ) C ) where C s cluster, ad Τ s C) = exp ( s μ ) Σ ( ) s μ d (π ) Σ r P ( C ) = N m m r m s the umber of the sgatures of model m cluster, ad N m s the total umber of the sgatures of model m all clusters. Sce the same probablty ca be assumed for all obects, the equato (6) becomes s C ) C ) O s) = (7) s C ) C ) Sce there are three dfferet types of sgatures ad clusters, the probablty of model O gve sgature foc s equal to 3 O foc ) = log O s ) = log O s ) (8) s= where s s oe of the f, o, ad c sgatures of sample. Sce oe outler wth the probablty equal to zero equato (8) affects all further calculatos, the zero probablty was substtuted by the lowest probablty calculated for all clusters equato (8). For N sample pots selected o the surface of the test obect, the probablty of obect O gve sgature set foc s equal to N = k P ( O ) = log O foc ) = log( O foc )) Sce the lowest probable sample pot was elmated as a possble outler for each model, the umber of the sample pots reduced to N-. The model wth the hghest probablty was chose as the possble caddate model. Fg. 6 Top: Number of the clusters versus umber of the sgatures. Mddle: Hstogram of umber of the sgatures per clusters. Bottom: Number of the clusters vs. umber of the models shared each cluster D. Geometrc Verfcato Because geometrc cosstecy betwee the obect ad the caddate model s ot beg checked, there s possblty that the postve caddate model selected the prevous secto may ot be the correct match. To make the recogto more flexble, the top M most probable caddates were selected ad passed o for geometrc verfcato. For the geometrc verfcato process, correspodg oreted pots o the surface of the caddate models are eeded for each set of foc sgatures. Sce the flatess sgatures are more descrptve tha the oretato ad covexty sgatures, the correspodg oreted pots were selected from the two most probable clusters of the flatess sgatures. I ths way, there may be a hadful of correspodg pots (r m may be large) for each caddate Iteratoal Scholarly ad Scetfc Research & Iovato (5)

6 World Academy of Scece, Egeerg ad Techology Iteratoal Joural of Computer, Electrcal, Automato, Cotrol ad Iformato Egeerg Vol:, No:5, 008 Iteratoal Scece Idex, Computer ad Iformato Egeerg Vol:, No:5, 008 waset.org/publcato/9067 model, so the correspodg oreted pots were clustered based o ther dstace ad the agle of ther oreted pots for geometrc verfcato process. Ths process s ecessary to lmt umber of the samples partcularly f the models were sampled wth a fe resoluto. The clusterg process reduces the umber of the sample pots to a maageable umber. I our expermets, we have clustered the sample pots oly f ther dstace was less tha 5 uts ad ther agle was less tha π / 8. To verfy the oreted pots geometrcally, the same method used to verfy sp mages [6 s used. Assume that s x ad m y represet a match betwee the pot x o the scee to the pot y o the surface of a model. Now, let C = [ s, m ad C = [ s, m be two correspodeces betwee the scee ad the model. The geometrc cosstecy betwee two sets of the matches s calculated as follows: Sm ( m ) S ( ) s s d( P, P ) = S ( m ) + S ( s ) where S p m ( q) = ( α, β ) = s ( p q ( ( p q)), ( p q) ) where p ad q are two pots 3D, ad s ormal at pot p. Sce d s ot symmetrc, the maxmum of the d(p, p ) s used to defe the geometrc cosstecy, D. D = max( d( P, P ), d( P, P )) Whe D s small, P ad P are geometrcally cosstet that scee ad model pots P ad P are the same dstace apart, ad ther surface ormal form the same agle wth each other. All the matched oreted pots wth ther D greater tha a threshold, k, were elmated as outlers. I our expermets k was set from 0.5 to 0.. E. Regstrato For regstrato process, oly three oreted pots are eeded to estmate the trasformato matrx, sce three pots ad three depedet ormal are eough to calculate trasformato [7. The remag matches betwee test obect ad model were the tragulated usg ther D value from the prevous secto. Sce, ths s a smple look-up table, the process s fast ad effcet. The oreted pots, whch are tragulated, are sorted based o ther D value. Dtragulat ed = D + D3+ D3 Several radom tragulated matches wth lowest D values were selected ad used for tal estmato of trasformato matrx. Sce calculato of trasformato whle keepg R = ad RR τ = s a kow problem, quatero [8 was used to calculate the rotato matrx, R. From there, the k- earest eghbor method [9 was used to regster the test obect o the scee to the model. Fg. 7 Average of recogto results for osy test models. Top: Postve caddate selecto. Bottom: True-postve recogto rate Fg. 8 Average of recogto results for cluttered obects. Top: Postve caddate selecto. Bottom: True-postve recogto rate Iteratoal Scholarly ad Scetfc Research & Iovato (5)

