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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, L"akV,w4p  
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, MW^,l=kqW)  
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? ^nYS @  
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? !?o661+b  
v^a. b  
8()L}@y  
*.UM[Wo  
\)=X=yn2  
function z = zernfun(n,m,r,theta,nflag) yE(>R(^  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. s 9,?"\0Zm  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N rTiW&#  
%   and angular frequency M, evaluated at positions (R,THETA) on the %8)GuxG*  
%   unit circle.  N is a vector of positive integers (including 0), and "0F =txduS  
%   M is a vector with the same number of elements as N.  Each element U}55;4^LX  
%   k of M must be a positive integer, with possible values M(k) = -N(k) aD aQ 7i  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, <Q06<{]R8  
%   and THETA is a vector of angles.  R and THETA must have the same +)#d+@-  
%   length.  The output Z is a matrix with one column for every (N,M) MVW2 %6  
%   pair, and one row for every (R,THETA) pair. !4 4)=xW  
% =gCv`SFW  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike J 00%,Ju_  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), =rV*iLy  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral =y; tOdj  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, QfuKpcT &  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized NJG-~ w  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. AR i_m  
% P#/k5]g  
%   The Zernike functions are an orthogonal basis on the unit circle. #<X+)B6t  
%   They are used in disciplines such as astronomy, optics, and k#8,:B2  
%   optometry to describe functions on a circular domain. FnN@W^/z  
% qm-G=EX  
%   The following table lists the first 15 Zernike functions. aHosu=NK  
% Iz/o|o]#  
%       n    m    Zernike function           Normalization #{)=%5=c  
%       -------------------------------------------------- j$h.V#1z  
%       0    0    1                                 1 3%?01$k  
%       1    1    r * cos(theta)                    2 J G xuB*}  
%       1   -1    r * sin(theta)                    2 *]Nd I  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) U[/k=}76  
%       2    0    (2*r^2 - 1)                    sqrt(3) sgdxr!1?y  
%       2    2    r^2 * sin(2*theta)             sqrt(6) -hav/7g  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) \$Xo5f<  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) cD&53FPXC  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) 'u }|~u?m  
%       3    3    r^3 * sin(3*theta)             sqrt(8) F6*n,[5(  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) b !FX]d1~k  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) c <8s \2  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) @EZ@X/8{&  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ^EGe%Fq*x]  
%       4    4    r^4 * sin(4*theta)             sqrt(10) D2o,K&V  
%       -------------------------------------------------- YGP.LR7  
% 9Xb,Swo~  
%   Example 1: isaDIl;L/  
% ) -+u8#  
%       % Display the Zernike function Z(n=5,m=1) z;6 Tp  
%       x = -1:0.01:1; jM8e2z3  
%       [X,Y] = meshgrid(x,x); 8A{n9>jrb  
%       [theta,r] = cart2pol(X,Y); hqW4.|&\c  
%       idx = r<=1; 5mwtlC':l?  
%       z = nan(size(X)); p\]Mf#B  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); R}MdBE  
%       figure fZK&h.  
%       pcolor(x,x,z), shading interp 4,CQJ  
%       axis square, colorbar hj@< wU  
%       title('Zernike function Z_5^1(r,\theta)') P?GHcq$\  
% 1Wd?AyTY,  
%   Example 2: QiB ^U^f  
% <aJdm!6  
%       % Display the first 10 Zernike functions { -*+G]  
%       x = -1:0.01:1; E/mp.f2!  
%       [X,Y] = meshgrid(x,x); D_oGhQYY4  
%       [theta,r] = cart2pol(X,Y); cn&\q.!fh  
%       idx = r<=1; _-aQ.p ?T  
%       z = nan(size(X)); lub(chCE[  
%       n = [0  1  1  2  2  2  3  3  3  3]; QXZjsa_|  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3];  gBQK  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; HvSKR1wL\  
%       y = zernfun(n,m,r(idx),theta(idx)); u_[^gS7  
%       figure('Units','normalized') FB {4& ;  
%       for k = 1:10 yyke"D  
%           z(idx) = y(:,k); f/t1@d!  
%           subplot(4,7,Nplot(k)) <11pk  
%           pcolor(x,x,z), shading interp va \ 5  
%           set(gca,'XTick',[],'YTick',[]) iM;7V*u  
%           axis square O,(p><k$/  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) o<@b]ukl&  
%       end  PZZTRgVc  
% o@TxDG  
%   See also ZERNPOL, ZERNFUN2. 1"J\iwN3  
N1rBpt  
Fy!u xT-\  
%   Paul Fricker 11/13/2006 R/8>^6  
t]?u<KD<  
'f0*~Wq|  
B0Ql1x#x  
V?U->0>Z4  
% Check and prepare the inputs: gJn|G#!  
