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

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130 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 45, NO. 2, FEBRUARY 1998results in a spatio–temporal filter tuned to velocities with thesame magnitude in the opposite direction.CNN spatio–temporal Gabor-type filters constructed in thisway are separable in space and time. 3-D spatio–temporalGabor filters are also space–time separable if their covariancematrices have the formAn important difference between Gabor filters and the CNNfilters is that Gabor filters are noncausal in space and time,while the CNN filters are noncausal in space, but causal intime.In order to ensure space–time separability, the spatial filteringCNN’s must settle much faster than the time scales of theimage motion so that their outputs at any time are essentiallythe result of spatial filtering the input at time . Fortunately, itis easy to design a VLSI implementation of the CNN spatialfilters with settling times on the order of microseconds orfaster. The time scale of image motion is usually on the orderof milliseconds. For example, video frame rates are about oneframe every 30–40 ms.There are several advantages to this spatio–temporal filteringarchitecture. First, multiple filters tuned to different velocitiescan be obtained by cascading the same spatio–temporalfiltering stage with different temporal filters. Since all of thefilters share the same spatial filtering stage, the differencesbetween their outputs are purely a function of the temporalvariation of the image, i.e., the motion. Second, the temporalfiltering at each pixel is independent of the filtering at the restof the pixels. Since the output of the spatial frequency stageis bandlimited, it can be subsampled without distortion due toaliasing. Therefore, it is not necessary to build a temporal filterat every pixel of the output. This can result in a significantsaving in chip area.Example 4: Denote the output of the spatial filtering stageat a pixel and time by . One possible continuoustime temporal filter [38] tuned to has outputwhich satisfiesFig. 7.(a)(b)(a) Contour plot and (b) mesh plot of the magnitude of the CNNspatio–temporal filter’s frequency response for 1-D images. The parametersused were = = 0:3; ! xo =2atan0:5, and ! to =0:3:This filter istuned to velocities v = 00:32.impulse response is the product of the spatial and temporalimpulse responseswhere. It has frequency responseThe combined spatio–temporal frequency response is the productof the spatial and temporal frequency responses. Using thespatial filtering stage of Example 1, the frequency response iswhose magnitude is plotted in Fig. 7. Since the correspondingimpulse response is complex valued, the frequency responseis not symmetric with respect to the origin. Due to spatialsampling, the spatio–temporal frequency response is periodicin with period . There is no periodicity in sincethe temporal filter operates in continuous time. Similarly, theIntuitive insight into the operation of the spatio–temporal filtercan be gained by considering the effect of the temporal filteron one pixel in isolation since the temporal filtering stageprocesses each output pixel of the spatial filter independently.Assuming uniform translation, the output of the spatial filteringstage at pixel and time can be written aswhere is constant and varies slowly in and .The output of the spatial filter at a fixed pixel rotates aroundthe origin of the complex plane with frequency . Thespeed of rotation is proportional to the speed of translation. Thedirection of rotation depends on the direction of translation.Assuming , the direction of rotation is clockwisefor positive velocities and counterclockwise for negative (seeFig. 8).
SHI: GABOR-TYPE FILTERING IN SPACE AND TIME WITH CELLULAR NEURAL NETWORKS 131Fig. 8.The output of the spatial filtering stage at a given pixel rotates around the origin of the complex plane. This figure shows the outputs at pixel n of thespatial filtering stage in Example 1 with = 0:3 and ! xo =0:93 for image inputs consisting of impulses located at different pixels near n. If the impulseis translating uniformly, the output rotates around the origin clockwise for positive velocities and counterclockwise for negative velocities.Fig. 10. A circuit implementation of the second-order differential (9) usingthree operational amplifiers. Resistors are labeled by their conductances.and imaginary parts of the temporal filter output. These satisfythe following real valued differential equation:(9)Fig. 9. The derivative of along trajectories of the unforced temporal filterfor = 00:7 and ! to =1illustrates that the system is a damped linearoscillator. Trajectories naturally tend to rotate counterclockwise about theorigin.The differential equation satisfied by the temporal filtercorresponds to a damped linear oscillator forced by the outputof the spatial filter. The derivative along different trajectoriesof the unforced system are shown in Fig. 9 for .Ifthe velocity is negative, the input rotates counterclockwise,facilitating the natural motion of output trajectories and leadingto a large response. For positive velocity, the input rotationopposes the natural motion, leading to a small response.In an analog circuit, the filter can be implemented with twocapacitors whose voltages represent the real and imaginaryparts of the output. Let and denote the realWriting KCL at the inverting inputs of the first and thirdoperational amplifiers and assuming unit capacitance, it canreadily be shown that the circuit shown in Fig. 10 implementsthis differential equation. To cascade this temporal filter withthe spatial filter in Example 1, the inputs andshould be equal to the voltages and from thebottom and top rows of Fig. 4.IV. CONCLUSIONThis paper has defined a class of Gabor-type spatial andspatio–temporal filters which posses the important propertiesof filters used in phase-based algorithms for computer vision.Gabor-type filters can be implemented using CNN’s. It ispossible to convert any low-pass spatial filtering CNN into acorresponding Gabor-type filtering CNN by a simple transformationof the cloning template coefficients. Spatio–temporal
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