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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, Qqp_(5S|>  
    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, & XS2q0-x  
    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? c"nowbf  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? )K=%s%3h<  
    bc+~g>o  
    dC&OjBQ  
    .G ^-. p  
    D=B$ Pv9%  
    function z = zernfun(n,m,r,theta,nflag) W0zRV9"P  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. <7U\@si4  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 3q$[r_   
    %   and angular frequency M, evaluated at positions (R,THETA) on the ]lX`[HX7  
    %   unit circle.  N is a vector of positive integers (including 0), and >9WJa5{  
    %   M is a vector with the same number of elements as N.  Each element >i6sJ)2?>  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) fX ^h O+f  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1,  X.q,  
    %   and THETA is a vector of angles.  R and THETA must have the same u-8b,$@Z>'  
    %   length.  The output Z is a matrix with one column for every (N,M) q=EHB5!q  
    %   pair, and one row for every (R,THETA) pair. & bKl(,  
    % J?oI%r7^  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike _1c0pQ^}3  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), W2$MH: j  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral 6KvoHo  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, Nldy76|g  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized &["s/!O1R  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. Z<yLu'48)A  
    % lQ8h-Tz  
    %   The Zernike functions are an orthogonal basis on the unit circle. GiZv0>*x  
    %   They are used in disciplines such as astronomy, optics, and #Nv^F  
    %   optometry to describe functions on a circular domain. K@f@vyw]  
    % 6-fdfU  
    %   The following table lists the first 15 Zernike functions. Gu#Vc.e  
    % 8Q{"W"]O7  
    %       n    m    Zernike function           Normalization tj' xjX  
    %       -------------------------------------------------- {Vw\#/,  
    %       0    0    1                                 1 -ho%9LW%|  
    %       1    1    r * cos(theta)                    2 1*aO2dOq  
    %       1   -1    r * sin(theta)                    2 a-cLy*W,~  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) rexNsKRK_  
    %       2    0    (2*r^2 - 1)                    sqrt(3) r_x|2 A oO  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) Qm"&=<  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) [$Dzf<0  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) {4 y#+[  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) rW P -Rm  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) tk5zq-/ d  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) < dD)>Y.  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) X([8TR  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) @^2?97i c  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) l%_r3W  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) tl=H9w&@  
    %       -------------------------------------------------- t@;r~S b  
    % yrF"`/zv6|  
    %   Example 1: ;4'pucq5/  
    % m]?C @ina  
    %       % Display the Zernike function Z(n=5,m=1) W"v"mjYud  
    %       x = -1:0.01:1; +_T`tmQ  
    %       [X,Y] = meshgrid(x,x); m ]h<y  
    %       [theta,r] = cart2pol(X,Y); MQY}}a-oug  
    %       idx = r<=1; <k'%rz  
    %       z = nan(size(X)); rqi/nW  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); \-W|)H  
    %       figure tRCz[M&  
    %       pcolor(x,x,z), shading interp Yo*.? Mq'  
    %       axis square, colorbar ~PtIq.BY  
    %       title('Zernike function Z_5^1(r,\theta)') W7` fI*lc  
    % -z~;f<+I`  
    %   Example 2: k d9<&.y{  
    % -<{;.~nI.  
    %       % Display the first 10 Zernike functions _)U.5f<   
    %       x = -1:0.01:1; h]jy):9L  
    %       [X,Y] = meshgrid(x,x); b6?&h:{k  
    %       [theta,r] = cart2pol(X,Y); v,d bto0  
    %       idx = r<=1; >+FaPym  
    %       z = nan(size(X)); vveL|j  
    %       n = [0  1  1  2  2  2  3  3  3  3]; Rn_FYP  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; >X_5o^s2s  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; d=DQS>Nz  
    %       y = zernfun(n,m,r(idx),theta(idx)); _'0C70  
    %       figure('Units','normalized') p9-s'F|@i  
    %       for k = 1:10 NiSH$ MJ_  
    %           z(idx) = y(:,k); %F}i2!\<L  
    %           subplot(4,7,Nplot(k)) -(lCM/h  
    %           pcolor(x,x,z), shading interp EXEB A&*  
    %           set(gca,'XTick',[],'YTick',[]) ' 4.T1i,  
    %           axis square !dV2:`|+  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) -d4|EtN  
    %       end })y B2Q0  
    % !T"jvDYH  
    %   See also ZERNPOL, ZERNFUN2. 8)ykXx/f@  
    x(+H1D\W   
    `LOW)|6r`  
    %   Paul Fricker 11/13/2006 X.GK5Phd  
    VCX^D)[-  
    E}g)q;0v|2  
    JFu9_=%+  
    A&S n^mw  
    % Check and prepare the inputs: `kYcTFk  
    % ----------------------------- 7V2xg h!W  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) rHp2I6.0a  
        error('zernfun:NMvectors','N and M must be vectors.') )?;+<,  
    end 'Bwv-J  
    K"jS,a?s 6  
    dCA! R"HD  
    if length(n)~=length(m) .$ X|96~$  
        error('zernfun:NMlength','N and M must be the same length.') tF:AqR: (~  
    end FWW*f _L  
    =`ECM7  
    T h!;zu^t  
    n = n(:); /8wfI_P>M"  
    m = m(:); slQEAqG)B  
    if any(mod(n-m,2)) 57Bxx__S4`  
        error('zernfun:NMmultiplesof2', ... fb8)jd'~}O  
              'All N and M must differ by multiples of 2 (including 0).') zG)vmysJf  
    end @xeJ$ rlu  
    ]oLyvG  
    V-9\@'gc  
    if any(m>n) DJb9] ,=a  
        error('zernfun:MlessthanN', ... wpg7xx!  
