A KERNEL MACHINE BASED APPROACH FOR MULTI- VIEW FACE RECOGNITION

Transcription

1 A KERNEL MACHINE BASED APPROACH FOR MULI- VIEW FACE RECOGNIION Mohammad Alwawi erm project report submitted in partial fulfillment of the requirement for the course of estimation and detection Supervised by Assist. Prof. Dr Aykut Hocanin Electrical and Electronics Engineering Department Eastern Mediterranean University July

2 able of Contents Abstract 2 1.Introduction what is face recognition Methods.4 2.1Generalized Discriminant Analysis Direct LDA Kernel Direct Discriminant Analysis Eigen-analysis of SBW in the Feature Space...5, Eigen-analysis of SWH in the Feature Space Dimensionality Reduction and Feature Extraction Comments Experimental Results Distribution of Multi-view Face Patterns Comparison with KPCA and GDA.8,9 4. Conclusion..10 Appendix 1 and II ,12,13 References.. 14 List of ables...15 List of Figures

3 Abstract echniques that can introduce low-dimensional feature representation with enhanced discriminatory power is of paramount importance in face recognition (FR) systems. It is well known that the distribution of face images, under a perceivable variation in viewpoint, illumination or facial expression, is highly nonlinear and complex. It is therefore not surprising that linear techniques, such as those based on Principle Component Analysis (PCA) or Linear Discriminant Analysis (LDA), cannot provide reliable and robust solutions to those FR problems with complex face variations. In this paper, we propose a kernel machine based discriminant analysis method, which deals with the nonlinearity of the face patterns distribution. he proposed method also effectively solves the so-called small sample size (SSS) problem which exists in most FR tasks. Keywords Face Recognition (FR), Kernel Direct Discriminant Analysis (KDDA), Linear Discriminant Analysis (LDA), Principle Component Analysis (PCA), Small Sample Size Problem (SSS), Kernel Methods. 2

4 1- Introduction Within the last decade, face recognition (FR) has found a wide range of applications, from identity authentication, access control, and face-based video indexing/browsing, to human-computer interaction/communication. As a result, numerous FR algorithms have been proposed. wo issues are central to all these algorithms: (i) feature selection for face representation, and (ii) classification of a new face image based on the chosen feature representation. his work focuses on the issue of feature selection. he main objective is to find techniques that can introduce low-dimensional feature representation of face objects with enhanced discriminatory power. he most successful techniques are those appearance-based approaches, which generally operate directly on images or appearances of face objects and process the images as 2D holistic patterns, to avoid difficulties associated with 3D modeling, and shape or landmark detection. In this paper, motivated by the success that Support Vector Machines (SVM), Kernel PCA (KPCA) and Generalized Discriminant Analysis (GDA) have in pattern regression and classi.cation tasks, we propose a new kernel discriminant analysis algorithm for face recognition. he algorithm generalizes the strengths of the recently presented D-LDA and the kernel techniques while at the same time overcomes many of their shortcomings and limitations. he rest of the paper is organized as follows. Since KDDA is built on D-LDA and GDA, we start the analysis by briefly reviewing the two latter methods. Following that, KDDA is introduced and analyzed. he relationship of KDDA to D-LDA and GDA is also discussed. Finally two sets of experiments are presented to demonstrate the effectiveness of the KDDA algorithm on highly nonlinear, highly complex face pattern distribution. 1.1 What is Face Recognition Face recognition is defined as the identification of a person from an image of their face. It falls into a bracket of technology aptly named biometrics. We, as humans, have an innate ability to recognise faces and take for granted our abilities that allow us to quickly identify a person often with very limited information. Creating a computer system to try and compete with the human visual system is extremely complex and so far unsolved. 3

5 Face recognition is a very complex problem, as there are numerous factors that influence the appearance of ones facial features. here are two groups of influences, intrinsic and extrinsic factors. Intrinsic factors are independent of the surroundings and are only concerned with the changes in the three dimensional profile of the face. Extrinsic factors are the effect on the appearance of a person s face due to external factors such as lighting conditions. 2- Methods 2.1 Generalized Discriminant Analysis (GDA) For solving nonlinear problems, the classic LDA has been generalized to its kernel version, namely GDA.he idea behind GDA is to perform a classic LDA in the feature space F instead of the input space R^N, the feature space F could be considered as a linearization space. Let SBW and SWH be the between and within class scatter matrices in the feature space f respectively, expressed as follows: C 1 SBW = Ci ( φ i φ )( φ i φ ) L i= 1 (1) SWH = 1 L C i 1 Ci j= 1 ( φ ij φ i)( φij φ i) (2) Ci C Ci 1 1 where φ ij = φ( Zij), φi = φ( Zij) is the mean of class Zi, φ = φ ( Zij Ci j= 1 L i = 1 j = 1 ) is the average of the ensemble, and Ci is the element number in Zi. Ψ = [ Ψ 1,..., Ψ M ] which represents the optimal discriminant basis vectors, in such away that the ratio of SBW and SWH is maximized. he maximization is equivalent to solve the following eigenvalue problem, 4

