Inversion of convolution by small kernels

Abstract

Discrete convolution is a very important operation for filter design, image restoration, and applications [4,8, 10,11,16].In this paper, we investigate the existence of inverse elements in respect to convolution, and we derive a method to compute inverse elements without using Fourier transform. Furthermore, we get a simple condition for existence of inverse elements which is easy to verify. We describe the computation of small convolution kernels. These small convolution kernels are least squares optimal [15]. By a simple idea, we can regularize the problem, and the computational effort can be reduced drastically. Furthermore, we can use this theory to restore noise pictures, and we can develop a regularization theory. For the clarity of presentation, the approach given here is one-dimensional, but the multidimensional generalization is straightforward.