Efficient synthesis of parameterized Gaussian-like filters by approximation

Abstract

Gaussian filters and their various extensions such as Gaussian derivative filters and Gabor filters have found many applications in image representation, compression, segmentation and stereo vision. In these applications, the images are usually convolved with a bank of parameterized Gaussian-like filters. Unless there is some a priori knowledge about images to be processed, one has to use a large number of filters so as to capture useful information contained in the images as much as possible, resulting in high computation expense. In this paper, we propose an efficient way to synthesize these filters. The idea of our method is to approximate the parameterized Gaussian-like filters by a finite sum of products of functions in which spatial variables are separated from the parameters. Thus, the responses of filters with any values of parameters within the range of interest can be synthesized by the responses of a finite number of fixed basis filters. The approximation is optimal with respect to the L 1 norm of the difference between the filter and its approximation. This optimality criterion allows us to explore a new property of Gaussian-like filters to give explicit solutions to such approximation problems. It is interesting that the resulting basis filters preserve the forms of the original filters; therefore the existing algorithms developed for implementing Gaussian-like filters for a fixed parameter are applicable to the basis filters. The approximation results together with applications to scale-space construction and texture segmentation show the effectiveness of this approach.