Simple Binary Restoration

Start with an image composed of pixels at sites. Let I be the set of pixels in an image. The image we are interested in is then given by

.
We cannot measure x, only y where

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At each pixel site there is a neighborhood, N, of size R >= 1. There is a finite set of possible patterns in a neighborhood. Each pattern has an associated central pixel value and a probability with which it can occur.

The probabilities, Q, encapsulate knowledge about the local structure of the image. Binary restoration restores x from the noisy measurement y using

We call this the average mean-square error or best AMSE estimate because of all the local estimators, it is the one resulting in the smallest AMSE.

Weighted Mean Square Error

Restoration using the AMSE estimate works well when the image to be restored does not have critical but rare features, for example, a pattern that is distributed over the image. However, when the image contains rare, thin objects this is not the case. Q for the rare event of a thin object is so low that the AMSE can ignore the thin objects and incur an insignificant penalty for doing so.

To solve that problem we introduce the weighted mean-square error or WMSE estimate, given by:

w is a weighting function that weights neighbourhoods by their contents. If we are intersested in thin objects we apply a higher weight to neighborhoods with those objects. Although the derivation of the WMSE is a little longer than for AMSE, the result is simple enough. Just replace Q in the AMSE with wQ to get the WMSE. We can think of this as a dictionary-based method [Hancock and Kittler, 1990] where the dictionary is used to look up the wQ for wQ not equal to zero.

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