Error bound for multi-stage synthesis of narrow bandwidth Gabor filters

This paper develops an error bound for narrow bandwidth Gabor filters synthesized using multiple stages. It is shown that the error introduced by approximating narrow bandwidth Gabor kernels by a weighted sum of spatially offset, separable kernels is a function of the frequency offset and the reduction in bandwidth of the desired kernel compared to the basis values, as well as the spatial subsampling rate between filter stages. This error bound should prove useful in the design of a general basis filter set for multi-stage filtering because the maximum frequency offset is largely determined by the spacing of the basis filters. Bir Bhanu College of Engineering University of California Riverside, CA 92521 as periodic patterns. When the basis filters are adjusted together as a set, the effects of scale changes and image rotations, as well as the effect of foreshortening due to aspect changes, can be simulated (which could be exploited to simplify object and model matching for object recognition). The multi-stage filter is eficient becauses it allows the choice of a convenient set of basis filters. That is, the size and shape of the basis kernel can be selected independent of the desired two-stage response, thereby exploiting desirable implementation properties such as kernel separability. In addition, the basis filters can be selected for dual use, so that the basis output is used for detecting initial interesting features, as well as the first stage of the two-stage filter. 1. MULTI-STAGE FILTERING 2. GABOR THEORY A direct spatial implementation of a narrow bandwidth filter requires a large size kernel. Multi-stage filtering makes it possible to synthesize a narrow bandwidth response in two or more stages, each comprising wide bandwidth filters. In this work, a two-stage implementation is used. The first stage performs the bulk of the image processing and is designed for computational efficiency. The second stage filter adjusts the frequency and bandwidth of the image response to the desired values. Since the bandwidth of the basis filter output is limited, the second stage kernel can be subsampled relative to the image, allowing for computational savings. The primary restriction in this approach is that the synthesized response must have a narrower bandwidth than the basis filters. The primary benefits of two-stage filtering, over filtering with a single kernel, are flexibility and eficiency. The second stage provides flexibility in the sense that the center frequency and bandwidth of a given filter can be adjusted in a continuous manner. This allows individual basis filters (or a set of basis filters) to adjust to the image data, tuning Gabor filter responses to match certain narrow-bandwidth image features, such The kernels of the basis set and the desired impulse response of the two-stage filter are 2D Gabor functions. The general form of the Gabor function Gn(x, y) is given by [a] [3] where z and y are the horizontal and vertical image coordinates, respectively; i and i denote the axes that are rotated by 4 ~ ~ ; w, and $, are the modulation frequency and orientation, respectively; and un and a, are the scale and the aspect ratio of the elliptical Gaussian window, respectively. The “bandwidths” of the Gabor function, measured along the principal axes of the Gaussian window, are defined as a;’ and a; ’ c;’. In this work, the basis filter set is comprised of spatially separable Gabor kernels with log-polar frequency spacing. Each quadrature pair of basis kernels is given 33 0-8186-7310-9/95 $4.00