% ----------------------------- `$j"nP F_  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) CAg\-*P|  
    error('zernfun:NMvectors','N and M must be vectors.') MQc|j'vEY  
end .]+Z<5Fo  
:A%|'HxH3  
Vy-N3L  
if length(n)~=length(m) d<mj=V@bd  
    error('zernfun:NMlength','N and M must be the same length.') !~5;Jb>s[/  
end L'k )  
(=:9pbP  
X;JptF^  
n = n(:); b X.S`  
m = m(:); 9Z}Y2:l'  
if any(mod(n-m,2)) 4qq+7B  
    error('zernfun:NMmultiplesof2', ... kL;sA'I:S  
          'All N and M must differ by multiples of 2 (including 0).') &oJ=   
end \Z0-o&;w  
tRU+6D <w  
P_11N9C  
if any(m>n) 7FL!([S5i  
    error('zernfun:MlessthanN', ... s5 ? 1w   
          'Each M must be less than or equal to its corresponding N.') PLDg'4DMg  
end rUjK1A{V  
SP][xdN7  
f\CJ |tKX  
if any( r>1 | r<0 ) hES_JbX}]  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') J7:VRf|,?(  
end Vae}:8'}  
H8d%_jCr  
3|4jS"t{f  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) PveY8[i  
    error('zernfun:RTHvector','R and THETA must be vectors.')  qW8sJ=  
end f0rM 4"1  
rEwEdyK  
61e)SIRz9I  
r = r(:); -*8|J;  
theta = theta(:); ?+-uF }  
length_r = length(r); @~pIyy\_  
if length_r~=length(theta) MY>mP  
    error('zernfun:RTHlength', ... HGqT"N Jr  
          'The number of R- and THETA-values must be equal.') 2pR+2p`  
end y8"8QH  
ut8v&i1?  
&NbhQY`k  
% Check normalization: Q)eYJP=W  
% -------------------- -xg$qvK  
if nargin==5 && ischar(nflag) Db"jzMW.  
    isnorm = strcmpi(nflag,'norm'); ,l-tLc  
    if ~isnorm x6Q,$B  
        error('zernfun:normalization','Unrecognized normalization flag.') (> O'^W\3p  
    end yhzC 9nTH  
else ziUEA>m */  
    isnorm = false; sPMCN's  
end gA 0:qEL\  
)C^ZzmB  
]PWK^-4P  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Wk1o H  
% Compute the Zernike Polynomials ] %A mX-U  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %JUD54bBt  
W&E?#=*X  
m-V_J`9"  
% Determine the required powers of r: Nl~'W  
% ----------------------------------- J~.8.]gXW  
m_abs = abs(m); Ph@hk0dgr/  
rpowers = []; {4B{~Qe;  
for j = 1:length(n) TmI~P+5w  
    rpowers = [rpowers m_abs(j):2:n(j)]; $tKz|H)  
end (jj=CLe  
rpowers = unique(rpowers); "u#,#z_  
WdQR^'b$   
7p"4rL  
% Pre-compute the values of r raised to the required powers, y5>X0tT  
% and compile them in a matrix: 0hJ,l.  
% ----------------------------- .g Z1}2GF=  
if rpowers(1)==0 ^FO&GM2a  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); dM n0nc+  
    rpowern = cat(2,rpowern{:}); A7 U]wW9  
    rpowern = [ones(length_r,1) rpowern]; =3K}]3f  
else :SBB3G)|  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 6ZvGD}/  
    rpowern = cat(2,rpowern{:}); FU]jI[  
end C/34K(  
jU~q~e7Te  
HLYog+?  
% Compute the values of the polynomials: ~`Uil=  
% -------------------------------------- <L!9as]w  
y = zeros(length_r,length(n)); 94uAt&&b(  
for j = 1:length(n) } O:Y?Wq^  
    s = 0:(n(j)-m_abs(j))/2; &k\`!T1  
    pows = n(j):-2:m_abs(j); 5?] Dn k.o  
    for k = length(s):-1:1 a!?JVhD&  
        p = (1-2*mod(s(k),2))* ... =}F}XSvXH  
                   prod(2:(n(j)-s(k)))/              ... c&> S  
                   prod(2:s(k))/                     ... %s&"gWi  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... wqx9  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); (FVHtZi7  
        idx = (pows(k)==rpowers); ya`Z eQ-p  
        y(:,j) = y(:,j) + p*rpowern(:,idx); zE,1zBS<  
    end 3UR'*5|'  
     CdZS"I  
    if isnorm gUa-6@  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); Z'!Ii+'6  
    end J?R\qEq%  
end a_z1S Z2[  
% END: Compute the Zernike Polynomials \+iZdZD  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Z;'5A2  
!-tP\%'  
Zb&5)&'X  
% Compute the Zernike functions: #[odjSb  
% ------------------------------ xj< K6  
idx_pos = m>0; w ]%EJ|'  
idx_neg = m<0; BV"l;&F[  
f%[0}.wp  
\281X  
z = y; y8.3tp  
if any(idx_pos) RKb{QAK!v  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); "=Xky,k  
end JHJIjYG>P  
if any(idx_neg) Y@l>4q")  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 16-1&WuY@  
end r?%,#1|$$  
ZCAg)/  
O-uf^ S4  
% EOF zernfun E51'TT9  
phoenixzqy 2012-04-23 20:38
慢慢研究,这个专业性很强的。用的人又少。
sansummer 2012-04-27 10:22
这个太牛了,我目前只能把zygo中的zernike的36项参数带入到zemax中,但是我目前对其结果的可信性表示质疑,以后多交流啊
jssylttc 2012-05-14 11:28
sansummer:这个太牛了,我目前只能把zygo中的zernike的36项参数带入到zemax中,但是我目前对其结果的可信性表示质疑,以后多交流啊 (2012-04-27 10:22)  g E$@:j  
evE$$# 6R  
DDE还是手动输入的呢? CuA A)Bj  
."h>I @MH  
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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