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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, 0B1*N_.L@  
    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, S 8h/AW6l  
    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? <;SMczR  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? xdp{y =,[  
    gwR ^Z{  
    `h :&H,N  
    (!{_O_&  
    1 dI  
    function z = zernfun(n,m,r,theta,nflag) w doA>a?q  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. pk(<],0]X  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N A^%z;( 0p  
    %   and angular frequency M, evaluated at positions (R,THETA) on the OsvAm'B  
    %   unit circle.  N is a vector of positive integers (including 0), and D OPOzh  
    %   M is a vector with the same number of elements as N.  Each element >0:h(,?V  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) BI,K?D&W-  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, (/Z~0hA[Q  
    %   and THETA is a vector of angles.  R and THETA must have the same az0( 54M  
    %   length.  The output Z is a matrix with one column for every (N,M) ~F>oNbJIv  
    %   pair, and one row for every (R,THETA) pair. B>#zrCD  
    % 8uS1HE\%  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike #C4  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), VLu_SXlo*  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral M)Tv(7  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, D-A#{e _  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized m7^a4  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. Lm:O vVVB  
    % GAtK1%nPD  
    %   The Zernike functions are an orthogonal basis on the unit circle. u\&oiwSIP  
    %   They are used in disciplines such as astronomy, optics, and $* 8c0.{U  
    %   optometry to describe functions on a circular domain. lb`P9mbr+  
    % sVaWg?=qs'  
    %   The following table lists the first 15 Zernike functions. JB''Ujyi  
    % ^fXNeBj  
    %       n    m    Zernike function           Normalization ~$!eB/6ty  
    %       -------------------------------------------------- SU2 (XP]5  
    %       0    0    1                                 1 1:q55!b  
    %       1    1    r * cos(theta)                    2 RAXqRP,iw  
    %       1   -1    r * sin(theta)                    2 -!(3fO:  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) B;hc|v{(  
    %       2    0    (2*r^2 - 1)                    sqrt(3) zO9|s}J8q  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) f1hi\p0q  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) +J_A *B  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) 1\kOjF)l  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) zZki9P   
    %       3    3    r^3 * sin(3*theta)             sqrt(8) u%VO'}Gz  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) 0MrtJNF]_O  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ?VS {,"X  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) wToz{!n  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) _6^vxlF  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) dGP*bMCT  
    %       -------------------------------------------------- 4U C/pGZY  
    % \qV5mD]"M  
    %   Example 1: /$&~0pk  
    % T* -*U /  
    %       % Display the Zernike function Z(n=5,m=1) 4xe:+sA.N  
    %       x = -1:0.01:1; </:f-J%U/  
    %       [X,Y] = meshgrid(x,x); /=,^fCCN  
    %       [theta,r] = cart2pol(X,Y); 9SC#N 5V  
    %       idx = r<=1; @ g~kp  
    %       z = nan(size(X)); G/2@ Mn-  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); P}DrUND  
    %       figure Uu>YE0/)  
    %       pcolor(x,x,z), shading interp !ny; YV  
    %       axis square, colorbar $-M1<?5  
    %       title('Zernike function Z_5^1(r,\theta)') XuoI19V[  
    % kh^AH6{2  
    %   Example 2: 6(D K\58  
    % s2b!Nib  
    %       % Display the first 10 Zernike functions *z` {$hc  
    %       x = -1:0.01:1; :}UWy?F  
    %       [X,Y] = meshgrid(x,x); 5(u7b  
    %       [theta,r] = cart2pol(X,Y); QbxjfW"/+  
    %       idx = r<=1; ;9=9D{-4+  
    %       z = nan(size(X)); c^A3|tCi  
    %       n = [0  1  1  2  2  2  3  3  3  3]; IOvYvFUUJ  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; *G'zES0x  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; <kPU*P,  
    %       y = zernfun(n,m,r(idx),theta(idx)); )1~4Tl,S  
    %       figure('Units','normalized') zRsT6u  
    %       for k = 1:10 scJ`oc: <J  
    %           z(idx) = y(:,k); E I)Pfx"0  
    %           subplot(4,7,Nplot(k)) 2=(=Wjk.  
    %           pcolor(x,x,z), shading interp ehO F@IA_  
    %           set(gca,'XTick',[],'YTick',[]) }I#;~|v~<  
    %           axis square HP*x?|4  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 0*B_$E06  
    %       end [-s0'z  
    % e`<=& w  
    %   See also ZERNPOL, ZERNFUN2. s:jr/ j!  
