Legendre moment

In mathematics, Legendre moments are a type of image moment and are achieved by using the Legendre polynomial. Legendre moments are used in areas of image processing including: pattern and object recognition, image indexing, line fitting, feature extraction, edge detection, and texture analysis. Legendre moments have been studied as a means to reduce image moment calculation complexity by limiting the amount of information redundancy through approximation.
Legendre moments.
With order of "m" + "n", and object intensity function "f"("x","y"):
where "m","n" = 1, 2, 3, ... with the "n"th-order Legendre polynomials being:
which can also be written:
where "D"("n") = floor("n"/2). The set of Legendre polynomials {"P""n"("x")} form an orthogonal set on the interval [−1,1]:
A recurrence relation can be used to compute the Legendre polynomial:
"f"("x","y") can be written as an infinite series expansion in terms of Legendre polynomials [−1 ≤ "x","y" ≤ 1.]: