Zernike polynomials

Zernike polynomials and functions (orthogonal basis on the unit circle).

Description

ZERNFUN.m and ZERNFUN2.m compute the Zernike functions Znm(r,theta). These functions, which form an orthogonal basis on the unit circle, are used in disciplines such as astronomy, optics, optometry, and ophthalmology to characterize functions and data on a circular domain. ZERNPOL.m computes the Zernike polynomials Znm(r), which are the radial portion of the Zernike functions.

A MATLAB Digest article describing the use of the Zernike functions for analyzing optics data (using a LASIK surgery data as an example) also is available, on the File Exchange as a PDF,

http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=19133&objectType=file

and in HTML at:

http://www.mathworks.com/company/newsletters/digest/2008/jan/zernike.html

• Acknowledgements: This submission has inspired the following:; Zernike decomposition, Zernike Polynomial Coefficients for a given Wavefront using Matrix Inversion in Matlab, zernike3
• MATLAB release: MATLAB 7.3 (R2006b)

[Jose Flores]

 

[Udai Kiran]

this the function i was looking for past few weeks

[shijun xiang]

Thanks

[hou xiaobing]

thank you very much

[ravi bakaraju]

thank u

[Bijan Nemati]

Very helpful, fast, and complete. Examples in the comments are very helpful too.

[smiled fisher]

 

[Dong Liu]

Why this error? Thank you!

??? Error: File: E:\zernike\zernpol.m Line: 151 Column: 25
"identifier" expected, "(" found.

[Mike lu]

very helpful!It will be perfect if include the zernike annular polynomials

[Alex Cht.]

Valuable work, thank you.

[sadegh sadeghian]

 

[sadeq sadeghian]

 

[hua housheng]

 

[Ping Su]

 

[Michael Black]

Looks promising but I'm having a problem duplicating the LASIK example...I copied the lasik image and edited out the the lasik portion -- but all I get from the following is a big red blob -- not a decent reconstructed image...anybody know what's wrong?
    image=imread('zk_fig2_w.jpg');
    figure(1);
    I=im2double(image);
    imagesc(image);
    % make grid coordinate matrices expressed in polar coordinates
    L = size(I,1);
    X = -1:2/(L-1):1;
    [x,y] = meshgrid(X);
    x=x(:);
    y=y(:);
    [theta,r] = cart2pol(x,y);
    % Compute the required degree and order values from n=0-7, inclusive
    N = [];
    M = [];
    for n=0:7
        N = [N n*ones(1,n+1)];
        M = [M -n:2:n];
    end
    is_in_circle = ( r <= 1);
    Z = zernfun(N,M,r(is_in_circle),theta(is_in_circle));
    a = Z\I(is_in_circle);
    % Reconstruct image using Zernike coefficients
    r=NaN(size(I));
    r(is_in_circle) = Z*a;
    % rescale to 0-255 to display image
    figure(2);
    r = im2uint8(im2double(r));
    imshow(r);

[Alex Kararg]

I am interested in extracting the 3D Zernike descriptors of 3D shapes? Do you happen to know if there is any Matlab code?

[Jeff]

Paul, Thanks very much for this code. It is very much appreciated.

I think, however, that you have a bug in how you are normalizing your Zernike polynomials. Noll's paper shows that for terms with m=0 the normalization term should be (n+1)**0.5 and for terms with m!=0 the normalization term should be (2*(n+1))**0.5. In the routine zernpol.m you are normalizing all terms (including those with m=0) by (2*(n+1))**0.5.

Perhaps you have already fixed this? I have just downloaded it on 1-21-10 from this site. The creation date for the file I'm looking at is on Nov 13, 2006 12:56:48 PM.

[Elia]

I am still becoming accustomed to MATLAB. When I run the zernfun.m by itself without any modifications, I get this error

??? Input argument "n" is undefined.
Error in ==> zernfun at 88
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ).

However, when I initialized n, m, r, & theta, I get this error

??? Maximum recursion limit of 500 reached. Use set(0,'RecursionLimit',N)
to change the limit. Be aware that exceeding your available stack space can
crash MATLAB and/or your computer.

Error in ==> meshgrid

Does anybody have any thoughts about making zernfun.m to work? Any help would be greatly appreciated.

[Bob]

Very great functions! But the normalization with the 'norm' - option seems to be wrong. I've changed in:

zernfun.m

% For the normalized polynomials the line 177:
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi);

% I have to replaced by:
if m(j)==0
        y(:,j) = y(:,j)*sqrt((n(j)+1));
else
        y(:,j) = y(:,j)*sqrt(2*(n(j)+1));
end

[Chauncey Graetzel]

I was running into problems when using my fitted terms in another program: the same coefficients did not give the same surface.
I found the cause: the cosine and sine terms are inverted in zernfun. M>0 should be a cosine, while M<0 should be a sine.

I've corrected this, starting at line 193

if any(idx_pos)
    z(:,idx_pos) = y(:,idx_pos).*cos(theta*m(idx_pos)');
end
if any(idx_neg)
    z(:,idx_neg) = y(:,idx_neg).*sin(theta*-m(idx_neg)');
end

% note the required sign change "-m" in the sine term

[Vic]

Thanks for the code! I have a question about the unit of the zernike polynomials generated by this code, is it in [number of waves] or [micrometer]?

Thanks again!

[Shahab]

Hi,

Does anybody know about the discrete Zernike Transform (DZT)?
I am so eager to implement that.

thank you in advance