MLTSP: New 3D Framework, Based on the Multilayer Tensor Spectrum Pyramid

Abstract

A tensor representation structure based on the multilayer tensor spectrum pyramid (MLTSP) is introduced in this work. The structure is “truncated”, i.e., part of the high-frequency spectrum coefficients is cut-off, and on the retained low-frequency coefficients, obtained at the output of each pyramid layer, a hierarchical tensor SVD (HTSVD) is applied. This ensures a high concentration of the input tensor energy into a small number of decomposition components of the tensors obtained at the coder output. The implementation of this idea is based on a symmetrical coder/decoder. An example structure for a cubical tensor of size 8 × 8 × 8, which is represented as a two-layer tensor spectrum pyramid, where 3D frequency-ordered fast Walsh–Hadamard transform and HTSVD are used, is given in this paper. The analysis of the needed mathematical operations proved the low computational complexity of the new approach, due to a lack of iterative calculations. The high flexibility of the structure in respect to the number of pyramid layers, the kind of used orthogonal transforms, the number of retained spectrum coefficients, and HTSVD components, permits us to achieve the desired accuracy of the restored output tensor, imposed by the application. Furthermore, this paper presents one possible application for 3D object searches in a tensor database. In this case, to obtain the invariant representation of the 3D objects, in the spectrum pyramid, the 3D modified Mellin–Fourier transform is embedded, and the corresponding algorithm is shown.