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Multidimensional Gabor-Like Filters Derived from Gaussian Functions on Logarithmic Frequency Axes

Dherik Devakumar, Ole Christian Eidheim

TL;DR

The paper addresses the mismatch between conventional wavelets and the exponential scaling of frequency content by introducing a multidimensional wavelet-like family derived from Gaussian functions on logarithmic frequency axes. The method yields an inverse-Fourier-form filter family with tunable center and width parameters, enabling compact, octave-aware filter banks. Key contributions include continuous and discrete formulations, built-in low-pass content at the frequency-origin, and a practical radial-angular design for covering frequency space. This approach offers a principled, octave-aware alternative to Log-Gabor for multidimensional filter banks with controlled coverage and filter behavior.

Abstract

A novel wavelet-like function is presented that makes it convenient to create filter banks given mainly two parameters that influence the focus area and the filter count. This is accomplished by computing the inverse Fourier transform of Gaussian functions on logarithmic frequency axes in the frequency domain. The resulting filters are similar to Gabor filters and represent oriented brief signal oscillations of different sizes. The wavelet-like function can be thought of as a generalized Log-Gabor filter that is multidimensional, always uses Gaussian functions on logarithmic frequency axes, and innately includes low-pass filters from Gaussian functions located at the frequency domain origin.

Multidimensional Gabor-Like Filters Derived from Gaussian Functions on Logarithmic Frequency Axes

TL;DR

The paper addresses the mismatch between conventional wavelets and the exponential scaling of frequency content by introducing a multidimensional wavelet-like family derived from Gaussian functions on logarithmic frequency axes. The method yields an inverse-Fourier-form filter family with tunable center and width parameters, enabling compact, octave-aware filter banks. Key contributions include continuous and discrete formulations, built-in low-pass content at the frequency-origin, and a practical radial-angular design for covering frequency space. This approach offers a principled, octave-aware alternative to Log-Gabor for multidimensional filter banks with controlled coverage and filter behavior.

Abstract

A novel wavelet-like function is presented that makes it convenient to create filter banks given mainly two parameters that influence the focus area and the filter count. This is accomplished by computing the inverse Fourier transform of Gaussian functions on logarithmic frequency axes in the frequency domain. The resulting filters are similar to Gabor filters and represent oriented brief signal oscillations of different sizes. The wavelet-like function can be thought of as a generalized Log-Gabor filter that is multidimensional, always uses Gaussian functions on logarithmic frequency axes, and innately includes low-pass filters from Gaussian functions located at the frequency domain origin.
Paper Structure (4 sections, 7 equations, 2 figures)

This paper contains 4 sections, 7 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Continuous Function
  3. Discrete Function
  4. Filter Bank

Figures (2)

  • Figure 1: A demonstration of Equation \ref{['equation1']} with centered domain origins. The Gaussian function shown in a) is placed on logarithmic frequency axes as can be seen in b) using regular axes. The results of $\Uppsi$ shown in c) and d) are approximately orthogonal and sum to roughly zero.
  • Figure 2: The real parts of an example filter bank from Equation \ref{['equation1']}, where the top left image shows a low-pass filter from $\boldsymbol{\mu} = \boldsymbol{0}$. The bottom left image displays the sum of the Gaussian functions in the frequency domain over $\boldsymbol{\mu} \in \{0, 0\} \cup \{(r \cos \theta, r \sin \theta)\ |\ r \in \{6, 12, \ldots, 48\}$, $\theta \in \{0, \frac{\pi}{22}, \frac{2\pi}{22}, \ldots, \frac{43\pi}{22}\}\}$, which shows the potential frequency domain coverage of the utilized $r$ and $\theta$ intervals. Moreover, the normalized real parts of the inverse Fourier transform of the bottom left image approximate an identity filter, i.e. $1$ at the origin and $0$ elsewhere.