Optimum, Nonlinear Processing of Noisy Images

Abstract

It has been traditional to constrain image processing to linear operations upon the image. This is a realistic limitation of analog processing. In this paper, we find the optimum restoration of a noisy image by the criterion that expectation 〈θj-θ¯|K.〉 be a minimum. Subscript j denotes the spatial frequency ωj at which the unknown object spectrum is to be restored, θ¯ denotes the optimum restoration by this criterion, and K is any positive number at the user’s discretion. In general, such processing is nonlinear and requires the use of an electronic computer. Processor θ¯ uses the presence of known, Markov-image statistics to enhance the restoration quality and permits the image-forming phenomenon to obey an arbitrary law Ij = £(τj, θj, Nj). Here, τj denotes the intrinsic system characteristic (usually the optical transfer function), and Nj represents a noise function. When restored values θ¯j, j= 1, 2, …, are used as inputs to the band-unlimited restoration procedure (derived in a previous paper), the latter is optimized for the presence of noise. The optimum θ¯j is found to be the root of a finite polynomial. When the particular value K=2 is used, the root θ¯j is known analytically. Particular restorations θ¯j are found for the case of additive, independent, gaussian detection noise and a white object region. These restorations are graphically compared with that due to conventional, linear processing.