Talk:Zernike polynomials

Self-promoting citations

In the section Application under breast masses paragraph, this user Amir has been adding citations to promote his work. I don't find it disturbing as much as sorry, but really what's Wikipedia stand on such activities? --A 11:58, 27 July 2012 (UTC)

Visual representation

{{reqdiagram}}

well I just was wondering if someone could give a visual representation of the polynome. In mathworld the expression of the first polynome are given.

It has been used in machine vision to obtain invariant features (rotation and translation and reflexion invariant) under the name of Zernike moment. That why I would be interested in being able to visualize them to understand from where these invariance properties are coming from. They are apparently good because the lower moments (under degree 4) are alledgedly robust to noise and they provide low information redundancy.

There are some expansions at Aberration in optical systems —The preceding unsigned comment was added by 15.235.249.72 (talk) 23:02, August 21, 2007 (UTC)

I have added some plots of the polymials, but none is done for the invariance properties. Rocchini (talk) 07:37, 7 May 2008 (UTC)

Picture

Zernike polynomials are better displayed using a pyramid scheme, like here. This is because the number of possible variations increase with polynomial order. The picture should be replaced. --Anna Lincoln (talk) 11:19, 10 September 2009 (UTC)

I made it into the pyramid shape, though Wikimedia (curse you) didn't let me update the file, so I had to re-upload it under a different name. --Zom-B (talk) 22:22, 30 September 2009 (UTC)

GUYS if anyone could explain this topic to a layman it would be highly apreciated . Some say inteligence is the ability convay complex terms to a layman ... think about it .. can you do it ? moriszen — Preceding unsigned comment added by 37.19.121.12 (talk) 22:43, 17 May 2012 (UTC)

Image accurate?

user:Neurotype raised the issue whether the discs in the image of even base are accurate or if the colors should be inverted.

Original comment by Neurotype: "(I am worried that this image may be wrong, and that the colours should be reversed for polynomials with base 2 and base 4.)"

I didn't make this picture (although I piramidyzed it), but I did implement the algorithm, and the results of my algorithm all agree with the discs in the image.

Can someone else verify this? Zom-B (talk) 11:25, 8 September 2010 (UTC)

Supposed that the x-axis points to the right and the y-axis up, which is the expected orientation because the image does not tell us that an unusual coordinate setup is used, some of the colors are wrong. Blue is obviously +1, as seen in Z_0^0. So in Z_2^(-2), the first quadrant where we see red, phi is around 45 degrees. In this area, Z_2^(-2) is rho^2*sin(+2*phi) , and both rho and the sin are positive. So the colors are reversed in that circle compared to the formulas. Image discussion R. J. Mathar (talk) 17:36, 8 September 2010 (UTC)

Oops sorry, I didn't realize these talk pages exist or I would have written here... I assumed that red was higher valued than blue, and first noticed a problem with Z_2^(0). I haven't tried running the code on the site but have written my own and it doesn't match up, instead looks like this: http://www.alpao.fr/al/products_DM241.html --Neurotype (talk) 19:29, 8 September 2010 (UTC)

I re-did the image adding the axes and colourbar to prevent confusion, and I added the Zernike indices in m,n- and Noll-format. I used a colour-blind compatible red-blue scheme for amplitude. 2pem (talk) 14:28, 19 November 2012 (UTC)

The Noll indices Z_2 and Z_3 are swapped and do not correspond to the respectively shown images. See table "Zernike polynomials": Noll index 2 is (n,m)=(1,1), tilt in X direction (i.e., around Y axis); Noll index 3 is (n,m)=(1,-1), tilt in Y direction (i.e., around X axis). -- Please swap Noll indices 2 and 3.

