The Community for Technology Leaders
RSS Icon
Subscribe
Issue No.08 - August (2011 vol.33)
pp: 1590-1602
Roman Sandler , Yahoo! Research, Haifa
Michael Lindenbaum , Technion, Israel Institute of Technology, Haifa,
ABSTRACT
Nonnegative matrix factorization (NMF) approximates a given data matrix as a product of two low-rank nonnegative matrices, usually by minimizing the L_2 or the KL distance between the data matrix and the matrix product. This factorization was shown to be useful for several important computer vision applications. We propose here two new NMF algorithms that minimize the Earth mover's distance (EMD) error between the data and the matrix product. The algorithms (EMD NMF and bilateral EMD NMF) are iterative and based on linear programming methods. We prove their convergence, discuss their numerical difficulties, and propose efficient approximations. Naturally, the matrices obtained with EMD NMF are different from those obtained with L_2-NMF. We discuss these differences in the context of two challenging computer vision tasks, texture classification and face recognition, perform actual NMF-based image segmentation for the first time, and demonstrate the advantages of the new methods with common benchmarks.
INDEX TERMS
Nonnegative matrix factorization, earth mover's distance, image segmentation.
CITATION
Roman Sandler, Michael Lindenbaum, "Nonnegative Matrix Factorization with Earth Mover's Distance Metric for Image Analysis", IEEE Transactions on Pattern Analysis & Machine Intelligence, vol.33, no. 8, pp. 1590-1602, August 2011, doi:10.1109/TPAMI.2011.18
REFERENCES
[1] S. Alpert, M. Galun, R. Basri, and A. Brandt, "Image Segmentation by Probabilistic Bottom-Up Aggregation and Cue Integration," Proc. IEEE Conf. Computer Vision Pattern Recognition, June 2007.
[2] P. Arbelaez, M. Maire, C. Fowlkes, and J. Malik, "From Contours to Regions: An Empirical Evaluation," Proc. IEEE Conf. Computer Vision Pattern Recognition, 2009.
[3] M. Berry, M. Browne, A. Langville, P. Pauca, and R. Plemmons, "Algorithms and Applications for Approximate Nonnegative Matrix Factorization," Computational Statistics and Data Analysis, vol. 52, no. 1, pp. 155-173, Sept. 2007.
[4] P.N. Bellhumer, J. Hespanha, and D. Kriegman, "Eigenfaces vs Fisherfaces: Recognition Using Class Specific Linear Projection," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 19, no. 7, pp. 711-720, July 1997.
[5] R.E. Broadhurst, "Statistical Estimation of Histogram Variation for Texture Classification," Proc. Texture Analysis and Synthesis Workshop, 2005.
[6] H.H. Bülthoff and S. Edelman, "Psychophysical Support for a Two-Dimensional View Interpolation Theory of Object Recognition," Proc. Nat'l Academy of Sciences USA., vol. 89, pp. 60-64, Jan. 1992.
[7] H. Cevikalp, M. Neamtu, M. Wilkes, and A. Barkana, "Discriminative Common Vectors for Face Recognition," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 27, no. 1, pp. 4-13, Jan. 2005.
[8] D. Comanicu and P. Meer, "Mean Shift: A Robust Approach Toward Feature Space Analysis," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 24, no. 5, pp. 603-619, May 2002.
[9] T. Cour, F. Benezit, and J. Shi, "Spectral Segmentation with Multiscale Graph Decomposition," Proc. IEEE CS Conf. Computer Vision and Pattern Recognition, vol. 2, pp. 1124-1131, 2005.
[10] I. Dhillon and S. Sra, "Generalized Nonnegative Matrix Approximations with Bregman Divergences," Proc. Neural Information Processing Systems, vol. 18, pp. 283-290, 2006.
[11] D. Donoho and V. Stodden, "When Does Non-Negative Matrix Factorization Give a Correct Decomposition into Parts?" Proc. Neural Information Processing Systems, 2003.
[12] K. Grauman and T. Darrel, "Fast Contour Matching Using Approximate Earth Mover's Distance," Proc. IEEE CS Conf. Computer Vision and Pattern, 2004.
[13] D. Guillamet and J. Vitria, "Determining a Suitable Metric When Using Non-Negative Matrix Factorization," Proc. Int'l Conf. Pattern Recognition, 2002.
[14] T. Hazan and A. Shashua, "Analysis of l2-Loss for Probabilistically Valid Factorizations under General Additive Noise," Technical Report 2007-13, The Hebrew Univ., 2007.
[15] M. Heiler and C. Schnörr, "Learning Sparse Representations by Non-Negative Matrix Factorization and Sequential Cone Programming," J. Machine Learning Research, vol. 7, pp. 1385-1407, 2006.
[16] F.S. Hillier and G.J. Lieberman, Introduction to Operations Research. McGraw-Hill Science/Engineering/Math, 2005.
[17] P. Hoyer, "Non-Negative Matrix Factorization with Sparseness Constraints," J. Machine Learning Research, vol. 5, pp. 1457-1469, 2004.
