Mixed exponentials filter for edge detection and the Fredholm integral equation of the first kind

We try to answer the following set of questions: "is it possible to design step edge detection filters given the desired response to an input step edge signal?"; "the method extends easily to two dimensions?" and "do we improve edge detection using this method?". Answers to these questions are given by formulating the step edge detection problem in terms of a linear first kind Fredholm integral equation. The unidimensional solution to the problem and the extension to the bidimensional case is presented. Several filter designs are presented for different types of filter responses. We show the filters obtained for abrupt and exponential type responses. The first derivative of Gaussian (DG) and the exponential filter (ISEF) are immediately obtained after solving the integral equation. A novel mixed exponential filter (MEXP) is proposed that combines the behaviors of the ISEF and the DG filters. Experimental results are presented for the proposed MFXP filter.