Starl8gazer (talk) 16:59, 2 January 2014 (UTC)

The images are still wrong. First of all, red is commonly larger than blue, as in the jet color map. The first image is blue because it has only one value, and the software than shows the lowest color. (This is what usually happens.) No matter the orientation, according to the formula, the first three polynomials are 1, y, and x. What's shown is 1, -y, x (or y, -x). Nschloe (talk) 16:17, 5 June 2020 (UTC)

The July 9, 2020 picture edits by Nschloe are a step backward. The updated picture does not list the Zernike indices. As a result, this picture is very difficult to understand and use as a reference. Removing the labels may remove any chance of inaccurate labels, but it also makes the figure less useful. — Preceding unsigned comment added by 67.1.12.218 (talk) 17:31, 14 November 2020 (UTC)

Would this help?

Hi,

I have (almost) no idea how to add it to the wiki-page in the right way... but I have a nice animated gif (both in color and black/white), separate wavefronts, the general equations and the specific ones up to Z64.

If anyone could add it for me, I'd appreciate (but with source-mentioning of course ;-) Take a look at my Zernike page: http://www.debrouwer.org/zernike/

Cheers, Steven (zernike -at- debrouwer.org) — Preceding unsigned comment added by 77.248.193.246 (talk) 22:02, 21 May 2012 (UTC)

Zernike polynomials examples section error

The Zernike polynomials in example section are incoherent with the radial polynomials, also in example section. Looking into the reference I found this text:

“Because of this normalization difference, the polynomials used in this paper are technically a modified set of Zernike polynomials. For convenience, in this paper the modified Zernike polynomials are simply called Zernike polynomials.”

Could someone confirm this? What could I do? — Preceding unsigned comment added by Albrosa (talk • contribs) 16:17, 27 November 2012 (UTC)

-- yes, there is a normalization factor, that has not been mentioned before. probably these polynomials form an orthonormal set and not just orthogonal. The factor takes the shape \sqrt(2(n+1)/(1+\delta_{m,0}))

— Preceding unsigned comment added by 2001:638:902:2002:91A0:42A6:958:E646 (talk) 13:08, 8 May 2013 (UTC)

Indices

Brief internet research just yielded evidence that the Noll scheme is only one of several single-index schemes in use, and probably not even the most common one. A list with an explanation, or at least some references, would be nice. The ones I've found referenced on the internet (but not all of them documented) are ANSI, Noll, Zemax/Fringe, and Zygo. András (talk) 09:05, 10 September 2013 (UTC)

I've added the OSA and ANSI single-indexing scheme here. FerrousCathode (talk) 23:39, 18 December 2017 (UTC)

Which ANSI standard is meant? There is no reference in the article. — Preceding unsigned comment added by 109.233.145.74 (talk) 07:00, 11 December 2019 (UTC)

Plain English

This article needs to be rewritten - in its present form it is virtually incomprehensible to anyone without a degree in physics or mathematics. I appreciate that this is a highly advanced subject which requires considerable background knowledge but that is the challenge of Wikipedia - to provide such knowledge in a manner that a reasonably intelligent but non expert reader could understand. I think this article, though probably succeeding in providing the information fails completely in providing it in a comprehensible manner. — Preceding unsigned comment added by 151.224.96.247 (talk) 16:32, 14 August 2015 (UTC)

Error in Summation Limit of the Radial Formula?

The upper limit of the summation given in the Definitions section is shown as (n-m)/2. This does not produce the results shown in the Examples section when, for instance, (m,n) is (1,-1), (2,-2), (3,-3), (3,-1), etc. Nor does it seem consistent with the symmetry in m that is explained throughout the article. Changing the upper limit to (n- Abs(m))/2 would seem to correct the situation. --m.w. — Preceding unsigned comment added by 2600:1700:BFA1:8690:FCA2:1382:48A8:AB7B (talk) 02:19, 20 May 2018 (UTC)

The definition sections says that n and m are non-negative integers. So according to the KISS principle there is no need to introduce the absolute value of m in the formulas. - R. J. Mathar (talk) 10:02, 11 February 2019 (UTC)

m,n are always >= 0. But m' (m prime) is a different notation where m can be negative. In the article it appears that if the formula starts with R then m,n are positive and if it starts with Z then (m',n) (m prime) notation is used where m' is negative for "odd" polynomials. It looks like m=abs(m') Gr5555 (talk) 17:17, 20 January 2021 (UTC)

Page vandalized? Change in formula

An anonymous editor with an ip address near Amsterdam Netherlands changed the multipliers in equation for R35 changing the multiplies of 5 and 4 to 6 and 5. I don't know if this was an honest mistake or a jokester vandalizing. I triple checked the formula and checked a few other of the formulas but "5 and 4" are the correct multipliers. I restored the previous page. Hopefully it was an honest mistake. Gr5555 (talk) 17:01, 20 January 2021 (UTC)

Incomplete Sentence?

