Image deconvolution using 2-D non-causal fast Kalman filters

Restoration of an image distorted by a linear shift-invariant system is a 2-D deconvolution problem which is treated here in a Bayesian framework to stabilize the solution. The usual introduction of dynamics into the state equation to reduce the problem dimensions requires an artificial causality assumption. Thus we propose state-space models where the state is constant and equal to the entire object to be restored, and where dynamics appear only in the observation equation which may be either vectorial or scalar. When the image is scanned row by row, the shift properties of the convolution summation allow derivation of a fast optimal Kalman filter through factorization techniques. When the image is scanned pixel by pixel, the computational requirement can be further reduced at the expense of an extra assumption. Finally, a sub-optimal asymptotic filter with a reduced update is derived.