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.[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]
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.]:
References
- Lakshmi Deepika, C. et al. "Palmprint authentication using modified legendre moments", Procedia Computer Science, 2010, Vol.2, pp. 164–172
- Huazhong Shu, et al. "An Efficient Method for Computation of Legendre Moments", Academic Press, 2000
- 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 )