[18] D.D. Lee and H.S. Seung, "Learning the Parts of Objects by Non-Negative Matrix Factorization," Nature, vol. 401, no. 6755, pp. 788-791, Oct. 1999.
[19] D.D. Lee and H.S. Seung, "Algorithms for Non-Negative Matrix Factorization," Proc. Conf. Advances in Neural Information Processing Systems, vol. 13, pp. 556-562, 2001.
[20] T. Leung and J. Malik, "Representing and Recognizing the Visual Appearance of Materials Using Three-Dimensional Textons," Int'l J. Computer Vision, vol. 43, no. 1, pp. 29-44, June 2001.
[21] E. Levina and P. Bickel, "The Earth Mover's Distance Is the Mallows Distance: Some Insights from Statistics," Proc. Eighth IEEE Int'l Conf. Computer Vision, vol. 2, pp. 251-256, 2001.
[22] S. Li, X. Hou, H. Zhang, and Q. Cheng, "Learning Spatially Localized, Parts-Based Representation," Proc. IEEE CS Conf. Computer Vision and Pattern Recognition, vol. 1, 2001.
[23] D.G. Lowe, "Distinctive Image Features from Scale-Invariant Keypoints," Proc. Int'l J. Computer Vision, vol. 60, no. 2, pp. 91-110, 2004.
[24] D. Martin, C. Fowlkes, D. Tal, and J. Malik, "A Database of Human Segmented Natural Images and Its Application to Evaluating Segmentation Algorithms and Measuring Ecological Statistics," Proc. IEEE Int'l Conf. Computer Vision, 2001.
[25] S. Mikeš and M. Haindl, "Prague Texture Segmentation Data Generator and Benchmark," Proc. 19th IEEE CS Int'l Conf. Pattern Recognition, 2008.
[26] P. Paatero and U. Tapper, "Positive Matrix Factorization: A Non-Negative Factor Model with Optimal Utilization of Error Estimates of Data Values," Environmetrics, vol. 5, no. 2, pp. 111-126, 1994.
[27] O. Pele and M. Werman, "Fast and Robust Earth Mover's Distances," Proc. IEEE Int'l Conf. Computer Vision, 2009.
[28] Y. Rubner, Perceptual Metrics for Image Database Navigation. PhD thesis, Stanford Univ., 1999.
[29] L. Rudin, S. Osher, and E. Fatemi, "Nonlinear Total Variation Based Noise Removal Algorithms," Physica D, vol. 60, nos. 1-4, pp. 259-268, 1992.
[30] F. Samaria and A. Harter, "Parameterisation of a Stochastic Model for Human Face Identification," Proc. Second IEEE Workshop Applications of Computer Vision, Dec. 1994.
[31] R. Sandler and M. Lindenbaum, "Unsupervised Estimation of Segmentation Quality Using Nonnegative Factorization," Proc. IEEE CS Conf. Computer Vision and Pattern Recognition, 2008.
[32] R. Sandler and M. Lindenbaum, "Unsupervised Estimation of Segmentation Quality Using Nonnegative Factorization," IEEE Conf. Computer Vision and Pattern Recognition, 2008.
[33] R. Sandler and M. Lindenbaum, "Nonnegative Matrix Factorization with Earth Moverś Distance Metric," Proc. IEEE CS Conf. Computer Vision and Pattern Recognition, 2009.
[34] R. Sandler and M. Lindenbaum, "Optimizing Gabor Filter Design for Texture Edge Detection and Classification," Int'l J. Computer Vision, vol. 84, no. 3, pp. 308-324, 2009.
[35] J. Shi and J. Malik, "Normalized Cuts and Image Segmentation," Proc. IEEE CS Conf. Computer Vision and Pattern Recognition, 1997.
[36] S. Shirdhonkar and D. Jacobs, "Approximate Earth Mover's Distance in Linear Time," Proc. IEEE CS Conf. Computer Vision and Pattern Recognition, 2008.
[37] N. Sochen, R. Kimmel, and R. Malladi, "A General Framework for Low Level Vision," IEEE Trans. Image Processing, vol. 7, no. 3, pp. 310-318, Mar. 1998.
[38] C. Thurau and V. Hlavac, "Pose Primitive Based Human Action Recognition in Videos or Still Images," Proc. IEEE CS Conf. Computer Vision and Pattern Recognition, 2008.
[39] S. Ullman, High-Level Vision: Object Recognition and Visual Cognition. The MIT Press, 1996.
[40] M. Werman, S. Peleg, and A. Rosenfeld, "A Distance Metric for Multidimensional Histograms," Computer Vision, Graphics, and Image Processing, vol. 32, pp. 328-336, 1985.
[41] J. Yang, S. Yang, Y. Fu, X. Li, and T. Huang, "Non-Negative Graph Embedding," Proc. IEEE CS Conf. Computer Vision and Pattern Recognition, 2008.
14 ms
(Ver 2.0)

Marketing Automation Platform Marketing Automation Tool