Abstract
In this work, new approaches are proposed for the 3D decomposition of a cubical tensor of size N × N × N for N = 2n through hierarchical deterministic orthogonal transforms with low computational complexity, whose kernels are based on the Walsh-Hadamard Transform (WHT) and the Complex Hadamard Transform (CHT). On the basis of the symmetrical properties of the real and complex Walsh-Hadamard matrices are developed fast computational algorithms whose computational complexity is compared with that of the famous deterministic transforms: the 3D Fast Fourier Transform, the 3D Discrete Wavelet Transform and the statistical Hierarchical Tucker decomposition. The comparison results show the lower computational complexity of the offered algorithms. Additionally, they ensure the high energy concentration of the original tensor into a small number of coefficients of the so calculated transformed spectrum tensor. The main advantage of the proposed algorithms is the reduction of the needed calculations due to the low number of hierarchical levels compared to the significant number of iterations needed to achieve the required decomposition accuracy based on the statistical methods. The choice of the 3D hierarchical decomposition is defined by the requirements and limitations related to the corresponding application area.
Highlights
The famous tensor decompositions—Canonical Polyadic Decomposition (CPD), Higher-OrderSingular Value Decomposition (HOSVD) [1,2,3], Tensor Trains Decomposition (TTD) [4], and HierarchicalTucker decomposition (H Tucker) [5] - and their modifications [6] are based on the calculation of the eigen values and eigen vectors of the decomposed tensor
Their basic advantage is that they are optimum with respect to the Mean Square Error (MSE) of the approximation in the case of the truncation of the low-energy decomposition components
The comparison results permit the choosing of the number of hierarchical decomposition levels n for which the developed new algorithms 3D-FWHT and 3D Hierarchical Fast Complex Walsh-Hadamard Transform (3D-FCHT) are more efficient than 3D Fast Fourier Transform (3D-FFT), 3D Discrete Wavelet Transform (3D-DWT) and H-Tucker
Summary
The famous tensor decompositions—Canonical Polyadic Decomposition (CPD), Higher-Order. Tucker decomposition (H Tucker) [5] - and their modifications [6] are based on the calculation of the eigen values and eigen vectors of the decomposed tensor Their basic advantage is that they are optimum with respect to the Mean Square Error (MSE) of the approximation in the case of the truncation of the low-energy decomposition components. Each single 2D image is divided into N2 blocks of size N × N for N = 2n , from which is obtained a sequence of N cubical tensors, each of size N × N × N.
Sign-in/Register to view highlights & summary 
Published Version (
Free)
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call