The sentence "Orthogonality in the angular part is represented by the elementary [...]" in Section "Properties," subsection "Orthogonality" seems to be either incomplete ("elementary" what?) or incorrect (perhaps "integrals" instead of "elementary"?). Granted that the integrals that follow may be indeed described as "elementary," but I at best awkward as it is. Would it be OK if I replaced "elementary" with "integrals"? anonymous (talk) 18:38, 9 April 2021 (UTC)

Fringe/University of Arizona Indices

The formula for j gives fractional values when i = 0 because sgn i = 0 in this case. The sgn function should probably be replaced by another function that is the same as sgn except it is 1 when its argument is 0. Example: n=6, i = 0 gives 33/2 not 16.

Additionally, the Fringe/University of Arizona indices range from 1 to 37 and this is not clear in the article. See R.W. Gray, PhD Dissertation, University of Rochester (2015), pp. 37, 39-40..

How do I delete this comment? I was making a stupid math error...

Mapping of Zernike polynomials and Fringe indices

Mapping of Zernike polynomials and Fringe indices

$\operatorname {j} \left(n,\ m\right)=\left(1+{\frac {n+\left|m\right|}{2}}\right)^{2}-2\left|m\right|+{\begin{cases}0\leq m:\ 0\\0>m:\ 1\end{cases}}$

$\operatorname {g} \left(j\right)=\left\lceil {\sqrt {j}}-1\right\rceil$

$\operatorname {n} \left(j\right)=\operatorname {g} \left(j\right)-1+\left\lceil {\frac {j-\operatorname {g} \left(j\right)^{2}}{2}}\right\rceil$

$\operatorname {m} \left(j\right)=\left(-1\right)^{\left(j-\operatorname {g} \left(j\right)^{2}\right)\mathbin {\%} 2+1}\left(2\operatorname {g} \left(j\right)-\operatorname {n} \left(j\right)\right)$

ISO 14999-2

https://oeis.org/A375510 — Preceding unsigned comment added by Asebian (talk • contribs) 11:20, 18 September 2024 (UTC)

Asebian (talk) 22:53, 21 August 2024 (UTC)

Mapping of Zernike polynomials and Noll indices

Mapping of Zernike polynomials and ANSI / OSA indices

$\operatorname {j} \left(n,\ m\right)=n{\frac {n+1}{2}}+\left|m\right|+{\begin{cases}n\mathbin {\%} 4\leq 1:{\begin{cases}m>0:\ 0\\m\leq 0:\ 1\end{cases}}\\n\mathbin {\%} 4>1:{\begin{cases}m<0:\ 0\\m\geq 0:\ 1\end{cases}}\end{cases}}$

$\operatorname {n} \left(j\right)=\left\lfloor {\frac {{\sqrt {8\left(j-1\right)+1}}-1}{2}}\right\rfloor$

$\operatorname {m} \left(j\right)=\left(-1\right)^{j}\left(\operatorname {n} \left(j\right)\mathbin {\%} 2+2\left\lfloor {\frac {j-\operatorname {n} \left(j\right){\frac {\operatorname {n} \left(j\right)+1}{2}}-1+\left(\operatorname {n} \left(j\right)+1\right)\mathbin {\%} 2}{2}}\right\rfloor \right)$

ANSI Z80.28
ISO 24157

Asebian (talk) 23:54, 21 August 2024 (UTC)

