论文研究-Statistical Modeling for Multiple Modes Facial Images using GND-PCA.pdf

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基于GND-PCA的高维度人脸图像的统计建模,谯旭,,在本文中,我们提出了使用广义N维主成分分析(GND-PCA)的方法对于高维度的人脸数据进行统计建模。高维度的人脸数据的多个模式包括不同�
山国武获论文在丝 http:/www.paper.edu.cn In lincar algebra, Singular Valuc Dccomposition (SvD) is an important factorization of a rectangular real or complex matrix, with several applications in signal processing and statistics 80 SVD computes the low-rank approximation ofa set ofID vectors This can be generalized to a two dimensional singular value Decomposition(2DS VD) to do low-rank approximation of a set of matrices such as a set of images. Higher Order Singular value Decomposition(HOSVD)is a generalization of SvD for high dimensional tensor [6]. In the case of a 3D tensor executing HOS Vd in 2 dimensions gives the same result as 2DSVD 85 ND-PCA is proposed for the modeling of higher-dimensional data. This method is based on the HOSVD and the data is treated as the higher order tensor. After unfolding the tensor into a ' mode-n matrix", we can use the traditional PCa method to deal with the matrix and get the bases of this mode-subspace [5, 8 90 2. GND-PCA GND-PCA is formalized as follows [10]: Given a series of the N-order tensors with ero-means 写∈R2x…xx 1=1,2, ., M, M is the number of samples. We hope to get another series of low rank-(J1, J2, ..., JN) tensors A, which accurately approximate the original tensors, 95 where J, sI, The new series is decomposed by the matrices UmER'x, with orthogonal columns according to Turker Model[7] which is shown by 丌=×U0x2Ux…,Ux…xU Where B, ERXJ2xI-K/N are the core tensors. The illustration of reconstructing a third order tensor by three orthogonal bases is shown in Fig2 T(3) Um×1h1B2 Opt A 100 Fig 2 Reconstruction of a Three order tensor by the Three Orthogonal Bases of Mode Subspace The orthogonal matrices u() can be determined by minimizing the cost function as C∑x=∑xUxU2x×U×…U Supposing the rank of the N matrices UnI are known, we use an iteration algorithm to get the N 105 optimal matrices, UOn UOm UOm which are able to minimize the cost function C. The algorithm is shown in fig 3 山国武获论文在丝 http:/www.paper.edu.cn Algorithm 1 Iteration Algorithm to Compute the N Matrices U).U(2)..U(N) oI N-Lh ordler l A;∈R OUT: N Matrices U(n F RinxJn(Jn<In, n-1, 2,N)with orthogonal column vecto: s Initial valucs: h=0 and ul' whose columns are determincd as the first Jn leading eigenvec- cors of the matrices =l(Ai(n).Am)) T · Maximize s-∑∠1l‖C 卩2.C2-A×2t Solution: U()whose columns are determnined as the first J, leading eigenvectors of T(1) Maximize s=∑l1|C:x2U(2T1,c=×;U(21x3U(32x…xUg)2 Solution: U(2)whose columns are determined as the first J2 leading eigenvectors of =1 (2)·C C(2)·C U(2) e Maximize s U ××nU N Solution: U() whose colum. s are determined as the first J n. leading eigenvectors of () vTi( Maximize s=∑1c.xUN7p,c=4x;12x….m=UN Solution: UN)whose columns are determined as the first JN leading eigenvectors of Sot U=u( =h+1 3.stU-Ug),U2)-U2), Fig 3 The Iteration algorithm 110 3. Experimental Results 4.1 Mavic database The database we used is MavIC(KAo-Ritsumeikan Multi-angle View, Illumination and 1 15 Cosmetic Facial Database)[ll]. The facial images in the database are captured from various angles of view and various illuminations with a"multi-angle image capturing system(MICs) In Ma vIC, there are 170 Japanese womens natural facial images and 250 Japanese womens cosmetic facial images. Each subject was photographed in 13 different view-points under 14 illuminations. Some sample images from MavIC Database are shown in Fig 4 120 4.2 Statistical Appcarance Models by GND-PCA The proposed GND-PCA is used to construct statistical appearance models for the 3D-Tensor data. In our experiment, we used a portion of the mavic database. 80 samples( subjects) with 125 natural facial images are used and each sample has 13 viewpoints(poses) and 13 illuminations. The size of each image is 60 by 50 pixels. The 2D image is unfolded into a vector with the dimension of 4 山国武获论文在丝 http:/www.paper.edu.cn 3000. All images of each sample are written together as a 3D-Tensor. Then we put the vectors in different directions according to the following rules: one for different view-points, one for different illuminations and the other for image textures (pixels). Each sample is corresponding to a 3D tensor 130 with the dimension of 3000x 13 x 13 a 135 (b) Fig4 Example images of MaVIC Database.