6 Ψ = arg max ( Ψ S ( Ψ S BW WH Ψ ) Ψ ) (3) Number of training samples is much smaller than the dimensionality leading to a degenerated SWH which produces the SSS problem, GDA technique solves this problem by removing null space from SWH. 2.2 Direct LDA (D-LDA) Its an algorithm that attempts to avoid the shortcomings existing in traditional solutions to the SSS problem. he idea behind the algorithm is that the null space of SWH may contain significant discriminant information if the projection of SBW is not zero in that direction, and that no significant information will be lost if the null space of SBW is discarded. Assume that A and B represent the null spaces of SBW and SWH respectively, the N N complement space of A can be written as R -A and for B, R -B, therefore the optimal discriminant subspace sought by the D-LDA algorithm is the intersection space (complement space of A intersection B).he intersection space can be obtained by solving the null space of projection of SWH into complement of A. 2.3 Kernel Direct Discriminant Analysis (KDDA) Eigen-analysis of SBW in the Feature Space he analysis starts by solving the eigenvalue problem of SBW, which can be rewritten as follows: SBW c Ci Ci = i i ( ( φ φ))( ( φ φ )) = φ φ (4) L b b i = 1 L computing φ b φ requires dot product evaluation in F.his can be done by utilizing kernel b methods. he introduction of the kernel function allows us to avoid the explicit evaluation of the mapping. 5

7 Using the kernel function for two arbitrary classes can be defined as K lh = ( K ij ) i= c j= 1 1..,.., c where K ij = K( Z, Z ) = φ. φ ij hj ij hj For all C classes the K kernel matrix can be defined as K = ( K lh ) l= c h= 1 1..,.., c which implies φ φ b b = 1 B.( A. K. A L LC LC 1 1 ( A. K.1 ) (1. K. A ) + L LC LC L LC LC L 1 2 (1. K.1 )). B LC LC where B = diag[ C1... Cc], and A = diag a... a ] (see appendix 1 and 2 for a detailed derivations) LC [ c1 Cc Eigen-analysis of SWH in the Feature space his method can be summarized by the following equation; 1 2 Let U = VΛ.projecting SBW and SWH into the subspace spanned by U. b U SBWU = I U SWHU = E Λ m b 1 1 ) ( φ SWHφ )( E Λ ) (5) b b m b ( 2 2 by using the kernel matrix K the following expression can be obtained; φ 1 S ( J1 J 2) b WHφ b = L (6) J1 and J2 are defined in the appendix Dimensionality Reduction and Feature Extraction his method can be represented by the following equations; For any input pattern z, its projection into the set of feature vectors,γ,can be calculated by 6

8 1 2 2 y = Γ φ( Z) = ( E. Λ. p. Λ ) ( φ ϕ( Z)) (7) m b 1 w b Where φ φ( Z) = [ φ 1... φc] φ( Z), given that b 1 1 φb φ( Z) = B.( ALC. γ ( φ( Z)) 1LC. γ ( φ( Z))) (8) L L Where γ ( φ( Z )) = [ φ11φ ( Z)... φ φ( Z)] CC By combining eq.7 and eq.8 we obtain y = Θγ. ( φ( Z)) (9) 1 1 Where (... 2 Θ = Em Λb p Λ w ) ( B.( ALC 1LC )), eq.9 alow dimensional representation L L y on z with enhanced discriminant power has been introduced Comments he KDDA method implements an improved D-LDA in a high-dimensional feature space using a kernel approach. Its main advantages can be summarized as follows: 1. KDDA introduces a nonlinear mapping from the input space to an implicit high dimensional feature space, where the nonlinear and complex distribution of patterns in the input space is linearized and simplified so that conventional LDA can be applied. 2. KDDA effectively solves the SSS problem in the high-dimensional feature space by employing an improved D-LDA algorithm. Unlike the original D-LDA method of zero eigenvalues of the within-class scatter matrix are never used as divisors in the improved one. In this way, the optimal discriminant features can be exactly extracted from both of inside and outside of SWH s null space. 3. In GDA, to remove the null space of SWH, it is required to compute the pseudo inverse of the kernel matrix K, which could be extremely ill-conditioned when certain kernels or kernel parameters are used. Pseudo inversion is based on inversion of the 7

