Polyhedron Volume-Ratio-based Classification for Image Recognition

Transcription

1 Polyhedron Volume-Ratio-based Classifiation for Image Reognition Qingxiang Feng, Jeng-Shyang Pan, Senior Member, IEEE, Jar-Ferr Yang, Fellow, IEEE and Yang-ing Chou Abstrat In this paper, a novel method, alled polyhedron volume ratio lassifiation (PVRC), is proposed for image reognition. he linear regression based lassifiation (LRC) and lass mean (CM) approahes aim to find the losest projetion to the subspae formed by the prototype data vetors in eah lass. Better than LRC and CM lassifiers, the PVRC lassifier omputes the ratio of two polyhedron s volumes, where the first inlusive polyhedron is enlosed by the test plus lass prototype data vetors while the seond exlusive polyhedron is enlosed by lass prototype data vetors. he PVRC tries to find the least ratio of lass-based inlusive and exlusive polyhedron volumes among all lasses. With mathematial optimization, the PVRC lassifier ahieves better reognition rate than the existing statistial methods for objet and fae reognition. With the proposed fast algorithm, it is noted that the omputational ost of the proposed lassifier is very small. A large number of experiments on Coil00 objet database, Eth80 objet database, Soil47 objet database, G fae database and UMIS fae database are used to evaluate the proposed algorithm. he experimental results demonstrate that the proposed method ahieves better reognition rate and less omputational ost than the existing methods, suh as LRC, CM, nearest neighbor (NN), sparse representation based lassifiation (SRC) and two phase test sample sparse representation (PSSR) lassifiers. Index erms Fae Reognition, Objet Reognition, Linear Regression Classifiation, Sparse Representation based Classifiation, Nearest Neighbor. F I. INRODUCION ACE and objet reognition systems are known to be ritially dependent on lassifiers. Nearest neighbor (NN) [] and lass mean (CM) [] methods, whih are designed to minimize the distane from the test vetor to the subspae spanned by all training vetors of eah objet lass, are the well-known approahes in pattern reognition area. he NN lassifies the test sample based on the best representation in terms of a single training sample, while the CM lassifies it based on the best mean representation in terms of all the training samples in eah lass. Samples from a speifi objet lass are known to lie on a linear subspae [3]-[4]. Borrowing the above subspae onept, the loally linear regression (LLR) [] is proposed speifially to solve the problem of pose identifiation. Linear regression-based lassifiation (LRC) [], whih is proposed for the problem of fae identifiation, uses the similar onept to develop lass-speifi models of the registered users and renovate the task of fae reognition as a problem of linear regression. For fae reognition, the LRC approahes, inluding kernel LRC, Improved PCA-LRC, LDA-LRC and Unitary-LRC [7]-[0], have been proposed to further improve the reognition performane under different situations suh as variable illumination and faial expression. Instead of the lass-model suggested in the LRC approahes, sparse representation based lassifiation (SRC) [-] uses all-lass-model to lassify the test sample. Based on the SRC lassifier, some improved methods are further presented for fae reognition. hese improved lassifiers ould be lassified into two ategories. he first ategory utilizes the novel representations of eah lass, in whih all use the first phase to hoose some prototype samples while the seond phase is different [3]-[6]. Chang et. al utilizes the ollaborative representation instead of the sparse representation [7]. he seond ategory uses the sparse representation for the subspae learning/disriminant analysis, whih employs the disriminant projetion tensor disriminant projetion [8], [9] or the sparse eigenfae [0] to selet the features. SRC lassifier gains better performane for fae reognition under variations of illumination, noise, and orruption. However, SRC lassifier has two drawbaks. Firstly, SRC lassifier ould not diretly apply for objet reognition [4]-[7] and fae gesture. Seondly, the SRC lassifier aquires large omputational omplexity. hus, we need a low omputation lassifier, whih an be used for fae and objet reognition, In this paper, the polyhedron volume ratio lassifiation (PVRC) method is proposed for image reognition [7]. With the minimum distane metri, the LRC and CM lassifiers test the distane between the test sample vetor and eah lass subspae, then the losest distane is identified as the best mathed lass. With the similar lass-based lassifiation, the PVRC lassifier tests the ratio of the polyhedron volume enlosed by the test sample plus the lass prototype samples over the polyhedron volume enlosed by the lass prototype samples. he objetive of LRC and PVRC are oneptually same, however, the differene between PVRC and LRC an be found in Setion V-B. With Cayley-Menger matrix [], in this paper, we first define the polyhedron volume and propose a new metri, alled polyhedron volume ratio for inlusive and exlusive test sample to a speifi lass-subspae, for image

2 lassifiation. Based on mathematial (linear algebra) derivations, the PVRC is separated into training phase and test phase (P&P), where the omputational ost in the test phase of PVRC is very small. In the test phase, the omputation of the PVRC is almost equal to that of NN lassifier and is muh less than those of SRC, CRC and LRC. he rest of the paper is organized as the follows. First, we review several well-known image lassifiers in Setion II. he omputation of polyhedron volume and the PVRC lassifier will be introdued in Setion III. o redue the omputation, the fast PVRC lassifiation proedure is desribed in Setion IV. he analyses of the proposed PVRC are disussed in Setion V. In Setion VI, a number of experiments to show the effetiveness of the proposed lassifier are present. he onlusions and future work are finally addressed in Setion VII. II. REVIEW OF CLASS-SPECIFIC CLASSIFICAIONS Let Y { yi, i,,..., N,,,..., M} R denote the prototype image set, where denotes the i th prototype image of the th lass, M denotes the number of lasses, and is the number of prototypes belonging to the th lass. y i II-A Linear Regression Classifiation Algorithm For linear regression, eah a b image is transformed to vetor ab q by olumn onatenation as y i R xi R, where q = a b. By using the onept that data vetors from the same lass lie on a linear subspae, the LRC develops a lass-speifi model by staking the q-dimensional image vetors as X X [ x x... x ] R qn N N. () Let y be an unlabeled a b test image and our problem is to lassify y as one of the lasses. Similarly, we an transform y into the vetor form x R q. If y belongs to the th lass, it shall be well represented as a linear ombination of the training images of the lass as ( x x X X X ) X x. () X X If X X is singular in (), will be replaed by X X 0.0I, where I is a unit matrix. In (), the predited vetor an be treated as the projetion of x onto the th lass subspae. he LRC now alulates the distane measure between the predited response vetor x and the original response vetor x as d( x) x x, (3) x where * means L -norm. he LRC lassifiation rule is in favor of the lass with the minimum distane of the lass as min * d ( x),,,..., M. (4) II-B. Collaborative representation based lassifiation (CRC) Suppose that we have M lasses of subjets, we an ollet all lass-speifi models X, =,,, M, defined in () to form the omplete data model as qmn X [ X X... X M ] R. (5) If the vetor of all-lass weighing parameters is denoted as R MN, it an be alulated as follows. ( X X ) X x. (6) For (6), if is singular, will be replaed by X X 0.0I. he regularized residual of the th lass, is given as x X r, (7) X X X X where orresponding to the oeffiient of the sample of lass is a setioned olumn of. And the CRC lassifiation rule in favor of the lass with the minimum distane beomes min * r,,,..., M. (8) II-C Sparse representation based lassifiation (SRC) For the sparse representation-based lassifiation (SRC), we first normalize the olumns of X stated into (5) to have unit L -norm. We an solve the L -norm minimization problem as: gˆ argmin g subjet to Xg x. (9) Compute the regularized residuals g r x X ˆ g, (0) the SRC lassifiation rule in favor of the lass with the minimum distane an be expressed by minr,,,..., M. () * r as, III. POLYHEDRON VOLUME RAIO CLASSIFICAION Before the introdution of the polyhedron volume ratio lassifiation (PVRC), we first review the onept of polyhedron volume of n sample points. In [], Cayley-Menger matrix onstruted by n (n ) sample points is given by: where 0 ( b ) ( b3)... ( b n) ( b) 0 ( b3)... ( bn ) ( ) ( ) 0... ( ) b3 b3 b3n Q n, () ( bn) ( bn ) ( bn3) b ij denotes Eulidian distane of ij b ji i j x i and x j as b x x. (3) he squared polyhedron volume of n sample points an be defined as the determinant of Cayley-Menger matrix as, with n n n v det( Q ), for n, (4) n ( ) n, (5) n (( n )!) r