(a)One sample with different poses(different columns )and different illuminations(different rows). (b)Different samples lumination AlAIn Pose Texture Fig. 5 A tensor corresponding to a sample As shown in Fig. 6, the iterated convergence of GND-PCA is very fast, we usually iterate twic 145 in our experiments 山国武获论文在丝 http:/www.paper.edu.cn 3.2720+06 3.2715+06 3.2710+06 3270+06 g s3.2700+06 3.2695+06 8910 Iterations Fig 6 Convergence of GND-PCA(when 300x5x5 mode subspace bases are trained) We choose one sample as the template and use the others for training The leave-one-out experiment is used to test the generalization ability of the constructed models 150 The accuracy of the reconstructed tensor from the reduced mode-subspace is measured by a normalized correlation. Normalized correlation of two tensor-formed data, the original tori (x, v, z) and the reconstructed tRe (x, y, z), is defined as ∑to(x,y,z)ts(x,y,2) NC=-. 2 ,0≤MC≤1. ∑÷2(x,y,2)∑1(x,y,2) The more similar the two data the larger the value of nc 155 Fig 7 shows the reconstructed results from different mode-subspace bases respectively. Since the dimension of the original data is 3000 x 13X13, the compress rates are as follows 100×5×5 300×5×5 0.49% 1.48%;1000×5×5 493%and30×10×10 ≈592% It can be seen that the 3000×13×13 300×13×13 300)×13×13 300×13×13 quality of reconstructed rcsults bccomc better and better when the modc-subspacc bascs arc increased a (b) C (d) Fig. 7 Some images of Reconstructed results 山国武获论文在丝 http:/www.paper.edu.cn 165 3.3 Comparison with ND-PCA Since ND-PCA can only compress the data in one mode-subspace, it represents data not as efficient as GND-PCA method under the same compression rate. This can be demonstrated by the following experiments With the quite similar compression rate, where 1000x8x8 ≈12.62% for GNd-PCa and 3000x13×13 170465×13×13300×2×1330003X21538% for mode-l, mode-2, mode-3 of ND-PCA respeclively, 3000×13×133000×13×133000×13×13 we use leave-one-out experiment to compare the quality of reconstruction of GND-PCA with the ND-PCA method by normalized correlation. It can be seen clearly that the results of GND-PCA are beller than those of ND-PCA as shown in Fig 8 0.9 ND-PCA uf Mode-1 DND-PCA ur Moule-2 NPCA of Mude-3 175 Fig8 Comparison of the results in the leave-one-out tasting for ND-PCA and proposed GND-PCa 4. Conclusion In this paper we proposed GNd-PCA for statistical appearance modeling of facial images with 180 multiple modes including different people, different pose and different illumination. Compared with ND-PCa method, it can represent the data more efficiently and the constructed models have good performance on generalization. In the future, we will apply the proposed method to statistical analysis of cosmetic facial images in order to design and control various types of facial appearance using cosmetic foundations. The proposed method can also be used for generating various types 185 of facial images in computer graphics References 1]IT. Jolliffe. Principal Component Analysis [M]. Springer, 2002 [2]M. Turk, A. Pentland. Eigenfaces for Recognition[J] Journal of Cognitive Neuroscience, 1991, 3(1): 71-86 3J. Yang, D. Zhang, A.F. Frangi and J.Y. Yang. To-Dimensional PCA: A New Approach to Appearance-based 190 Face Representation and Recognition[J]. IEEE Trans. on PAML, 2004, 26(1):131-137 14Il.Kong, XLi, L Wang, EK.Teoh, J. G. Wang, and Rvenkateswarlu Generalized 2d principal component analysis for face image representation and recognition[J]. 2005, 18: 585-594 [511.C. Yu and M.Bennamoun ID-PCA, 2D-PCA to nD-PCAUJ. The 1 8th International Conference on Pattern Recognition(ICPR06), 2006 95 [6]L D. Lathauwer, B D Moor and J Vandewalle. A Multilinear Singular Value Decomposition[J]. SIAM Journal of Matrix Analysis and Application, 2001, 21(4): 1253-1278 I7 L.R.Tucker. Some mathematical notes of three-mode factor analysis[ J]. Psychometrika, 1996, 31: 279-322 [8 MAOVasilescu and DTerzopoulos Multilinear Analysis of Image Ensembles: Tensorface[ ] Proceedings of 山国武获论文在丝 http:/www.paper.edu.cn European Conference on Computer Vision, 2002, 447-460 IEEE Conf On Computer Vision and Pattern Recognition, 2003, 2: 93-gp of Image Ensemble[J]. Proceedings of 200 [9]MA0. Vasilescu and D Terzopoulos Multilinear Subspace Analysis do]RXu and Y WChen Appearance models for medical volumes with few samples by generalized 3D-PCA[] ccturc Notes in Computcr Scicncc, Springer, 2007 [11]YWChen, T Fukui, XQiao. T Igarashi, K Nakao and A Kashimoto. Multi-angle view, Illumination and 205 Cosmetic Facial Image Database(MavIC) and its statistical analysis[J]. Lecture Notes in Computer Science, 2008 基于 GND-PCA的高维度人脸图像的统 计建模 210 谯旭 (控制科学与工程学院,山东大学,济南250061) 摘要:在木文中,我们提出了使用广义N维主成分分析( GND-PCA)的方法对」高维度的人脸 数据进行统计建模。高维度的人脸数据的多个模式包括不同的人,不同的姿势和不同的照明 多种模式的面部图像可以被认为是高维数据,可以看做是一个张量。GND-PCA通过张量分 215解来对于各个维度的数据进行压缩,并得到近似的中心张量。通过在我们建立的MaVC数 据库( KAO-Ritsumeikan多视点,多照明和美容面部数据库)的压缩重建试验,证明GND-PCA 对于图像的压缩和特征提取比传统的PCA和ND-PCA方法相比更有效率。 关键词:主成分分析;统计建模;张量分解 中图分类号:TP18人工智能理论 220

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