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Pseudo-Zernike polynomials

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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 shape descriptors.

Definition

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

where and orthogonality on the unit disk is given as

where the star means complex conjugation, and , , are the standard transformations between polar and Cartesian coordinates.

The radial polynomials are defined as[1]

with integer coefficients

Examples

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Examples are:

Moments

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The pseudo-Zernike Moments (PZM) of order and repetition are defined as

where , and takes on positive and negative integer values subject to .

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

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.[1]

See also

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References

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  1. ^ a b Teh, C.-H.; Chin, R. (1988). "On image analysis by the methods of moments". IEEE Transactions on Pattern Analysis and Machine Intelligence. 10 (4): 496–513. doi:10.1109/34.3913.