3 where det(*) denotes the determinant of the argument and n represents the unifying fator in the omputation of polyhedron volume. o verify (4), we an hek the following interesting examples. For n =, the Cayley-Menger matrix in () for two sample points, i.e., x and x, is given as 0 ( b ) Q n ( b) 0. (6) 0 he determinant of Q is Q ( b) and = / with b x x. It is interested that the volume of -point polyhedron is the true distane between these two points. Aording to Heron s formula [], the true area of triangle of three points,,, and, an be expressed by S x x x [( b) ( b3) ( b3) ( b) ( b3) ( b3) 3 3 ( b ) ( b ) ( b ) ( b ) ( b ) ( b ) ]/6 (7) 3 3 Similarly, the Cayley-Menger matrix of three points, and is given as x 3 he determinant of Q x, x 0 ( b ) ( b3) ( b) 0 ( b3) Q 3. (8) ( b ) ( ) 0 3 b3 0 Q 3 stated in (8) is given as [( b) ( b3) ( b3) ( b) ( b3) ( b3) ( b) ( b3) ( b) ( b3) ( b3) ( b3) ]. (9) From (5), 3 = /6 and (9), the polyhedron volume defined in (4) is equal to the true area of triangle enlosed by three points, x, x, and x 3 stated in (7). Similarly, we an find 4-point Cayley-Menger matrix as 0 ( b ) ( b3) ( b4) ( b) 0 ( b3) ( b4) Q ( ) ( ) 0 ( ) 4 b3 b3 b. (0) 34 ( b4) ( b4) ( b43) 0 0 With 4=/88, the square volume of tetrahedron beomes 4 Q v det( 4 )/ 88. In [5], the omputation of n-point polyhedron volume an be interpreted as the iterative omputation of the shortest distane to the (n)-point surfae while we treat (n)-point polyhedron volume as the base volume. he multipliation of the shortest distane to the base (n)-point volume will be the n-point polyhedron volume. If the number of sample data points are less than the dimension of data vetors, it an be easily, inferred that the ratio of the n-point (with one test sample) polyhedron volume over the (n)-point polyhedron volume (with samples in a lass) will be the perpendiular distane of the test sample to the (n)-point polyhedron. Based on the above interpretation of n-point and (n)-point polyhedron volumes, we introdue a new lassifier, alled polyhedron volume ratio lassifiation (PVRC) method. For images with a b pixels, they are originally represented ab as yi R by labeling the th lass while y is an unlabeled test image. In the training phase, these (n) prototype images are transformed to prototype vetors by onatenation of olumns ab q as y i R xi R, with q=a b for i=, 3,, n. hus, eah lass ontains (n) prototype samples, whih is { x, x3,, x } q n R for =,,, M. he test image y is also transferred to a test vetor x and renamed as x =. hus, the metri of PVRC an be omputed as v n R q x / n )] / n vn [ n det( Q, () vn [ n det( Q )] where denotes the polyhedron volume of the test sample and (n) prototype samples of the -lass and represents the polyhedron volume of (n) prototype samples of the -lass. he deision rule of PVRC is simply to find the least ratio as vn *= min,,,..., M. () he above minimum polyhedron volume ratio means the least perpendiular distane between the test sample and the *-lass. However, the omputation of diret PVRC method involving the alulation of two determinates is too high for pratial appliations. hus, a fast omputation method for the PVRC is needed. IV. FAS COMPUAION MEHOD AND DEAILED CLASSIFICAION PROCEDURE OF PVRC he PVRC lassifier omputes the ratio of two polyhedron volumes, where the first polyhedron is enlosed by the test sample with (n) prototype images in eah lass and the seond polyhedron is enlosed by the (n) prototype images in eah lass. o redue the omputation of the PVRC method, we should first explore the matrix segmentation and its determinant omputation in heorem. heorem : Matrix Segmentation and Its Determinant Computation Let P be a square ( m n) ( m n) matrix to be segmented into four submatries, A, B, C and D are with the sizes of m m, m n, n m and n n, respetively. If D is a reversible square matrix, we an have the equality as, det( ) det A B P det( A BD C) det( D) C D, (3) A B where P C D. he detailed proof of heorem an be found in Appendix A.

