A fast Kalman filter for images degraded by both blur and noise

An optimal line-by-line recursive Kalman filter is derived for restoring images which are degraded in a deterministic way by linear blur and in a stochastic way by additive white noise. To reduce the computational and storage burden imposed by this line-by-line recursive Kalman filter circulant matrix approximations are made in order to diagonalize - by means of the fast Fourier transform (FFT) - both the model matrices and the distortion matrix in the dynamical model of the total image-recording system. Then the dynamical model reduces to a set of N decoupled equations and the line-by-line recursive Kalman filter based on this model reduces to a set of N scalar Kalman filters suitable for parallel processing of the data in the Fourier domain. Finally, via an inverse FFT the filtered data is presented in the data domain. The total number of computations for an N×N image reduces from the order of 0(N
⁴
) to
0(N^{2}\log_{2}N)
.