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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, 0_]aF8j  
    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, V GvOwd)E  
    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? S?3{G@!  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? a|s=d  
    ,}2j Fb9z4  
    H>7!+&M  
    t3s}U@(C  
     zIAMM  
    function z = zernfun(n,m,r,theta,nflag) ~r>UjC_ B:  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. 41uS r 1  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N @pS[_!EqYz  
    %   and angular frequency M, evaluated at positions (R,THETA) on the (/KF;J^M  
    %   unit circle.  N is a vector of positive integers (including 0), and mMjVbeh[  
    %   M is a vector with the same number of elements as N.  Each element }E1Eq  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) v'@LuF'e8  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, 7I44BC*R~  
    %   and THETA is a vector of angles.  R and THETA must have the same ah<f&2f  
    %   length.  The output Z is a matrix with one column for every (N,M) [cW  
    %   pair, and one row for every (R,THETA) pair. ^X;>?_Bk  
    % *{Z!m@?  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 87>Qw,r  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), -"nYCF  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral +es6c')  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, `fV$'u  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized 6?iP z?5  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. <A&R%5Vs  
    % VN".NEL  
    %   The Zernike functions are an orthogonal basis on the unit circle. S8cFD):q  
    %   They are used in disciplines such as astronomy, optics, and o{Ep/O`  
    %   optometry to describe functions on a circular domain. iGlZFA  
    % 1lQ1 0J  
    %   The following table lists the first 15 Zernike functions. W P&zF$  
    % {2Ibd i  
    %       n    m    Zernike function           Normalization ;C<A }  
    %       -------------------------------------------------- CKur$$B  
    %       0    0    1                                 1 W!^=)Qs  
    %       1    1    r * cos(theta)                    2 l`]!)j|+  
    %       1   -1    r * sin(theta)                    2 qs b4@jt+  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) _L72Ae(_  
    %       2    0    (2*r^2 - 1)                    sqrt(3) igL^k`&5^"  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) CUG<v3\  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) 1GdgF?4  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) s#fmGe"8  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) TDGzXJf[  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) ~>R)H#mP7  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) 5~F0'tb|}  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) b#U%aPH  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) c 1GP3  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) *~L]n4-  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) BYf"l8^,  
    %       -------------------------------------------------- lTP02|eK  
    % e- CW4x  
    %   Example 1: iD`XD\.?  
    % k}.nH"AQ  
    %       % Display the Zernike function Z(n=5,m=1) u2Obb`p S  
    %       x = -1:0.01:1; q}i87a;m  
    %       [X,Y] = meshgrid(x,x); (jG$M=q-  
    %       [theta,r] = cart2pol(X,Y); F)w83[5_d  
    %       idx = r<=1; _JDr?Kg  
    %       z = nan(size(X)); D=vq<X'  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); .#J3UZ  
    %       figure CQ[-Cp7  
    %       pcolor(x,x,z), shading interp 6hq)yUvo4  
    %       axis square, colorbar 1aG}-:$t'  
    %       title('Zernike function Z_5^1(r,\theta)') %R>S"  
    % OEW,[d  
    %   Example 2: >cb gL%  
    % A'8K^,<  
    %       % Display the first 10 Zernike functions (c2\:hvy  
    %       x = -1:0.01:1; L />GYx  
    %       [X,Y] = meshgrid(x,x); #VE$C3<  
    %       [theta,r] = cart2pol(X,Y); Se`N5hQ  
    %       idx = r<=1; z-G (!]:  
    %       z = nan(size(X)); R.T-Ptene  
    %       n = [0  1  1  2  2  2  3  3  3  3]; 1n"X?K5;A  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; Se8y-AL6x>  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; 6%#'X  
    %       y = zernfun(n,m,r(idx),theta(idx)); #^ #i]{g  
    %       figure('Units','normalized') )M 0O=Cl1  
    %       for k = 1:10 yFo5pKF.J  
    %           z(idx) = y(:,k); jYz3(mM'J  
    %           subplot(4,7,Nplot(k)) eb\`)MI/  
    %           pcolor(x,x,z), shading interp QO/7p]$_  
    %           set(gca,'XTick',[],'YTick',[]) xk8p,>/  
    %           axis square \k_3IP?o=  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) >}*jsqaVU  
    %       end OvG0UXRU  
    % %U7f9  
    %   See also ZERNPOL, ZERNFUN2. Wp3l>:  
    @\8gzvkt  
    8-ssiiJ}gh  
    %   Paul Fricker 11/13/2006 jt--w"|-r  
    o7XRa]O  
    EIQy?ig86  
    sLp LY1X  
    YO0x68  
    % Check and prepare the inputs: |l(lrJ{  
    % ----------------------------- ]#.&f]6l  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) t|QMS M?s  
        error('zernfun:NMvectors','N and M must be vectors.') (Nb1R"J `  
    end U&$]?3?  
    MToQ8qKs  
    [frD L)  
    if length(n)~=length(m) 9z/_`Xd_  
        error('zernfun:NMlength','N and M must be the same length.') 5q`)jd!*)  
    end {Y%=/ba W  
    0\nhg5]?  
