Self-Invertible 2D Log-Gabor Wavelets

Abstract

Abstract. Orthogonal and biorthogonal wavelets became very popular image processing tools but exhibit major drawbacks, namely a poor resolution in orientation and the lack of translation invariance due to aliasing between subbands. Alternative multiresolution transforms which specifically solve these drawbacks have been proposed. These transforms are generally overcomplete and consequently offer large degrees of freedom in their design. At the same time their optimization gets a challenging task. We propose here the construction of log-Gabor wavelet transforms which allow exact reconstruction and strengthen the excellent mathematical properties of the Gabor filters. Two major improvements on the previous Gabor wavelet schemes are proposed: first the highest frequency bands are covered by narrowly localized oriented filters. Secondly, the set of filters cover uniformly the Fourier domain including the highest and lowest frequencies and thus exact reconstruction is achieved using the same filters in both the direct and the inverse transforms (which means that the transform is self-invertible). The present transform not only achieves important mathematical properties, it also follows as much as possible the knowledge on the receptive field properties of the simple cells of the Primary Visual Cortex (V1) and on the statistics of natural images. Compared to the state of the art, the log-Gabor wavelets show excellent ability to segregate the image information (e.g. the contrast edges) from spatially incoherent Gaussian noise by hard thresholding, and then to represent image features through a reduced set of large magnitude coefficients. Such characteristics make the transform a promising tool for processing natural images.

Multi-spectral palmprint recognition based on oriented multiscale log-Gabor filters

“…Our choice of both bandwidths σ ρ and σ θ was inspired by [31] where the purpose was to be as close as possible…”

Biologically-inspired characterization of sparseness in natural images

“…We set the range of these parameters to match what has been reported for the responses in primates' V1. In particular, we set the bandwidth of the Fourier representation of the filters to 1 and π/8 respectively in log-frequency and polar coordinates to get a family of elongated and thus orientation-selective filters (see [3] , see Supplementary Figure 4). Prior to the analysis of each image, we used the spectral whitening filter [11] to provide a good balance of the energy of output coefficients [2,12].…”

“…Tests with a range of different numbers of orientations and scales yielded similar results [15] This transform is linear and can be performed by a simple convolution repeated for every edge type. Following [3], convolutions were performed in the Fourier (frequency) domain for computational efficiency. The Fourier transform allows for a convenient definition of the edge filter characteristics, and convolution in the spatial domain is equivalent to a simple multiplication in the frequency domain.…”

“…By multiplying the envelope of the filter and the Fourier transform of the image, one may obtain a filtered spectral image that may be converted to a filtered spatial image using the inverse Fourier transform. We exploited the fact that by omitting the symmetrical lobe of the envelope of the filter in the frequency domain, the output of this procedure gives a complex number whose real part corresponds to the response to the symmetrical part of the edge, while the imaginary part corresponds to the asymmetrical part of the edge (see [3] for more details). More generally, the modulus of this complex number gives the energy response to the edge (comparable to the response of complex cells in area V1), while its argument gives the exact phase.…”

Riesz-based Volume Local Binary Pattern and A Novel Group Expression Model for Group Happiness Intensity Analysis

“…So far, many candidates on the band-pass filter have been proposed in the literature. In this paper, we choose the commonly utilized multi-orientation and multi-scale log-Gabor filter [10], which is defined in the frequency domain as:…”