A Modification of Deriche ́ s Approach to Edge Detection

In 1983 Canny presented criteria for measuring the quality of edge detectors and derived an optimal FIR-Filter for step edges by optimizing them. Four years later Deriche proposed an approach to edge detection based on Canny´s design utilizing IIR-Filters which can be implemented very efficiently recursively. Unfortunately, using the Deriche-Filter leads to a distortion of the amplitudes of the edges depending on their direction. In this paper, it will be shown that these distortions are systematic errors which can be eliminated by a simple modification of the edge detection procedure. Because of its obvious "kinship" to the Deriche-filter the Shen-filter has been included in the investigations. Deriche´s approach to the detection of step edges ([4]-[6]) is based on Canny´s filter design ([1],[2]). Canny had derived an optimal one-dimensional Finite Impulse Response filter for step edges by optimizing specific criteria for the quality of edge detectors. Deriche used the same method to construct an Infinite Impulse Response edge filter. His edge detector performs the following three steps: (i) Calculation of E x (x,y) by folding the image with the separable 2D-filter f(x)g(y). (ii) Calculation of E y (x,y) by folding the image with the separable 2D-filter g(x)f(y). (iii) Calculation of the amplitude and the direction of the edge as the length and the orientation of the vector [E x (x,y),E y (x,y)] T. In the algorithm above f(z) stands for a one-dimensional IIR-edge filter and g(z) for the one-dimensional smoothing filter obtained by integrating f(z). [E x (x,y),E y (x,y)] T is the gradient of the image after folding with the 2D-smoothing filter G(x,y) = g(x)g(y) (as shown in [7]). Thus Deriche´s approach can be seen as a gradient scheme. Deriche´s optimal 1D-edge filter has the following impulse response function: f(z) = K e-α|z| sin(ωz), α,ω > 0. The two most interesting embodiments of this filter, f D1 (z) and f D2 (z), have been investigated in our project: f D1 (z) is optimal according to Canny´s criteria of quality. It has been used by Deriche himself in later works, too ([6]). The filter f D2 (z) is similar to Canny´s edge filter in terms of the values of the criterions of quality obtained. Its application leads to far less distortions of amplitudes in the two-dimensional case, which will be shown later. Plugging in f D1 (z) and f D2 (z) in the sketch of the algorithm for edge detection above, …