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So the central problem of the local approach is finding the right window size and shape. Researchers have understood the need for variable window since the very beginning of applying the local method for stereo, as early as 1973. However only simple naïve optimization strategies have been tried. 2 methods have been proposed previously. First method is a local greedy search. Start with some small, say 3 by 3 window. Then add some set of pixels, say a column. If this improves the window cost function leave the new pixels in the window. Then try another set of pixels, say a row. If it improves the cost, leave them in a window again. However if adding some set of pixels worsens the window cost, do not include them from the window. Keep doing this until no more improvement can be made. Obviously this is a greedy not optimal strategy. Furthermore it is very inefficient.
The second method that previous approaches have tried is simply to try a few window shapes, say 8 different shapes. Since there is a need for efficiency, only a very limited number of window shapes can be tried, and you can’t possibly try all good shape candidates under any reasonable time constraints. There must be some efficient optimization method for window shape search.