Title: Hierarchical SVD-based image decomposition with tree structure

Authors: Roumen K. Kountchev; Roumiana A. Kountcheva

Addresses: Department of Radio Communications and Video Technologies, Technical University of Sofia, Bul. Kliment Ohridski No. 8, Sofia 1000, Bulgaria ' T&K Engineering Co., Mladost 3, Sofia 1712, Bulgaria

Abstract: This work is devoted to one new approach for decomposition of images represented by matrices of size 2n × 2n or 3n × 3n, based on the multiple application of the singular value decomposition (SVD) over image blocks of relatively small size (2 × 2 or 3 × 3), obtained after division of the original image matrix. The new decomposition, called hierarchical singular value decomposition (SVD), has tree structure of the kind binary or three nodes tree of n hierarchical levels. Its basic advantages over the famous SVD are: the reduced computational complexity, the opportunity for parallel and recursive processing of the image blocks, based on relatively simple algebraic relations, the high concentration of the image energy in the first decomposition components, and the ability to accelerate the calculations through cutting off the tree branches in the decomposition levels, where the corresponding eigenvalues are very small. The HSVD algorithm is generalised for images of unspecified size. The offered decomposition opens new opportunities for fast image processing in various application areas: image compression, filtering, segmentation, merging, digital watermarking, dimensionality reduction, etc.

Keywords: singular value decomposition; block SVD; hierarchical SVD; HSVD; binary trees; 3-node trees; computational complexity; image blocks; image processing; parallel processing; recursive processing.

DOI: 10.1504/IJRIS.2015.070906

International Journal of Reasoning-based Intelligent Systems, 2015 Vol.7 No.1/2, pp.114 - 129

Received: 19 Nov 2013
Accepted: 24 Aug 2014

Published online: 31 Jul 2015 *

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