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The basic task of optimal filtering is to operate on an observed image in a manner that produces a filtered image that is a good estimate of a desired image. In the language of image restoration, we wish to filter a degraded document image to produce a restored image close to an ideal document image. Implicit in the formulation of the problem is the existence of a goodness criterion by which to measure the similarity of the filtered and ideal images. In defining a goodness criterion we take into account that the images are modeled as random processes and recognize that restoration is relative to the stated goodness criterion. An optimal filter is one that, among some class of filters, achieves the least error as measured by the goodness criterion. A chosen goodness criterion should result in optimal filters that perform satisfactorily relative to a desired end of the processing (visual quality, character recognition, compression, etc.); nevertheless, it must also be mathematically and computationally tractable, in the sense that either optimal or good suboptimal filters can be found by a computational scheme (usually off-line) that can be performed within an acceptable time frame on the computational machinery at hand. In the present chapter we consider the design of optimal translation-invariant nonincreasing (not necessarily increasing) binary filters relative to the mean-absolute-error criterion. Research and practice have shown this criterion is satisfactory for the kind of binary document processing with which we are concerned.
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