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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, c`soVqT$?  
    我输入10阶的n、m,r,theta为38025*1向量,最后得到的z是29525*10阶的矩阵, pASX-rb  
    这个,跟我们用zygo干涉仪直接拟合出的36项zernike系数,有何关系呢? 4T31<wk  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? aOH|[  
    C< 9x\JY%  
    M@R"-$Z  
    j:h}ka/!p  
    i'm<{ v  
    function z = zernfun(n,m,r,theta,nflag) N[p o)}hp  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. G IN|cv=  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N w}gmVJ#p  
    %   and angular frequency M, evaluated at positions (R,THETA) on the !l9{R8m>eJ  
    %   unit circle.  N is a vector of positive integers (including 0), and ^+SE_-+]  
    %   M is a vector with the same number of elements as N.  Each element Z^_qXerjP  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) kJJT`Ba&/  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, TI'v /=;)  
    %   and THETA is a vector of angles.  R and THETA must have the same _K o#36.S  
    %   length.  The output Z is a matrix with one column for every (N,M) $D1ha CL  
    %   pair, and one row for every (R,THETA) pair. B n7uKa{P  
    % ECOJ .^  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike +nE>)ZH  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), KLyRb0V  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral D5,]E`jwu  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, lS4rpbU_  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized VHxBs  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ,AP0*Ln  
    % ~w? 02FU  
    %   The Zernike functions are an orthogonal basis on the unit circle. =6u@ JpOl  
    %   They are used in disciplines such as astronomy, optics, and Zz0bd473k?  
    %   optometry to describe functions on a circular domain. J#I RbO)  
    % ; Z]Wj9iY  
    %   The following table lists the first 15 Zernike functions. G&ck98  
    % (QDKw}O2b  
    %       n    m    Zernike function           Normalization 7%y$^B7{  
    %       -------------------------------------------------- J].Oxch&y  
    %       0    0    1                                 1 =rA?,74  
    %       1    1    r * cos(theta)                    2 'X;cgAq8(  
    %       1   -1    r * sin(theta)                    2  >Uw:cq  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) AELj"=RA  
    %       2    0    (2*r^2 - 1)                    sqrt(3) 8K,X3a9  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) l =E86"m  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) ev4[4T-( @  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) k =5k)}i  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) +V4)><  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) z`wIb  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) tF:AnNp=  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) )9hqd  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) fz(YP=@ZnP  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) &t= :xVn-M  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) `HX:U3/  
    %       -------------------------------------------------- iXDG-_K  
    % ~CNB3r5R  
    %   Example 1: L7$f01*  
    % I L*B@E8  
    %       % Display the Zernike function Z(n=5,m=1) csy6_q(  
    %       x = -1:0.01:1; ("8Hku?  
    %       [X,Y] = meshgrid(x,x); @7Ec(]yp  
    %       [theta,r] = cart2pol(X,Y); ^Hx}.?1  
    %       idx = r<=1; 2lTt  
    %       z = nan(size(X)); "wgPPop  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); OG5{oH#K  
    %       figure J :O!4gI  
    %       pcolor(x,x,z), shading interp 8,U~ p<Gz  
    %       axis square, colorbar #_DpiiS,.Q  
    %       title('Zernike function Z_5^1(r,\theta)') Fi i(dmn  
    % riIubX#  
    %   Example 2: ~<[+!&<U  
    % }j/\OY _&  
    %       % Display the first 10 Zernike functions #Zdh<.   
    %       x = -1:0.01:1; GHsDZ(d3.  
    %       [X,Y] = meshgrid(x,x); UD-+BUV  
    %       [theta,r] = cart2pol(X,Y); Ok!P~2J  
    %       idx = r<=1; " .7@  
    %       z = nan(size(X)); ]3 "0#Y  
    %       n = [0  1  1  2  2  2  3  3  3  3]; %p 6Ms  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; zDvV%+RW)  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; _}F& ^  
    %       y = zernfun(n,m,r(idx),theta(idx)); j9fBl:Fr  
    %       figure('Units','normalized') f Fi=/}  
    %       for k = 1:10 tK3$,9+  
    %           z(idx) = y(:,k);  "9;  
    %           subplot(4,7,Nplot(k)) j,OA>{-$  
    %           pcolor(x,x,z), shading interp Q`k;E}x_-  
    %           set(gca,'XTick',[],'YTick',[]) JLd%rM\m  
    %           axis square zqA>eDx  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 7J);{ &x9h  
    %       end z4YDngf=4  
    % mnA_$W3~I  
    %   See also ZERNPOL, ZERNFUN2. && ]ix3  
    lV'?X%  
    EB3/o7)L  
    %   Paul Fricker 11/13/2006 #6M |T+ =  
    :Racu;xf  
    z};|.N}  
    _WS8I>  
    ew\:&"@2]w  
    % Check and prepare the inputs: y3;M$Jr  
    % ----------------------------- Uh.swBC n  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) |sGJum&=  
        error('zernfun:NMvectors','N and M must be vectors.') .i;.5)shsu  
    end fq>{5ODO  
    "~VKUvDu  
    `+Nv =vk  
    if length(n)~=length(m) !$NK7-  
        error('zernfun:NMlength','N and M must be the same length.') 9wx]xg4l"  
    end &J/EBmY[  
    M[qhy.  
