Pseudo-Zernike polynomials

In mathematics, 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.

Definition

They are an orthogonal set of complex-valued polynomialsdefined as :

$V\_\{nm\}(x,y)\; =\; R\_\{nm\}(x,y)e^\{jmarctan(frac\{y\}\{x\})\}$

where $x^2+y^2leq\; 1,\; ngeq\; 0,\; |m|leq\; n$ and orthogonality on the unit disk isgiven as:

$int\_0^\{2pi\}int\_0^\{infty\}\; [V\_\{nl\}(rcos\; heta,rsin\; heta)]\; ^*.\; [V\_\{mk\}(rcos\; heta,rsin\; heta)]\; drd\; heta\; =\; frac\{pi\}\{n+1\}delta\_\{mn\}delta\_\{kl\}$

The radial polynomials $\{R\_\{nm\}\}$ are defined as:

$R\_\{nm\}(x,y)\; =\; sum\_\{s=0\}^\{n-|mD\_\{n,|m|,s\}(x^2+y^2)^\{(n-s)/2\}$

where

$D\_\{n,|m|,s\}\; =\; (-1)^sfrac\{(2n+1-s)!\}\{s!(n-|m|-s)!(n-|m|-s+1)!\}$

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

$A\_\{nl\}=frac\{n+1\}\{pi\}int\_0^\{2pi\}int\_0^\{infty\}\; [V\_\{nl\}(rcos\; heta,rsin\; heta)]\; ^*f(rcos\; heta,rsin\; heta)rdrd\; heta$

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

The image function can be reconstructed by expansion of the Pseudo-Zernikecoefficients 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 shownto be more robust and less sensitive to image noise than the Zernike moments.

ee also

*Zernike polynomials

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).
# [http://www.cosc.canterbury.ac.nz/mukundan/ivcnz01.pdf An Efficient Algorithm for Fast Computation of Pseudo-Zernike Moments]
# [http://homepages.inf.ed.ac.uk/rbf/CVonline/LOCAL_COPIES/SHUTLER3/node11.html Complex Zernike Moments]