4 o further explore the relationship of the first and seond polyhedron volumes, we an first divide Q n into four sub matries as where and Q n 0 B Q B n, (4) 0 ( b 3) ( b n ) ( b3) 0 ( b3n ) Q n, (5) ( bn ) ( bn3) n B ( b ) ( b ) ( b ), (6) From (3), we know that the omputation of determinant of Q n beomes as det( Q n ) n det( Q )det( B ( Q ) B ). (7) n By substituting (7) into (), the omputation of the PVRC metri an be simply expressed by v / n ( n ) det( B ( Qn ) B ). (8) vn It is noted that B ( Qn ) B is a salar, the determinant operation will not hange it. One we pre-ompute ( ) Q n in advane, the omputation of ratio of two polyhedron volumes stated in (8) is less than any of the existing subspae-based lassifiations. Finally, the lassifiation proedure of the PVRC using fast omputation method an be addressed as follows. PVRC raining Phase: By using (5), in the PVRC training phase, we should use all (n) prototype samples, { x, x3,, xn} to ompute and store ( Q for =,,, M; n ) PVRC esting Phase: For any test sample, x, we should use all prototype samples { x, x3,, x } q n R for =,,, M to. ompute all distanes of the test sample to all prototype samples as: b i x xi, for i=, 3,, n and =,,, M;. onstrut B by using (6) for =,,, M; 3. ompute absolute value of square volume ratio as det( B ( Qn ) B ) ; (9) 4. pik *= min,,,..., M. * Atually, for true polyhedron volume, we need to inlude unifying fator n and square root operation as (5). In Step 3, the absolute value of square volume ratio with the fator related n is intentionally ignored due to fixed n. Besides, the square root operation is also skipped without affeting the minimization proess. When the numbers of samples in the lasses are various, we should further inlude a fator of 4( n ) 4 in (9) for orret lassifiation. It is noted that the prototype samples are very similar or the number of prototype samples is less than the dimension of sample vetors, the linear regression solutions stated in (), (6), (9) might not be stable due to the possible singular problem in matrix inversion. o solve the singular problems, we an add a small identity matrix for most linear regression problem. For the PVRC omputation, involving either (5) or (7), the singularity of matrix inversion only exists for similar prototype samples. Without sarifying the detetion performane, the addition of a small identify matrix to Qn V. ANALYSIS OF PVRC CLASSIFIER is also possible. In this setion, we analyze the lassifiation rule of PVRC lassifier firstly. hen, the interpretation among the PVRC, CM, NN, LRC, CRC, PSSR and SRC is introdued. In the last, the omputational ost of PVRC is desribed. V-A. Classifiation rule of PVRC From (8) and (9), we know that the deision rule of the PVRC is based on two matrixes and ( Qn ). he matrix ( Qn ) is unrelated to the test sample while the matrix is formed by the distanes between the test sample and the prototype samples of eah lass subspae. So, we learn that the lassifiation rule of the PVRC an be seen as the ombination of the distanes between the test sample and the samples of eah lass subspae. hat is to say, the PVRC lassifier an be treated as the extension of the NN lassifier. However, the ombination distane metri of PVRC lassifier is superior to the single distane metri of NN lassifier. V-B. Computation of PVRC he most omputational ost of the PVRC lassifier will be in the training phase by onstruting Qn and omputing its matrix inversion, ( Qn ) for all lasses. he omputational omplexity will be in the order of O(n 3 ) for eah lass. he omputational ost in the testing phase of the PVRC lassifier is very small. he detailed omputational omplexity of the testing phase of the PVRC lassifier is disussed as follows. From (9), the omputation in the testing phase mainly ontains two parts. he first part is the omputation ost in onstrution of the vetor, B stated in (6). If dimensionality of the sample vetor is with q, Its omputational omplexity is about in the order of O(nq) for eah lass, he total omputation omplexity will be in the order of O(nq)+O(n ). Sine the size of sample vetors, q is muh greater than the number of sample vetors, n, hus, we an say that the omputational omplexity of the PVRC lassifier is approximately equal to in the order of O(nq), i.e., the ost in onstrution of the vetor B. It is easy to know that the omputation in onstituting the matrix B is approximately equal to the ost of the NN lassifier. In summary, the omputation ost of the PVRC is approximately equal to the B B

