Non-Orthogonal HOSVD Applied to 3-D Image Compression

In this paper we introduce an iterative approach to decomposing tensors into a sum of rank-one non-orthogonal components. The approach uses the repeated application of an idea from the Higher Order Singular Value Decomposition (HOSVD) method that uses power iterations to obtain a rank-one approximation to a tensor. At each step, that rank-one approximation is subtracted from the tensor and the process is repeated. We abbreviate our method NOTSVD for Non-Orthogonal Tensor Singular Value Decomposition. For 2-D tensors (i.e., matrices) the process converges to the usual SVD of the matrix and orthogonality is preserved. We present numerical evidence of convergence of the iterative algorithm for approximating 3-D tensors despite the lack of orthogonality. We apply the NOTSVD method to compression of RGB images which can be regarded as 3-D tensors, each color represented by a matrix. We compress the image two ways, one using the matrix SVD applied to each of the three color matrices, and the other applying NOTSVD to the full 3-D tensor. For matrices, we use L-2 norms to characterize differences. We find that the same level of accuracy in reconstructing the image can be achieved with lower storage requirements (higher compression factor) for the NOTSVD method. We also apply the method to a generic 3-D (volumetric) image and show good reconstruction with relatively few components.