Band-pass filtering

A band-pass filter is a device that passes frequencies within a certain range and rejects (attenuates) frequencies outside that range.

An ideal bandpass filter would have a completely flat passband (e.g. with no gain/attenuation throughout) and would completely attenuate all frequencies outside the passband. Additionally, the transition out of the passband would be instantaneous in frequency. In practice, no bandpass filter is ideal. The filter does not attenuate all frequencies outside the desired frequency range completely; in particular, there is a region just outside the intended passband where frequencies are attenuated, but not rejected. This is known as the filter roll-off, and it is usually expressed in dB of attenuation per octave or decade of frequency. Generally, the design of a filter seeks to make the roll-off as narrow as possible, thus allowing the filter to perform as close as possible to its intended design. Often, this is achieved at the expense of pass-band or stop-band ripple.

The bandwidth of the filter is simply the difference between the upper and lower cutoff frequencies.

Gabor feature based classification using the enhanced Fisher linear discriminant model for face recognition

A good FR methodology should consider representation as well as classification issues, and a good representation method should require minimum manual annotaions.

Invariance Properties of Gabor Filter-based Features –Overview and Applications

The most important properties of Gabor filtering are related to invariance to illumination, rotation, scale, and translation.

Introduction

 By invariance, not only are features meant which are invariant to a set of geomeric transforms, but also methods to perform object detection regardless of pose and imaging conditions using features which are not invariant.

Confusing terminologies: Gabor filter, Gabor expansion, Gabor transform, Gabor jet, Gabor frame, or Gabor wavelet?

As Daugman pointed out, the 2-D Gabor filters are good models of the simple cells in the mammalian visual cortex system.

Survey and overview

The uncertainty can be measured by root mean square (rms) bandwidth: dt, df

The signal which occupies the minimum area dt*df=1/4π is the modulation product of a harmonic oscillation of any frequency with pulse of the form of a probability function.

Gabor expansion

It should be noted that the expansion functions do not have to constitute an orthogonal basis as typically assumed in wavelet or FT, but an unconditional basis, a frame, may succeed as well.

 

Invariant recognition

The Gabor filters are optimally joint localized in time and frequency, and thus, distortions and noise present in distinct locations, time or frequency, do not significantly interfere with the filter responses.

 

Examples

Symbol recognition

This first example utilizes a globally computed sum of Gabor responses, corresponding to edge histograms over different orientation, and thus, provides no elegance or novelty as a feature extraction method.

Simplified Gabor wavelets for human face recognition

Date:2008

Introduction

The Gabor wavelet (GW) is similar to the response of the 2-D receptive field of the mammalian simple cortical cell.

As the dimension of the feature vectors using GWs is very large, PCA/FLD are used to reduce the dimension. To further improve the performance, kernel methods are also used with the Gabor features. The improvement of both the linear and the kernel methods is due to the fact that GW features are robust to illumination, rotation, and scale.

Simplified GW (SGW)

Protected: Optimal sampling of Gabor features for face recognition

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EEG pattern discrimination betwwen salty and sweet taste using adaptive Gabor transform

Date:2005

Materials and methods

The problem consists of finding the Gabor function hp(t) among the set of normalized Gabor functions that are most similar to sp(t). In each step p, one Gabor function hp(t) is found and the next residual sp+1(t) is computed.

For each EEG raw data, 10 (p=10) AGR (adpative Gabor transform) coefficients are obtained.

Results and discussion

The advantage of this method is the increased spectral and temporal resolution of the signal. Several experiments have shown that the time-frequency analysis method provides narrow frequency peaks, permitting more precise frequency identification and enhanced ability in the determination of frequency changes at any EEG signal point.

Protected: new ideas

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Capitalize on Dimensionality Increasing Techniques for Improving Face Recognition Grand Challenge Performance

Author: Chengjun Liu

Date: May 2006

IEEE PAMI,

Abstract:

A novel pattern recognition framework which integrates Gabor image representation, a novel multiclass Kernel Fisher Analysis (KFA) method, and fractional power polynomial models for improving pattern performance is presented.

