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 〈θ<sub><i>j</i></sub>-θ¯|<sup><i>K</i></sup>.〉 be a minimum. Subscript <i>j</i> denotes the spatial frequency ω<sub><i>j</i></sub> at which the unknown object spectrum is to be restored, θ¯ denotes the optimum restoration by this criterion, and <i>K</i> 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 <i>I</i><sub>j</sub> = £(τ<sub><i>j</i></sub>, θ<sub>j</sub>, <i>N<sub>j</sub></i>). Here, τ<sub><i>j</i></sub> denotes the intrinsic system characteristic (usually the optical transfer function), and <i>N<sub>j</sub></i> represents a noise function. When restored values θ¯<sub><i>j</i></sub>, <i>j</i>= 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¯<i><sub>j</sub></i> is found to be the root of a finite polynomial. When the particular value <i>K</i>=2 is used, the root θ¯<i><sub>j</sub></i> is known analytically. Particular restorations θ¯<i><sub>j</sub></i> 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.
B. Roy FRIEDEN, "Optimum, Nonlinear Processing of Noisy Images," J. Opt. Soc. Am. 58, 1272-1275 (1968)