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

Source:

With order of m + n, and object intensity function f(x,y):

$L_{mn}=\frac{(2m+1)(2n+1)}{4}\underset{-1}{\overset{1}{\int }}\underset{-1}{\overset{1}{\int }}{P}_m(x)P_n(y)f(x,y)dxdy$

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

$P_n(x)=\sum _{k=0}^n{a}_{k,n}{x}^k=\frac{(-1{)}^n}{2^{n}n!}\left(\frac{d}{dx}\right)[(1-x^2{)}^n]$

which can also be written:

$\begin{array}{cc}{P}_n(x)& =\sum _{k=0}^{D(n)}(-1{)}^k\frac{(2n-2k)!}{2^{n}k!(n-k)!(n-2k)!}{x}^{n-2k}\\ & =\frac{(2n)!}{2^n(n!{)}^2}{x}^n-\frac{(2n-2)!}{2^{n}1!(n-1)!(n-2)!}{x}^{n-2}+\cdots \end{array}$

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

$\underset{-1}{\overset{1}{\int }}{P}_n(x)P_m(x)dx=\frac{2}{2n+1}{\delta }_{nm}$

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

$(n+1)P_{n+1}(x)-(2n+1)x{P}_n(x)+n{P}_{n-1}(x)=0$

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

$f(x,y)=\sum _{m=0}^{\mathrm{\infty }}\sum _{n=0}^{\mathrm{\infty }}{\lambda }_{mn}{P}_m(x)P_n(y)$

See also

• Image moment
• Legendre polynomial
• Zernike polynomials