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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, yU7I;]YP  
    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, 62kb2C  
    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? j[XYj6*d  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? >vujZw_0>  
    qS.)UaA  
    n3ZAF'  
    RtC'v";6  
    <MdGe1n  
    function z = zernfun(n,m,r,theta,nflag) `f)(Y1%.  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. ArzDI{1  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N h/<=u9J  
    %   and angular frequency M, evaluated at positions (R,THETA) on the os$nL'sq  
    %   unit circle.  N is a vector of positive integers (including 0), and NfwYDY  
    %   M is a vector with the same number of elements as N.  Each element \H4U8)l  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) 4x,hj  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, q4ipumy*  
    %   and THETA is a vector of angles.  R and THETA must have the same XoItV  
    %   length.  The output Z is a matrix with one column for every (N,M) 9?EVQ  
    %   pair, and one row for every (R,THETA) pair. |nY~ZVTt/  
    % mp\%M 1<  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike )~ z Z'^  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), V=}1[^  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral >F3.c%VU]w  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, A#DR9Eq  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized |RhM| i  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. r,xmEj0E  
    % 6GINmkA  
    %   The Zernike functions are an orthogonal basis on the unit circle. vM4<d>  
    %   They are used in disciplines such as astronomy, optics, and Bo r7]#  
    %   optometry to describe functions on a circular domain. {/}^D-  
    % r{[OJc!  
    %   The following table lists the first 15 Zernike functions. oT&m4I  
    % , 2`~ NPb  
    %       n    m    Zernike function           Normalization (C S8(C4[  
    %       -------------------------------------------------- SDBt @=Nl  
    %       0    0    1                                 1 #w>~u2W  
    %       1    1    r * cos(theta)                    2 )q3"t2-  
    %       1   -1    r * sin(theta)                    2 u7=T(4a  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) &5Y_>{,  
    %       2    0    (2*r^2 - 1)                    sqrt(3) - k`.j  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) it1/3y =]  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) s@!$='|  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) YG[w@u  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) &[j]Bp?  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) ?CY1]d  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) 8eyl,W=dn  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) [e e30ELn  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) NK/4OAt%  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) WY>Knp=  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) FtIa*j^G  
    %       -------------------------------------------------- fKkjn4&W  
    % /1fwl5\  
    %   Example 1: R^8{bP  
    % y=H@6$2EQ  
    %       % Display the Zernike function Z(n=5,m=1) U<bYFuS"  
    %       x = -1:0.01:1; l [%lE  
    %       [X,Y] = meshgrid(x,x); /fwgqFVk  
    %       [theta,r] = cart2pol(X,Y); =+oZtP-+o  
    %       idx = r<=1; gx;O6S{  
    %       z = nan(size(X)); tZho)[1  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); %Q4i%:Qi  
    %       figure {THqz$KN  
    %       pcolor(x,x,z), shading interp &s VadOBQ  
    %       axis square, colorbar G]*|H0j  
    %       title('Zernike function Z_5^1(r,\theta)') 6 bO;&  
    % U5+vN[ K  
    %   Example 2: 4JO@BV>t  
    % |_zO_Frtp  
    %       % Display the first 10 Zernike functions ;BBpN`T  
    %       x = -1:0.01:1; @8gEH+r  
    %       [X,Y] = meshgrid(x,x); ^:cRp9l"7  
    %       [theta,r] = cart2pol(X,Y); }5bM1h#z  
    %       idx = r<=1; |~e?,[-2`r  
    %       z = nan(size(X)); w~+aW(2  
    %       n = [0  1  1  2  2  2  3  3  3  3]; {#hVD4$b  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; t9u|iTY f!  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; 8MF2K6  
    %       y = zernfun(n,m,r(idx),theta(idx)); -s"0/)HD  
    %       figure('Units','normalized') ?<~WO?  
    %       for k = 1:10 b^Cfhy^RTq  
    %           z(idx) = y(:,k); n1J]p#nCa.  
    %           subplot(4,7,Nplot(k)) 2`Gv5}LfyR  
    %           pcolor(x,x,z), shading interp NFyMY#\]  
    %           set(gca,'XTick',[],'YTick',[]) !OE*z $\  
    %           axis square V4K'R2t  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) $>w/Cy  
    %       end &/WAZs$2n  
    % (tCib 4  
    %   See also ZERNPOL, ZERNFUN2. f/ahwz  
    ijW 7c+yd  
    4hAJ!7[A.  