              'Each M must be less than or equal to its corresponding N.') 9p,PWA  
    end CrB4%W:{  
    _9y! ,ST  
    "j8`)XXa(  
    if any( r>1 | r<0 ) SQJ +C%   
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') g?N^9B,$2  
    end #$;}-*  
    jAdZS\?w  
    EE-wi@  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) V8rS~'{\  
        error('zernfun:RTHvector','R and THETA must be vectors.') 6^)eW+  
    end q[(1zG%NbA  
    <k 'zz:[c!  
    / 5/m x  
    r = r(:); {f\{{JJ]  
    theta = theta(:); 7c!#e=W@B  
    length_r = length(r); XEBj=5sG  
    if length_r~=length(theta) #nq_R  
        error('zernfun:RTHlength', ... ZgfhNI\  
              'The number of R- and THETA-values must be equal.') YjiMUi\V  
    end &$ fyY:<\  
    sB5@6[VDI  
    Sd/7#  
    % Check normalization: v]#[bqB.b  
    % --------------------  F*_+k  
    if nargin==5 && ischar(nflag) ]&s@5<S[  
        isnorm = strcmpi(nflag,'norm'); rv c%[HfW;  
        if ~isnorm <cxe   
            error('zernfun:normalization','Unrecognized normalization flag.') &3Lhb}m  
        end UrO& K]Z  
    else ]X> I(p@  
        isnorm = false; ^kke  
    end \Hw*q|  
    p6&<eMwFA  
    ,/&|:PkS  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% DOOF--ua  
    % Compute the Zernike Polynomials j`#H%2W\;  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ] Upr<!  
    ix"BLn]YZ  
    "wCx]{Di  
    % Determine the required powers of r: -f*5lkO  
    % ----------------------------------- Pif-uhOk%  
    m_abs = abs(m); #/5eQTBD  
    rpowers = []; ]WK~`-3C^  
    for j = 1:length(n) egAYJK-,!  
        rpowers = [rpowers m_abs(j):2:n(j)]; Et y?/  
    end 2B^WZlx  
    rpowers = unique(rpowers); ,~/WYw<o  
    HL-'\wtl  
    }$[@*  
    % Pre-compute the values of r raised to the required powers, luW"|  
    % and compile them in a matrix:  uAs!5h  
    % ----------------------------- H^dw=kS  
    if rpowers(1)==0 U'u_'5 {  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); !MVf(y$  
        rpowern = cat(2,rpowern{:}); \Rs9B .  
        rpowern = [ones(length_r,1) rpowern]; hhz#I A6,  
    else i(;.Y  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); SAdo9m'  
        rpowern = cat(2,rpowern{:}); ;GKL[ tI"  
    end W>qu~ak?x  
    Z>l|R C  
    NwdrJw9  
    % Compute the values of the polynomials: 1CR\!?  
    % -------------------------------------- g W_E  
    y = zeros(length_r,length(n)); *sau['Ha  
    for j = 1:length(n) !p76I=H%  
        s = 0:(n(j)-m_abs(j))/2; DWEDL[{  
        pows = n(j):-2:m_abs(j); olr-oi`4C  
        for k = length(s):-1:1 ;kWWzg  
            p = (1-2*mod(s(k),2))* ... =k,?+h~  
                       prod(2:(n(j)-s(k)))/              ... E;9J7Q 4  
                       prod(2:s(k))/                     ... X{(?p=]  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... YQyI{  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); bxzx@sF2l  
            idx = (pows(k)==rpowers); @eutp`xoT\  
            y(:,j) = y(:,j) + p*rpowern(:,idx); Jd?qvE>Pp  
        end 6(x53 y__  
         3t9CN )*  
        if isnorm @.c[z D  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); lMG+,?<uK&  
        end `7'^y  
    end 1k^$:'  
    % END: Compute the Zernike Polynomials KUq7Oa !  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Onh R`  
    eo@8?>}{X  
    .C &kWM&j  
    % Compute the Zernike functions: fUfd5W1"  
    % ------------------------------ NFP h}D  
    idx_pos = m>0; E 0l&d  
    idx_neg = m<0; cK2;)&U7  
    :_]0 8  
    t: oQHhO?  
    z = y; .z=%3p8+  
    if any(idx_pos) ;(jL`L F  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); @t@B(1T  
    end Rkp +}@Y_  
    if any(idx_neg) }_F:]lI*R  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); d[5v A/8O  
    end _sZ&=-FR  
    , s otZT  
    7&/1K%x9;  
    % EOF zernfun edCVIY'1  
     
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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)  uh#E^~5S  
    <;>k[P'  
    DDE还是手动输入的呢? r4gLoHD)  
     r3OtQ  
    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