7 World Academy of Scece, Egeerg ad Techology Iteratoal Joural of Computer, Electrcal, Automato, Cotrol ad Iformato Egeerg Vol:, No:5, 008 Iteratoal Scece Idex, Computer ad Iformato Egeerg Vol:, No:5, 008 waset.org/publcato/9067 IV. EXPERIMENTS I our expermets the support dstace was set from 0 to 00 wth tervals of 0, the support agle was set to π / 3, ad the base-radus was set to 0. The sx radom oreted pots were selected o the surface of the osy models, ad ther FOC sgatures were created. The sgatures were the compared ad matched to the lbrary models show Fg. 0 that cossts of 69 toys cludg vectors for each F, O, ad C descrptve sgatures. The sgatures were grouped to 9906, 7060, ad 53 clusters for F, O, ad C sgatures respectvely. Each osy model was tested for 50 recogto tmes for each support dstace. A. Results The top-row of Fg. 7 shows the average of postve caddate selecto from frst to fourth caddates. As dcated the fgure, by extedg the caddate selecto to secod ad more, the postve recogto rate creases. The bottom-row of the fgure shows average true-postve recogto results whe the average algmet error after regstrato process falls below a threshold value whch was set to 0 uts our expermets. The false-postves are the cases where smlar obects were recogzed as the target model. Fg. 8 shows the same results for cluttered obects. The toprow of the fgure shows the average of postve caddate selecto for dfferet levels of clutter, ad the bottom-row of the fgure shows the average true-postve recogto results whe the algmet error falls below 0 uts. As dcated the fgure, by creasg clutter, the recogto rate decreases. Sce sample pots ear the cluttered area are ot relable for matchg purposes, recogto expermets of the cluttered obects, e sample pots were selected radomly to elmate the effect of the sample pots selected ear the cluttered area. I our expermets parameter M was set to 4. By creasg value of M, the recogto rate of the cluttered obects sgfcatly mproves. B. Comparso Sce sp mages ad orgal dfferetated method were used by a smlar approach for matchg ad recogto process (pleaser refer to [0 ad [), the sp mages of the test obects ad models were created wth exactly the same parameters for both modelg techques. The support dstace was set toπ / 3, ad the b of sp mages set to Δ R. Each sp mage was a hstogram of 0 x 40 bs. The recogto results of the proposed method were compared wth the recogto results of the orgal dfferetated method ad those of sp mages. The sample pots selected were the same all expermets, ad expermets coducted wth exactly the same parameters. The top-row of Fg. 9 shows the recogto rate gaed by the proposed method comparg wth the orgal dfferetated method ad the method used by sp mages. Fg. 9 Top-row: Recogto gaed by clustered sgatures. Secodrow: comparg recogto rate of the proposed techque ad sp mage for dfferet level of clutterg. Thrd-row: comparg recogto rate of the proposed techque ad the orgal model for dfferet level of clutterg. Bottom-row: Comparg model sze used by dfferet modelg techques Iteratoal Scholarly ad Scetfc Research & Iovato (5)

8 World Academy of Scece, Egeerg ad Techology Iteratoal Joural of Computer, Electrcal, Automato, Cotrol ad Iformato Egeerg Vol:, No:5, 008 Iteratoal Scece Idex, Computer ad Iformato Egeerg Vol:, No:5, 008 waset.org/publcato/9067 Each graph s the result of subtractg the recogto rate of the dfferetated method ad sp mages from recogto rate of the proposed method. The sold curve the fgure shows gaed acheved recogto rate comparg wth the recogto rate of the orgal dfferetated method, ad the dotted-curve shows the recogto rate gaed by proposed method comparg wth the recogto rate of the sp mages. The fgure the secod-row shows dfferece of the truepostve recogto rate betwee the proposed method ad the sp mage for dfferet level of clutter. As dcated the fgure, the recogto rate of the proposed method s cosderably better tha the recogto rate of the sp mage for low clutter level ad for small support dstaces. However, the recogto rate of the sp mage s better for hgher clutter level ad large support dstace. The