RESTORING IMAGES DEGRADED BY SPATIALLY VARIANT BLUR

Restoration of images that have been blurred by the effects of a Gauss ian blurring function is an ill-posed but well-studied problem. Any bl ur that is spatially invariant can be expressed as a convolution kerne l in an integral equation. Fast and effective algorithms then exist fo r determining the original image by preconditioned iterative methods. If the blurring function is spatially variant, however, then the probl em is more difficult. In this work we develop fast algorithms for form ing the convolution and for recovering the original image when the con volution functions are spatially variant but have a small domain of su pport. This assumption leads to a discrete problem involving a banded matrix. We devise an effective preconditioner and prove that the preco nditioned matrix differs from the identity by a matrix of small rank p lus a matrix of small norm. Numerical examples are given, related to t he Hubble Space Telescope (HST) Wide-Field/Planetary Camera. The algor ithms that we develop are applicable to other ill-posed integral equat ions as well.