Legendre moments

The Legendre moments [18] of order $(m+n)$ are defined as:

$$\lambda_{mn} = \frac{(2m+1)(2n+1)}{4}\int_{-1}^{1}\int_{-1}^{1}P_{m}(x)P_{n}(y)f(x,y)~dx~dy$$ (45)

where $m,n=0,1,2,...,\infty$, $P_{m}$ and $P_{n}$ are the Legendre polynomials and $f(x,y)$ is the continuous image function. The Legendre polynomials are a complete orthogonal basis set defined over the interval $[-1,1]$. For orthogonality to exist in the moments, the image function $f(x,y)$ is defined over the same interval as the basis set, where the $n^{th}$ order Legendre polynomial is defined as:

$$P_{n}(x) = \sum_{j=0}^{n} a_{nj}~x^{j} %%= \frac{1}{2^{n}n!} \frac{d^{n}}{dx^{n}}(x^{2} - 1)^{n}$$ (46)

and $a_{nj}$ are the Legendre coefficients given by:

$$a_{nj}=(-1)^{(n-j)/2}\frac{1}{2^n}\frac{(n+j)!}{(\frac{(n-j)}{2})!(\frac{(n+j)}{2})!j!} ~~~~~\mbox{where $n-j$~=~even}$$ (47)

So, for a discrete image with current pixel $P_{xy}$, Equation 1.45 becomes:

$$\lambda_{mn} = \frac{(2m+1)(2n+1)}{4} \sum_{x} \sum_{y} P_{m}(x)P_{n}(y)P_{xy}$$ (48)

and $x,y$ are defined over the interval $[-1,1]$.