Analytic inverse of discrete Gaussian blur

Abstract

The problem of removing degradation and blur is common in signal and image processing. While Gaussian convolution is often the model for the blur, no exact deblurring techniques for the Gaussian kernel have been given. Previously, a Gaussian deblurring kernel in the continuous domain has been presented. We present an exact deblurring method for the discrete domain where linear convolution is replaced by matrix multiplication, the Gaussian kernel is replaced by a highly structured Toeplitz matrix, and the deblurring kernel is replaced by the inverse of this blur matrix. To bypass numerical errors, the inverse is derived analytically and a closed-form solution is presented. In particular, the matrix is decomposed into the product of triangular matrices and a diagonal matrix, where numerically ill-conditioned elements are gathered and that allows for a direct accurate handling of numerical errors. This exact inverse is effective in degradation problems characterized by low noise and high representational accuracy and has potential application in the many areas where a Gaussian point spread function is relevant.