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Kernel based operations are common in a variety of applications, such as image processing (e.g., blurring), generating smoothed estimation maps, and so on. A common approach is to select four surrounding cells (see the following figure: A the green colored cells) then calculate the average value of them including the value of the central cell to be assigned as the estimate.

Some questions appear to me to have a discussion:

1- I propose two other cell selections i.e., B and C. The question is what kind of statistical justification could help to recommend one of them?

2- Which part of statistical analysis would fit to discuss such conceptual/practical questions?

enter image description here

my understanding is:
Because those cells (green ones in A) are the closest to the central cell (the cell of interest for estimation etc). For case B compared to A it is obvious so B is not a better choice. In comparison between A and C however the case A has a disadvantage, lacking diagonal supports while for case C it covers full orientations. I think the case A is good because of less computation required however with the expense of biases diagonally.
Here is an example implementation of those ideas:
enter image description here
Hope this helps a bit more in understanding the difference brought by applying different kernel averaging configurations.

Applications: A sample application can be found in "After Kernel Averaging" in which type C has been applied.

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In the absence of a particular application or context, this question appears overlay vague. Kernel operations are widely varied in nature and performed for many reasons. Accordingly, I am concerned about the possibility that you will get either no good responses or many different responses unless you narrow the question to address a particular problem you want solved (as suggested in our FAQ). –  whuber Oct 4 '11 at 23:14
At this stage the question is general. We can see in many applications however such as finite difference method that kernel (A) is used. I haven't seen any discussion so far why only that configuration is used! If these configurations are not already discussed why don't we try? It is a practice at least if not a discovery. –  Developer Oct 5 '11 at 11:46
Because as it stands the question is, imho, overly broad: it covers so many distinctly different applications that there cannot possibly be a single answer. It is unclear what the question really seeks in the nature of an answer. This is not a discussion site (even though the comments sometimes make it appear so...). Please read our FAQ and refine your question following its guidelines. –  whuber Oct 5 '11 at 14:46
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