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

SHI: GABOR-TYPE FILTERING IN SPACE AND TIME WITH CELLULAR NEURAL NETWORKS 127Fig. 5. This resistive network better approximating Gaussian filtering adds negative resistance connections to second nearest neighbors. Only the fullconnections for node n are shown. The resistor labels denote conductances.The circuit in Fig. 4 implements two cells of the CNN andtheir interconnections with their nearest neighbors. In thefollowing, we refer to each circuit node by its labeled nodalvoltage. The voltages across each pair of vertically alignedcapacitors represent the real and imaginary parts of the stateof a cell. The resistances and transconductance amplifiersimplement the interconnection between cells. The input to thenetwork is provided by current sources which supply currentsproportional to the input image intensity. By writing KCL atnodes and and assuming a unit capacitance, onecan verify that the time evolution of satisfies the topequation in (7) and that satisfies the bottom equation.The entire CNN array can be constructed by replicating thiscircuit to add more cells and connections. At steady state,the voltages across the lower capacitors are the result ofconvolving the spatial distribution of the input currents withthe real part of . The voltages across the upper capacitorscorrespond to convolving with the imaginary part. This mixedtransconductance amplifier and resistor implementation canbe proven to be more robust to parameter variations thanimplementations based on either resistors or transconductanceamplifiers alone [35].The circuit implementation also gives good intuitive understandingof the CNN’s operation. Assume that the inputimage is an impulse at pixel . In the circuit, this correspondsto setting the current source to amps and settingthe remaining current sources to zero. If the gains and conductanceswere chosen so that and , thenthe steady-state voltages across the lower capacitors wouldfollow the spatial distribution shown in Fig. 1(d) where thecenter peak occurs at cell and the voltages across the uppercapacitors would follow the distribution shown in Fig. 1(e).To see how this would arise in the circuit, consider thecurrent supplied by the source . Part of the currentflows through the conductance (which is positive) pushingthe voltage positive. As this voltage increases, the tworesistors with conductancecause a smoothingeffect which pulls the voltages and uptoward . Current also flows through the diagonal resistorwith conductance pulling positiveas well. At the same time, the transconductance amplifierwith input draws current from node pushingnegative. The larger , the more the voltages atnodes and are pushed negative and positive.On the other hand, the larger , the greater the smoothingbetween nodes. Thus, the larger the ratiothe higher the spatial frequency at which the impulseresponse oscillates. This is consistent with our theoreticalpredictions since tan increases with .Example 2: Based upon Kobayashi et al.’s resistor networkfor Gaussian filtering [36], it is possible to design a CNNwhich implements a filter with an impulse response which iscloser to the Gabor, albeit at the price of increased complexity.For 1-D images, the resistor network in Fig. 5 smooths theinput current distribution with an approximately Gaussianconvolution kernel. The corresponding CNN cloning templatesareThe filter implemented by this network is a discrete approximationto a continuous space filter with impulse and frequencyresponses given by [37]The width of the impulse response increases linearly with .The approximation improves as decreases.The template for the corresponding Gabor-type filter isFig. 1(g)–(i) plots the impulse and frequency responses of thisfilter.To compare how well the filters in the preceding examplesapproximate Gabor filters, we define the normalized squarederror (NSE) to be the total energy in the difference betweenthe Gabor impulse response and the impulse response of thecontinuous space filter approximated by the CNN filter dividedby the total energy in the Gabor impulse responsewhere is the Gabor impulse response in (1) and isgiven either by (5) or (8). The frequencies of the complexexponentials in the Gabor and Gabor-type filters are(8)