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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, ,& \&::R  
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, W? 6  
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? Z IGbwL  
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? 8Pnqmjjj  
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function z = zernfun(n,m,r,theta,nflag) {U+9,6.`  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. _Oaso >  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N "=v J }  
%   and angular frequency M, evaluated at positions (R,THETA) on the =_H*fhXS  
%   unit circle.  N is a vector of positive integers (including 0), and T{v<  
%   M is a vector with the same number of elements as N.  Each element M25z<Y  
%   k of M must be a positive integer, with possible values M(k) = -N(k) %3@RZe  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, '6Z/-V4k  
%   and THETA is a vector of angles.  R and THETA must have the same D; 35@gtj  
%   length.  The output Z is a matrix with one column for every (N,M) :a^,Ei-&  
%   pair, and one row for every (R,THETA) pair. =":V WHf  
% k*UR# z(I  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike w$<fSe7  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), aF4V|?+  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral sL[(cX?;2  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, Br.$L  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized R;Ix<y{U  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 2=UTH% 1D  
% gC,0+Y~  
%   The Zernike functions are an orthogonal basis on the unit circle. V }r_   
%   They are used in disciplines such as astronomy, optics, and {H7$uiq3:B  
%   optometry to describe functions on a circular domain. X G@>1/  
% v'2OHb#  
%   The following table lists the first 15 Zernike functions. CH R?i1e  
% $4ZDT]n  
%       n    m    Zernike function           Normalization   _c7  
%       -------------------------------------------------- H&>>]DD  
%       0    0    1                                 1 gWU(uBS  
%       1    1    r * cos(theta)                    2 yrv SbqR  
%       1   -1    r * sin(theta)                    2 E)Zd{9A5)  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) e^l+ #^fR  
%       2    0    (2*r^2 - 1)                    sqrt(3) O.40^u~  
%       2    2    r^2 * sin(2*theta)             sqrt(6) ]6c2[r?g{  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) >=q!!'$:  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) dQ2i{A"BKz  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) x.4)p6  
%       3    3    r^3 * sin(3*theta)             sqrt(8) @0@'6J04  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) GT`<jzAiQ  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) .qU%SmQ^  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) 6;=wuoJi  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) !E.l yz  
%       4    4    r^4 * sin(4*theta)             sqrt(10) `5 v51TpH  
%       -------------------------------------------------- ~>-;(YU"t  
% PFn[[~5V  
%   Example 1: }Us$y0W\  
% `LHfAXKN  
%       % Display the Zernike function Z(n=5,m=1) EpS8,[w  
%       x = -1:0.01:1; =rtA{g$)+  
%       [X,Y] = meshgrid(x,x); +H^V},dBp!  
%       [theta,r] = cart2pol(X,Y); $4~}_phi  
%       idx = r<=1; M&\?)yG  
%       z = nan(size(X)); j!8+|eA kk  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); s$y#Ufz  
%       figure Cot\i\]jv  
%       pcolor(x,x,z), shading interp PHL@1K{)  
%       axis square, colorbar J,M5<s[Xqt  
%       title('Zernike function Z_5^1(r,\theta)') IF?B`TmZ  
% aiX;D/t?  
%   Example 2: O?J:+L(  
% qA\kx#v]P  
%       % Display the first 10 Zernike functions -v+^x`HR  
%       x = -1:0.01:1; pxnUe1=  
%       [X,Y] = meshgrid(x,x); Y,Zv0-"  
%       [theta,r] = cart2pol(X,Y); )PATz #  
%       idx = r<=1; CH+&  
%       z = nan(size(X)); 7wEG<,D  
%       n = [0  1  1  2  2  2  3  3  3  3]; Jt, 4@  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; HaN _}UMP  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; DczF0Ow  
%       y = zernfun(n,m,r(idx),theta(idx)); M[N.H9  
%       figure('Units','normalized') eu|q {p  
%       for k = 1:10 iBW6<2@oZF  
%           z(idx) = y(:,k); J0W).mD_H  
%           subplot(4,7,Nplot(k)) g~D6.OZU  
%           pcolor(x,x,z), shading interp w=>mG-  
%           set(gca,'XTick',[],'YTick',[]) s ^@Cq=  
%           axis square (eE}W~Z  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) /<pQ!'/G  
%       end zi[M{bm  
% =v=!x  
%   See also ZERNPOL, ZERNFUN2. ]<z(Rmn`Q  
+( (31l  
\ OINzfbr  
%   Paul Fricker 11/13/2006 (SVr>|Db  
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=0yJ2[R7Do  