5 ost of NN lassifier, whih is muh less than the osts of the SRC, CRC, LRC and PSSR lassifiers. he detailed advantages in various lassifiation performanes of the PVRC lassifier will be verified by experimental results. V-C Interpretation of well-known lassifiers Fig. also shows the onept of the other lassifiation methods based on lass linear regression and lass mean. he NN lassifier uses the distane between the test sample and eah prototype sample to lassify the test sample. he CM lassifier uses the distane between the test sample and the mean sample of eah lass to lassify the test sample. he other lassifiers suh as the CRC, PSSR and SRC lassifiers try to find the better approximations of the subspae of the lass to ahieve better reognition performanes. hough the NN, CM, LRC, CRC, PSSR and SRC lassifier are different, they have one thing in ommon, whih is that they all need to ompute the distane between the test sample and the approximated lass enter or the subspae, whih is derived from the prototype samples or the re-sampled prototype samples, for lassifiations. he performanes of the NN, CM, LRC, CRC, PSSR and SRC lassifiers only depend on the loseness of their approximated lass enters or subspaes. Approximated Subspae st lass Class Mean x Approximated Subspae x i m Approximated Subspae m x x x i nd lass Class Mean x est sample m x i th lass Class Mean : lass samples : lass mean : linear subspae Fig.. Coneptual relationships among the proposed PVRC and the projetion-based lassifiation methods V-D. Comparisons of PVRC and LRC lassifiers he ratio of polyhedron volumes with/without extra the test sample to the lass samples an be interpreted as the omputation of the perpendiular distane to the prototype samples. From Fig., the PVRC find the minimum distane from the polyhedron of the enlosed lass data vetors while the LRC find the least distane from spanned subspae formed by lass data vetors, he differenes between the LRC and the PVRC are addressed as follows. From omputation point of view, the PVRC is different from the LRC, where the former is derived through a mathematial (linear algebra) approah as stated in () while the latter is obtained from a statistial way to minimize the statistial preditin error as addressed in (). By simulations, Fig. shows the ratio of the omputed distanes obtained by the PVRC and the LRC if they use the same randomly-generated data points. o ompute the ratio between PVRC s distane and LRC s distane, the orresponding MALAB ode an be downloaded from [ he ratio between the PVRC s distane and LRC s distane is variation from 5 to 9. herefore, we learnt that the results of PVRC and LPC are different. Fig.. he ratio of PVRC s and LRC s distanes for 300 tests. For omputation analyses, if we separate the LRC into training phase and test phase (P&P) suh that we ould ahieve a faster lassifiation. In the test phase, the omputational omplexity of the LRC is O(Q ) while the PVRC and NN is about O(QN ). hus, the LRC aquires muh more omputation than the PVRC and NN lassifiers sine Q >> N. he similar results also verified from ode [ that the omputational time of PVRC is almost equal to that of the NN, whih is muh simpler than the original LRC and P&P LRC. Besides, the LRC needs about O(MQQ) storage spae to store pre-omputed results, while the PVRC needs about O(MN N ) storage spae. VI. EXPERIMENAL RESULS he lassifiation performanes of the PVRC lassifier are ompared to those of the LRC, SRC, CRC, PSSR, CM and NN lassifiers. wo test shemes are taken for omparisons. Firstly leave one out sheme: All images within prototype database are taken as the test samples. When an image is used as the test sample, it is not used as a prototype and it is removed from the prototype set during the lassifiations; Seondly First N sheme: the first N fae images of eah lass are used as the prototype set. he rest fae images of test database are used as test samples. he reognition rate (RR) is used to evaluate the performane of new algorithms. All experiments are implemented using the MALAB R009a under Intel (R) Core (M) i5 CPU 760 with a lok speed of.80ghz,.80ghz and 3GB RAM.

