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):

[math]\displaystyle{ L_{mn}=\frac{(2m+1)(2n+1)} 4 \int\limits_{-1}^1 \int\limits_{-1}^1 P_m(x) P_n(y)f(x,y) \,dx\, dy }[/math]

where m,n = 1, 2, 3, ...∞ with the nth-order Legendre polynomials being:

[math]\displaystyle{ P_n(x)=\sum_{k=0}^n a_{k,n}x^k=\frac{(-1)^n}{2^n n!} \left( \frac{d}{dx} \right) [(1-x^2)^n] }[/math]

which can also be written:

[math]\displaystyle{ \begin{align} P_n(x) & =\sum_{k=0}^{D(n)}(-1)^k \frac{(2n-2k)!}{2^n k!(n-k)!(n-2k)!} x^{n-2k} \\[5pt] & = \frac{(2n)!}{2^n(n!)^2}x^n-\frac{(2n-2)!}{2^n 1!(n-1)!(n-2)!} x^{n-2} + \cdots \end{align} }[/math]

where D(n) = floor(n/2). The set of Legendre polynomials {P_n(x)} form an orthogonal set on the interval [−1,1]:

[math]\displaystyle{ \int_{-1}^1 P_n(x)P_m(x) \, dx = \frac{2}{2n+1}\delta_{nm} }[/math]

A recurrence relation can be used to compute the Legendre polynomial:

[math]\displaystyle{ (n+1)P_{n+1}(x)-(2n+1)xP_n(x)+nP_{n-1}(x)=0 }[/math]

f(x,y) can be written as an infinite series expansion in terms of Legendre polynomials [−1 ≤ x,y ≤ 1.]:

[math]\displaystyle{ f(x,y)=\sum_{m=0}^\infty \sum_{n=0}^\infty \lambda_{mn}P_m(x)P_n(y) }[/math]

See also

• Image moment
• Legendre polynomial
• Zernike polynomials