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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, /K#t$O4  
    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, b&Dc DX  
    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? . PzlhTL7  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? ^:b%Q O  
    8:BPXdiK  
    5UFR^\e  
    I+) Acy;  
    -&#H@Gyw  
    function z = zernfun(n,m,r,theta,nflag) QU&b5!;&  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. Jy,Dcl  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N Wcgy:4K3  
    %   and angular frequency M, evaluated at positions (R,THETA) on the H:~41f[  
    %   unit circle.  N is a vector of positive integers (including 0), and (IbT5  
    %   M is a vector with the same number of elements as N.  Each element ]FJpe^ ua  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) AT#&`Ew  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, SI:+I4i  
    %   and THETA is a vector of angles.  R and THETA must have the same =Vgj=19X(  
    %   length.  The output Z is a matrix with one column for every (N,M) 0FDfB;  
    %   pair, and one row for every (R,THETA) pair. </K"\EU  
    % `_IgH  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike k5>K/;*9  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), KcGM=z?:  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral EZm6WvlxSI  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, x)X=sX.  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized GC{)3)_ t  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 5"f')MKUV9  
    % ,j4 ;:F  
    %   The Zernike functions are an orthogonal basis on the unit circle. py,B6UB5  
    %   They are used in disciplines such as astronomy, optics, and ^-CQ9r*  
    %   optometry to describe functions on a circular domain. ))M; .b.D  
    % ^9})@,(D  
    %   The following table lists the first 15 Zernike functions. ]-o0HY2  
    % 49o5"M(  
    %       n    m    Zernike function           Normalization rb+&]  
    %       -------------------------------------------------- q@;z((45  
    %       0    0    1                                 1 P1f?'i ?J  
    %       1    1    r * cos(theta)                    2 axTvA(k9  
    %       1   -1    r * sin(theta)                    2 bDLPA27  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) a0Y/,S*K  
    %       2    0    (2*r^2 - 1)                    sqrt(3) 3{mu7 7  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) 2 {lo  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) : "[dr~.  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) Wcy N, 5  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) v{ F/Bifo  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) L0_qHLY  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) [u_-x3`  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) :y)'_p *l/  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) mVYLI!n}0#  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) *@'\4OO  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) zt1Pu /e  
    %       -------------------------------------------------- 1* _wJ  
    % 0K -jF5i$`  
    %   Example 1: `>@n6>f  
    % 33O@jb s@  
    %       % Display the Zernike function Z(n=5,m=1) |w(@a:2 kw  
    %       x = -1:0.01:1; c&Pgz~iP  
    %       [X,Y] = meshgrid(x,x); ,+0>p  
    %       [theta,r] = cart2pol(X,Y); N?d4Pu1m  
    %       idx = r<=1; YuWsE4$  
    %       z = nan(size(X)); "{0 o"k  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); tqY)  
    %       figure &H!#jh\w  
    %       pcolor(x,x,z), shading interp W s!N%%g  
    %       axis square, colorbar 1mw<$'pm0  
    %       title('Zernike function Z_5^1(r,\theta)') '-F }(9M  
    % \lVX~r4  
    %   Example 2: M[ea!an  
    % u$c)B<.UR  
    %       % Display the first 10 Zernike functions t:m2[U_}  
    %       x = -1:0.01:1; utq*<,^  
    %       [X,Y] = meshgrid(x,x); B]K@'#  
    %       [theta,r] = cart2pol(X,Y); /? n 9c;w  
    %       idx = r<=1; $x&\9CRM  
    %       z = nan(size(X)); g->cgExj  
    %       n = [0  1  1  2  2  2  3  3  3  3]; * %p6+D-C  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; !=(~e':Gv  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; {0fQ"))"  
    %       y = zernfun(n,m,r(idx),theta(idx)); cGw*edgp6  
    %       figure('Units','normalized') pU`4bT(w%  
    %       for k = 1:10 28L3"c  
    %           z(idx) = y(:,k); Cc:m~e6r  
    %           subplot(4,7,Nplot(k)) ZbJUOa?WF  
    %           pcolor(x,x,z), shading interp y%CaaK=V3  
    %           set(gca,'XTick',[],'YTick',[]) oI9Jp`  
    %           axis square Ws[[Me, =  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) NJb5HoYZ  
    %       end fL7ym,?  
    % irNGURLm  
    %   See also ZERNPOL, ZERNFUN2. DiF=<} >x  
    S8+Xk= x  
    L 6){wQ%c  
    %   Paul Fricker 11/13/2006 ]*+ozAG4  
    8H_3.MK  
    Pm]6E[zC  
    C% <[mM  
    {5U;9: sO6  
    % Check and prepare the inputs: I(r^q"  
    % ----------------------------- K;2tY+I  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) O$B]#]L+  
        error('zernfun:NMvectors','N and M must be vectors.') rvRtR/*?j  
    end 9V&%_.Z  
    JcxhI]E  
    P)~PrTa%  
    if length(n)~=length(m) ,,r%Y&:`6  
        error('zernfun:NMlength','N and M must be the same length.') )2mi6[qs0l  
    end T`46\KkN  
    fSdv%$;Hc  
     m?hC!n>  
    n = n(:); ;p%a!Im_ <  
    m = m(:); F9]j{'#  
    if any(mod(n-m,2)) Fs7/3  
        error('zernfun:NMmultiplesof2', ... /OaLkENgvf  
              'All N and M must differ by multiples of 2 (including 0).') xKz^J SF  
    end DUiqt09`~  
    :Vq gmn  
    9I/o;Js  
    if any(m>n) HPs$R [  
        error('zernfun:MlessthanN', ... v`B7[B4K3  
              'Each M must be less than or equal to its corresponding N.') +O:Qw[BL/Z  
    end Ftj3`Mu  
    $H^hK0?'  
