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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, y{@P 1{  
    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, (Nm}3p  
    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? /#:Rd^  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? cBg,k[,  
    dCa}ITg  
    S`ax*`  
    3Ne9% "  
    TS\9<L9S  
    function z = zernfun(n,m,r,theta,nflag) (~q#\  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. -3C* P  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N GS$ZvO  
    %   and angular frequency M, evaluated at positions (R,THETA) on the ?BWHr(J  
    %   unit circle.  N is a vector of positive integers (including 0), and .jvSAV5B  
    %   M is a vector with the same number of elements as N.  Each element +vSCR (n  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) "Czz,;0  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, #citwMW  
    %   and THETA is a vector of angles.  R and THETA must have the same dE 3i=  
    %   length.  The output Z is a matrix with one column for every (N,M) X{5v?4wI  
    %   pair, and one row for every (R,THETA) pair. _F}IF9{?G  
    % L4\SB O  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike B rez&3[  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), [$hptQv  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral ,:0Q1~8  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, u@GRN`yn  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized p2pTs&}S  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. Ymwx (Pm  
    % TSc~$Q]  
    %   The Zernike functions are an orthogonal basis on the unit circle. hEyX~f  
    %   They are used in disciplines such as astronomy, optics, and Y{%4F%Oy  
    %   optometry to describe functions on a circular domain. UgF)J  
    % ]&3s6{R  
    %   The following table lists the first 15 Zernike functions. W HlD %u  
    % K[iY{  
    %       n    m    Zernike function           Normalization g\ 8#:@at  
    %       -------------------------------------------------- &Iv\jhq  
    %       0    0    1                                 1 ki[;ZmQq Y  
    %       1    1    r * cos(theta)                    2 }V1DyLg :  
    %       1   -1    r * sin(theta)                    2 hN>('S-cq  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) H B::0l<  
    %       2    0    (2*r^2 - 1)                    sqrt(3) %f_)<NP9=  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) LV}UBao5n  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) m NUN6qVP~  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) BxSk%$J  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) '0'"k2"vC  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) }Q{ =:X9  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) pl jV|.?  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) r6O7&Me<  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) syWv'Y[k?  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) SX_kr^#  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) Y(#d8o}}#  
    %       -------------------------------------------------- (5f5P84x  
    % %0ll4"  
    %   Example 1: |x _ -I#H  
    % /tId#/Y  
    %       % Display the Zernike function Z(n=5,m=1) *tq|x[<  
    %       x = -1:0.01:1; ;55tf l  
    %       [X,Y] = meshgrid(x,x); w*&n(zJF>  
    %       [theta,r] = cart2pol(X,Y); 1+16i=BF)  
    %       idx = r<=1; tj"v0u?zW  
    %       z = nan(size(X)); y]z)jqX<  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); +(QMy&DtS  
    %       figure Mm>zpB`qP  
    %       pcolor(x,x,z), shading interp zVc7q7E  
    %       axis square, colorbar g6[/F-3Qlf  
    %       title('Zernike function Z_5^1(r,\theta)') ZbZAx:L  
    % 2;Y@3d:z  
    %   Example 2: aIn)']  
    % ?c=R"Yg$  
    %       % Display the first 10 Zernike functions w]o:c(x@  
    %       x = -1:0.01:1; /JK-}E  
    %       [X,Y] = meshgrid(x,x); %U=S6<lbj;  
    %       [theta,r] = cart2pol(X,Y); r2E>sHw  
    %       idx = r<=1; dCoi>PO  
    %       z = nan(size(X)); RAD4q"}k  
    %       n = [0  1  1  2  2  2  3  3  3  3]; t9f4P^V`  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; ZZ]OR;8  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; ={mPg+Ei'  
    %       y = zernfun(n,m,r(idx),theta(idx)); t]u(jX)  
    %       figure('Units','normalized') PtPGi^  
    %       for k = 1:10 % L %1g  
    %           z(idx) = y(:,k); = h<? /Krs  
    %           subplot(4,7,Nplot(k)) XoH[MJC  
    %           pcolor(x,x,z), shading interp 0w'y#U)&8  
    %           set(gca,'XTick',[],'YTick',[]) {d?4;Kd  
    %           axis square n&3iv ^  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 'n>3`1E,  
    %       end .qqb> 7|q  
    % RIVL 0Ig  
    %   See also ZERNPOL, ZERNFUN2. :ET3&J L  
    _Pfx_+  
    v8'`gY  
    %   Paul Fricker 11/13/2006 [MQJ71(3  
    >arO$|W  
    Ch \ed|u  
    )/+eL RN5G  
    sjkKaid  
    % Check and prepare the inputs: a' >$88tl  
    % ----------------------------- 9 .&Or4>  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) G0 nH Z6  
        error('zernfun:NMvectors','N and M must be vectors.') [! dnm1   
    end R.2KYhp ,  
     +,F= -  
    c~pUhx1(  
    if length(n)~=length(m) 8x^H<y=O  
        error('zernfun:NMlength','N and M must be the same length.') LO$#DHPt  
    end ?%za:{  
    xXY)KI N[  
    xo)?XFM2  
    n = n(:); 6(<~1{ X%  
    m = m(:); qK6  uU9z  
    if any(mod(n-m,2)) Lm*LJ_+ B  
        error('zernfun:NMmultiplesof2', ... "-j@GCme  
              'All N and M must differ by multiples of 2 (including 0).') xeP;"J}  
    end N5w]2xz!  
