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jssylttc 2012-04-23 19:23

如何从zernike矩中提取出zernike系数啊

下面这个函数大家都不会陌生，计算zernike函数值的，并根据此可以还原出图像来， @uA=v/>+  
我输入10阶的n、m，r,theta为38025*1向量，最后得到的z是29525*10阶的矩阵， !|4fww  
这个，跟我们用zygo干涉仪直接拟合出的36项zernike系数，有何关系呢？ )S?.YCv?  
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ SB~HHx09  
~5`p/.L)ZD  
 UU\wP(f  
[0c7fH`8V  
 5rp,xk!  
function z = zernfun(n,m,r,theta,nflag)  -c%#Hd  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. cdd6*+E  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N HGycF|]2  
%   and angular frequency M, evaluated at positions (R,THETA) on the m q#8 [D  
%   unit circle.  N is a vector of positive integers (including 0), and RJ}%pA4I  
%   M is a vector with the same number of elements as N.  Each element xXb7/.*qE  
%   k of M must be a positive integer, with possible values M(k) = -N(k) a|}v?z\  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, 7`xeuK  
%   and THETA is a vector of angles.  R and THETA must have the same WAq)1gwN  
%   length.  The output Z is a matrix with one column for every (N,M) hk*@<ff  
%   pair, and one row for every (R,THETA) pair. ohEIr2  
% 7Eb | AR  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike %u=b_4K"j  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), QC\r|RXW  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral m8}c(GwcP  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, I{nrOb1G(  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized .)(5F45Wg  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. \d-H+t]  
% !LI 8Xk  
%   The Zernike functions are an orthogonal basis on the unit circle. @kst G3@  
%   They are used in disciplines such as astronomy, optics, and `@<>"ff#F  
%   optometry to describe functions on a circular domain. wWYo\WH'  
% t[e`wj+qz  
%   The following table lists the first 15 Zernike functions. m!Y4+KTwD`  
% idzc4jR6BT  
%       n    m    Zernike function           Normalization x*'2%3C~  
%       -------------------------------------------------- E;1QD/E$  
%       0    0    1                                 1 WRCf [5  
%       1    1    r * cos(theta)                    2 =Ju%3ptH0  
%       1   -1    r * sin(theta)                    2 :,Zs {\oI3  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) Jb;@'o6  
%       2    0    (2*r^2 - 1)                    sqrt(3) Z\[6 'R4.#  
%       2    2    r^2 * sin(2*theta)             sqrt(6) "J(T?|t  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) O'rz  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) x@*RF:\}  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) ,7:? Du}  
%       3    3    r^3 * sin(3*theta)             sqrt(8) jjT)3 c:J[  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) Cl>|*h+m  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 'Lu d=u{  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) m!- R}PQC  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) E"<-To  
%       4    4    r^4 * sin(4*theta)             sqrt(10) Mg~62u  
%       -------------------------------------------------- |S6L[Uo  
% zhZ!!b^6<  
%   Example 1: Mni@@W  
% >PS`;S!(  
%       % Display the Zernike function Z(n=5,m=1) 2F&VG|"  
%       x = -1:0.01:1; Xk>YiV",?  
%       [X,Y] = meshgrid(x,x); )+ (GE  
%       [theta,r] = cart2pol(X,Y); ]q`'l_O  
%       idx = r<=1; MA,7 |s  
%       z = nan(size(X)); ^ *1hz<  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); 'O^<i`8U]  
%       figure S*l=FRFI  
%       pcolor(x,x,z), shading interp #O1%k;BL  
%       axis square, colorbar dy.U;  
%       title('Zernike function Z_5^1(r,\theta)') _aP 2gH  
% 45 B |U  
%   Example 2: 'H8b+  
% GR,gCtG+L  
%       % Display the first 10 Zernike functions ' e:rL.  
%       x = -1:0.01:1; e3.q8r  
%       [X,Y] = meshgrid(x,x); &{e:6t  
%       [theta,r] = cart2pol(X,Y); .b|!FWHNS  
%       idx = r<=1; .+A2\F.^  
%       z = nan(size(X)); YZy%]i=1  
%       n = [0  1  1  2  2  2  3  3  3  3]; LjySO2  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; ' OXL'_Xl  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; v*JXrB&x  
%       y = zernfun(n,m,r(idx),theta(idx)); yvCX is  
%       figure('Units','normalized') ?_`X8Ok  
%       for k = 1:10 e ~ %=H 0n  
%           z(idx) = y(:,k); K3^2;j1F Q  
%           subplot(4,7,Nplot(k)) #_kV o3  
%           pcolor(x,x,z), shading interp Yn5a4  
%           set(gca,'XTick',[],'YTick',[]) Ex+E66bE  
%           axis square /5Tp)h|  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) Oj1B @QE  
%       end a}g <<{  
% [_-[S  
%   See also ZERNPOL, ZERNFUN2. $O^"O Q_@  
9P& \2/ {  
 ('6sW/F*ab  
%   Paul Fricker 11/13/2006 MZvxcr{x  
PUEEfq!%  
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 f {j`d&|  
