Restoring Images Degraded by Spatially Variant Blur
J. Nagy and D. O'Leary
Restoration of images that have been blurred by the effects
of a Gaussian blurring function is an ill-posed but well-studied
problem. Any blur that is spatially invariant can be expressed
as a convolution kernel in an integral equation. Fast and effective
algorithms then exist for determining the original image by preconditioned
iterative methods. If the blurring function is spatially variant,
however, then the problem is more difficult. In this work we develop
fast algorithms for forming the convolution and for recovering
the original image when the convolution functions are spatially
variant but have a small
domain of support. This assumption leads to a discrete problem involving
a banded matrix. We devise an effective preconditioner and prove
that the preconditioned matrix differs from the identity by a matrix
of small rank plus a matrix of small norm.
Numerical examples are given,
related to the Hubble Space Telescope Wide-Field / Planetary Camera.
The algorithms that we develop are applicable to other ill-posed
integral equations as well.