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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, %_aMl  
    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, rJ'I>Q~x6  
    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? ZRUhAp'<qj  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? MZSxQ8  
    }T0K^Oe+eS  
    7ji=E";.w  
    }5O>EXE0R  
    v)kEyX'K2d  
    function z = zernfun(n,m,r,theta,nflag)  ql&*6KZ"  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. 0ZPV' `KGp  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ! sA_?2$  
    %   and angular frequency M, evaluated at positions (R,THETA) on the t.hm9}UQ  
    %   unit circle.  N is a vector of positive integers (including 0), and rt+..t\  
    %   M is a vector with the same number of elements as N.  Each element ])#\_' fg  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) MuEy>dl  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, QldzQ%4c\  
    %   and THETA is a vector of angles.  R and THETA must have the same x q-$\#O  
    %   length.  The output Z is a matrix with one column for every (N,M) %YlTF\-  
    %   pair, and one row for every (R,THETA) pair. ? {F{;r  
    % D 0]a\,aZ  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 2 vKx]w  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ]`w}+B'/  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral <B&R6<]T  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, snp v z1iS  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized Q\J,}1<`6  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. fd8#Ng"1  
    % 4R) |->"  
    %   The Zernike functions are an orthogonal basis on the unit circle. w3D]~&]  
    %   They are used in disciplines such as astronomy, optics, and 3rf#Q }"  
    %   optometry to describe functions on a circular domain. 9-bG<`v\E  
    % #G,XDW2"w  
    %   The following table lists the first 15 Zernike functions. mf|pNiQ,  
    % g>7Y~_}  
    %       n    m    Zernike function           Normalization Oz:ZQ M  
    %       -------------------------------------------------- Yi rC*  
    %       0    0    1                                 1 ; a/cty0Ch  
    %       1    1    r * cos(theta)                    2 X`\:_|  
    %       1   -1    r * sin(theta)                    2 kJ: 2;t=  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) K{ }4zuZ  
    %       2    0    (2*r^2 - 1)                    sqrt(3) "t&{yBQ0u  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) JFqf;3R  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) *"G8  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) VKLU0*2R  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) ]s|lxqP  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) CYB=Uq,  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) +~|AT+|iI  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) >e8JK*Blz  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) %f[Ep 3D  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ?:|YGLaB  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) ,\h YEup  
    %       -------------------------------------------------- |r~ uos  
    % 6|;0ax4:P  
    %   Example 1: z-0:m|=yH  
    % j=.g :&r)  
    %       % Display the Zernike function Z(n=5,m=1) qCJ=Z  
    %       x = -1:0.01:1; yCM{M  
    %       [X,Y] = meshgrid(x,x); "L~@.W!@  
    %       [theta,r] = cart2pol(X,Y); 9Nl* 4  
    %       idx = r<=1; 8g5V,3_6  
    %       z = nan(size(X)); ^)cM&Bx t%  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); U \Dca&=  
    %       figure T~Yg5J  
    %       pcolor(x,x,z), shading interp y-`I) w%  
    %       axis square, colorbar C"T ,MH  
    %       title('Zernike function Z_5^1(r,\theta)') rqvU8T7A  
    % .g-3e"@  
    %   Example 2: cy:;)E>/  
    % owMuT^x?  
    %       % Display the first 10 Zernike functions L/k40cEI^z  
    %       x = -1:0.01:1; C/+nSe.  
    %       [X,Y] = meshgrid(x,x); fJ :jk6@  
    %       [theta,r] = cart2pol(X,Y); S.fXHtSx  
    %       idx = r<=1; c57bf  
    %       z = nan(size(X)); A. Nz_!  
    %       n = [0  1  1  2  2  2  3  3  3  3]; +IsWI;lp  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; [n<.fw8$b  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; > I%zd/q?  
    %       y = zernfun(n,m,r(idx),theta(idx)); <jL#>L%%  
    %       figure('Units','normalized') h2}am:%mC  
    %       for k = 1:10 "X?LAo  
    %           z(idx) = y(:,k); A1!:BC  
    %           subplot(4,7,Nplot(k)) `Wwh`]#"~d  
    %           pcolor(x,x,z), shading interp O&P>x#w  
    %           set(gca,'XTick',[],'YTick',[]) >DmRP7v   
    %           axis square )n7)}xy#z  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) cJ4S!  
    %       end bf^ly6ml  
    % xXa#J)'  
    %   See also ZERNPOL, ZERNFUN2. lWl-@ *'  
    xDe47&qKM  
    8c?8X=|D7  
    %   Paul Fricker 11/13/2006 l_sg)Vr/b  
    2vXGO|W  
    (E]"Srwh  
    ,yi2O]5e>!  