    C( C4R+U  
    if any( r>1 | r<0 ) XiI@Px?FL  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') Kx6_Vp  
    end kEWC  
     L's_lC  
    ~DcX}VCm  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) $@q)IK%FDL  
        error('zernfun:RTHvector','R and THETA must be vectors.') 39?iX'*p  
    end }Tn]cL{]C  
    72} MspzUt  
    CDei+ q  
    r = r(:); [Fe`}F}Co8  
    theta = theta(:); d;|Pp;dc  
    length_r = length(r); KcP86H52I  
    if length_r~=length(theta) z (rQ6  
        error('zernfun:RTHlength', ... =kohQ d.n  
              'The number of R- and THETA-values must be equal.') zLue j'  
    end )DuOo83n["  
    l"!.aIY"e  
    RH^8"%\  
    % Check normalization: zzy%dc  
    % -------------------- ro7\}O:I  
    if nargin==5 && ischar(nflag) {$4fRxj  
        isnorm = strcmpi(nflag,'norm'); T>d-f=(9KH  
        if ~isnorm o <8L, u(U  
            error('zernfun:normalization','Unrecognized normalization flag.')  Aki8#  
        end !0ySS {/  
    else 31k.{dnm  
        isnorm = false; <9YRSE [Ed  
    end MzsDWx;eJ  
    d@pD5n=m;  
    Ab -uK|<  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% draY /  
    % Compute the Zernike Polynomials azz6_qk8  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% '~%1p_0dq  
    D_D<N(O  
    )(b]-  )  
    % Determine the required powers of r: !HM{imT  
    % ----------------------------------- Q/r9r*>z  
    m_abs = abs(m); Rer\='  
    rpowers = []; 6~V$0Y>]  
    for j = 1:length(n) FkR9-X<  
        rpowers = [rpowers m_abs(j):2:n(j)]; Hb=4k)-/]  
    end #rqLuqw  
    rpowers = unique(rpowers); " GkBX  
    a_w# ,^/P  
    5`*S'W}\>  
    % Pre-compute the values of r raised to the required powers, & ,2XrXiFu  
    % and compile them in a matrix: IIUoB!`  
    % ----------------------------- {LVii}<  
    if rpowers(1)==0 "zJ1vIZY  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 9a"[-B:  
        rpowern = cat(2,rpowern{:}); pJ"Wg@+  
        rpowern = [ones(length_r,1) rpowern]; gI6./;;x  
    else ko*Ir@SDv  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); _n@#Lufx  
        rpowern = cat(2,rpowern{:}); 3iJ4VL7  
    end L|EvI.f  
    ]re1$ W#*  
    _dVzvk`_R  
    % Compute the values of the polynomials: E$=!l{Ms  
    % -------------------------------------- w4<1*u@${  
    y = zeros(length_r,length(n)); b;`gxXeL  
    for j = 1:length(n) '@i/?rNi%N  
        s = 0:(n(j)-m_abs(j))/2; 03L+[F&"?  
        pows = n(j):-2:m_abs(j); nAG2!2_8  
        for k = length(s):-1:1 $(K[W}  
            p = (1-2*mod(s(k),2))* ... SwpS6  
                       prod(2:(n(j)-s(k)))/              ... 4,!#E0  
                       prod(2:s(k))/                     ... _@;t^j+l  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... }p$>V,u  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); A 'rfoA6  
            idx = (pows(k)==rpowers); a` A V  
            y(:,j) = y(:,j) + p*rpowern(:,idx); b/HhGA0  
        end 4\a KC%5  
         v\PqhIy"  
        if isnorm @ U xO!  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); n=WwB(}q  
        end P!3)-apP\  
    end NK;%c-r0v7  
    % END: Compute the Zernike Polynomials FY+0r67]  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% /J )MW{;O  
    ER<LP@3k  
    rg64f'+Eug  
    % Compute the Zernike functions: $!%/Kk4M  
    % ------------------------------ 9`]Gosz  
    idx_pos = m>0; N]udZhkn  
    idx_neg = m<0; ^0p y  
    uOUgU$%zqH  
    d*$$E  
    z = y; bYQvh/(J  
    if any(idx_pos) =+;l>mn?O  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ?XN=Er^  
    end $_IvzbOh  
    if any(idx_neg) o|O730"2F  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); _rt+OzZ*L  
    end zrVw l\&  
    2%P{fJbwd  
    yIy'"BCxM  
    % EOF zernfun :@#9P,"  
     
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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)  jd`h)4  
    EVlj#~mV  
    DDE还是手动输入的呢? 9Iu"DOxX%  
    baoyU#X9  
    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