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
.
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.
[Back]
[Next]
[Contents]