    F$ p*G][  
    n = n(:); >%dAqYi $  
    m = m(:); B1#>$"_0}=  
    if any(mod(n-m,2)) G3^]Wwu  
        error('zernfun:NMmultiplesof2', ... mm<iT59  
              'All N and M must differ by multiples of 2 (including 0).') u>6/_^iq  
    end 1>x@1Mo+K  
    -xIhN?r)  
    D@W3;T^  
    if any(m>n) _hL4@ C  
        error('zernfun:MlessthanN', ... ,nRwwFd.  
              'Each M must be less than or equal to its corresponding N.') XPo'iI-  
    end k]9>V@C  
    @M^Qh Hs  
    VhIIW"1  
    if any( r>1 | r<0 ) %]$p ^m  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') T)tHN#6I  
    end Pt0}9Q  
    ^GaPpm  
    DB'KIw  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) @/NZ>.  
        error('zernfun:RTHvector','R and THETA must be vectors.') [mzF)/[_2  
    end LEnP"o9ZW  
    Mqf}Aiqk;  
    O[p^lr(B7  
    r = r(:); _m?(O/BTx  
    theta = theta(:); x&oBO{LNK,  
    length_r = length(r); L8xprHgL  
    if length_r~=length(theta) AaC1 ||?R  
        error('zernfun:RTHlength', ... M #=5u`h  
              'The number of R- and THETA-values must be equal.') 4U;XqUY /  
    end C*6)Ut '  
    2$W,R/CLh  
    'Qq_Xn8  
    % Check normalization: UMi`u6#  
    % -------------------- iA{jKk=  
    if nargin==5 && ischar(nflag) 7RC096 ?}  
        isnorm = strcmpi(nflag,'norm'); ~nc([%!=  
        if ~isnorm *[~o~e/YCb  
            error('zernfun:normalization','Unrecognized normalization flag.') 4FE@s0M,  
        end t:s q*d  
    else =*:_swd  
        isnorm = false; bKMR7&e.Ep  
    end bd P,Zqd  
    !5SQN5K  
    <eFAI}=s  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Y{2d4VoW6  
    % Compute the Zernike Polynomials 5h=TV  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% q(tG bhQ  
    OC>_=i$ '  
    r{2].31'  
    % Determine the required powers of r: $EGRaps{j>  
    % ----------------------------------- e=jT]i*cU  
    m_abs = abs(m); [H:GKhPC`  
    rpowers = []; 3) c K*8#  
    for j = 1:length(n) ssyd8LC#  
        rpowers = [rpowers m_abs(j):2:n(j)]; ]F* a PV  
    end +=~%S)9F  
    rpowers = unique(rpowers); @?7{%j*  
    i$^B-  
    X*9N[#wu6  
    % Pre-compute the values of r raised to the required powers, SM /ykk  
    % and compile them in a matrix: fxoi<!|iGY  
    % ----------------------------- dbuJ~?D,  
    if rpowers(1)==0 |*c\6 :  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 7kX$wQZ_  
        rpowern = cat(2,rpowern{:}); Am4^v?q  
        rpowern = [ones(length_r,1) rpowern]; zIm_7\e  
    else vG<pc_ak  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 7Cd_zZ  
        rpowern = cat(2,rpowern{:}); g;!@DVF$  
    end @6 ;oN  
    MYw8wwX0kJ  
    5%Oyvt]}2  
    % Compute the values of the polynomials: Ph#F<e(9  
    % -------------------------------------- 24jtJC,7  
    y = zeros(length_r,length(n)); ImV]}M~_  
    for j = 1:length(n) K%(XgXb(</  
        s = 0:(n(j)-m_abs(j))/2; I' ! r  
        pows = n(j):-2:m_abs(j); Z2rzb{oS}  
        for k = length(s):-1:1 T4~`e_  
            p = (1-2*mod(s(k),2))* ... mYh5#E41J  
                       prod(2:(n(j)-s(k)))/              ... U7B/t3,=U  
                       prod(2:s(k))/                     ... M,t*nG  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... p>:ef<.i  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); K4k~r!&OU  
            idx = (pows(k)==rpowers); Gh2Q$w:  
            y(:,j) = y(:,j) + p*rpowern(:,idx); Zv|p>q`R2  
        end 4j@i%  
         ) Kc%8hBv  
        if isnorm @ 2!C^}d3F  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); fzS`dL5,W  
        end -!|WZ   
    end Z%+BWS3YqY  
    % END: Compute the Zernike Polynomials `D)Lzm R  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -.{oqs$  
    BeI;#m0  
    %0yS98']g  
    % Compute the Zernike functions: $Ei o$TI  
    % ------------------------------ +:>JZ$  
    idx_pos = m>0; Ts(t:^  
    idx_neg = m<0; oe!:|ck<  
    y7JZKtsFA  
    `k(u:yGK  
    z = y; l801` ~*gO  
    if any(idx_pos) Sk/#J!T8{  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); \T`["<  
    end U!c]_q  
    if any(idx_neg) ,M) k7t:  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 6x_ T@  
    end WHUT/:?f  
    ?C &x/2lt  
    I#U>5"%\a  
    % EOF zernfun {)V?R  
     
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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)  3y ryeS  
    =h|wwQE  
    DDE还是手动输入的呢? g`[`P@  
    >Q=Ukn;k  
    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