    %   Paul Fricker 11/13/2006 S; /. %  
    oXgdLtsu  
    OJ3UE(,I=  
    Eipp ~GD  
    qd6fU^)i  
    % Check and prepare the inputs: =m tY  
    % ----------------------------- n-afDV  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) + ^ yq;z  
        error('zernfun:NMvectors','N and M must be vectors.') id,NONb\  
    end )K0i@hM(n  
    wCb(>pL0  
    =G${[V \  
    if length(n)~=length(m) hIU(P Dl4  
        error('zernfun:NMlength','N and M must be the same length.') Yl({)qK{  
    end ;YH[G;aJ  
    qqOFr!)g  
    Z4q~@|+%  
    n = n(:); HW6.O|3  
    m = m(:); j1U 5~%^  
    if any(mod(n-m,2)) r"wtZ]69  
        error('zernfun:NMmultiplesof2', ... mP^SS Je  
              'All N and M must differ by multiples of 2 (including 0).') p3]Q^KFS  
    end ]<trA$ 0  
    T|){<  
    I.V:q!4*  
    if any(m>n) h @/;`E[  
        error('zernfun:MlessthanN', ... V3sL;  
              'Each M must be less than or equal to its corresponding N.') i[nF.I5*f  
    end WES#ZYtT  
    6mPm=I[oh  
    S~yR5cb  
    if any( r>1 | r<0 ) ejePDgi_[  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') }31Z X  
    end r4{<Z3*N  
    2F+gF~znQ  
    9 [Y-M  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) N4{nG,Mo]  
        error('zernfun:RTHvector','R and THETA must be vectors.') P3o @gkXP  
    end (q;bg1\UK  
    -Zc![cAlO  
    Op}ZB:  
    r = r(:); !xK=#pa  
    theta = theta(:); PuCc2'#  
    length_r = length(r); m&Yi!7@(  
    if length_r~=length(theta) x]4Kkpqm  
        error('zernfun:RTHlength', ... +t!S'|C  
              'The number of R- and THETA-values must be equal.') %s=Dj2+  
    end 8OFj0S1r`  
    `jsEN ;<  
    (Tvcq  
    % Check normalization: o(G"k  
    % -------------------- M\oVA=d\0  
    if nargin==5 && ischar(nflag) l54 m22pfv  
        isnorm = strcmpi(nflag,'norm'); dl|gG9u4Q  
        if ~isnorm W`)<vGn=Y  
            error('zernfun:normalization','Unrecognized normalization flag.') Le#spvV3J|  
        end ([E]_Q  
    else /iQ(3F  
        isnorm = false; ^twivNB  
    end hv)8K'u  
    l|"6yB |  
    "IjI'c  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% :8f[|XR4\N  
    % Compute the Zernike Polynomials %,V YiW0  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% K~6e5D7.  
    b>=_*nw9  
    [c&B|h=>  
    % Determine the required powers of r: %JL]; 4'  
    % ----------------------------------- `: |@Zln  
    m_abs = abs(m); tY/vL^mi  
    rpowers = []; "VUYh$=[  
    for j = 1:length(n) )b4$A:  
        rpowers = [rpowers m_abs(j):2:n(j)]; 4,P bg|  
    end IApT'QNM  
    rpowers = unique(rpowers); j;nb?;  
    n=sXSxl  
    v/Pw9j!r;m  
    % Pre-compute the values of r raised to the required powers, h0|}TV^UJ  
    % and compile them in a matrix: /n5n )P@L  
    % ----------------------------- `N8 7 h"  
    if rpowers(1)==0 `C72sA{M.  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 1=VJ&D;  
        rpowern = cat(2,rpowern{:}); Z|m`7xeCy  
        rpowern = [ones(length_r,1) rpowern]; >)nS2b OE  
    else '+y_\  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); fw-\|fP  
        rpowern = cat(2,rpowern{:}); E1V^}dn  
    end Mt>oI SN&d  
    Zj9c9  
    k5BXirB  
    % Compute the values of the polynomials: ?+CV1 ]  
    % -------------------------------------- qYB~VE03  
    y = zeros(length_r,length(n)); nD6G  
    for j = 1:length(n) ](0mjE04<d  
        s = 0:(n(j)-m_abs(j))/2; 4`v!Z#e/aX  
        pows = n(j):-2:m_abs(j); @tT-JwU  
        for k = length(s):-1:1 d5m`Bm-{  
            p = (1-2*mod(s(k),2))* ... 3{7T4p.G  
                       prod(2:(n(j)-s(k)))/              ... ^4Uw8-/9  
                       prod(2:s(k))/                     ... k3Cz9Vt%  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... .IrNa>J~  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); H=c`&N7E  
            idx = (pows(k)==rpowers); mLbN/M  
            y(:,j) = y(:,j) + p*rpowern(:,idx); *|:Q%xr-  
        end v4vf }.L]  
         n> w`26MMp  
        if isnorm B;#J"6w  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); s95F#>dr  
        end :py\ |  
    end IVvtX}  
    % END: Compute the Zernike Polynomials epD?K  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ,_v|#g@{  
    lx0 ~>K]  
    #CUz uk&  
    % Compute the Zernike functions: ,^s0</v e  
    % ------------------------------ 2?7(A  
    idx_pos = m>0; Y$ Fj2nk+  
    idx_neg = m<0; k#>hg#G  
    zd %rs~*c  
    #(;<-7M2  
    z = y; dbdM"z 4  
    if any(idx_pos) } (FPV*mS  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); k-LEI}h  
    end ~W0(1# i  
    if any(idx_neg) aE VsU|  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ,T{<vRj7_  
    end vRQOs0F;  
    yJx?M  
    Zl/< w(f_  
    % EOF zernfun ^V*-1r1  
     
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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)  ^i)hm  
    lS^(&<{  
    DDE还是手动输入的呢? \vfBrN  
    /2M.~3gQ  
    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