Hierarchical Decomposition of 2D/3D Images, Based on SVD 2×2

The famous Singular Value Decomposition (SVD) is very efficient in the processing of multidimensional images, when efficient compression, and reduction of the features, used for objects recognition, are needed. The basic obstacle for the wide use of SVD is its high computational complexity. To solve the problem, here is offered the new approach for hierarchical image decomposition through SVD (2×2)-based algorithm. In this case, the multidimensional image is represented as a 3D tensor, divided into sub-tensors of size 2×2×2, called kernels. Each such kernel is then decomposed through Hierarchical SVD (HSVD), based on the SVD for a matrix of size 22. In the paper are given the HSVD algorithms for a 4×4 matrix, and for a tensor of size 4×4×4. The tensor decomposition is generalized for a tensor of size N×N×N, when N=2. The computational complexity of HSVD is evaluated, and compared to that of the iterative SVD algorithm for 2D matrices, and 3D tensors. The basic advantages of the new approach for decomposition of multidimensional images are the low computational complexity and the tree-like structure, which permits the low-energy branches to be cut-off through threshold selection. The algorithm is suitable for parallel processing of multi-dimensional images through multiprocessor systems, where the basic cell processor executes the decomposition of the kernel tensor, of size 2×2×2.