Wavelets on L-groups
Andrey Popoff · 2022
Chapter 8 discusses the algorithms of discrete wavelet transform based on L-group operations. Starting from the consideration of classic continuous and discrete wavelet transforms, as well as their main properties, discrete wavelet transforms (DWT) is defined in the sample space with L-group properties. The chapter considers L-group discrete wavelet transforms defined in the sample spaces with pseudometric, l 1 -metric, and semimetric; L-group discrete wavelet transforms of harmonic signals in such sample spaces; L-group discrete wavelet transforms of BPSK, QPSK, V-LFM, FSK signals in sample space with l 1 -metric. The chapter explores the features of L-group DWTs of the signals with finite duration in sample spaces with pseudometric and semimetric. It is found out that L-group DWTs do not require performing operation of multiplication, i.e. they are multiplication-free. It is shown that data processing rate when performing discrete wavelet transforms in sample spaces with L-group properties, unlike their analogue from linear space, does not depend on data width. Multiscale image decomposition on L-groups is introduced. The chapter considers multiscale image decompositions based on Hadamard matrix defined in both linear space with scalar product and space with L-group properties. The chapter considers fast 2D discrete wavelet transforms using both known linear algorithm and algorithm based on L-group operations.