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.
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With order of m + n, and object intensity function f(x,y):
where m,n = 1, 2, 3, ... with the nth-order Legendre polynomials being:
which can also be written:
where D(n) = floor(n/2). The set of Legendre polynomials {P<sub>n</sub>(x)} form an orthogonal set on the interval [âÂÂ1,1]:
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 [âÂÂ1 ⤠x,y ⤠1.]: