Truncated Hierarchical SVD for image sequences, represented as third order tensor

Abstract

In this work is presented new algorithm, called Truncated Hierarchical SVD (THSVD), aimed at the processing of sequences of correlated images, represented as third-order tensors. The algorithm is based on the multiple calculation of the matrix SVD for elementary tensors (ET) of size 2×2×2, which build the tensor of size N×N×N, when N=2n. The new approach is compared to closest famous hierarchical SVD methods for ET: the Sequential Unfolding SVD (SUSVD) and the Radix 2×2Hierarchical SVD (Radix 2×2 HSVD). New two-level algorithm is developed for ET decomposition, with lower computational complexity than these of Radix 2×2 HSVD and SUSVD. In the paper is presented the THSVD algorithm for tensor of size 4×4×4, which is generalized for a tensor of size N×N×N. Adaptive new algorithm is offered for the “truncation” of the tensor decomposition components with small weights. The multiple execution of similar operations for the SVD calculation for matrices of size 2×2 in each THSVD level, permits its parallel implementation by using processors with relatively simple structures. As a result of the „truncation“ and of the parallel calculations of THSVD, the processing of image sequences represented by third-order tensors, is significantly accelerated. This advantage of the algorithm opens new abilities for its application in real-time image processing systems in various areas: compression of image sequences, digital watermarking, computer vision, machine learning, processing of multidimensional signals, etc.