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    [讨论]如何从zernike矩中提取出zernike系数啊 [复制链接]

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    离线jssylttc
     
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    只看楼主 倒序阅读 楼主  发表于: 2012-04-23
    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, %6i=lyH-  
    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, IG|\:Xz  
    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? `bqzg  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? #LWg"i  
    M/B/b<['  
    H,|YLKg-|  
    g1V)$s 7  
    +^gO/ 0  
    function z = zernfun(n,m,r,theta,nflag) !Uy>eji}  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. ^PQM;"  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N or.\)(m#(  
    %   and angular frequency M, evaluated at positions (R,THETA) on the ,8VXA +'_  
    %   unit circle.  N is a vector of positive integers (including 0), and }Vl^EAR  
    %   M is a vector with the same number of elements as N.  Each element e5OVq ,  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) FL&dv  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, P` ]ps?l  
    %   and THETA is a vector of angles.  R and THETA must have the same j_c+.iET  
    %   length.  The output Z is a matrix with one column for every (N,M) VDn:SGj5  
    %   pair, and one row for every (R,THETA) pair. JqEb;NiP)5  
    % a_%>CD${t  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike UkfA}b^@v  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), Hirr=a3  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral ~U%j{8uH  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, /7vE>mSY  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized O 6]u!NqG  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. (9'be\  
    % L*^ V5^-  
    %   The Zernike functions are an orthogonal basis on the unit circle. !gJzg*{u@  
    %   They are used in disciplines such as astronomy, optics, and rKIRNc#d  
    %   optometry to describe functions on a circular domain. bd{\{[^S!  
    % m1y `v"  
    %   The following table lists the first 15 Zernike functions. 3'^S3W%  
    % mu>] 9ZW  
    %       n    m    Zernike function           Normalization A:)sg!Lt  
    %       -------------------------------------------------- zq=&4afOE  
    %       0    0    1                                 1 e5L 1er;6  
    %       1    1    r * cos(theta)                    2 A^L?_\e6  
    %       1   -1    r * sin(theta)                    2 %rXexy!V  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) 8_ X.c  
    %       2    0    (2*r^2 - 1)                    sqrt(3) cNeiD@t3V&  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) vv* |F  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) :`5;nl63  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) r\RFDj  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) U!NI_uk  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) ;-Ado8  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) 5p{25N_t  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Gw`/.0  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) OPLl*bnf  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Ys%'#f  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) -#OwJ*-U  
    %       -------------------------------------------------- =h7[E./U1  
    % !mae^A1  
    %   Example 1: 5\3 swP_7  
    % E4Zxv*  
    %       % Display the Zernike function Z(n=5,m=1) AoU_;B\b%  
    %       x = -1:0.01:1; ``6{T1fQS  
    %       [X,Y] = meshgrid(x,x); UQnBqkE  
    %       [theta,r] = cart2pol(X,Y); PY\W  
    %       idx = r<=1; j@CKO cn2  
    %       z = nan(size(X)); R. O  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); [9J:bD  
    %       figure $$\V 2%v  
    %       pcolor(x,x,z), shading interp OOfy Gvs  
    %       axis square, colorbar }pK v.  
    %       title('Zernike function Z_5^1(r,\theta)') ~W3:xnBEk  
    % FvAbh]/4  
    %   Example 2: 8XlU%a6x  
    % X*)?LxTj  
    %       % Display the first 10 Zernike functions 9u?Eb~#$  
    %       x = -1:0.01:1; |+u+)C  
    %       [X,Y] = meshgrid(x,x); Yfe'#MKfL  
    %       [theta,r] = cart2pol(X,Y); @wMQC\Z  
    %       idx = r<=1;  M$F{N  
    %       z = nan(size(X)); Enu!u~1]F  
    %       n = [0  1  1  2  2  2  3  3  3  3]; r:73uRk  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; %6N)G!P  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; HmW=t}!  
    %       y = zernfun(n,m,r(idx),theta(idx)); ^glX1 )  
    %       figure('Units','normalized') "A]?M<R  
    %       for k = 1:10 }a' cm!"  
    %           z(idx) = y(:,k); )O9fhj)  
    %           subplot(4,7,Nplot(k)) ~z&0qQ  
    %           pcolor(x,x,z), shading interp 1*L^^% w  
    %           set(gca,'XTick',[],'YTick',[]) tg3zXJ4k_  
    %           axis square */4tJ G1U  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) '!AT  
    %       end 2G ZF/9}  
    % $,.3&zsy  
    %   See also ZERNPOL, ZERNFUN2. 4Q@\h=r  
    D/e&7^iK  
    ;4l-M2  
    %   Paul Fricker 11/13/2006 X=JFWzC  
    lx`q *&E  
    R08&cd#$  
    R9Ldl97'  
    d3og?{i<}&  
    % Check and prepare the inputs: )sRN!~  
    % ----------------------------- ^)Smv\Md  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 7,f:Qi@g  
        error('zernfun:NMvectors','N and M must be vectors.') !;TR2Zcn  