Background

Feature extraction in pattern recognition needs to consider the effectiveness on both data representation and class separability.

Kernel Fisher Analysis

Biologically motivated computationally intensive approaches to image pattern recognition

Author: Nikolay Petkov

Date: 1995

Abstract:

The concerned approaches are biologically motivated, in that we try to mimic and use mechanisms employed by natural vision systems, more specifically the visual system of primates.  Visual information representations which are motivated by the function of the primary visual cortex, more specifically by the function of so-called simple cells, are computed.

Cortical filters and images

Computational models of visual neurons with linear spatial summation

The receptive field of a visual neuron is the part of the visual field within which a stimulus can influence the response of the concerned neuron.

The response r  of a neuron to an input image s(x,y) can be modelled as follows:

1. linear spatial summation. The receptive field function is like the impulse response of a linear system.
2. thresholding and non-linear local contrast normalization. (this part not understood!!!)

Simple cells

In this study we are concerned with computer simulations of so-called simple cells in the primary visual cortex.

Neurophysiological research has shown that, on the population of all simple cells, the receptive field sizes vary considerably with the diameters of smallest to the largest receptive fields being in a ratio of at least 1:30.

It has been found that the spatial aspect ratio vary in a very limited range of 0.23<γ<0.92. The value γ=0.5 is used in our simulations.

The ratio σ/λ determines the number of parallel excitatory and inhibitory zones which can be observed in a receptive field. Neurophysiological research shows that the paramters λ and σ are closely correlated; on the set of all cells, the ratio σ/λ which determines the spatial-frequency bandwidth of a cell varies in a very limited range of 0.4-0.9 which corresponds to two to five excitatory and inhibitory stripe zones in a receptive field. The value σ/λ=0.5 is used in our simulations.

In our simulations, we use for φ the following values: 0 (symmetric receptive fields to which we refer as ‘center-on’ in analogy with retinal ganglion cell receptive fields whose central areas are excitatory), π (symmetric receptive fields to which we refer to as ‘center-off’, since their central lobe are inhibtory) and -0.5π and 0.5π (antisymmetric receptive fields with opposite polarity).

For technical applications one might wish to have filters with a more distinct effect, in that such a filter should enhance edges or bars, but not both at the same time. This would  simplify the interpretation of the resulting cortical images and the subsequent processing steps. We propose a mechanism for the elimination of so-called ‘shadow’ lines in cortical images obtained from filters with antisymmetric spatial summation functions. An edge which is strongly enhanced by a cortical filter with an antisymmetric receptive field function of orientation θ and phase φ will give rise to a pair of parallel lines in the cortical image produced by a cortical filter with an antisymmetric receptive field function of the same orientation θ and phase φ+π. We call these lines ‘shadow’ lines and proposed to eliminate them by a mechanism we called lateral inhibitation. Roughly speaking, this mechanism acts as follows: the value of a pixel in a cortical image corresponding to orientation  θ and phase φ is set to zero, if a pixel of higher value is found in a certain vicinity of the concerned pixel in the cortical image corresponding to orientation θ and phase φ+π. Here, it is proposed to extend the lateral inhibition mechanism applying it to all cortical images with the same set of values of the receptive field function parameters, but φ.

Using cortical images

Extracting lower dimension representations

Although the sets of cortical images computed  according to the above model deliver usefully structured information, they themselves do not give an ultimate solution to the image pattern recognition problem.

All pixel values in a cortical image are summed together to build a quantity which is (partially) characteristic of the input image. (A positive aspect of this scheme is that the result does not depend on the precise position of the object to be recognized.) Since a number of different cortical channels are used, each of them computing a different cortical image from the input image,  a number of such quantities are computed, one per cortical channel. Together these quantities form a descriptor vector which is considered as a projection of the input image onto a point in a lower-dimension space. This representation is then used to discriminate among different images.

Computing optic flow

Gabor Wavelet

Gabor filters: link from wikipedia