% Check and prepare the inputs: gaU(ebsE  
% ----------------------------- ,uL}O]L  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) bScW<DZJ-  
    error('zernfun:NMvectors','N and M must be vectors.') a8 .x=j<  
end *?~&O.R"  
LMaY}m>  
 mvu$  
if length(n)~=length(m) &?*H`5#?G  
    error('zernfun:NMlength','N and M must be the same length.') i4\DSQJ  
end ~j yl  
Qe;R3D=T;  
 5Ve T8/7Q  
n = n(:); d`3>@*NR<  
m = m(:); YhO-ecN  
if any(mod(n-m,2)) | $D`*  
    error('zernfun:NMmultiplesof2', ... \t ^9UN  
          'All N and M must differ by multiples of 2 (including 0).') e=Q{CsP  
end !h70<Q^  
|3C5"R3ZGO  
 P"a9+ti+'  
if any(m>n) 5@^['S4%8*  
    error('zernfun:MlessthanN', ... fzr0dcNgM  
          'Each M must be less than or equal to its corresponding N.') P;K <P  
end lbm ,#  
J0 UF(  
 -"5r-qq*  
if any( r>1 | r<0 ) LLPbZ9q  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') /-Y.A<ieN8  
end CD pLV:  
pwmH(94$0  
 Gg3< }(  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) wZb7 7  
    error('zernfun:RTHvector','R and THETA must be vectors.') "$U!1  
end kqZ+e/o>O9  
s!?T$@a=  
 :xZ^Jq91  
r = r(:); u&r+ylbs I  
theta = theta(:); cm`x;[e6l  
length_r = length(r); 7;SI=  
if length_r~=length(theta) 'Omj-o'tn9  
    error('zernfun:RTHlength', ... W$`p ,$.n  
          'The number of R- and THETA-values must be equal.') -op(26:W<  
end  lx&;?QQ  
[ Mp8"  
 b$N&sZ  
% Check normalization: sW0<f& 3  
% -------------------- ]::g-&%Um  
if nargin==5 && ischar(nflag) h`pXUnEZ  
    isnorm = strcmpi(nflag,'norm'); b 2XUZ5  
    if ~isnorm p]x9hZ  
        error('zernfun:normalization','Unrecognized normalization flag.') ,XYtoZa  
    end gJ?Vk<hp  
else wg=-&-  
    isnorm = false; +t*Ks_V,*  
end CYZ0F5+t  
,_I#+XiXY  
 E\vW>g*W  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ~c${?uf   
% Compute the Zernike Polynomials BL H~`N3U  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ehyCAp0oI  
a$]i8AeG  
 L]{1@~E:q  
% Determine the required powers of r: &'oZ]}^ 0  
% ----------------------------------- ~ }?*v}  
m_abs = abs(m); &nfGRb  
rpowers = []; O^sOv!!RH/  
for j = 1:length(n) ObSRd$M  
    rpowers = [rpowers m_abs(j):2:n(j)]; 3oMhsQz~z  
end ;}4^WzmK^(  
rpowers = unique(rpowers); o>o! -uf  
3pjK`"Nmz\  
 y28 e=i  
% Pre-compute the values of r raised to the required powers, P:fcbfH+  
% and compile them in a matrix: yR[htD`  
% ----------------------------- N_R(i3c6U!  
if rpowers(1)==0 G$-[(eu -  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ^e(*{K;8  
    rpowern = cat(2,rpowern{:}); <L+y 6B  
    rpowern = [ones(length_r,1) rpowern]; /eY}0q%  
else i?B(I4a!G  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); S94S[j0D  
    rpowern = cat(2,rpowern{:}); 1XJLGMW,  
end eKW^\  
 I6rB_~]h  
 wOF";0EN  
% Compute the values of the polynomials: )=%TIkeF  
% -------------------------------------- _7b' i6-  
y = zeros(length_r,length(n)); i2X%xYv ^  
for j = 1:length(n) `} S; _g!  
    s = 0:(n(j)-m_abs(j))/2; Lb}$)AcC  
    pows = n(j):-2:m_abs(j); oP_}C[  
    for k = length(s):-1:1 XxLauJP K  
        p = (1-2*mod(s(k),2))* ... NM4b]>   
                   prod(2:(n(j)-s(k)))/              ... ayr CLv  
                   prod(2:s(k))/                     ... `XrF ,  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ?XB[awTD~  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); ]La~Bh6;m  
        idx = (pows(k)==rpowers); JXq l=/%  
        y(:,j) = y(:,j) + p*rpowern(:,idx); 6D/K=-   
    end M6XpauR-  
     c|<E~_ .w@  
    if isnorm v`Yj)  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); % 9} ?*U  
    end _p;=]#+c&  
end D]+]Br8  
% END: Compute the Zernike Polynomials FgnPh%[u  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% xP-\)d-.aN  
Eal*){"<,?  
 si#1sdR  
% Compute the Zernike functions: b E6bx6=u  
% ------------------------------ L)X[$:  
idx_pos = m>0; ZK5 wZU  
idx_neg = m<0; ?g9oiOhnG  
Dy|)u1?  
 /&+*X)#v  
z = y; x&*2R#Ai  
if any(idx_pos) 8 DPn5E#M1  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); C9^C4   
end  i)= \-C  
if any(idx_neg) v \dP  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); W9:(P  
end ;VS$xnZ  
2x!cblo  
 Zmz $ hr  
% EOF zernfun f"R'Q|7D  
phoenixzqy 2012-04-23 20:38
慢慢研究，这个专业性很强的。用的人又少。
sansummer 2012-04-27 10:22
这个太牛了，我目前只能把zygo中的zernike的36项参数带入到zemax中，但是我目前对其结果的可信性表示质疑，以后多交流啊
jssylttc 2012-05-14 11:28
sansummer:这个太牛了，我目前只能把zygo中的zernike的36项参数带入到zemax中，但是我目前对其结果的可信性表示质疑，以后多交流啊 (2012-04-27 10:22)  HL%|DCo  
Bhk@0\a  
DDE还是手动输入的呢？ !+ (H(,gI  
F#a'N c9  
zygo和zemax的zernike系数，类型对应好就没问题了吧
jssylttc 2012-05-14 11:37
顶顶·········
18257342135 2016-12-13 10:03
支持一下，慢慢研究
查看本帖完整版本: [-- 如何从zernike矩中提取出zernike系数啊 --] [-- top --]

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