    end  ccRlql(  
    EG%I1F%  
    DQ%`v =  
    if length(n)~=length(m) ix:2Z-  
        error('zernfun:NMlength','N and M must be the same length.') '^8g9E .4K  
    end Rq"VB.ef&{  
    E2h(w_l  
    [TP  
    n = n(:); [+y &HNf  
    m = m(:); ,|6Y\L  
    if any(mod(n-m,2)) "pOqd8>]  
        error('zernfun:NMmultiplesof2', ... ?0 HR(N(z!  
              'All N and M must differ by multiples of 2 (including 0).') w8G7Jy  
    end :wFb5"  
    e jP,29  
    R_t~UTfI;  
    if any(m>n) +d.u##$  
        error('zernfun:MlessthanN', ... Rk}\)r\  
              'Each M must be less than or equal to its corresponding N.') 2TE\4j  
    end G!nl'5|y  
    [SK2x4  
    ur?d6 a  
    if any( r>1 | r<0 ) XAw2X;F%  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') ~azF+}x90N  
    end _2wAaJvA  
    ^cB49s+{e  
    ${wU+E*  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 0Ulxp  
        error('zernfun:RTHvector','R and THETA must be vectors.') Cq-hPa}2  
    end ~ &t!$  
    $$k7_rs  
    >?^~s(t  
    r = r(:); h1n*WQ-  
    theta = theta(:); mYntU^4f  
    length_r = length(r); yb[{aL^4%  
    if length_r~=length(theta) FX{ ~"  
        error('zernfun:RTHlength', ... YI L'YNH  
              'The number of R- and THETA-values must be equal.') d^ 2u}^kG  
    end vEu Ka<5  
    <l* agH-.3  
    jn.R.}TT  
    % Check normalization: 7h(HG?2Y  
    % -------------------- ?lu_}t]  
    if nargin==5 && ischar(nflag) &r&;<Q  
        isnorm = strcmpi(nflag,'norm'); Mr$# e  
        if ~isnorm <E D8"~_  
            error('zernfun:normalization','Unrecognized normalization flag.') ~sZqa+jB0  
        end lF0K=L  
    else GwTT+  
        isnorm = false; 8dV.nO  
    end 6\; 4 4,3  
    }rO?5  
    5oVLv4Z9u  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% RpBiE8F4  
    % Compute the Zernike Polynomials $KoPGgC[  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% x, G6\QmA  
    i58ZV`Rk`  
    .}IK}A/-  
    % Determine the required powers of r: A ~qW.  
    % ----------------------------------- r~ZS1Tp  
    m_abs = abs(m); K<$wz/\  
    rpowers = []; /X(@|tk:  
    for j = 1:length(n) hB|H9+  
        rpowers = [rpowers m_abs(j):2:n(j)]; clh3  
    end p:DL:^zx  
    rpowers = unique(rpowers); )B -MPuB  
    FZ[@])B  
    Xz;et>UD*B  
    % Pre-compute the values of r raised to the required powers, ^n\9AE3  
    % and compile them in a matrix: \(.nPW]9  
    % ----------------------------- P5'iYahCq_  
    if rpowers(1)==0 <_##YSGh,  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); !yoSMI-  
        rpowern = cat(2,rpowern{:}); Ha46U6_'h  
        rpowern = [ones(length_r,1) rpowern]; ti$oZ4PpF  
    else u Y?/B~  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); A[sM{i~Z  
        rpowern = cat(2,rpowern{:}); b &\3ps  
    end oUW )H  
    tIz<+T_  
    ek<PISlci  
    % Compute the values of the polynomials: tYI ]LL  
    % -------------------------------------- AzLbD2Pl  
    y = zeros(length_r,length(n)); +-Z"H)  
    for j = 1:length(n) *u|lmALs  
        s = 0:(n(j)-m_abs(j))/2; Cfv L)f  
        pows = n(j):-2:m_abs(j); h(C#\{V  
        for k = length(s):-1:1 0EL\Hd  
            p = (1-2*mod(s(k),2))* ... Hs:4I  
                       prod(2:(n(j)-s(k)))/              ... K7 t&fDI  
                       prod(2:s(k))/                     ... 6%\7.h  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... Hmz=/.$  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); e5*5.AB6&  
            idx = (pows(k)==rpowers); (PCimT=5  
            y(:,j) = y(:,j) + p*rpowern(:,idx); { ()p%#*  
        end `^ieT#(O  
         N0y;PVAGu  
        if isnorm -XS+Uv  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); nUI63?  
        end uR06&SaA>  
    end _H~pH7WU  
    % END: Compute the Zernike Polynomials baUEsg[~V  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% KKeb ioW  
    Bi9 N  
    9TYw@o5V  
    % Compute the Zernike functions: >< <$  
    % ------------------------------ f7EIDFX>pt  
    idx_pos = m>0; 8Pr&F  
    idx_neg = m<0; ?6gDbE%  
    A%NK0j$;}  
    EmtDrx4!(f  
    z = y; "?2  
    if any(idx_pos) p6&LZ=tL3  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); p0}+071o%  
    end zh#OD{  
    if any(idx_neg) X_-Hrp!h  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); jQ.>2-;H9  
    end )1ZJ  
    5t"bCzp  
    mJ6t.%'d  
    % EOF zernfun 9[t]]  
     
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    离线phoenixzqy
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    只看该作者 1楼 发表于: 2012-04-23
    慢慢研究,这个专业性很强的。用的人又少。
    2024年6月28-30日于上海组织线下成像光学设计培训,欢迎报名参加。请关注子在川上光学公众号。详细内容请咨询13661915143(同微信号)
    离线sansummer
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    只看该作者 2楼 发表于: 2012-04-27
    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
    离线jssylttc
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    只看该作者 3楼 发表于: 2012-05-14
    回 sansummer 的帖子
    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)  I8-&.RE  
    *>&N t  
    DDE还是手动输入的呢? rY_C3;B  
    k>z-Zg  
    zygo和zemax的zernike系数,类型对应好就没问题了吧
    离线jssylttc
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    只看该作者 4楼 发表于: 2012-05-14
    顶顶·········
    离线18257342135
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    只看该作者 5楼 发表于: 2016-12-13
    支持一下,慢慢研究