Pseudo zernike polynomials

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Pseudo-Zernike polynomials are well known and widely used in the analysis of optical systems. They are also widely used in image analysis as region descriptors.

[edit] Definition

They are an orthogonal set of complex-valued polynomials defined as :

$V_{nm}(x,y) = R_{nm}(x,y)e^{jm\arctan(\frac{y}{x})}$

where $x^2+y^2\leq 1, n\geq 0, |m|\leq n$ and orthogonality on the unit disk is given as:

$\int_0^{2\pi}\int_0^{\infty}[V_{nl}(r\cos\theta,r\sin\theta)]^*. [V_{mk}(r\cos\theta,r\sin\theta)]drd\theta = \frac{\pi}{n+1}\delta_{mn}\delta{kl}$

The radial polynomials ${R n m}$ are defined as:

$R_{nm}(x,y) = \sum_{s=0}^{n-|m|}D_{n,|m|,s}(x^2+y^2)^{(n-s)/2}$

where

$D_{n,|m|,s} = (-1)^s\frac{(2n+1-s)!}{s!(n-|m|-s)!(n-|m|-s+1)!}$

The PZM of order $n$ and repetition $l$ are defined as:

$A_{nl}=\frac{n+1}{\pi}\int_0^{2\pi}\int_0^{\infty}[V_{nl}(r\cos\theta,r\sin\theta)]^* f(r\cos\theta,r\sin\theta)rdrd\theta$

where $n = 0, \ldots \infty$ and $l$ takes on positive and negative integer values subject to $|l|\leq n$ .

The image function can be reconstructed by expansion of the Pseudo-Zernike coefficients on the unit disk as:

$f(x,y) = \sum_{n=0}^{\infty}\sum_{l=-n}^{+n}A_{nl}V_{nl}(x,y)$

Pseudo-Zernike moments are derived from conventional Zernike moments and shown to be more robust and less sensitive to image noise than the Zernike moments.

[edit] References

TEH C.-H., CHIN R.: On image analysis by the methods of moments. Pattern Analysis and Machine Intelligence, IEEE Transactions on 10, 4 (1998), 496–513.
BELKASIM S., AHMADI M., SHIRDHAR M.: Efficient algorithm for the fast computation of zernike moments.
HADDADNIA J., AHMADI M., FAEZ K.: An efficient feature extraction method with pseudo-zernike moment in rbf neural network-based human face recognition system. EURASIP Journal on Applied Signal Processing (2003), 890–901.
LIN T.-W., CHOU Y.-F.: A comparative study of zernike moments. Proceedings of the IEEE/WIC International Conference on Web Intelligence (2003).
An Efficient Algorithm for Fast Computation of Pseudo-Zernike Moments
Complex Zernike Moments