3. Comparative Analysis of the Hierarchical 3D-SVD and Reduced Inverse Tensor Pyramid in Regard to Famous 3D Orthogonal Transforms

In this work are presented two new approaches for hierarchical decomposition represented as tensors of size N × N × N for N = 2ⁿ, based on algorithms which (unlike the famous similar approaches) do not require iterative calculations. Instead, they use repetitive simple calculations in each hierarchical decomposition level. As a result, the computational complexity (CC) of the new hierarchical algorithms is lower than that of the iteration-based. In general, hierarchical decompositions are divided into two basic groups: statistical and deterministic. To the first group is assigned the algorithm hierarchical tensor SVD (HTSVD) based on the multiple calculation of the two-level SVD for the elementary tensor of size 2 × 2 × 2. The decomposition is executed by using the HTSVD in three orthogonal spatial directions simultaneously. The deterministic decompositions have lower CC than the statistical, but they do not ensure full decorrelation between the components of the 3D decompositions. In this group are the famous orthogonal transforms 3D fast Fourier transform (3D-FFT), 3D discrete cosine transform (3D-DCT), 3D discrete wavelet transform (3D-DWT), 3D contourlet discrete transform (3D-CDT), 3D shearlet discrete transform (3D-SDT), etc., and also, the algorithm 3D reduced inverse spectrum pyramid (3D-RISP). The last is distinguished by its lower CC and the high energy concentration in the first decomposition components. To achieve this, for the basic tensor of size 2ⁿ × 2ⁿ × 2ⁿ is executed the 3D fast truncated Walsh–Hadamard transform (3D-FTWHT). Significant advantage of 3D-RISP compared to the famous pyramidal decompositions of the kind 3D-DWT, 3D-CDT, 3D-SDT, etc., is the absence of 3D decimation and 3D interpolation which produce distortions in the restored tensor.