What Is a Logic-Based Filter? - Logic-based Nonlinear Image Processing - page 13

2.3 Window Constraint

An assumption is made that the noise and signal statistics of the processed image
are localized . In simple terms, a pixel is more likely to be related to its immediate
neighbors than to pixels a large distance away. This means that it is not necessary to
consider every location of the input image when estimating the value of a pixel. The
filter is therefore influenced mainly by local structure.
The true value of a pixel may therefore be estimated from the noise-corrupted
version of the image by considering only a finite collection of pixels within a local
neighborhood centered on the pixel. Considering pixels outside of the neighbor-
hood will add little further information. If this assumption is not true or only par-
tially true, the filter obtained will be suboptimal. If the size of the window is
increased, the resulting filter will be closer to the optimal. The images in this topic
will therefore be processed using a sliding window (or mask) of values centered on
the pixel to be estimated.

2.4 Translation Invariance

An assumption is usually made that the statistics of the image detail and the cor-
rupting noise process are wide-sense stationary. This means that the same filter
may be used at every location of the image. If this assumption is not true, the filter
produced will be a weighted average of the different filters that would be optimum
at each location. In this case the solution would be suboptimal. In practice, the re-
sults obtained from filters based on this assumption have been acceptable for a wide
variety of applications.
Therefore, adopting these two constraints means that not only will the images
be processed using a sliding window, but the filter characteristics within the win-
dow will be the same for all locations in the image.
Note: It is true that a window-based filter is unable to determine pixel values at
the edges and corners of the image. For image-restoration purposes, the edge pixels
are usually simply omitted from the process. In applications such as image coding
where processed pixels are required, a smaller asymmetrical version of the window
is used at the extreme locations. 2

2.5 Filter Windows

A number of different windows of increasing size have been commonly used in im-
age restoration. Some examples are shown in Fig. 2.4. For the same data, the best
possible filter for a large window will always be better (or the same) than the best
filter that may be found within a smaller sub-window.
Therefore, if MAE (

i

) is the mean-absolute error that results from filtering
an image with the optimum filter using a window containing i pixels, then

ψ opt