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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2//EN"><html><head><title>Log-Gabor Filters</title></head><body bgcolor="#ffffff" vlink="#ff0000"><h2>What Are Log-Gabor Filters and Why Are They Good?</h2><p>Gabor filters are a traditional choice for obtaining localisedfrequency information. They offer the best simultaneous localizationof spatial and frequency information. However they have two mainlimitations. The maximum bandwidth of a Gabor filter is limited toapproximately one octave and Gabor filters are not optimal if one isseeking broad spectral information with maximal spatial localization.<p>An alternative to the Gabor function is the Log-Gabor functionproposed by Field [1987]. Log-Gabor filters can be constructed witharbitrary bandwidth and the bandwidth can be optimised to produce afilter with minimal spatial extent.<h4>Bandwidth limitations of Gabor filters</h4><p>One cannot construct Gabor functions of arbitrarily wide bandwidth andstill maintain a reasonably small DC component in the even-symmetricfilter. This difficulty can be seen if we look at the transferfunction of an even-symmetric Gabor filter in the frequency domain.The transfer function is the sum of two Gaussians centred at plus andminus the centre frequency. If the standard deviation ofthese Gaussians becomes more than about one third of the centrefrequency the tails of the two Gaussians will start to overlapexcessively at the origin, resulting in a nonzero DC component. <center><table width=50%><tr><td><img src=GaborDCproblem.png><tr><td>Transfer function of a high bandwidth even-symmetric Gabor filter. The two Gaussians that make up the function overlapat the origin, resulting in a significant DC component.</table></center><p>At the limiting situation where the centre frequency is equal tothree standard deviations, the bandwidth will be approximately oneoctave. This can be seen as follows: For a Gaussian, the points whereits value falls to half the maximum are at approximately plus andminus one standard deviation, these points defining the cutofffrequencies. Thus the upper and lower cut-off frequencies will be atapproximately 4σ and 2σ respectively, giving a bandwidthof one octave. This limitation on bandwidth means that we need manyGabor filters to obtain wide coverage of the spectrum.<h4>The Log-Gabor Filter</h4><p>An alternative to the Gabor function is the log-Gabor functionproposed by Field [1987]. Field suggests that natural images arebetter coded by filters that have Gaussian transfer functions whenviewed on the <em>logarithmic</em> frequency scale. (Gabor functionshave Gaussian transfer functions when viewed on the <em>linear</em>frequency scale). On the linear frequency scale the log-Gaborfunction has a transfer function of the form<center>G(w) = e<sup> (-log(w/w<sub>o</sub>)<sup>2</sup>) / ( 2 (log(k/w<sub>o</sub>)<sup>2</sup> ) </sup></center><!--\begin{equation}{\cal G}(\omega) = e^{\frac{- (log (\omega / \omega_{o}))^{2}} {2 (log (\kappa/ \omega_{o}))^{2}}} \ ,\end{equation}--><p>where w<sub>o</sub> is the filter's centre frequency. To obtainconstant shape ratio filters the term k/w<sub>o</sub> must also beheld constant for varying w<sub>o</sub>. For example, ak/w<sub>o</sub> value of .74 will result in a filter bandwidth ofapproximately one octave, .55 will result in two octaves, and .41 willproduce three octaves.<center><table width=70%><tr><td><img src=LOGGs_f.png> <td> <img src=LOGGs_lf.png><tr><td colspan=2> An example of a log-Gabor transfer function viewed on both linear and logarithmic frequency scales.</table></center><p>There are two important characteristics to note. Firstly, log-Gaborfunctions, by definition, always have no DC component, and secondly,the transfer function of the log Gabor function has an extended tailat the high frequency end. Field's studies of the statistics ofnatural images indicate that natural images have amplitude spectrathat fall off at approximately 1/w. To encode images havingsuch spectral characteristics one should use filters having spectrathat are similar. Field suggests that log Gabor functions, havingextended tails, should be able to encode natural images moreefficiently than, say, ordinary Gabor functions, which wouldover-represent the low frequency components and under-represent thehigh frequency components in any encoding. Another point in supportof the log Gabor function is that it is consistent with measurementson mammalian visual systems which indicate we have cell responses thatare symmetric on the log frequency scale.<p>What do log Gabor functions look like in the spatial domain?Unfortunately due to the singularity in the log function at the originone cannot construct an analytic expression for the shape of the logGabor function in the spatial domain. One is reduced to designing thefilters in the frequency domain and then performing a numericalinverse Fourier Transform to see what they look like. Theirappearance is similar to Gabor functions though their shape becomesmuch `sharper' as the bandwidth is increased. The shapes of log Gaborand Gabor functions are almost identical for bandwidths less than oneoctave. Shown below are three log Gabor filters of differentbandwidths all tuned to the same centre frequency.<center> <table width=70%><tr> <td> <img src=logg1e.png><td> <img src=logg2e.png><td> <img src=logg3e.png><tr><td> <img src=logg1o.png><td> <img src=logg2o.png><td> <img src=logg3o.png><tr><td colspan = 3>Three quadrature pairs of log Gabor wavelets all tuned to thesame frequency, but having bandwidths of 1, 2 and 3 octaves respectively.