Talk:Pseudo-Zernike polynomials

Hi,

In the definition of $D_{n,|m|,s}$ , i think that the term $(n-|m|-s+1)!$ should be replaced by $(n+|m|-s+1)!$ . Could someone else confirm there is an issue in the definition of $D_{n,|m|,s}$ ? J.F. 77.194.224.177 (talk) 15:12, 6 June 2008 (UTC)[reply]

There are quite a lot of errors in this page. The restriction on the (n,m) pairs for which the polynomials are defined should be not only that |m|<=n but also that n-m is even, the polynomial for R should be a sum over only even values of s in that range, the formula for the coefficients D is wrong, and most of the example polynomials need deleting or correcting.Fathead99 (talk) 15:00, 6 June 2011 (UTC)[reply]

"On image analysis by the methods of moments" states that unlike for the Zernike radial polynomials, which are defined for |m|<=n only where n-m is even, pseudo-Zernike radial polynomials are defined for |m|<=n with no additional conditions on n-m. Cafet (talk) 20:11, 13 December 2011 (UTC)[reply]

I'm not sure how to correct it, but the proper link for the refernce to "An Efficient Algorithm for Fast Computation of Pseudo-Zernike Moments" is: http://ir.canterbury.ac.nz/bitstream/10092/448/1/12584534_ivcnz01.pdf —Preceding unsigned comment added by 147.197.140.136 (talk) 09:39, 24 November 2009 (UTC)[reply]

Used in Optics ?

I am an optical engineer and I have never heard of anyone using these "pseudo" Zernike polynomials. We use the Zernike polynomials, which are not defined the same way have their own page on wikipedia: https://en.wikipedia.org/wiki/Zernike_polynomials. I think that assertion needs a reference. — Preceding unsigned comment added by 212.203.64.194 (talk) 16:31, 6 October 2016 (UTC)[reply]

Original Definition of the pseudo-Zernike Polynomials

The pseudo-Zernike polynomials is similar than the Zernike polynomials. The differences between them are two. These was derived by Bhatia and Wolf ^[1]. The first is that it must to eliminate the condition n − |m| = even regarding the Zernike polynomials. The second is that the radial polynomial equation changes slightly from the Zernike polynomials. Since the set of pseudo-Zernike orthogonal polynomials is analogous to that of Zernike polynomials, most of the previous discussion for the Zernike Moments(ZM) can be adapted to the case of pseudo-Zernike Moments (PZM).

References

^ https://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/article/abs/on-the-circle-polynomials-of-zernike-and-related-orthogonal-sets/962F0EA6CA54D87C9263B7FE6D22FE24#

[1] ttps://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/article/abs/on-the-circle-polynomials-of-zernike-and-related-orthogonal-sets/962F0EA6CA54D87C9263B7FE6D22FE24#

[1]