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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， vQE` c@^{  
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， 15RI(BN   
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？ $XtV8  
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ NO%|c|B|  
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function z = zernfun(n,m,r,theta,nflag) @|JPE%T   
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. aA!@;rR<yU  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N C8O7i[uc  
%   and angular frequency M, evaluated at positions (R,THETA) on the gogl[gHO  
%   unit circle.  N is a vector of positive integers (including 0), and EVby 9!  
%   M is a vector with the same number of elements as N.  Each element lU >)n  
%   k of M must be a positive integer, with possible values M(k) = -N(k) ) >-D={  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, f[w jur  
%   and THETA is a vector of angles.  R and THETA must have the same 89?3,k  
%   length.  The output Z is a matrix with one column for every (N,M) h/fb<jIP1  
%   pair, and one row for every (R,THETA) pair. ^*j[&:d  
% _CYmG"mY  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike :K a^  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), {3_Ffsg`  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral 4'7 v!I9  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, vUA)#z<  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized uk>q\j  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. LL6ON }  
% 6ba2^3GH  
%   The Zernike functions are an orthogonal basis on the unit circle. i:NJ>b  
%   They are used in disciplines such as astronomy, optics, and 0Te)s3X  
%   optometry to describe functions on a circular domain. !ds"88:5^  
% Q7OnhGA  
%   The following table lists the first 15 Zernike functions. QqT6P`0u  
% 3:z4M9f  
%       n    m    Zernike function           Normalization >*ha#PE  
%       -------------------------------------------------- s0`]!7D<  
%       0    0    1                                 1 ` :B  
%       1    1    r * cos(theta)                    2 3<Pyr-z h  
%       1   -1    r * sin(theta)                    2 H@OrX  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) _ cHV3cz  
%       2    0    (2*r^2 - 1)                    sqrt(3) bHlDm~5  
%       2    2    r^2 * sin(2*theta)             sqrt(6) a`GN@ 8  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) D{3 x}5  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) %<bG%V(  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) nATfmUN L  
%       3    3    r^3 * sin(3*theta)             sqrt(8) %^)JaEUC  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) ~ L i%  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 6O[wVaC1u  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) Oujlm|  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) !Sr0Im0  
%       4    4    r^4 * sin(4*theta)             sqrt(10) : p*ojl|  
%       -------------------------------------------------- #E~WVTO w  
% yScov)dp(  
%   Example 1: OL6xMToP  
% 1zEZ\G  
%       % Display the Zernike function Z(n=5,m=1) u" NIG  
%       x = -1:0.01:1; CzDR%vx  
%       [X,Y] = meshgrid(x,x); SBYMDKZ  
%       [theta,r] = cart2pol(X,Y); u3v6$CD?  
%       idx = r<=1; 3T.M?UG>  
%       z = nan(size(X)); 3Wtv+L7Br  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); X?k V1  
%       figure s5Bmv\e.i5  
%       pcolor(x,x,z), shading interp y:|Xg0Kp  
%       axis square, colorbar fuIv,lDA  
%       title('Zernike function Z_5^1(r,\theta)') e8ig[:B>+  
% # |*,zIYo  
%   Example 2: V?L$ ys  
% $'mB8 S  
%       % Display the first 10 Zernike functions wDC/w[4:  
%       x = -1:0.01:1; #Ot*jb1  
%       [X,Y] = meshgrid(x,x); %?9r(&  
%       [theta,r] = cart2pol(X,Y); *Yk8Mj^_h  
%       idx = r<=1; %JA&O  
%       z = nan(size(X)); & 4Iqm(  
%       n = [0  1  1  2  2  2  3  3  3  3]; 1p "EE~ v  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; +68K[s,FD  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; Cx3m\ \c  
%       y = zernfun(n,m,r(idx),theta(idx)); -aeo7C  
%       figure('Units','normalized') {-7yZ]OO$  
%       for k = 1:10 y:6'&`L  
%           z(idx) = y(:,k); ^*UfCoj9Z  
%           subplot(4,7,Nplot(k)) D A)0Y_  
%           pcolor(x,x,z), shading interp J7xT6Q=  
%           set(gca,'XTick',[],'YTick',[]) %F]9^C+  
%           axis square BQJ`vIa  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) EwBN+v;)  
%       end "VVR#H}{  
% _=^hnv  
%   See also ZERNPOL, ZERNFUN2. 5`{;hFl  
: R*^Izs=  
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%   Paul Fricker 11/13/2006 )D-.7m.v]  
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 )PoI~km  
% Check and prepare the inputs: jvQ+u L  
% ----------------------------- JE:n`l/p  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) !}Ou|r4_  
    error('zernfun:NMvectors','N and M must be vectors.') Xgth|C}k  
end /$.vHt 5nt  
"M#`y!__  
 HF=C8ZtlL  
if length(n)~=length(m) a4qpnr]0  
    error('zernfun:NMlength','N and M must be the same length.') 'GdlqbX(%  
end xS-nO_t 'E  
}br<2?y,  
 &*:)5F5  
n = n(:); x^#{2}4u  
m = m(:); qsRfG~Cg  
if any(mod(n-m,2)) C`T5d  
    error('zernfun:NMmultiplesof2', ... V7'x? pt  
          'All N and M must differ by multiples of 2 (including 0).') gsq[ 9  
end >,]e[/p  
*]| JX&  
 714nUA872  
if any(m>n) it|:P  
    error('zernfun:MlessthanN', ... vKOn7  
          'Each M must be less than or equal to its corresponding N.') b|@op>UZ  
end S^`9[$KH0  
ieLN;)Iy^  
 W9m[>-Ew  
if any( r>1 | r<0 ) H_f2:Za  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') 4k?JxA)  
end ./*,Thc  
^F0jI5j).  
