2-Dimensional Singular Value Decomposition for 2D Maps and Images

For a set of 1D vectors, standard singular value decomposition (SVD) is frequently applied. For a set of 2D objects such as images or weather maps, we form 2dSVD, which computes principal eigenvectors of rowrow and column-column covariance matrices, exactly as in the standard SVD. We study optimality properties of 2dSVD as low-rank approximation and show that it provides a framework unifying two recent approaches. Experiments on images and weather maps illustrate the usefulness of 2dSVD.

[1]  Lawrence Sirovich,et al.  Application of the Karhunen-Loeve Procedure for the Characterization of Human Faces , 1990, IEEE Trans. Pattern Anal. Mach. Intell..

[2]  David G. Stork,et al.  Pattern Classification , 1973 .

[3]  Richard A. Harshman,et al.  Indexing by Latent Semantic Analysis , 1990, J. Am. Soc. Inf. Sci..

[4]  Golub Gene H. Et.Al.  Matrix Computations, 3rd Edition , 2007 .

[5]  Jieping Ye,et al.  Generalized Low Rank Approximations of Matrices , 2004, Machine Learning.

[6]  R. Preisendorfer,et al.  Principal Component Analysis in Meteorology and Oceanography , 1988 .

[7]  Alejandro F. Frangi,et al.  Two-dimensional PCA: a new approach to appearance-based face representation and recognition , 2004 .

[8]  Jieping Ye,et al.  GPCA: an efficient dimension reduction scheme for image compression and retrieval , 2004, KDD.

[9]  Tommi S. Jaakkola,et al.  Weighted Low-Rank Approximations , 2003, ICML.

[10]  Inderjit S. Dhillon,et al.  Concept Decompositions for Large Sparse Text Data Using Clustering , 2004, Machine Learning.

[11]  Shigeo Abe DrEng.  Pattern Classification , 2001, Springer London.

[12]  Tamara G. Kolda,et al.  Orthogonal Tensor Decompositions , 2000, SIAM J. Matrix Anal. Appl..

[13]  C. Eckart,et al.  The approximation of one matrix by another of lower rank , 1936 .

[14]  Susan T. Dumais,et al.  Using Linear Algebra for Intelligent Information Retrieval , 1995, SIAM Rev..

[15]  Gene H. Golub,et al.  Rank-One Approximation to High Order Tensors , 2001, SIAM J. Matrix Anal. Appl..