[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， %Q080Ltet  
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， !{+a2wi  
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？ 9*2Q'z}_  
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ Y6[O s1  
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function z = zernfun(n,m,r,theta,nflag) pSl4^$2XR  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. 20 Z/Y\  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N JKF/z@Vbe\  
%   and angular frequency M, evaluated at positions (R,THETA) on the  X@Bg_9\i  
%   unit circle.  N is a vector of positive integers (including 0), and CklIrD{  
%   M is a vector with the same number of elements as N.  Each element |4j'KM;U  
%   k of M must be a positive integer, with possible values M(k) = -N(k) |%g)H,6c  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, ANRZQpnXQ  
%   and THETA is a vector of angles.  R and THETA must have the same dAr=X4LE  
%   length.  The output Z is a matrix with one column for every (N,M) +7mUX  
%   pair, and one row for every (R,THETA) pair. 6ltV}Wt-  
% J(Fk@{!F.*  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike z^o7&\:  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), C*stj  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral `$Y%c1;  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, mM2DZ^"j(  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized "!R*f $  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. w&>*4=^a  
% 8 +mW  
%   The Zernike functions are an orthogonal basis on the unit circle. = G>Y9Sc  
%   They are used in disciplines such as astronomy, optics, and +TC##}Zmb  
%   optometry to describe functions on a circular domain. U.Fs9F4M#  
% P#9Pq,I  
%   The following table lists the first 15 Zernike functions. tI<6TE'!p#  
% 4*9BAv  
%       n    m    Zernike function           Normalization wWVB'MRXB,  
%       -------------------------------------------------- nH
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%       0    0    1                                 1 MP p   
%       1    1    r * cos(theta)                    2 `4,]Mr1b
 
%       1   -1    r * sin(theta)                    2 ge]Z5E(1  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) _LFABG=  
%       2    0    (2*r^2 - 1)                    sqrt(3) |*g\-2j{  
%       2    2    r^2 * sin(2*theta)             sqrt(6) u`"Y!*[ -  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) uBw[|,yn2*  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) ^[VEr"X  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) 0v|qP  
%       3    3    r^3 * sin(3*theta)             sqrt(8) ]Na;b  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) N>w+YFM  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) i(4.7{*  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) XCT3:db  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) r_MP[]f|0  
%       4    4    r^4 * sin(4*theta)             sqrt(10) I9h{fB  
%       -------------------------------------------------- 3uL$+F  
% y]g
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%   Example 1: U45-R-  
% LhSXz>AX  
%       % Display the Zernike function Z(n=5,m=1) em2Tet  
%       x = -1:0.01:1; *i"Mu00b  
%       [X,Y] = meshgrid(x,x); t$PJ*F67M  
%       [theta,r] = cart2pol(X,Y); ;?Q0mXr  
%       idx = r<=1; `)NTJc$):  
%       z = nan(size(X)); PHMp,z8  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); 3}B-n!|*  
%       figure p2gu@!  
%       pcolor(x,x,z), shading interp 9hgIQl  
%       axis square, colorbar @h\i<sh!^  
%       title('Zernike function Z_5^1(r,\theta)') }tJMnq/m($  
% \==Mgy2J8  
%   Example 2: ;\]DZV4?)r  
% <9x|)2P  
%       % Display the first 10 Zernike functions (L!u[e0[#  
%       x = -1:0.01:1; N6v*X+4JH  
%       [X,Y] = meshgrid(x,x); O]l-4X#8F  
%       [theta,r] = cart2pol(X,Y); _zLEHEZ-  
%       idx = r<=1; qv`:o `  
%       z = nan(size(X)); w$`u_P|@E:  
%       n = [0  1  1  2  2  2  3  3  3  3]; *7qa]i^]  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; ;1k_J~Qei  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; OA7=kH@3c  
%       y = zernfun(n,m,r(idx),theta(idx)); wKJK!P  
%       figure('Units','normalized') ]0pI6"  
%       for k = 1:10 qz 29f  
%           z(idx) = y(:,k); akQb%Wq  
%           subplot(4,7,Nplot(k)) \\/ !I   
%           pcolor(x,x,z), shading interp hP/uS%X   
%           set(gca,'XTick',[],'YTick',[]) R=W$3Ue~,  
%           axis square z.W1Za  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) s%
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%       end 3ep L'My$  
% we?t/YB=  
%   See also ZERNPOL, ZERNFUN2. M+4S>Sjw  
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%   Paul Fricker 11/13/2006 '=IuwCB|;  
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% Check and prepare the inputs: 'YEiT#+/  
% ----------------------------- P2)g%$ME  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) %;`3I$  
    error('zernfun:NMvectors','N and M must be vectors.') 5JZZvc$au  
end 94XRf"^  
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if length(n)~=length(m) uMBb=   
    error('zernfun:NMlength','N and M must be the same length.') U7G|4(  
end Q1 vse  
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n = n(:); <zu)=W'R]  
m = m(:); BimM)4g  
if any(mod(n-m,2)) ||?wRMV  
    error('zernfun:NMmultiplesof2', ... td+[Na0d  
          'All N and M must differ by multiples of 2 (including 0).') hpticW|  
end 2K'}Vm+  
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=`%%
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if any(m>n) ,@2d4eg4  
    error('zernfun:MlessthanN', ... 5xG/>fn  
          'Each M must be less than or equal to its corresponding N.') }Z\+Qc<<  
end 5TdI  
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if any( r>1 | r<0 ) #/pZ#ny  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') Ia)
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end Y'%_--  
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) fI0L\^b%  
    error('zernfun:RTHvector','R and THETA must be vectors.') 
#kGxX@0  
end on1mu't_;  
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r = r(:); ~LzTqMHM  
theta = theta(:); ';7|H|,F
 
