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]
With order of m + n, and object intensity function f(x,y):
Lmn=
| (2m+1)(2n+1) | |
| 4 |
| 1 | |
| \int\limits | |
| -1 |
| 1 | |
| \int\limits | |
| -1 |
Pm(x)Pn(y)f(x,y)dxdy
where m,n = 1, 2, 3, ... with the nth-order Legendre polynomials being:
Pn(x)=\sum
| n | |
| k=0 |
ak,n
| ||||
| x |
\left(
| d | |
| dx |
\right)[(1-x2)n]
which can also be written:
\begin{align} Pn(x)&
| D(n) | |
| =\sum | |
| k=0 |
(-1)k
| (2n-2k)! | |
| 2nk!(n-k)!(n-2k)! |
xn-2k\\[5pt] &=
| (2n)! | |
| 2n(n!)2 |
| ||||
| x |
xn-2+ … \end{align}
where D(n) = floor(n/2). The set of Legendre polynomials form an orthogonal set on the interval [−1,1]:
| 1 | |
| \int | |
| -1 |
Pn(x)Pm(x)dx=
| 2 | |
| 2n+1 |
\deltanm
A recurrence relation can be used to compute the Legendre polynomial:
(n+1)Pn+1(x)-(2n+1)xPn(x)+nPn-1(x)=0
f(x,y) can be written as an infinite series expansion in terms of Legendre polynomials [−1 ≤ ''x'',''y'' ≤ 1.]:
| infty | |
| f(x,y)=\sum | |
| m=0 |
| infty | |
| \sum | |
| n=0 |
λmnPm(x)Pn(y)