6 VI-A Computational Cost he omputational ost of the lassifiation proedure of several lassifier is depended on the number of samples, is not depended on the speifi value of samples. So, we only need provide the result of one database. he other databases are similar. In able I, the omputational ost of eah query of several lassifiers on oil00 objet database [3] is desribed, whih is orresponding to the run time of the seond experiments in the part B. From able I, after omputer MALAB simulations, we learn that the omputational osts (unit: seonds) of the PVRC, NN and CM lassifiers are similar, whih are less than those of the CRC and LRC lassifiers and muh less than that of the SRC lassifier. It is noted that the omputational ost of the first phase of the PSSR is similar to that of CRC, the omputational ost of the seond phase of the PSSR is aording to the number of samples hosen in the first phase. So, the total omputational ost of the PSSR is more than that of the CRC, whih is also more than that of the proposed PVRC. the prototypes. From Fig.4, we an know that the RR of the PVRC lassifier outperforms the best RR of the PSSR with 0.% (8.50%) improvement when the first 4 (8) samples of eah lass are used as the prototypes. he horizontal axis of Fig. 4 is the number of nearest neighbors of the PSSR. Fig. 3. Some images of the subset of Coil-00 database able I: COMPUAIONAL COS OF EACH QUERY ON COIL OBJEC DAABASE USING HE FIRS N SCHEME (UNI: SECONDS) Classifier RR (4) RR (8) NN CM LRC CRC SRC PVRC VI-B Objet reognition on oil objet database he Coil-00 data set [3] was widely used as an objet-reognition benhmark [4]-[7]. In this data set, there are 00 objets and eah objet has 7 different views (images) that are taken every 5 around an axis passing through the objet. Eah image is a 8 8 olor one with R, G, B hannels. We use only a limited number of views per objets for experiments. In our experiments, different views per objet (0, 30, 60, 90, 0, 50, 80, 0, 40, 70, 300 and 330 ) were used, shown in Fig. 3. So the subset of Coil-00 data set ontains 00 images, and all images in subset of Coil-00 database were manually ropped into a 3 3 olor one with R, G, B hannels. In the first experiment, we adopt the leave one out sheme on oil objet database. he results are listed in able II. In the seond experiment, we test the reognition rate on oil objet database using first N sheme. he results are desribed in Fig. 4 and able III. In able II, the reognition rate (RR) of PVRC lassifier outperforms the RRs of the SRC, LRC, CRC, NN and CM lassifiers with 7.08%,.50%, 3.5%, 6.75%, and 5.83% improvements, respetively. In able III, the RR of the PVRC lassifier outperforms the RRs of the SRC, LRC, CRC, NN and CM lassifiers, with 7.00%,.00%, 30.00%, 4.88%, and 4.00% improvements, respetively while the first 4 samples are used as prototype. he RR of the PVRC lassifier outperforms the RRs of them with 4.7%,.95%, 9.5%, 6.45%, and 3.5% improvements, respetively while the first 8 samples are used as Fig. 4. he RR of the PSSR lassifier on oil objet database using the first N sheme able II: RECOGNIION RAES (RRS) OF SEVERAL CLASSIFIERS ON COIL OBJEC DAABASE USING HE LEAVE ONE OU SCHEME Classifier NN CM LRC RR 86.5% 77.7% 9.50% Classifier CRC SRC PVRC RR 79.75% 85.9% 93.00% able III: RECOGNIION RAES (RRS) OF SEVERAL CLASSIFIERS ON COIL OBJEC DAABASE USING HE FIRS N SCHEME Classifier RR (4) RR (8) NN 69.87% 85.55% CM 70.75% 78.75% LRC 73.75% 90.05% CRC 65.00% 8.75% SRC 67.75% 87.5% PVRC 74.75% 9.00% VI-C Objet reognition on eth80 objet database In the eth80-ropped-lose8 objet database [8] [9], all images are ropped, so that they ontain only the objet without any border area. In addition, they are resaled to a size

7 of 8 8 pixels. Again, the sale is left the same for all images of the same objet. his dataset is useful when no derivatives are needed. In our experimental, all images are resized to 3 3 gray images. Some images of eth80 objet database are shown in Fig. 5. Fig. 5. Some sampled images of eth80 database In the third experiment, we adopt the leave one out sheme on eth80 objet database. he result is listed in able IV. In the fourth experiment, we test the reognition rate on eth80 objet database using first N sheme. he results are desribed in Fig. 6 and able V. In able IV, the reognition rate (RR) of the PVRC lassifier outperforms the RRs of the SRC, LRC, CRC, NN and CM lassifiers with.0%, 4.75%, 4.45%,.5%, and 9.05% improvements, respetively. In able V, the RR of the PVRC lassifier outperforms the RRs of the SRC, LRC, CRC, NN and CM lassifiers with 7.56%, 4.