Recursively implementating the Gaussian and its derivatives

Abstract

Gaussian filtering is one of the most successfully operation in computer vision in order to reduce noise, calculating the gradient intensity change or performing Laplacian or the second directional derivative of an image. However, it is well known that in a multi-resolution context, where the need for large filters is required, this technique suffers from the fact it is a computationally expensive since the number of operations per point in convolving an image with a Gaussian filter is directly proportional to the width of the operator. We propose in this paper a technique in order to use Gaussian filtering with a reduced and fixed number of operations per output independently of the size of the filter. The key of our approach is to approximate in a mean square sense the prototype Gaussian filters with an exponentially based filter family depending on the same scale factor than the Gaussian filters (i.e. s) and then to implement in an exact and recursive way the approximate filters. An important point of the design presented in this paper is that dealing with Gaussian filters having different scale factor (i.e. s) will not require a new design algorithm as. The coefficients looked for in the recursive realization are determined function of the scale factor of each considered prototype filter, namely the Gaussian filter, its first and second derivative. Some experimental results will be shown to illustrate the efficiency of the approximation process and some applications to edge detection problems and multi-resolution techniques will be considered and discussed.

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