Double Integrals by Summing Values of a Cumulative Distribution Function

Let be a function, and suppose that its "cumulative distribution function" , is known. is the integral of over the rectangle below and to the left of , and the double integral of over a rectangle can be computed easily in terms of the values of at the corners via:

.

Checking boxes causes a region to be shaded such that the combination values at the checked corners is the integral of over the shaded region.

Snapshot 1: only the vertex is chosen in the linear combination

Snapshot 2: both vertices and are chosen in the linear combination; light green regions are where the function's double integral was added only once in the linear combination with a positive coefficient; dark green regions are where the function's double integral was added twice in the linear combination

Snapshot 3: Both vertices C and D are chosen in the linear combination; light red regions are where the function's double integral was subtracted only once in the linear combination; dark green regions are where the function's double integral was subtracted twice in the linear combination

More information regarding this algorithm is available in Wikipedia, and in the following papers:

[1] F. C. Crow, "Summed-Area Tables for Texture Mapping," in SIGGRAPH '84: Proceedings of the 11th Annual Conference on Computer Graphics and Interactive Techniques, 1984 pp. 207–212.

[2] P. Viola and M. Jones, "Robust Real-Time Object Detection," International Journal of Computer Vision, 57(2), 2002 pp. 137–154. http:\\research.microsoft.com/en-us/um/people/viola/pubs/detect/violajones_ijcv.pdf