Optimal Real-Time Image Processing with Imperfect Information on Convolution-Type Distortion

Abstract

We study a problem of designing an optimal two-dimensional circularly symmetric convolution kernel (or point spread function (PSF)) with a circular support of a chosen radius \( R \). Such function will be optimal for estimating an unknown signal (image) from an observation obtained through a convolution-type distortion with the additive random noise. This technique is then generalized to the case of an imprecisely known or random PSF of the measurement distortion. It is shown that the construction of the optimal convolution kernel reduces to a one-dimensional Fredholm equation of the first or a second kind on the interval \( [0,R] \). If the reconstruction PSF is sought in a finite-dimensional class of functions, the problem naturally reduces to a finite-dimensional optimization problem or even a system of linear equations. We also analyze how reconstruction quality depends on the radius of the convolution kernel. It allows finding a good balance between computational complexity and quality of the image reconstruction.