Double Integrals by Summing Values of a Cumulative Distribution Function

​
A
B
C
D
Let
f
be a function, and suppose that its "cumulative distribution function"
F(x,y)=
y
∫
u=0
x
∫
v=0
f(u,v)dudv
, is known.
F(P)
is the integral of
f
over the rectangle below and to the left of
P
, and the double integral of
f
over a rectangle can be computed easily in terms of the values of
F
at the corners via:
∫∫
ABCD
f(x,y)dxdyF(A)-F(B)+F(C)-F(D)
.
Checking boxes causes a region to be shaded such that the combination
F
values at the checked corners is the integral of
f
over the shaded region.

Details

Snapshot 1: only the vertex
A
is chosen in the linear combination
Snapshot 2: both vertices
A
and
B
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

Permanent Citation

Amir Finkelstein
​
​"Double Integrals by Summing Values of a Cumulative Distribution Function"​
​http://demonstrations.wolfram.com/DoubleIntegralsBySummingValuesOfACumulativeDistributionFunct/​
​Wolfram Demonstrations Project​
​Published: March 7, 2011