    9t\ [N/  
    % Check and prepare the inputs: #y`k$20"  
    % ----------------------------- o;'4c  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) K-Y* T}?  
        error('zernfun:NMvectors','N and M must be vectors.') j) <[j&OWw  
    end ]J~g'">  
    7#/|VQX<A  
    R^?PAHE 7  
    if length(n)~=length(m) {G]`1Q1DR  
        error('zernfun:NMlength','N and M must be the same length.') 'v`~(9'Rcj  
    end Rmgxf/  
    9w$7VW;  
    jM[f[  
    n = n(:); dTgM"k  
    m = m(:); 4jD\]Q="1  
    if any(mod(n-m,2)) !%)L&W_  
        error('zernfun:NMmultiplesof2', ... 1o)=GV1  
              'All N and M must differ by multiples of 2 (including 0).') z+2u-jG  
    end oYGUjI  
    Cg?I'1]o6  
    FrgV@4'2G  
    if any(m>n) Fj48quW1\P  
        error('zernfun:MlessthanN', ... _/8y1) I  
              'Each M must be less than or equal to its corresponding N.') zh hGqz[K  
    end 2tlO"c:_/  
    \j/}rzo]  
    s=}~Q&8  
    if any( r>1 | r<0 ) gtl;P_  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') I[a%a!QO  
    end /!o1l\i=5  
    z4 nou>  
    olslzXn7o  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) (h%|;9tF  
        error('zernfun:RTHvector','R and THETA must be vectors.') =`ywd]\7  
    end s:G [Em1  
    U0ns3LirP  
    cKSfqqPm$"  
    r = r(:); nN!vgn j  
    theta = theta(:); =54Vs8.  
    length_r = length(r); Ty(yh(oYF`  
    if length_r~=length(theta) {m>~`   
        error('zernfun:RTHlength', ... re2Fv:4{  
              'The number of R- and THETA-values must be equal.') @ICejB<  
    end KX$qM g1j  
    aslNlH6  
    '&1  
    % Check normalization: QJniM"8v  
    % -------------------- FDZeIj9uF  
    if nargin==5 && ischar(nflag) dW:w<{a!R  
        isnorm = strcmpi(nflag,'norm'); oT$(<$&<  
        if ~isnorm @DUN;L 4  
            error('zernfun:normalization','Unrecognized normalization flag.') @5JLjCN  
        end {: Am9B  
    else $a)J CErN  
        isnorm = false; {EZFx,@t  
    end 0:PH[\Z  
    ?T$*5d  
    m7weR>aS4  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% dY48S{  
    % Compute the Zernike Polynomials X=-gAutfE=  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% _wIBm2UO  
    ~t1O]aO(  
    0m)-7@  
    % Determine the required powers of r: h0&>GY;i  
    % ----------------------------------- n$}R/*  
    m_abs = abs(m); K*J4&5?/  
    rpowers = []; A} x_zt  
    for j = 1:length(n) ..v@Q%  
        rpowers = [rpowers m_abs(j):2:n(j)]; 8T!fGzHx  
    end d "QM;9  
    rpowers = unique(rpowers); FCUVP,"T  
    B Lsdx }  
    S&gKgQD"Q  
    % Pre-compute the values of r raised to the required powers, ;HD 4~3   
    % and compile them in a matrix: 5#N"WHz!  
    % ----------------------------- ir( -$*J  
    if rpowers(1)==0 {Zd)U "  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); F}VS)  
        rpowern = cat(2,rpowern{:}); ^59YfC<f  
        rpowern = [ones(length_r,1) rpowern]; YL0WUD_>  
    else (25^r  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); )VV4HoH]8  
        rpowern = cat(2,rpowern{:}); +8?R+0P  
    end %M4XbSN|  
    qcpG}o+&D  
    sM);gI14  
    % Compute the values of the polynomials: =0jmm(:Jh  
    % -------------------------------------- |e.3FjTH  
    y = zeros(length_r,length(n)); '? !7 Be  
    for j = 1:length(n) w[J (E  
        s = 0:(n(j)-m_abs(j))/2; }+QhW]nO{F  
        pows = n(j):-2:m_abs(j); Q8M:7#ySji  
        for k = length(s):-1:1 Ah8^^h|TPJ  
            p = (1-2*mod(s(k),2))* ... r P<d[u  
                       prod(2:(n(j)-s(k)))/              ... `CTkx?e[  
                       prod(2:s(k))/                     ... Y3sNr)qss  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 6@,'m  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); DLg`Q0`M5  
            idx = (pows(k)==rpowers); zO7lsx2 =  
            y(:,j) = y(:,j) + p*rpowern(:,idx); 2s]]!{Z#  
        end *h5ldP  
         y~#R:&d"  
        if isnorm H *z0xxa  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); hhh: rmEZl  
        end ;_Of`C+  
    end )0 42?emn  
    % END: Compute the Zernike Polynomials fjz2m   
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% zd*W5~xKg  
    }. Na{]<gh  
    ] _]6&PZXk  
    % Compute the Zernike functions: OJC*|kN-#^  
    % ------------------------------ Jte:l:yjtA  
    idx_pos = m>0; [/#k$-  
    idx_neg = m<0; <or>bo^  
    b|V4Fp  
    ,& pF:ql F  
    z = y; g)zn.]  
    if any(idx_pos) hjm .Ath  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); x:&L?eOT  
    end F%ylR^H>  
    if any(idx_neg) !_/8!95  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ck4T#g;=  
    end Sv^'CpQ  
    1rvf\[  
    51FK~ 5  
    % EOF zernfun =+sIX3  
     
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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)  MxA'T(Ay  
    qvT+d l3#[  
    DDE还是手动输入的呢? tZ24}~da  
    6i=wAkn_J  
    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