9 nonzero eigenvalues. Due to round-of errors, it is not easy to identify the true null eigenvalues. 3- Experimental Results wo sets of experiments are included in this paper to illustrate the effectiveness of the KDDA algorithm. 3.1 Distribution of Multi-view Face Patterns he first experiment aims to provide insights on how the KDDA algorithm linearizes and simplifies the face pattern distribution. A subset of database used in this experiment because of the simplicity in visualization, in Fig.2 the visualized projections are the first tow most significant principle components extracted by PCA and KPCA, and they provide a low dimensional representation for the samples, which can be used to capture the structure of data. Fig.3 depicts the first two most discriminant features extracted by utilizing D-LDA and KDDA respectively. 3.2 Comparison with KPCA and GDA he second experiment compares the classification error rate performance of the KDDA algorithm to two other commonly used kernel FR algorithms, KPCA and GDA. he FR procedure is completed in two stages. 1. Feature extraction. he training set is composed of 120 images: 6 images per person are randomly chosen. he remaining 455 images are used to form the test set. After training is over, both sets are projected into the feature spaces derived from the KPCA, GDA and KDDA methods. 2. Classification. Since the focus in this paper is on feature extraction, a simple classifier is always preferred so that the FR performance is not mainly contributed by the classifier but the feature selection algorithm. herefore using classifier like SVM instead of the nearest neighbor will improve the classification accuracy. ow typical kernel functions were used to evaluate the overall performance of the three method, the RBF and the polynomial function. Analysis is performed with respect to the kernel parameters and the number of used feature vectors M. Figs.4-5 8

10 depict the average error rates of the three methods compared when RBF and polynomial kernels were used. he optimal feature number M is a result of the existence of the peaking effect in the feature selection procedure.fig.4.a, M =99 is the value used for KPCA, while M=19 is used for GDA and KDDA.Fig.4.B,depicts the error rates as functions of M within the range from 5 to Conclusion 9

11 A new FR method has been introduced in this paper. he proposed method combines kernel-based methodologies with discriminant analysis techniques. he kernel function is utilized to map the original face patterns to a high-dimensional feature space, where the highly non-convex and complex distribution of face patterns is linearized and simplified, so that linear discriminant techniques can be used for feature extraction. he small sample size problem caused by high dimensionality of mapped patterns is addressed by an improved D-LDA technique which exactly finds the optimal discriminant subspace of the feature space without any loss of significant discriminant information. In conclusion, the KDDA algorithm is a general pattern recognition method for nonlinearly feature extraction from high-dimensional input patterns without suffering from the SSS problem. We expect that in addition to face recognition, KDDA will provide excellent performance in applications where classification tasks are routinely performed, such as content-based image indexing and retrieval, video and audio classification. 10

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15 References [1] R. Chellappa, C.L. Wilson, and S. Sirohey, Human and machine recognition of faces: A survey, Proceedings of the IEEE, vol. 83, pp , [2] Matthew A. urk and Alex P. Pentland, Eigenfaces for recognition, Journal of Cognitive Neuroscience, vol. 3, no. 1, pp , [3] Bernhard Sch olkopf, Chris Burges, and Alex J. Smola, Advances in Kernel Methods - Support Vector Learning, MI Press, Cambridge, MA, [4] G. Baudat and F. Anouar, Generalized discriminant analysis using a kernel approach, Neural Computation, vol. 12, pp , [5] Sarunas J. Raudys and Anil K. Jain, Small sample size e.ects in statistical pattern recognition: Recommendations for practitioners, IEEE ransactions on Pattern Analysis and Machine Intelligence, vol. 13, no.3, pp ,

16 LIS OF ABLES 1 he average percentages of the error rate of KDDA over those of others

17 ABLE 1 he average percentages of the error rate of KDDA over those of others. Kernel RBF Polynomial (RBF + Polynomial)/2 KDDA/KPCA % % % KDDA/GDA % % % 16

18 LIS OF FIGURES 1- Some face samples of one subject from the UMIS face database Distribution of 170 samples of 5 subjects in PCA and KPCA based subspaces Distribution of 170 samples of 5 subjects in D-LDA and KDDA based subspaces Comparison of error rates based on RBF kernel function Comparison of error rates based on Polynomial kernel function.20 17

19 Fig.1. Some face samples of one subject from the UMIS face database. (A) Fig.2. Distribution of 170 samples of 5 subjects in PCA and KPCA based subspaces. (B) 18

20 (A) (B) Fig.3. Distribution of 170 samples of 5 subjects in D-LDA and KDDA based subspaces. (A) (B) Fig.4. Comparison of error rates based on RBF kernel function. 19

21 (A) Fig.5. Comparison of error rates based on Polynomial kernel function. (B) 20

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