    T 7Lk4cU  
    .fU qsq  
    %   Paul Fricker 11/13/2006 K )KE0/ n  
    s/`4]B;2U  
    Uc<B)7{'  
    ',*I=JW;  
    i*9eU*i|H  
    % Check and prepare the inputs: a!Z,~ V8  
    % ----------------------------- $T1 D ?X  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 7:mM`0g!  
        error('zernfun:NMvectors','N and M must be vectors.') 04WKAP'c N  
    end PX\}lTJ  
    Rj^bZ%t  
    M\e%GJ0  
    if length(n)~=length(m) 9i,QCA  
        error('zernfun:NMlength','N and M must be the same length.') ]1abz:  
    end r,[vXxMy(;  
    6LNm>O  
    7 82NiVed  
    n = n(:); 9.#\GI ;  
    m = m(:); Lo7R^>  
    if any(mod(n-m,2)) `"A\8)6-  
        error('zernfun:NMmultiplesof2', ... @6h=O`X>  
              'All N and M must differ by multiples of 2 (including 0).') y9Yh%M(  
    end Uu }ai."iB  
    S>*i^If  
    jW?.>(  
    if any(m>n) .~ZNlI {K  
        error('zernfun:MlessthanN', ... -[0)n{AVvU  
              'Each M must be less than or equal to its corresponding N.') ldI;DoE#U1  
    end 4K[U*-\"  
    Ct$e`H!;  
    Ks8S^77  
    if any( r>1 | r<0 )  {hZ_f3o  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') QmT]~4PqS  
    end -UUP hGC  
    }"Hf/{E$_"  
    N}>`Xm 5'  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) )Qp?N<&'  
        error('zernfun:RTHvector','R and THETA must be vectors.') \qNj?;B  
    end Y;xVB" (  
    {xr4CDP  
    #RlI([f|&  
    r = r(:); v)okVyv  
    theta = theta(:); 3MNo&0M9  
    length_r = length(r); .OX.z~":y  
    if length_r~=length(theta) \Ao M'+  
        error('zernfun:RTHlength', ... xh_6@}D2J  
              'The number of R- and THETA-values must be equal.') +\\,FO_  
    end |v[{k>7f  
    h+t{z"Ic=  
    |a3)U%rUEQ  
    % Check normalization: nFX8:fZ$>  
    % -------------------- ~O 65=8  
    if nargin==5 && ischar(nflag) EAj2uV  
        isnorm = strcmpi(nflag,'norm'); `fY~Lv{4d_  
        if ~isnorm iW.8+?Xq&  
            error('zernfun:normalization','Unrecognized normalization flag.') [fxAj]  
        end qZ6P(5X  
    else o*'J8El\y^  
        isnorm = false; @m1vB!  
    end H2E!A2\m  
    |XLx6E2F  
    5?kF'yksR  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% zw7=:<z=  
    % Compute the Zernike Polynomials V78QV3  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% C8-4 m68"  
    t?QR27cs$  
    $X9-0-  
    % Determine the required powers of r: xzz[!yJjG  
    % ----------------------------------- ]y2(ZTNTs  
    m_abs = abs(m); ;ZFn~!V  
    rpowers = []; RUlM""@b  
    for j = 1:length(n) |A 8xy#  
        rpowers = [rpowers m_abs(j):2:n(j)]; hg]\~#&-  
    end l {\~I  
    rpowers = unique(rpowers); dAm( uJ  
    `.#e4 FBW  
    ^z "90-V^  
    % Pre-compute the values of r raised to the required powers, 8ooj)  
    % and compile them in a matrix: XB50>??NE  
    % ----------------------------- P%ev8]2  
    if rpowers(1)==0 kzbgy)PK3  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); bJx{mq  
        rpowern = cat(2,rpowern{:}); M})2y+  
        rpowern = [ones(length_r,1) rpowern]; WG1Uv PK  
    else 5owUQg,W  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); K0g<11}(Yg  
        rpowern = cat(2,rpowern{:}); y4C_G?  
    end oz(<e  
    ,xn+T)2I  
    *h-_   
    % Compute the values of the polynomials: zq8 z#FN  
    % -------------------------------------- 4IG'T m  
    y = zeros(length_r,length(n)); y9=/kFPRm  
    for j = 1:length(n) B&0-~o3WP  
        s = 0:(n(j)-m_abs(j))/2; BBnj}XP*4  
        pows = n(j):-2:m_abs(j); ZgcA[P  
        for k = length(s):-1:1 Yih^ZTf]O?  
            p = (1-2*mod(s(k),2))* ... z%hB=V!~91  
                       prod(2:(n(j)-s(k)))/              ... ]mn(lK  
                       prod(2:s(k))/                     ... Fm#4;'x5E  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... pV=X  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); vAy`8Q  
            idx = (pows(k)==rpowers); #?@k=e\  
            y(:,j) = y(:,j) + p*rpowern(:,idx); ujXC#r&  
        end L @_IGH  
         bO>Mvf  
        if isnorm  =SRp  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); S"!nM]2L  
        end ([qw#!;w;  
    end #6 e  
    % END: Compute the Zernike Polynomials Ja4O*C<  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% '%. lY9D  
    zF>| 9JU  
    &\F`M|c  
    % Compute the Zernike functions: XTG*56IzL  
    % ------------------------------ h:Q*T*py  
    idx_pos = m>0; :K#'?tH  
    idx_neg = m<0; $*Njvr7  
    nBgksB*A  
    ^.&2-#i  
    z = y; CSN]k)\N(  
    if any(idx_pos) N32!*TsWs  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); Sy6Y3 ~7  
    end O'Lgb9  
    if any(idx_neg) SaH0YxnY+  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); S#/[>Cb  
    end ;$ D*,W *  
    nr Jl>H  
    m*6C *M  
    % EOF zernfun 4N[8LC;MH  
     
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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)  9,8}4Y=GVI  
    C 8 [W  
    DDE还是手动输入的呢? GddP)l{uCF  
    !U,W; R  
    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