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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, ]w>o=<?b  
    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, ^wWbW&<Tg  
    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? W;=Ae~  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? 2<B'PR-??y  
    3%5YUG@  
    FHU6o910  
    P~{8L.w!>W  
    gZ^Qt.6Z  
    function z = zernfun(n,m,r,theta,nflag) (o IGp  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. V6P-?Nd  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N cQhr{W,Un  
    %   and angular frequency M, evaluated at positions (R,THETA) on the :p}8#rb  
    %   unit circle.  N is a vector of positive integers (including 0), and CR'%=N04^  
    %   M is a vector with the same number of elements as N.  Each element "g5{NjimY  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) f%.Ngf9  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, xrvM}Il  
    %   and THETA is a vector of angles.  R and THETA must have the same g|]HS4y  
    %   length.  The output Z is a matrix with one column for every (N,M) I4jRz*Ufe?  
    %   pair, and one row for every (R,THETA) pair. pml33^*<U  
    % e^\e;>Dh>  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike y& yf&p  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), V($V8P/  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral *'{-!Y  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, lh'S_p8g  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized <$e|'}>A  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 24#qg '  
    % @T\n@M]  
    %   The Zernike functions are an orthogonal basis on the unit circle. #}y8hzS$  
    %   They are used in disciplines such as astronomy, optics, and JXJ+lZmsz  
    %   optometry to describe functions on a circular domain. :CE4< {V  
    % a)ry}E =f  
    %   The following table lists the first 15 Zernike functions. 70 7( LG  
    % '+_>PBOc  
    %       n    m    Zernike function           Normalization gEj#>=s  
    %       -------------------------------------------------- WuU wd#e  
    %       0    0    1                                 1 5_'lu  
    %       1    1    r * cos(theta)                    2 J;obh.}u"{  
    %       1   -1    r * sin(theta)                    2 To>,8E+GAb  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) RX>P-vp  
    %       2    0    (2*r^2 - 1)                    sqrt(3) }?9&xVh?\  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) *z VN6wG{  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) zw+aZDcV(  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) =p'+kS+  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) QKj0~ia 5  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) RJ3oI+gI  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) t>cGfA  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Ldjz-  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) /dYv@OU?  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) VdK%m`;2  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) 3>1^$0iq  
    %       -------------------------------------------------- pjFO0h_Y  
    % Y'|,vG  
    %   Example 1: xW`y7Q}p  
    % z/{X{+Z  
    %       % Display the Zernike function Z(n=5,m=1) e7U\gtZ.  
    %       x = -1:0.01:1; v~Q'm1!O4\  
    %       [X,Y] = meshgrid(x,x); uAPVR  
    %       [theta,r] = cart2pol(X,Y); 7l69SQo]?  
    %       idx = r<=1; vt#;j;liG  
    %       z = nan(size(X)); B}d&tH2^s  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); w2nReB z  
    %       figure ,_3hbT8Q  
    %       pcolor(x,x,z), shading interp @zg}x0]  
    %       axis square, colorbar tON>wmN  
    %       title('Zernike function Z_5^1(r,\theta)') R(~wSL*R>  
    % ^OY]Y+S`Ox  
    %   Example 2: 2cYBm^o|x  
    % >u$8Z  
    %       % Display the first 10 Zernike functions s7Agr!>f  
    %       x = -1:0.01:1; C.jWT1  
    %       [X,Y] = meshgrid(x,x); sP(+Z^/  
    %       [theta,r] = cart2pol(X,Y); #Lhv=0op  
    %       idx = r<=1; rR.It,,  
    %       z = nan(size(X)); Xi&J%N'  
    %       n = [0  1  1  2  2  2  3  3  3  3]; bT.q@oU  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; y'_8b=*  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; -@#w)  
    %       y = zernfun(n,m,r(idx),theta(idx)); .hat!Tt9  
    %       figure('Units','normalized') Yb+A{`  
    %       for k = 1:10 T0w_d_aS  
    %           z(idx) = y(:,k); D`LBv,n  
    %           subplot(4,7,Nplot(k)) P"vrYom  
    %           pcolor(x,x,z), shading interp n[ B~C  
    %           set(gca,'XTick',[],'YTick',[]) sT\:**  
    %           axis square yn62NyK  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) T`EV uRJ  
    %       end (9'^T.J  
    % 7N9NeSH  
    %   See also ZERNPOL, ZERNFUN2. }g}Eh>U  
    CFaY=Cy  
    !$Nj!  
