[讨论]如何从zernike矩中提取出zernike系数啊 [复制链接]

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， BK6oW3wD/  
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， @#;~_?$?C  
这个，跟我们用zygo干涉仪直接拟合出的36项zernike系数，有何关系呢？ 0(HUy`]>  
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ &@nI(PXv  
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function z = zernfun(n,m,r,theta,nflag) 1;S@XC>  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. 7oK!!Qd^
w  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N "){"{~  
%   and angular frequency M, evaluated at positions (R,THETA) on the arRbq!mO  
%   unit circle.  N is a vector of positive integers (including 0), and ?>DN7je  
%   M is a vector with the same number of elements as N.  Each element E%2]c?N5  
%   k of M must be a positive integer, with possible values M(k) = -N(k) qy/xJ>:  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, :[,-wZiT~6  
%   and THETA is a vector of angles.  R and THETA must have the same 8FU8E2zo  
%   length.  The output Z is a matrix with one column for every (N,M) `Z0FQ( r_  
%   pair, and one row for every (R,THETA) pair. <U$x')W  
% 1Sx2c  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike bRfac/:}  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), UM3}7|  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral ?7*.S Lt  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, /*i[MB
 
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized _?CyKk\I  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. (gQP_Oa(  
% RG0kOw0  
%   The Zernike functions are an orthogonal basis on the unit circle. 2.qEy6  
%   They are used in disciplines such as astronomy, optics, and *3d+ !#;rG  
%   optometry to describe functions on a circular domain. O,x[6P5
4P  
% ?^n),mR  
%   The following table lists the first 15 Zernike functions. Vo"Wr>F  
% r roI  
%       n    m    Zernike function           Normalization gE\&[;)DB  
%       -------------------------------------------------- 9$$dSN\&  
%       0    0    1                                 1 h'jc4mu0  
%       1    1    r * cos(theta)                    2 )%dxfwd6  
%       1   -1    r * sin(theta)                    2
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%       2   -2    r^2 * cos(2*theta)             sqrt(6) _(l?gj  
%       2    0    (2*r^2 - 1)                    sqrt(3) qHaH=g%  
%       2    2    r^2 * sin(2*theta)             sqrt(6) nl5A{ s
 
%       3   -3    r^3 * cos(3*theta)             sqrt(8) XXPn)kmWR  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) <hvs{}TS  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) k<Qhw)M8  
%       3    3    r^3 * sin(3*theta)             sqrt(8) . |%n"{  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) ' Dcj\=8
 
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) x{4{.s%+:  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) SO4?3wg7  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 6I2`oag  
%       4    4    r^4 * sin(4*theta)             sqrt(10) ^F,sV*  
%       -------------------------------------------------- G%iTL"6  
% & 6'Rc#\P  
%   Example 1: x<5ARK6\=  
% }@x!r=O)I  
%       % Display the Zernike function Z(n=5,m=1) u}3D'h  
%       x = -1:0.01:1; *IX<&u#  
%       [X,Y] = meshgrid(x,x); h?[|1.lJx(  
%       [theta,r] = cart2pol(X,Y); 6S`0<Z;;/  
%       idx = r<=1; ~jC+6v  
%       z = nan(size(X)); =' uePM")  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); *:bexDH  
%       figure bd]9kRq1K  
%       pcolor(x,x,z), shading interp 0vX4v)-^u  
%       axis square, colorbar >3ax `8  
%       title('Zernike function Z_5^1(r,\theta)') Xii>?sA5Z"  
% "i#aII+T  
%   Example 2: 0civXZgj  
% [?%q,>F  
%       % Display the first 10 Zernike functions Lq|>nY  
%       x = -1:0.01:1; Q2/65$nW  
%       [X,Y] = meshgrid(x,x); XeX\u3<D  
%       [theta,r] = cart2pol(X,Y); m/z,MT74*J  
%       idx = r<=1; mG"xo^1_H  
%       z = nan(size(X)); H2H`7 +I,  
%       n = [0  1  1  2  2  2  3  3  3  3]; XNgcBSD  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; +F-EgF+J  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; !O,Sq/=.  
%       y = zernfun(n,m,r(idx),theta(idx)); K!]a+M]>  
%       figure('Units','normalized') _f'v>"K  
%       for k = 1:10 > vdmN]  
%           z(idx) = y(:,k); gg>QXui  
%           subplot(4,7,Nplot(k)) DQT'OZ:w  
%           pcolor(x,x,z), shading interp 8Qo'[+4;  
%           set(gca,'XTick',[],'YTick',[]) d]poUN~x  
%           axis square h2 KI  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) nl qn:[BU  
%       end NMe{1RM  
% w lH\w?  
%   See also ZERNPOL, ZERNFUN2. (`S^6-^  
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%   Paul Fricker 11/13/2006 XEQTTD<  
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% Check and prepare the inputs: Hs*["zFc  
% ----------------------------- ,Cb3R|L8  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) #
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    error('zernfun:NMvectors','N and M must be vectors.') ?u|@,tQ[  
end ]I[~0PCSX  
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if length(n)~=length(m) pZ&,YX  
    error('zernfun:NMlength','N and M must be the same length.') "!~o  
end ^Jp,&  
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n = n(:); =eDIvNps  
m = m(:); .E<nQWz8  
if any(mod(n-m,2)) J0?kEr  
    error('zernfun:NMmultiplesof2', ... Ut;`6t  
          'All N and M must differ by multiples of 2 (including 0).') Zz0e4C  
end BH">#&j[  
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if any(m>n) ZEAUoC1E1  
    error('zernfun:MlessthanN', ... M2O_kOeZ  
          'Each M must be less than or equal to its corresponding N.') u.ggN=Z  
end xWxc1tT`  
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if any( r>1 | r<0 ) c2