%, 9.73%, 3.07%, and 5.0% improvements, respetively while the first 4 samples are used as prototype. he RR of the PVRC lassifier outperforms the RRs of them with 6.5%, 4.40%,.4%, 3.5%, and 6.5% improvements, respetively while the first 6 samples are used as prototype. From Fig. 6, we learnt that the RR of PVRC lassifier outperforms the best RR of the PSSR.03% (3.86%) when the first 4 (6) samples of eah lass are used as prototype. he horizontal axis of Fig. 6 is the number of nearest neighbors of the PSSR. able IV: HE RR OF SEVERAL CLASSIFIERS ON EH80 OBJEC DAABASE USING HE LEAVE ONE OU SCHEME Classifier NN CM LRC RR 64.60% 46.80% 7.0% Classifier CRC SRC PVRC RR 5.40% 64.8% 75.85% Fig. 6. he RR of PSSR lassifier on eth80 objet database using the first N sheme VI-D Objet reognition on soil47 objet database he Soil-47 data set [30] was widely used as an objet-reognition benhmark. In the data set, there are 47 objets and eah objet has different views (images) that are taken every 9 around an axis passing through the objet. In the experimental, the subset of soil-47 inludes 966 images of 46 objets. Eah olor image is downsampled to a 4 3 gray image. Fig. 7 shows some sampled images of these objets. Fig. 7 Some sampled images of SOIL-47 data set In the fifth experiment, we adopt the leave one out sheme on eth80 objet database. he result is listed in able VI. In the sixth experiment, we test the reognition rate on eth80 objet database using first N sheme. he results are desribed in Fig. 8 and able VII. able V: RECOGNIION RAES (RRS) OF SEVERAL CLASSIFIERS ON EH80 OBJEC DAABASE USING HE FIRS N SCHEME Classifier RR (4) RR (6) NN 9.46%.86% CM 7.43% 9.% LRC 8.3% 0.96% CRC.80% 4.% SRC 4.97% 9.% PVRC.53% 5.36% Fig. 8. he RR of PSSR lassifier on Soil47 objet database using the first N sheme

8 able VI : HE RR OF SEVERAL CLASSIFIERS ON SOIL47 OBJEC DAABASE USING HE LEAVE ONE OU SCHEME Classifier NN CM LRC RR 64.60% 44.6% 7.67% Classifier CRC SRC PVRC RR 64.8% 7.0% 75.85% able VII : RECOGNIION RAE (RR) OF SEVERAL CLASSIFIERS ON SOIL47 OBJEC DAABASE USING HE FIRS N SCHEME Classifier RR (4) RR (6) NN 4.46% 39.57% CM 39.39% 35.07% LRC 49.74% 50.5% CRC 45.0% 43.04% SRC 48.47% 48.70% PVRC 50.6% 5.3% In able VI, the reognition rate (RR) of the PVRC lassifier outperforms the RRs of the SRC, LRC, CRC, NN and CM lassifiers with 4.84%, 3.8%,.67%,.5%, and 3.3% improvement, respetively. In able VII, the RR of the PVRC lassifier outperforms the RRs of the SRC, LRC, CRC, NN and CM lassifiers with.79%, 0.5%, 5.5%, 7.80%, and 0.87% improvements, respetively while the first 4 samples are used as prototype. he RR of the PVRC lassifier outperforms the RR of them with.6%,.6%, 8.7%,.74%, and 6.4% improvements, respetively while the first 6 samples are used as prototype. From Fig. 8, we learnt that the RR of PVRC lassifier outperforms the best RR of the PSSR with 7.45% (0.7%) improvements when the first 4 (6) samples of eah lass are used as the prototypes. he horizontal axis of Fig. 8 is the number of nearest neighbors of the PSSR. VI-E Fae reognition on G fae database Georgia eh fae database [3] ontains images of 50 people taken in two or three sessions between 06/0/99 and /5/99 at the Center for Signal and Image Proessing at Georgia Institute of ehnology. All people in the database are represented by 5 olor JPEG images with luttered bakground taken at resolution pixels. he average size of the faes in these images is pixels. he pitures show frontal and/or tilted faes with different faial expressions, lighting onditions and sale. Eah image is manually of the G fae database is manually ropped in a gray image. Fig. 9 shows some seleted fae images of G fae database. In the seventh experiment, we adopt the leave one out sheme on G fae database. he results are listed in able VIII. In the eighth experiment, we test the reognition rate on G fae database using first N sheme. he results are desribed in Fig. 0 and able VIIII. Fig. 9. Some seleted fae images of G fae database ABLE VIII : HE RR OF SEVERAL CLASSIFIERS ON G FACE DAABASE USING HE LEAVE ONE OU SCHEME Classifier NN CM LRC RR 8.64% 73.60% 84.67% Classifier CRC SRC PVRC RR 74.7% 83.87% 87.0% able VIIII : HE RECOGNIION RAE (RR) OF SEVERAL CLASSIFIERS ON G FACE DAABASE USING HE FIRS N SCHEME Classifier RR (3) RR (6) NN 49.83% 67.% CM 48.50% 54.% LRC 5.83% 68.44% CRC 46.00% 6.33% SRC 53.7% 68.89% PVRC 53.83% 7.89% In able VIII, the reognition rate (RR) of the PVRC lassifier outperforms the RR of the SRC, LRC, CRC, NN and CM lassifiers with 3.33%,.53%,.93%, 4.56%, and 3.60% improvements, respetively. In able VIIII, ompared with SRC, LRC, CRC, NN and CM lassifiers, the RR of PVRC lassifier outperforms the RRs of them with 0.66%,.8%, 7.83%, 6.36%, and 6.8% improvements, respetively, when the first 3 samples are used as the prototypes. he RR of the PVRC lassifier outperforms the RRs of them with 4.00%, 4.45%,.56%, 5.75%, and 8.67% improvements, respetively, when the first 6 samples are used as the prototypes. From Fig.0, we learn that the RR of the PVRC lassifier outperforms the best RR of the PSSR with 0.83% (6.45%) improvement when the first 3 (6) samples of eah lass are used as prototype. he horizontal axis of Fig. 0 is the number of nearest neighbors of the PSSR. Fig.0. he RR of PSSR lassifier on G fae database using the first N sheme VI-F Fae reognition on UMIS fae database he Sheffield (previously UMIS) Fae Database [3] onsists of 564 images of 0 individuals, whih are mixed with rae, gender, and appearane. Eah individual is shown in a