    %   Paul Fricker 11/13/2006 bU! v  
    <gp?}Lk  
    TLdlPBnr8  
    s\ -,RQ1  
    po*G`b;v  
    % Check and prepare the inputs: _VrY7Mz:r  
    % ----------------------------- \/NF??k,jk  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) T D _@0Rd  
        error('zernfun:NMvectors','N and M must be vectors.') Q7s@,c!m_  
    end  js_`L#t  
    [oLV,O|s|j  
    Gnkar[oa&  
    if length(n)~=length(m) Kw -SOFE  
        error('zernfun:NMlength','N and M must be the same length.') 5>x_G#W  
    end k +-w%  
    ?5C'9 V  
    }E 'r?N  
    n = n(:); #4^d#Gj  
    m = m(:); >@YefNX6  
    if any(mod(n-m,2)) _;1{feR_  
        error('zernfun:NMmultiplesof2', ... ,;)ZF  
              'All N and M must differ by multiples of 2 (including 0).') 7>E.0DP  
    end "z~ba>,-\  
     ?%,NOX  
    *M.xVUPr  
    if any(m>n) V0nQmsP1U  
        error('zernfun:MlessthanN', ... \|;\  
              'Each M must be less than or equal to its corresponding N.') +hxG!o?O  
    end Wq1>Bj$J8  
    NX @FUct;  
    ZaFt4#  
    if any( r>1 | r<0 ) %M(RV_R+6  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') ^#/FkEt7bp  
    end *%j$i_  
    4DA34m(  
    XjX  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 7;'33Bm*  
        error('zernfun:RTHvector','R and THETA must be vectors.') >L7s[vKn  
    end 'YL[s  
    _P;D.>?  
    (`P\nnb  
    r = r(:); yYG<tUG;  
    theta = theta(:); &gXh:.  
    length_r = length(r); c`a(  
    if length_r~=length(theta) R@vcS=m7  
        error('zernfun:RTHlength', ... %Sr+D{B  
              'The number of R- and THETA-values must be equal.') ]R__$fl`8  
    end Tg\bpLk0=  
    K@D\5s|1|  
    zsFzg.$3&  
    % Check normalization: Cm}2>eH  
    % -------------------- {MUB4-@?F$  
    if nargin==5 && ischar(nflag) %oZ:Awx  
        isnorm = strcmpi(nflag,'norm'); 0 'QWa{dS\  
        if ~isnorm Uzu6>yT  
            error('zernfun:normalization','Unrecognized normalization flag.')  <wH+\  
        end T<AT&4  
    else ]-fkmnmWX  
        isnorm = false; 4=zs&   
    end zkQ[<  
    _VtQMg|u  
    .HqFdsm  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% "}4%vZz  
    % Compute the Zernike Polynomials !rvEo =^  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% )Fw/Cu  
    V~J5x >O  
    / HTY>b  
    % Determine the required powers of r: 2-&EkF4p'  
    % ----------------------------------- `8:0x?X  
    m_abs = abs(m); $pGT1oF[E  
    rpowers = []; MK$u }G  
    for j = 1:length(n) .L'w/"O  
        rpowers = [rpowers m_abs(j):2:n(j)]; 8[^'PIz  
    end i!wU8 @  
    rpowers = unique(rpowers); }aCa2%  
    F L0uY0K  
    7nZPh3%  
    % Pre-compute the values of r raised to the required powers, q'2vE;z Kb  
    % and compile them in a matrix: yU?jmJ  
    % ----------------------------- !3ggQG!e  
    if rpowers(1)==0 NkE0S`Xf  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ,Kit@`P%  
        rpowern = cat(2,rpowern{:}); \bA Yic  
        rpowern = [ones(length_r,1) rpowern]; `?Rq44=  
    else (~T*yH ~  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); t^t% >9o  
        rpowern = cat(2,rpowern{:}); XR5KJl  
    end 2_o#Gx'  
    cs9^&N:w[  
    q}1ZuK`6  
    % Compute the values of the polynomials: @NHh- &;w  
    % -------------------------------------- {7o#Ve  
    y = zeros(length_r,length(n)); 4ls:BO;k]  
    for j = 1:length(n) Ic& h8vSU  
        s = 0:(n(j)-m_abs(j))/2; i;[y!U  
        pows = n(j):-2:m_abs(j); p7?  
        for k = length(s):-1:1 G)3I+uxn  
            p = (1-2*mod(s(k),2))* ... +2tQ FV;  
                       prod(2:(n(j)-s(k)))/              ... 5{Cz!ut;tE  
                       prod(2:s(k))/                     ... md!6@)S-p  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... +SJ.BmT  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); dWqn7+:  
            idx = (pows(k)==rpowers); |s|}u`(@9  
            y(:,j) = y(:,j) + p*rpowern(:,idx); X1L@ G  
        end ~z,o):q1 }  
         nK&]8"  
        if isnorm L9x-90'q,  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 5J5si<v25  
        end K*6"c.D  
    end 4<s.|W`  
    % END: Compute the Zernike Polynomials 9KSi-2?H  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% xad`-vw  
    @=J|%NO  
    '<Z[e`/  
    % Compute the Zernike functions: @Mk`Tl  
    % ------------------------------ ]B8 A  
    idx_pos = m>0; q76POytV|  
    idx_neg = m<0; MvVpp;bd  
    R>' %}|v/  
    h}b:-a  
    z = y; VYyija:  
    if any(idx_pos) Z`Yt~{,Q  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); x2"iZzQlD  
    end 50UdY9E_v}  
    if any(idx_neg) GW2\YU^{  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 18g_v"6o  
    end _03?XUKV  
    :t?B)  
    HL)!p8UHJ  
    % EOF zernfun 8^mE<  
     
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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)  $8^Hk xy  
    bY_'B5$.^2  
    DDE还是手动输入的呢? --h\tj\U  
    Dfs^W{YA  
    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