Mapping of Zernike polynomials and Phasics indices

Mapping of Zernike polynomials and Phasics indices

$\operatorname {j} \left(n,m\right)=n{\frac {n+1}{2}}+\left|m\right|+{\begin{cases}m>0:\ 0\\m\leq 0:\ 1\end{cases}}$

$\operatorname {n} \left(j\right)=\left\lfloor {\frac {{\sqrt {8\left(j-1\right)+1}}-1}{2}}\right\rfloor$

$\operatorname {r} \left(j\right)=j-\operatorname {n} \left(j\right){\frac {\operatorname {n} \left(j\right)+1}{2}}$

$\operatorname {m} \left(j\right)=\left(-1\right)^{\operatorname {n} \left(j\right)-\operatorname {r} \left(j\right)}\left(\operatorname {n} \left(j\right)\mathbin {\%2} +2\left\lfloor {\frac {\operatorname {r} \left(j\right)-1+\left(\operatorname {n} \left(j\right)+1\right)\mathbin {\%} 2}{2}}\right\rfloor \right)$

OEIS: A064789

Asebian (talk) 00:23, 22 August 2024 (UTC) Asebian

Mapping of Zernike polynomials and Wyant indices

Mapping of Zernike polynomials and Wyant indices

$\operatorname {j} \left(n,m\right)=\left(1+{\frac {n+\left|m\right|}{2}}\right)^{2}-2\left|m\right|-{\begin{cases}0\leq m:\ 1\\0>m:\ 0\end{cases}}$

$\operatorname {g} \left(j\right)=\left\lceil {\sqrt {j+1}}-1\right\rceil$

$\operatorname {n} \left(j\right)=\operatorname {g} \left(j\right)-1+\left\lceil {\frac {j+1-\operatorname {g} \left(j\right)^{2}}{2}}\right\rceil$

${m}\left(j\right)=\left(-1\right)^{\left(j+1-\operatorname {g} \left(j\right)^{2}\right)\mathbin {\%} 2+1}\left(2\operatorname {g} \left(j\right)-\operatorname {n} \left(j\right)\right)$

(talk) 15:08, 22 August 2024 (UTC)

— Preceding unsigned comment added by Asebian (talk • contribs) 11:16, 23 August 2024 (UTC)

Mapping of Zernike polynomials and OSA/ANSI/ISO indices

Mapping of Zernike polynomials and ANSI / OSA indices

$\operatorname {n} \left(j\right)=\left\lfloor {\frac {{\sqrt {8j+1}}-1}{2}}\right\rfloor$

$\operatorname {m} \left(j\right)=2j-\operatorname {n} \left(j\right)\left(\operatorname {n} \left(j\right)+2\right)$

Asebian (talk) 15:42, 22 August 2024 (UTC)

Contour line figures

• blue:-1; gray:0; red: +1; isoline distance: 0.05
• 
  $Z_{0}^{0}$
• 
  $Z_{1}^{-1}$
• 
  $Z_{1}^{1}$
• 
  $Z_{2}^{-2}$
• 
  $Z_{2}^{0}$
• 
  $Z_{2}^{2}$
• 
  $Z_{3}^{-3}$
• 
  $Z_{3}^{-1}$
• 
  $Z_{3}^{1}$
• 
  $Z_{3}^{3}$
• 
  $Z_{4}^{-4}$
• 
  $Z_{4}^{-2}$
• 
  $Z_{4}^{0}$
• 
  $Z_{4}^{2}$
• 
  $Z_{4}^{4}$
• 
  $Z_{5}^{-5}$
• 
  $Z_{5}^{-3}$
• 
  $Z_{5}^{-1}$
• 
  $Z_{5}^{1}$
• 
  $Z_{5}^{3}$
• 
  $Z_{5}^{5}$
• 
  $Z_{6}^{-6}$
• 
  $Z_{6}^{-4}$
• 
  $Z_{6}^{-2}$
• 
  $Z_{6}^{0}$
• 
  $Z_{6}^{2}$
• 
  $Z_{6}^{4}$
• 
  $Z_{6}^{6}$

Asebian (talk) 22:31, 23 August 2024 (UTC)