Gabor-Type Filtering in Space and Time - Department of Electronic ...

128 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 45, NO. 2, FEBRUARY 1998chosen to be identical. The results of analyzing the discretespacefilters will be more complex, but very similar to theresults here, especially as and approach zero.Using the fact that, it can be shown thatthe error is independent of and depends only upon thedifference between the Gaussian envelope of the Gabor filterand the low-pass envelope of the Gabor-type filter. Forthe filter in Example 1, the minimum error of 15 dB isachieved forTABLE ICOMPARISON OF PHASE MEASUREMENT PROPERTIESOF GABOR AND GABOR-TYPE FILTERSFor the filter in Example 2, the minimum error ofachieved for21 dB isThe values of and have been estimated numerically.Simulation results in Fig. 1 where the parameters chosenminimize the NSE provide visual confirmation that the impulseand frequency responses of the filter in Example 2 are closerto the Gabor filter’s than are those of the filter in Example 1.To compare the three filters with respect to the desirableproperties for computing phase discussed in Section I-B, wedefine the following criteria. The criteria are based upon theassumption that the input image is white.Negative Frequency Rejection (NFR): This is the ratio betweenthe energy in the output due to positive frequencies inthe input and the energy due to negative frequencies. Ifis the frequency response of the filter, thenDC Rejection (DCR): This is a measure of the extent towhich the dc component is suppressed compared with the peakresponse. If is the frequency response of the filterPhase Wraparound Rejection (PWR): The output at a givenpixel can be considered as the sum of two components: thefirst (desired) component due to convolving the image with thepart of the impulse response where the phase varies fromto and the second (undesired) component due to convolvingthe input image with the part of the impulse response wherethe phase has wrapped around. Since we assume the imageinput is white, these two components are uncorrelated. Wedefine the PWR to be the ratio of their energies. If is theimpulse response of the filter, thenwell as whether or not the phase of the impulse responseis monotonous. The results were estimated numerically usingthe same filter parameters used to generate the plots. Sincethe parameters were chosen to minimize the energy in thedifference between the filter impulse responses, their spatialsupport is similar. Since the PWR is defined in terms of theenergy in the filters, their PWR is also similar. However, thereare large differences between the NFR and DCR. As can beseen from Table I as well as the plots in Fig. 1, the NFR andDCR is largest for the Gabor filter and smallest for the filterin Example 1. The rejection can be improved at the cost ofdecreased PWR by decreasing the bandwidth of the filters.Only the filter in Example 2 does not have monotonous phase,since its low-pass modulating function is not strictly positive.Although the degree to which the filters exhibit the propertiesdescribed in Section I-B varies, we would expect any of thefilters to be effective in phase-based algorithms because thesealgorithms have been found to be fairly insensitive to the exactform of the filter.B. 2-D ImagesThe 2-D generalization of (2) simply extends the summationto two dimensions:The feedback cloning templatebe represented by acan nowmatrix, e.g.,If the CNN filter is stable, its frequency response isAs in the 1-D case, for any 2-D CNN low-pass filter therecorresponds a 2-D CNN Gabor filter tuned to frequencyobtained by replacing the feedback cloning templatewithTable I compares the Gabor filter and the Gabor-type filtersin Examples 1 and 2 with respect to these three criteria asIts impulse response is the impulse response of the low-passfilter modulated by. Its frequency response is.