 LuQ M$/i  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) PL_wa(}y]D  
    error('zernfun:RTHvector','R and THETA must be vectors.') e6xjlaKb  
end *_rGBW  
|nZB/YZt  
 v6Wf7)d/1  
r = r(:); 4^}PnU7z  
theta = theta(:); dQ~"b=  
length_r = length(r); sW3D ( n  
if length_r~=length(theta) N6 (w<b  
    error('zernfun:RTHlength', ... qa`(,iN  
          'The number of R- and THETA-values must be equal.') aYCzb7  
end kL2sJX+  
MCpK^7]k  
 ^M5uLm-_s  
% Check normalization: `rJ ~*7-  
% -------------------- gm$MEeC  
if nargin==5 && ischar(nflag) |Qm%G\oB?  
    isnorm = strcmpi(nflag,'norm'); QD VA*6F  
    if ~isnorm \gv x)S11  
        error('zernfun:normalization','Unrecognized normalization flag.') J|8YB3K,  
    end x&b-Na3Xi  
else !)3Su=*R  
    isnorm = false;  #X_M  
end 7+r5?h|  
k8>^dZub  
 :2gO) 'cD  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Mo|5)8_  
% Compute the Zernike Polynomials 1vudT&  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% b>._ r&.  
zb)SlR  
 F>R)~;Ja  
% Determine the required powers of r: E?o8'r  
% ----------------------------------- w.- i !Ls  
m_abs = abs(m); r(CL=[  
rpowers = []; \1_&?( pU  
for j = 1:length(n) [kkcV5I-  
    rpowers = [rpowers m_abs(j):2:n(j)]; {moNtzE;  
end gq &85([  
rpowers = unique(rpowers); ZWEzL$VWi  
,Gx=e!-N5  
 YZ P  
% Pre-compute the values of r raised to the required powers, *j:5  
% and compile them in a matrix: rWmi 'niu  
% ----------------------------- zz*[JIe  
if rpowers(1)==0 ;KN@v5`p  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 9DKB+K.1  
    rpowern = cat(2,rpowern{:}); r5wXuA,Um  
    rpowern = [ones(length_r,1) rpowern]; nEr, jd~f  
else  ?C\9lLX  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); Nuq/_x  
    rpowern = cat(2,rpowern{:}); jK1! \j  
end 8#B;nyGD1I  
th|Q NG  
 Sp:de,9@  
% Compute the values of the polynomials: ;r}<o?'RM  
% -------------------------------------- [wYQP6Cyy  
y = zeros(length_r,length(n)); IW*.B6Hw8  
for j = 1:length(n) &|*|  
    s = 0:(n(j)-m_abs(j))/2; 8G<.5!f7`N  
    pows = n(j):-2:m_abs(j); Pw.+DA  
    for k = length(s):-1:1 aQ\O ]gCE  
        p = (1-2*mod(s(k),2))* ... }$U6lh/Ep  
                   prod(2:(n(j)-s(k)))/              ... jX@9849@  
                   prod(2:s(k))/                     ... ,/9|j*9H  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... mj~CCokF{?  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); c?S402M}  
        idx = (pows(k)==rpowers); OD|&qsbL  
        y(:,j) = y(:,j) + p*rpowern(:,idx); '5\1uB PKW  
    end w+H=Xh4t  
     Lq^/Z4L  
    if isnorm @DR&e^Zz  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); [*v\X %+  
    end z ZQoY_UI  
end d0 az#Yg!  
% END: Compute the Zernike Polynomials d?.x./1[qi  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% B;Vl+}R  
,55`s#;  
 f^ qQ 5N  
% Compute the Zernike functions: RJM(+5xQ|  
% ------------------------------ XF6= xD  
idx_pos = m>0; 5N/;'ySAE_  
idx_neg = m<0; L_|Y_=r."  
In2D32"F  
 V>Cf 8>m  
z = y; eF@E|kK  
if any(idx_pos) P2kZi=0  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ]gZjV  
end g&V.o5jIhc  
if any(idx_neg) gK'MUZ()  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); {[+gM?  
end q[lqEc  
K(^x)w r-:  
 @@QU"8q  
% EOF zernfun /^"TMm   
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)  :A$6Y*s\  
]k Pco4  
DDE还是手动输入的呢？ z(ajR*\#  
$2.DZ  
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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