</table></center><p>Given that we are now able to construct filters of arbitrary bandwidthand zero DC component the following question arises: What is the bestbandwidth to use? One observation is that as bandwidth increases sotoo does the sharpness of the filter. Therefore, one constraint mightbe imposed by the maximum sharpness of the filter that we caneffectively represent. Of perhaps greater interest is to study thevariation of the spatial width of filters with bandwidth. A usefulobjective might be to minimize the spatial width of filters in orderto get maximal spatial localization of our frequency information.<p>Normally when a function is wide in the frequency domain it is narrowin the spatial domain, thus we expect broad bandwidth filters to benarrow in the spatial domain. However, changing the bandwidth of alog Gabor filter does not result in a simple linear stretch of itstransfer function in the frequency domain, so one's first intuitivethoughts about their behaviour in the spatial domain can bemisleading. Careful observation of the behaviour of broad bandwidthlog Gabor filters in the spatial domain reveals that while the centralspike(s) of the filter may become very narrow the tails of the filterbecome extended. To investigate this phenomenon further two measuresof filter 'width' were studied.<ul><li> The width required to represent 99% of the spatial filter's absolute area.<li> The second moment about the centre of the filter with respect to the absolute value of the filter.</ul>Analytical investigation of these quantities is hampered by thesingularity in the expression for the log Gabor function at theorigin. Thus, the variation of both these width measures with respectto bandwidth could only be investigated numerically, and the resultsare shown below.<center><table width=50%><tr><td> <img src=width_vs_bw.png><tr><td>Variation of the spatial width of log Gabor functions with bandwidth(evaluated numerically).</table></center><p>As one can see, both measures of width are minimized when thebandwidth is about two octaves. The troughs in the curves are verybroad with any value between one and three octaves achieving a nearminimal spatial width. The data shown above were for even-symmetricfilters. The results for odd-symmetric filters are similar thoughwith a more gradual increase in width for bandwidths above threeoctaves being observed. These results have to be treated with somecaution as they are vulnerable to numerical effects; the spatial formof the filters was calculated via the discrete Fourier transform, andthe width measures are also determined numerically. The systematicundulations in the measure of width to represent 99% of the area aretroubling; all attempts to eliminate them were unsuccessful. Themagnitude of these undulations would vary with filter centre frequencybut their locations would remain constant. A flaw in attempting tomeasure the width required to represent 99% of the filter's absolutearea is that one does not know the <em>actual area</em>, all one knowsis the total discrete area in the finite spatial window beingconsidered. The data above was obtained using an FFT applied over1024 points and with filters having a centre frequency of 0.05 (awavelength of 20 units). The aim was to achieve a good discreterepresentation of the filter in both spatial and frequency domains,and also to avoid truncation of the filter tails. Despite theconcerns one might have over the absolute accuracy of the data, it isfelt that the overall trends of the curves are valid. It isinteresting to note that the range of bandwidths over which filterspatial size is near minimum, 1 to 3 octaves, matches well withmeasurements obtained on mammalian visual cells. One should also notethat the spatial width of a 3 octave log Gabor function isapproximately the same as that of a 1 octave Gabor function, clearlyillustrating the ability of the log Gabor function to capture broadspectral information with a compact spatial filter.<hr><h2>Efficient Implementation of Convolution of Log-Gabor Filters inthe Frequency Domain</h2><p>I am often asked how the code I use in my functions works. The codefor constructing the filters and performing the convolution doesappear a bit mysterious. Here's an attempt at an explanation.</p><p>In the frequency domain the even symmetric filter is represented bytwo real-valued log-Gaussian 'bumps' symmetrically placed on each sideof the origin. The odd-symmetric filter is represented by twoimaginary valued log-Gaussian 'bumps' anti-symmetrically placed oneach side of the origin.</p><center><img src=evenfilter.png><p>Even symmetric filter transfer function</p><pre></pre></center><p><center><img src=oddfilter.png><p>Odd symmetric filter transfer function</p></center><p>One can combine the convolution of the even and odd symmetric filtersinto the one operation. Exploiting the linearity of the FourierTransform where FFT(A+B) = FFT(A) + FFT(B) we can do the following:Multiply the FFT of the odd-symmetric filter by i (to make it realvalued) and add it to the FFT of the even symmetric filter. Theanti-symmetric 'bump' from the odd-symmetric filter will cancel outthe corresponding symmetric bump from the even-symmetric filter. Thisleaves a single 'bump' (multiplied by 2) on the positive side of thefrequency spectrum.</p><p>Thus if we construct a filter in the frequency domain with a singlelog-Gabor 'bump' on the positive side of the frequency spectrum we canconsider this filter to be the sum of the FFTs of the even and odd⌨️ 快捷键说明
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