Legendre moment

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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.[1] Legendre moments have been studied as a means to reduce image moment calculation complexity by limiting the amount of information redundancy through approximation.[2]

Legendre moments[3][edit]

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 {Pn(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.]:

See also[edit]


  1. ^ Lakshmi Deepika, C. et al. "Palmprint authentication using modified legendre moments", Procedia Computer Science, 2010, Vol.2, pp. 164–172
  2. ^ Huazhong Shu, et al. "An Efficient Method for Computation of Legendre Moments", Academic Press, 2000
  3. ^ Pew-Thian Yap. "An Efficient Method for the Computation of Legendre Moments", IEEE Transactions on Pattern Analysis and Machine Intelligence ( Volume: 27, Issue: 12, Dec. 2005 )