How to construct log-Gabor Filters?

Abstract

Firstly proposed by Dennis Gabor in 1946 [1], the canonical coherent states of the Gabor filters are different versions of a Gaussian-shaped window shifted in time/space and frequency variables [2] [3], Gabor's work synthesizes the studies of Nyquist in Communication Theory in 1924 [4] and Heisenberg in Quantum Mechanics in 1927, by which he proposed the Gaussian shape as an optimal envelope for time-frequency representation turning the uncertainly principle from inequality into equality. Some important characteristics of Gabor wavelets are [5]:- Construction by a linear combination.- Energy preservation in transform domain (Parseval's theorem).- Non-orthogonality but an unconditional basis, a frame [6].- Symmetry of the Fourier domain.- Time/space and frequency shift-invariance.