    @x1cV_s[  
    n = n(:); 9,8/DW.K  
    m = m(:); kI"9T`owR  
    if any(mod(n-m,2)) y{M7kYWtHV  
        error('zernfun:NMmultiplesof2', ... ~C{:G;Iy0  
              'All N and M must differ by multiples of 2 (including 0).') E]Mx<7;\.  
    end ,|*Gr"Q=  
    Tv#d>ZSD  
    l$5nv5r  
    if any(m>n) 4V9BmVS|Th  
        error('zernfun:MlessthanN', ... m ^FKE:  
              'Each M must be less than or equal to its corresponding N.') ViW2q"4=  
    end *-ys}sX  
    W$X/8K bn  
    3D6&0xTq  
    if any( r>1 | r<0 ) &j~9{ C  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') `Ij EwKra  
    end d%I7OBBx@  
    |[~ S&  
    6Gg`ExcT5  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) "$N$:B@U  
        error('zernfun:RTHvector','R and THETA must be vectors.') UIU Pi gd  
    end &yP|t":HWX  
    *ELU">!}G  
    %KVmpWku  
    r = r(:); V]Te_ >E;w  
    theta = theta(:); (1cB Tf  
    length_r = length(r); E-1u_7  
    if length_r~=length(theta) RL&0?OT  
        error('zernfun:RTHlength', ... }bRn&)e  
              'The number of R- and THETA-values must be equal.') K bQXH!J  
    end z 8M\(<  
    9Tr ceL;  
    Qq6'[Od  
    % Check normalization: PCwc=  
    % -------------------- \5tG>>c i  
    if nargin==5 && ischar(nflag) VsTgK  
        isnorm = strcmpi(nflag,'norm'); $hc=H  
        if ~isnorm 4Y'Ne2M{  
            error('zernfun:normalization','Unrecognized normalization flag.') `StuUa  
        end y =sae  
    else 6|lsG6uf  
        isnorm = false; :YRHO|  
    end ;1yF[<a  
    @-K[@e/uwy  
    !4<D^ eh  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% WI&A+1CK-5  
    % Compute the Zernike Polynomials hlre eXv  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% WL(Y1>|j  
    tJ_Y6oFm=  
    CH(Y.Kj-  
    % Determine the required powers of r: #qk=R7" Q  
    % ----------------------------------- MA_YMxP.'  
    m_abs = abs(m); VMF?qT3Nd  
    rpowers = []; Q7e4MKy7  
    for j = 1:length(n) =}tomN(F~[  
        rpowers = [rpowers m_abs(j):2:n(j)]; Kn3Xn`P?  
    end 3=U#v<  
    rpowers = unique(rpowers); S]=.p-Am  
    q{G8 Po$z'  
    ~-NSIV:f  
    % Pre-compute the values of r raised to the required powers, NRG06M  
    % and compile them in a matrix: g?|Z/eVJ  
    % ----------------------------- SFh<>J^ 0a  
    if rpowers(1)==0 mW{uChHP  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); @"h4S*U  
        rpowern = cat(2,rpowern{:}); O13]H"O_  
        rpowern = [ones(length_r,1) rpowern]; O Lt0Q.{  
    else 5nBJj  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); t$,G%micj  
        rpowern = cat(2,rpowern{:}); L<oQKe7Q:  
    end A`M-N<T  
    &ZMQ]'&  
    MCTJ^g"D  
    % Compute the values of the polynomials: [z\baL|  
    % -------------------------------------- W4av?H  
    y = zeros(length_r,length(n)); \IC^z  
    for j = 1:length(n) \15'~ ]d  
        s = 0:(n(j)-m_abs(j))/2; %m/lPL  
        pows = n(j):-2:m_abs(j); W$wX[  
        for k = length(s):-1:1 UAz^P6iQ`~  
            p = (1-2*mod(s(k),2))* ... <uBRLe`)  
                       prod(2:(n(j)-s(k)))/              ... JFc, f  
                       prod(2:s(k))/                     ... #b&tNZ4!_  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ~9APc{"A  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); ts(u7CJd  
            idx = (pows(k)==rpowers); bBc<p{  
            y(:,j) = y(:,j) + p*rpowern(:,idx); *w. ":\P]  
        end 'Q=)-  
         "9^b1UH<  
        if isnorm y5=,q]Qjk[  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); b{-"GqMO  
        end ( ./MFf  
    end -1B.A  
    % END: Compute the Zernike Polynomials AfhJ6cSIE  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 8pA<1H%  
    I+twI&GS  
    2<OU)rVE4  
    % Compute the Zernike functions: a+J>  
    % ------------------------------ Hmm0H6&u  
    idx_pos = m>0; 4x-,l1NMR  
    idx_neg = m<0; H-&27?s^  
    oB!Y)f6H1  
    0U/[hG"DKN  
    z = y; &qPezyt  
    if any(idx_pos) un!v1g9O  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); V;RgO}  
    end 2V% z=  
    if any(idx_neg) %U}6(~  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); H;_Ce'oU(  
    end t\QLj&h}E  
    "3]}V=L<5  
    4qE4 i:b  
    % EOF zernfun MC,Qv9m  
     
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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)  <T.R%Jys  
    q\!"FDOl4  
    DDE还是手动输入的呢? tQ'R(H`  
    Qz{Vl> "  
    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