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    error('zernfun:Rlessthan1','All R must be between 0 and 1.') Ax;=Zh<DAv  
end :OG I|[  
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Y()"2CCV  
    error('zernfun:RTHvector','R and THETA must be vectors.') 1^!SuAA@  
end T$I_nxh[)L  
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/|IPBU 5  
r = r(:); VPe0\?!d  
theta = theta(:); FJ:^pROpm  
length_r = length(r); *yu}e)(0  
if length_r~=length(theta) l3
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    error('zernfun:RTHlength', ... JZ:@iI5>+  
          'The number of R- and THETA-values must be equal.') >]\I:T  
end ieFl4hh[G  
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% Check normalization: =dmr
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% -------------------- c$O8Rhx  
if nargin==5 && ischar(nflag) :?>7Z6  
    isnorm = strcmpi(nflag,'norm'); '<R>cN"  
    if ~isnorm ^"WVE["  
        error('zernfun:normalization','Unrecognized normalization flag.') e-nA>v  
    end 3v/B*M VI  
else \^x{NV@v42  
    isnorm = false; =p+y$  
end &mwd0%4  
/Mqhx_)>A  
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ?sf<cFF  
% Compute the Zernike Polynomials KdkA@>L!;  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 9)Fx;GxL  
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2 !s&|lI  
% Determine the required powers of r: CXa[%{[n  
% ----------------------------------- M/zO|-j&  
m_abs = abs(m); Zf'*pp T&q  
rpowers = []; IH]9%d)  
for j = 1:length(n) *
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    rpowers = [rpowers m_abs(j):2:n(j)]; %FO{:@CH  
end (l{
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rpowers = unique(rpowers); i5'&u:  
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% Pre-compute the values of r raised to the required powers, #z\ub5um  
% and compile them in a matrix: dzf2`@8#  
% ----------------------------- Ql*zl  
if rpowers(1)==0 T(b9b,ov)  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); EBj^4=b[  
    rpowern = cat(2,rpowern{:}); sV\_DP/l  
    rpowern = [ones(length_r,1) rpowern]; oBzl=N3<  
else !wAT`0<94F  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); jvzioFCt  
    rpowern = cat(2,rpowern{:}); $v^hzC  
end !?2)apM  
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% Compute the values of the polynomials: }A#IBqf5  
% -------------------------------------- _P>YG<*"kQ  
y = zeros(length_r,length(n)); ;_<R
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for j = 1:length(n) K7 e~%mY  
    s = 0:(n(j)-m_abs(j))/2; ).T&fa"  
    pows = n(j):-2:m_abs(j); 6TtB3;5  
    for k = length(s):-1:1 xoaO=7\io  
        p = (1-2*mod(s(k),2))* ... @<.@X*#I  
                   prod(2:(n(j)-s(k)))/              ... ,g*!NK_:5t  
                   prod(2:s(k))/                     ... \br!77  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... &V"o
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                   prod(2:((n(j)+m_abs(j))/2-s(k))); _Nx /<isdL  
        idx = (pows(k)==rpowers); V%Uj\cv  
        y(:,j) = y(:,j) + p*rpowern(:,idx); jr6_|(0 i6  
    end VYvfx  
     A1WUK=P  
    if isnorm c}(WniR-"  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); t@q'm.:uw<  
    end &!!*xv-z  
end Ndmt$(b  
% END: Compute the Zernike Polynomials |)-kUu  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% nm'l}/Ug  
+mQ5\14#  
|P|B"I<?  
% Compute the Zernike functions: ^^y eC|~N:  
% ------------------------------ c_lHj#A(l  
idx_pos = m>0; v^|U?  
idx_neg = m<0; i\R0+O{  
5
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7|rH9Bc{U  
z = y; 3h@]cWp  
if any(idx_pos) RNg?o[S  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); Lvk}%,S8t  
end nJDGNm,  
if any(idx_neg) fi+}hGj(r  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); j\>LJai"  
end Xn7G2Yp  
7& M-^Ev  
<izQ]\kL  
% EOF zernfun #&3,T1i`  

支持一下，慢慢研究

顶顶·········

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)  UB~K/r`.|
 

.ARYCTyG  
DDE还是手动输入的呢？ dCx63rF`G  
KQ~y;{h?b  
zygo和zemax的zernike系数，类型对应好就没问题了吧

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

慢慢研究，这个专业性很强的。用的人又少。