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Notes:


Here in this box we show some examples of our compact windows, just to give you an idea of their shape. We call them compact because they have compact (as opposed to spread out) shape. That is their ratio of perimeter to area is small. All possible rectangles are in our compact window class, and rectangles are a very first reasonable approximation for a window shape. However our compact windows are much more general than just the rectangles. There are O(n^4) rectangles, but there are o(2^n) compact windows, where n in the largest allowed window size. We can find the optimal window shape in n^3 time in theory, but in practice the time is o(n^2), which is a linear time, since the size of the problem here is n^2, the number of pixels in the largest allowed window. In practice we take largest window to be 31 by 31. So we search over a huge # of windows, 2^31 in time 31x31. Contrast this with the previous approaches which could search over a very small set of window shapes. This fast search over a huge set of window shapes is the key to why our local algorithm performs better than other local algorithms.