Relating Zernike and Cartesian moments

To help reduce computation complexity, it may prove useful to express the Zernike moments in terms of Cartesian moments. This removes the need for the polar mapping of the image, while also removing the dependence on the trigonometric functions. Alternatively, expressing Cartesian moments in this way would aid the selection of less correlated descriptors. This conversion can be achieved by slightly re-arranging the Zernike moment equation. If, as before, the Zernike polynomials are given by Equation 1.51 [22] and the radial polynomials $R_{mn}(r)$ are defined by Equation 1.52, re-arranging $F(m,n,s,r)$ gives:

$$F(m,n,s,r) ~ = ~ \frac{(m-s)!}{s!(\frac{m+\mid n \mid}{2}~-s)!~(\frac{m-\mid n \mid}{2}~-s)! }~r^{m-2s}$$

$$~ = ~\frac{(m-s)!}{s!(\frac{m-2s+\mid n \mid}{2})!~(\frac{m-2s-\mid n \mid}{2})! }~r^{m-2s}$$ (70)

then substituting $k=(m-2s)$ and re-arranging again, produces:

$$R_{mn}(r) ~=~ \sum_{k=n}^m ~B_{mnk}~r^k~~~~~~~~(m-k) ~is~even, n \geq0~,~$$ (71)

where:

$$B_{mnk} ~=~ \frac{ (-1)^{(m-k)/2} \left(\frac{m+k}{2}\right)... ...right)!\left(\frac{k+n}{2}\right)!\left(\frac{k-n}{2}\right)!}$$ (72)

Using this manipulated form of the radial polynomials produces Zernike moment definitions (in continuous form) of:

$$A_{mn} = \frac{m+1}{\pi} \sum_{k=n}^m B_{mnk} \int_0^{2\pi} ... ...0^1 r^k \exp(-jn\theta)f(r,\theta)~r~dr ~d\theta ~~~;~~~r\leq1$$ (73)

which when translated to Cartesian coordinates is:

$$A_{mn} = \frac{m+1}{\pi} \sum_{k=n}^m B_{mnk} \int_x \int_y (x-jy)^n (x^2 + y^2)^{(k-n)/2} f(x,y)~dx ~dy$$ (74)

bounded by $x^2+y^2\leq1$ and $j=\sqrt{-1}$. The double integral can now be expressed in terms of a series of summed Cartesian moments, of the form:

$$m_{pq}~=~\int_x \int_y x^p y^q f(x,y)~dx~dy$$ (75)

For example:

$$Z_{00}~ = ~\frac{1}{\pi} \sum_{k=0}^0 B_{00k} \int_x \int_y (x-jy)^0 (x^2+y^2)^{k/2} f(x,y) ~dx~dy$$

$$~ = ~\frac{1}{\pi} \int_x \int_y f(x,y) ~dx~dy$$

$$~ = ~\frac{1}{\pi}m_{00}$$ (76)

It must be noted that this comparison is only valid if the Cartesian moments are calculated on images confined to $[-1,1]$, which is due to the Zernike moments being calculated over the unit disc.