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As the integrand (in the answer you quote) is periodic and fairly smooth, you can evaluate it numerically using the trapezoidal rule. For such integrands, the convergence is exponential, so you shouldn’t need too many points. Here is a nice explanation. I think it is quite feasible to do this in Excel. Here's some matlab code that shows the rapid convergence ...


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(I do not claim any optimality, or that I have thought of every detail) 1. Downsample and brute force: You project your black and white map onto a low resolution. On this lower resolution, you may represent any rectangle by two points marking the upper left and lower right corner. If your resolution is low enough, you can iterate through all non-redundant ...


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