    uZ2v;]\Y6  
    &;@b&p+  
    if any(m>n) J,^pt Ql  
        error('zernfun:MlessthanN', ... \")YKN=W  
              'Each M must be less than or equal to its corresponding N.') |H+k?C-w  
    end k+Ma_H`  
    C1P t3  
    qLW-3W;WUH  
    if any( r>1 | r<0 ) y/sWy1P7  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') 9J;H.:WH  
    end fssL'DD  
    [vjkU7;7A  
    9 <kkzy  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) s<r.+zqW  
        error('zernfun:RTHvector','R and THETA must be vectors.') <T.3ZZ%  
    end CO%O<_C  
    "w|k\1D  
    $hE'b9qx  
    r = r(:); FO'. a  
    theta = theta(:); 'xrbg]b%  
    length_r = length(r); z5*O@_r+.b  
    if length_r~=length(theta) e~ 78'UH  
        error('zernfun:RTHlength', ... EPd.atA  
              'The number of R- and THETA-values must be equal.') P2:Q+j:PX  
    end <T_Nlar^^  
    ;k41+O:f@  
    >'1Q"$;  
    % Check normalization: v!'@NW_  
    % -------------------- "RJk7]p`*  
    if nargin==5 && ischar(nflag) 4#7@KhK}  
        isnorm = strcmpi(nflag,'norm'); O"-PNF,J  
        if ~isnorm em9]WSfZ@`  
            error('zernfun:normalization','Unrecognized normalization flag.') ?L#SnnE  
        end ~z1KD)^   
    else 9B;Sk]y  
        isnorm = false; q}A3"$-F  
    end }?qnwx.  
    ?>\]%$5o  
    . ;@) 5"  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% UUEDCtF)  
    % Compute the Zernike Polynomials zUgkY`]:BJ  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% l'{goyf  
    p*&LEjaVM4  
    3{L vKe  
    % Determine the required powers of r: ]jY)M<:J4  
    % ----------------------------------- I8%'Z>E(  
    m_abs = abs(m); yExyx?j.  
    rpowers = []; oD}FJvV  
    for j = 1:length(n) dSOn\+  
        rpowers = [rpowers m_abs(j):2:n(j)]; 'nDT.i  
    end BMj&*p8R  
    rpowers = unique(rpowers); gLxy RbVI  
    gGdYh.K&e5  
    F5Q. Vh  
    % Pre-compute the values of r raised to the required powers, K$vRk5U  
    % and compile them in a matrix: .p0n\ $r  
    % ----------------------------- Ay6rUN1ef  
    if rpowers(1)==0 yrYaKh  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); L8K3&[l%  
        rpowern = cat(2,rpowern{:}); !skWe~/  
        rpowern = [ones(length_r,1) rpowern]; 9*Twx&  
    else 6)<oO(  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); dZYJ(7%  
        rpowern = cat(2,rpowern{:}); VM|)\?Q  
    end z'K7J'(R  
    1 'pQ,  
    ^[z\KmUqt  
    % Compute the values of the polynomials: %7wzGtM]ps  
    % -------------------------------------- 5.HztNL  
    y = zeros(length_r,length(n)); 8A]q!To  
    for j = 1:length(n) W",jZ"7  
        s = 0:(n(j)-m_abs(j))/2; 61wG:  
        pows = n(j):-2:m_abs(j); g\nL n#  
        for k = length(s):-1:1 acZ|H  
            p = (1-2*mod(s(k),2))* ... +hhbp'%  
                       prod(2:(n(j)-s(k)))/              ... .7Bav5 ;  
                       prod(2:s(k))/                     ... , ZW.P`  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... pG=zGx4  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); +Ws}a  
            idx = (pows(k)==rpowers); \`9|~!,Ix7  
            y(:,j) = y(:,j) + p*rpowern(:,idx); Jpnp'  
        end DYk->)   
         iZ;jn8  
        if isnorm \/%Q PE8  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); (8F?yBu  
        end  cJ{P,K  
    end -* j;  
    % END: Compute the Zernike Polynomials a2)*tbM 9\  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% EHJc*WFPU-  
    ^w}Ib']X  
    yf>,oNIAg  
    % Compute the Zernike functions: o%Q'<0d  
    % ------------------------------ S%|' /cFo  
    idx_pos = m>0; GDe$p;#"9g  
    idx_neg = m<0; @d9*<>@:  
    2uB26SEIl  
    *Y>'v%  
    z = y; Jq@LZ2^  
    if any(idx_pos) tXGcwoOB  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); aq**w?l  
    end fP*C*4#X  
    if any(idx_neg) O4URr  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); N.J:Qn`(  
    end j}Mpc;XOc  
    Qd=/e pkm  
    (VR nv  
    % EOF zernfun v3]M;Y\  
     
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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)  u=5^xpI<D  
    4 0eNgm^  
    DDE还是手动输入的呢? te_D  ,  
    ]`x~v4JU  
    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