reaso s that, the FOC sgatures created ear cluttered area are ot descrptve eough for matchg purpose. Oe possble soluto s to select sample pots far from cluttered area rather that radom selecto. The fgure the thrd-row shows dfferece of the truepostve recogto rate betwee the proposed method ad the orgal method for dfferet level of clutterg. As dcated the fgure, the proposed method shows better recogto rate for cluttered area tha the orgal dfferetated method. Fally, the bottom-row of the Fg. 9 compares the model sze created ad used by each method. As dcated the fgure the sze of the orgal dfferetated method was 7.5% of the sze of the sp mage, ad the sze of the clustered sgatures was oly 0.88% of the sze of the sp mage. However, the recogto rates of the clustered sgatures are better tha the recogto rates of the dfferetated method ad those of the sp mage. V. CONCLUSION I ths paper t s show that a varety of 3D obects ca be modeled wth shared data, ad that obects ca be recogzed relablty durg the matchg process. We chose oly sx Fg. 0 Lbrary models used the expermets sample pots o the surface of the test obect ad e sample pots o the surface of the cluttered obects for recogto process. By creasg the umber of the sample pots o the surface of the test obects recogto rate creases. The method preseted ths paper s sutable for multple obect recogto. However, f the target obect s kow, the the recogto results creases further tha the results show ths paper. Our expermets dcate that by addg more obects to the lbrary models, the umber of the clusters creases as a polyomal of order, whch teds to reach to a saturated level as umber of the sgatures teds to fty. Our expermets also dcate that the FOC sgatures are sutable for rgd obect classfcato. However, ths fdg eeds further vestgato. REFERENCES [ Beamou, G.J. Mamc, Obect Recogto: Fudametals ad Case Studes, Sprger, 000. [ A.E. Johso, Sp Image: A Represetato for 3D Surface Matchg. PhD Thess, Carege Mello Uversty, 997. [3 S. Correa, L. Shapro, A New Sgature-Based Method for Effcet 3D Obect Recogto, Proc. IEEE Cof. Computer Vso ad Patter Recogto, : , 00. [4 Ste, F.; Medo, G.; Structural dexg: Effcet 3D obect recogto of a set of rage vews, IEEE trasactos o patter aalyss ad Mache Itellgece, 7(4): , 995. [5 S.M. Yamay, A. Farag, Freeform Surface Regstrato Usg Surface Sgatures, Proc. It. Cof. o Computer Vso, : , 999. [6 H.B. Darbad, M.R. Ito; J. Lttle, Flatess ad Oretato sgature for Modelg ad Matchg 3D Obects, Thrd Iteratoal Symposum o 3D Data Processg, Vsualzato ad Trasmsso, 006. [7 Ma, A.S.; Beamou, M.; Owes, R. Patter Three-Dmesoal Model-Based Obect Recogto ad Segmetato Cluttered Scees, Aalyss ad Mache Itellgece, IEEE Trasactos o Volume 8, Issue 0, Oct. 006 Page(s): [8 R.J. Campbell, P.J. Fly, A Survey of Free-Form Obect Represetato ad Recogto Techques, Computer Vso ad Uderstadg, vol. 8, pp. 66-0, 00. [9 B.M. Platz, A.J. Maeder, J.A. Wllams, The correspodece framework for 3D surface matchg algorthms, Computer Vso ad Image Uderstadg, 005. Iteratoal Scholarly ad Scetfc Research & Iovato (5)

9 World Academy of Scece, Egeerg ad Techology Iteratoal Joural of Computer, Electrcal, Automato, Cotrol ad Iformato Egeerg Vol:, No:5, 008 Iteratoal Scece Idex, Computer ad Iformato Egeerg Vol:, No:5, 008 waset.org/publcato/9067 [0 H.B. Darbad, M.R. Ito, J. Lttle, Surface Sgature-Based Method for Modelg ad Recogzg Free-Form Obects, 3 rd Iteratoal Sysposum o Vsual Computg, 007. [ [ A. Johso; M. Herbert, Usg sp mages for effcet obect recogto cluttered 3D scees, IEEE Tr. o Patter Aalyss ad Mache Itellgece. Vol., N0 5. May 999. [3 J.D. Foley, A. va Dam; S.K. Feer; J.F. Hughes, Itroducto to Computer Graphcs, Addso-Wesley, 993. [4 T. Zhag, R. Ramakrsha, M. Lvy, BIRCH: A Effcet Data Clusterg Method for Very Large Databases, SIGMOD Coferece 996: [5 E. Alpayd, Itroducto to Mache Learg, MIT Press, 004. [6 A. Johso, M. Herbert, Usg sp mages for effcet obect recogto cluttered 3D scees, IEEE Tr. o Patter Aalyss ad Mache Itellgece. Vol., N0 5. May 999. [7 D.A. Forsyth, J. Poce, Computer Vso, a Moder Approach, Pretce Hall, 003. [8 J. B. Kupers, Quateros ad Rotato Sequeces: A Prmer wth Applcatos to Orbts, Aerospace ad Vrtual Realty, Prceto Uversty Press, 993. [9 H. Samet, Depth-frst k-earest eghbor fdg usg the Maxmum Nearest Dstace estmator, IEEE Trasactos, Image Aalyss ad Processg, 003. Iteratoal Scholarly ad Scetfc Research & Iovato (5)