length_r = length(r); ({x<!5XL  
if length_r~=length(theta) ^SRa!8z$W  
    error('zernfun:RTHlength', ... z'X_s.9F  
          'The number of R- and THETA-values must be equal.') ? 5 V-D8k  
end l@YpgyqaL  
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% Check normalization: JXnPKAN  
% -------------------- gf2w@CVF>=  
if nargin==5 && ischar(nflag) $RSVN?  
    isnorm = strcmpi(nflag,'norm'); Onoi6^G  
    if ~isnorm o[ %Q&u  
        error('zernfun:normalization','Unrecognized normalization flag.') O"9f^y
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    end ,K6]Q|U@r  
else L=}U
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    isnorm = false; L7%'Y}1e.  
end ;h3*MR  

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |,&!Q$<un  
% Compute the Zernike Polynomials 7"JU)@ U]  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Fk(0q/b  
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% Determine the required powers of r: qkUr5^1  
% ----------------------------------- aLXA9?  
m_abs = abs(m); cuk2\> Xl  
rpowers = []; j)IK  
for j = 1:length(n) 7RD` *s  
    rpowers = [rpowers m_abs(j):2:n(j)]; Q84KU8?d  
end A1ebXXD)  
rpowers = unique(rpowers); $'FPst8Q<  
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% Pre-compute the values of r raised to the required powers, rrwsj`  
% and compile them in a matrix: 3Ob"r`  
% ----------------------------- \ bT]?.si  
if rpowers(1)==0 Z ''P5B;  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); g&E_|}u4  
    rpowern = cat(2,rpowern{:}); A
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    rpowern = [ones(length_r,1) rpowern]; 56_KB.Ww~  
else 4!}fC
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    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); b);}x1L.T  
    rpowern = cat(2,rpowern{:}); i
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end VD#^Xy4% r  
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% Compute the values of the polynomials: VD3[ko  
% -------------------------------------- %<muVRkB\  
y = zeros(length_r,length(n)); iRVLo~  
for j = 1:length(n) 1aT$07G0  
    s = 0:(n(j)-m_abs(j))/2; TQ2Tt"  
    pows = n(j):-2:m_abs(j); 99:L#0!.W  
    for k = length(s):-1:1 QF>[cdl?8  
        p = (1-2*mod(s(k),2))* ... t@HE.h  
                   prod(2:(n(j)-s(k)))/              ... ::`j@ ]  
                   prod(2:s(k))/                     ... 3z#;0n
}  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 1a!h&!$9  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); 7=AKQ7BB>b  
        idx = (pows(k)==rpowers); HYH!;  
        y(:,j) = y(:,j) + p*rpowern(:,idx); ~3YNHm6V  
    end d?P aZz{4  
     j&mL]'Zy  
    if isnorm =
% JDo  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); E>1USKxn  
    end ]1[;A$7  
end W[m_IY  
% END: Compute the Zernike Polynomials E{ ,O}  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% }Tef;8
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% Compute the Zernike functions: -ur]k]R  
% ------------------------------ ,'673PR  
idx_pos = m>0; h5gXYmk  
idx_neg = m<0; W*m[t&;  
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z = y; R)C+wTG;  
if any(idx_pos) <<1oc{i  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ;hvXFU  
end 31C]TdJ  
if any(idx_neg) ZkJM?Fzq  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); :qp"Ao{M  
end `IoX'|C[h  
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% EOF zernfun j5O*H_D  

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)  |1+(Ny.%k  

dH2]ZE0V  
DDE还是手动输入的呢？ fb"J Bc}X  
::OFW@dS  
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。