9 range of poses from profile to frontal views - eah in a separate diretory labelled a, b,... t and images are numbered onseutively as they were taken. he files are all in PGM format, approximately 0 0 pixels with 56-bit grey-sale. In the experiment, a subset of UMIS fae database ontains 480 fae images of 0 individuals. Eah individual has 4 fae images. All images of the subset are manually ropped in gray images. Fig. exhibits some sampled fae images of UMIS fae database. Fig.. Some sampled fae images of UMIS fae database able VV: HE RR OF SEVERAL CLASSIFIERS ON UMIS FACE DAABASE USING HE LEAVE ONE OU SCHEME Classifier NN CM LRC RR 99.5% 95.75% 99.75% Classifier CRC SRC PVRC RR 98.5% 99.75% 99.75% Fig.: he RR of the PSSR lassifier on UMIS fae database using the first N sheme In the ninth experiment, we adopt the leave one out sheme on UMIS fae database. he results are listed in able VV. In the tenth experiment, we test the reognition rate on G fae database using first N sheme. he results are desribed in Fig. and able VVI. able VVI : RECOGNIION RAE (RR) OF SEVERAL CLASSIFIERS ON UMIS FACE DAABASE USING HE FIRS N SCHEME Classifier RR (3) RR (5) NN 90.00% 89.67% CM 89.% 88.00% LRC 88.4% 87.33% CRC 84.4% 83.67% SRC 86.47% 88.33% PVRC 90.00% 90.00% In able VV, the reognition rate (RR) of the PVRC lassifier outperforms the RRs of the SRC, LRC, CRC, NN and CM lassifiers, with 0.00%, 0.00%,.50%, 0.50%, 4.00% improvements, respetively. In able VVI, the RR of PVRC lassifier outperforms the RRs of the SRC, LRC, CRC, NN and CM lassifiers with 3.53%,.76%, 5.59%, 0.00%, and 0.88%, improvements, respetively, when the first 3 samples are used as prototype. he RR of PVRC lassifier outperforms the RRs of the SRC, LRC, CRC, NN and CM lassifiers with.67%,.67%, 5.33%, 0.33%, and.00% improvements, respetively, when the first 5 samples are used as the prototypes. From Fig., we learnt that the RR of the PVRC lassifier outperforms the best RR of the PSSR 3.4% (0.00%) when the first 3 (5) samples of eah lass are used as prototype. he horizontal axis of Fig. is the number of nearest neighbors of the PSSR. VII. CONCLUSION In this paper, a novel lassifier alled polyhedron volume ratio lassifiation (PVRC) is proposed for image reognition. he PVRC lassifier uses the test sample vetor and the original lass prototype samples to alulate the distane between the test sample and the orresponding lass prototype samples. he lass-based distane omputation is oneptually similar to the LRC and CM lassifiers. However, the PVRC omputes the distane between the test sample and the lass subspae by the ratio of two polyhedron volumes instead of alulating the distane between the test sample and the simplified lass features, suh as the lass means or approximated lass subspaes. he proposed PVRC lassifier, whih omputes the perpendiular distane of the test sample to the lass polyhedron, ahieves the better reognition rate than the NN, CM, CRC, SRC, PSSR and LRC lassifiers. he omputation of the PVRC in the testing phase is similar to those of the NN and CM lassifiers. hus, the PRVR takes less omputation than the CRC, LRC and SRC lassifiers. All experimental results onfirm the effetiveness and analyses of the proposed lassifiation algorithm. Appendix A: Matrix Segmentation and Its Determinant Computation A B Let P C D be a square matrix ( m n ) ( m n ) to be segmented into four submatries, A, B, C and D are with sizes of m m, m n, n m and n n, respetively. It is noted that D should be a reversible square matrix. hen, we an prove:

10 A C B D det( P) det det( A BD C) det( D) Proof: Im O Give a square matrix, D C I, where are m m, n and n n identify matries, respetively, and O is zeros matrix. hen, it is easy to verify I, I A B Im O A BD C B C D D C I n O D m n. (A) By using the existing properties of determinant, det(ab) = E F det(a)det(b) and det det( E)det( G) O G, we an ompute the determinant of both sides of (A) to obtain, A B Im O det( )det( ) C D D C I n A BD C B det( ) O D det( D)det( A BD C) (A) In (A), it is also straightforward to prove that Im O det det( )det( ) I m I D C I n. End of Proof. n REFERENCES []. M. Cover and P. E. Hart, Nearest Neighbor Pattern Classifiation, IEEE rans. on Inform. heory, vol. 3, no., pp. 7, Jan [] R. C. Gonzalez, and R. E. Woods, Digital image proessing (Addison Wesley, 997). [3] P. N. Belhumeur, J. P. Hespanha, and D. J. 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