Radix-(2×2) Hierarchical SVD for multi-dimensional images

Abstract

The processing of multi-dimensional images is of high importance in contemporary communication and information technologies. It requires significant computational resources, and is an object of large number of scientific investigations. One of the most efficient methods used, is the well-known Singular Value Decomposition (SVD), which permits the achievement of high compression together with significant reduction of the features' space, used for objects recognition. The main problem with the SVD is its high computational complexity. One approach to overcome the problem is presented in this paper. It is based on new decomposition for multi-dimensional images, which are treated as sequences of single high-correlated images through the so-called radix-(2×2) Hierarchical SVD (HSVD) algorithm. In correspondence with this approach, the multi-dimensional image is represented as a third-order tensor, divided into sub-tensors of size 2×2×2, called kernels. Each kernel is decomposed through Hierarchical SVD, based on the SVD for matrices of size 2×2, binary two-level tree and rearrangement of the components in each level. After kernel unfolding are obtained 2 matrices of size 2×2 and on each is applies SVD, calculated by using simple mathematical relations. In the paper is given a HSVD algorithm for a matrix of size 4×4, whose computational structure is described as a binary two-level tree. Same algorithm is used for the tensor decomposition of size 4×4×4. The decomposition is generalized for tensors of size N×N×N for N=2n>4. The computational complexity of the algorithm is evaluated and compared to that of the iterative SVD. The basic advantages of the new approach are the low computational complexity and the tree-like structure of the algorithm, which permits the low-energy leaves to be cut-off through threshold-based selection. As a result, the new algorithm is suitable for parallel processing of multi-dimensional images.