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{Y TF]J $  
% Check and prepare the inputs: nv Gd:]Z  
% ----------------------------- O +}EE^*a  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Y rnqi-P  
    error('zernfun:NMvectors','N and M must be vectors.') &V{,D))6[  
end 9yAu<a  
(,y/nc=GN  
7f,W zvV  
if length(n)~=length(m) qyfxTQ5  
    error('zernfun:NMlength','N and M must be the same length.') * %BI*p  
end R*C+Yk)Tkt  
" CoR?[,x  
z5'nS&x  
n = n(:);  P;/wb /  
m = m(:); C>VZf,JE1  
if any(mod(n-m,2)) 4x=Y9w0?8  
    error('zernfun:NMmultiplesof2', ... 6+#cyKj  
          'All N and M must differ by multiples of 2 (including 0).') *lO+^\HXD  
end ?tQv|x  
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if any(m>n) =6YO!B>7  
    error('zernfun:MlessthanN', ... n9-[z2n  
          'Each M must be less than or equal to its corresponding N.') <ft9B05*  
end Y\\nJuJo  
H2cc).8"  
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if any( r>1 | r<0 ) =(\!,S'  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') X#<Sv>c^  
end 6LQO>k  
>L4$DKO  
bOKNWI   
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) _3[BS9  
    error('zernfun:RTHvector','R and THETA must be vectors.') FR"^?z?}p  
end Q6>( Z  
s$6#3%h  
u;}B4Rx  
r = r(:); tn1aH +  
theta = theta(:); RfRaWbn  
length_r = length(r); {NDP}UATw  
if length_r~=length(theta) _"V0vV   
    error('zernfun:RTHlength', ... rd{( E  
          'The number of R- and THETA-values must be equal.') wv-8\)oA  
end !o!04_  
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;Vt u8f  
% Check normalization: N/VIP0Kb  
% -------------------- 1[]cMyV  
if nargin==5 && ischar(nflag) 4[q * 7m  
    isnorm = strcmpi(nflag,'norm'); =T]OYk  
    if ~isnorm &@-glF5  
        error('zernfun:normalization','Unrecognized normalization flag.') 'h6RZKG T  
    end gId+hxFa:r  
else V  ""  
    isnorm = false; oMbCljUC  
end = xk@Q7$  
p3'+"sFU  
Q<pM tW  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  .@Cshj  
% Compute the Zernike Polynomials  tS7u#YMh  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% rge/jE,^~Z  
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M<Mr (z  
% Determine the required powers of r: 9iE66N>z  
% ----------------------------------- =Wa\yBj_;m  
m_abs = abs(m); L?fv5 S3  
rpowers = []; (1^(V)@  
for j = 1:length(n) -tQ|&fl  
    rpowers = [rpowers m_abs(j):2:n(j)]; i}19$x.D`  
end -E7\ .K3  
rpowers = unique(rpowers); T_WQzEL^  
}UrtDXhA  
|.A>0-']M  
% Pre-compute the values of r raised to the required powers, Qp ,l>k  
% and compile them in a matrix: vkK+ C~"  
% ----------------------------- 0bE_iu>f'  
if rpowers(1)==0 x3Uv&  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ?x @khzk  
    rpowern = cat(2,rpowern{:}); )[1m$>  
    rpowern = [ones(length_r,1) rpowern]; OBZj-`fqJ  
else "^H+A-R[  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); &%lhov  
    rpowern = cat(2,rpowern{:}); $H^6I8>  
end QVpZA,  
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o2~x'*A0I  
% Compute the values of the polynomials: FyEl@ }W  
% -------------------------------------- uOQ5.S+  
y = zeros(length_r,length(n)); CS/-:>s%  
for j = 1:length(n) TI332,eL  
    s = 0:(n(j)-m_abs(j))/2; NmQ]qv  
    pows = n(j):-2:m_abs(j); ($wYaw z  
    for k = length(s):-1:1 RC 48e._t  
        p = (1-2*mod(s(k),2))* ... 3jNcL{  
                   prod(2:(n(j)-s(k)))/              ... m"*:XfOL  
                   prod(2:s(k))/                     ... ezn>3?S  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 7XNfH@  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); X'c5s~9  
        idx = (pows(k)==rpowers); [3.rG!Na  
        y(:,j) = y(:,j) + p*rpowern(:,idx); OjN]mp-q  
    end jnTl%aQYc  
     89paR[  
    if isnorm {x8`gP\H  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); +K?h]v]%  
    end F,sT[C  
end 4Qv|Z+$i  
% END: Compute the Zernike Polynomials i"'k|TGW^  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% EVf'1^f  
TDoYp  
R/#*~tPi8  
% Compute the Zernike functions: 7,1idY%cy  
% ------------------------------ `G'V9Xs(  
idx_pos = m>0; Ur`v*LT}~  
idx_neg = m<0; 3 *G=U  
Lg^m?~{  
mT.F$Y9  
z = y; <n>< A+D  
if any(idx_pos) ct  ZW7  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); % ;<FfS  
end 0^m02\Li  
if any(idx_neg) /$n${M5!  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); E|>I/!{u7`  
end |3i~?] A  
"CaVT7L  
|0&S>%=  
% EOF zernfun C>+UZ  
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)  Ad"::&&Wk  
-du+iOe?  
DDE还是手动输入的呢? {_ #   
S4|)N,#  
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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