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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, 6 IvAs-%W  
    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, %n$f#Ml_r  
    这个,跟我们用zygo干涉仪直接拟合出的36项zernike系数,有何关系呢? uH\EV`@'  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? [8'?G5/n  
    wR_mJMk_  
    ;1&"]N%  
    V Rv4p5  
    JSUD$|RiJ  
    function z = zernfun(n,m,r,theta,nflag) i*$+>3Q-  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. .>W [  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N !/G}vu  
    %   and angular frequency M, evaluated at positions (R,THETA) on the $}vk+.!*1  
    %   unit circle.  N is a vector of positive integers (including 0), and i$kB6B#==  
    %   M is a vector with the same number of elements as N.  Each element oG)T>L[&  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) q 4Pv\YO  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, "rMfe>;FJ  
    %   and THETA is a vector of angles.  R and THETA must have the same `,4yGgD!4  
    %   length.  The output Z is a matrix with one column for every (N,M) x<I[?GT=  
    %   pair, and one row for every (R,THETA) pair. JWHsTnB  
    % +pYgh8w@  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike {XU!p: x  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), syu/"KY^!  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral M"*NV(".g  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, w6Gez~ 8  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized 4D&L]eJ  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ;?u cC@  
    % y],op G6  
    %   The Zernike functions are an orthogonal basis on the unit circle. |mMsU,*gB  
    %   They are used in disciplines such as astronomy, optics, and =mLp g4  
    %   optometry to describe functions on a circular domain. &en2t=a  
    % }"+"nf5h  
    %   The following table lists the first 15 Zernike functions. ]#NfH-T  
    % UXji$|ET6  
    %       n    m    Zernike function           Normalization (A=PDjP!  
    %       -------------------------------------------------- C9+rrc@4  
    %       0    0    1                                 1 z uNm !$  
    %       1    1    r * cos(theta)                    2 ~Bl,_?CBr  
    %       1   -1    r * sin(theta)                    2 cq>J]35  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) u.q3~~[=  
    %       2    0    (2*r^2 - 1)                    sqrt(3) {ccc[G?>.Q  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) 8b0j rt  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) 2<*"@Vj  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) z?13~e[D  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) &n,v@ gt  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) wdj?T`4  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) yl?LXc[)  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) -W6@[5c  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) %UdE2D'bC  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Mx w-f4j  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) +6>2= ,?Z  
    %       -------------------------------------------------- 'bRf>=  
    % $m ;p@#n  
    %   Example 1: @5&57R3>  
    % kKRu]0J~[  
    %       % Display the Zernike function Z(n=5,m=1) '{0O!y[H6  
    %       x = -1:0.01:1; X"3p/!W.4  
    %       [X,Y] = meshgrid(x,x); ]2L11" erP  
    %       [theta,r] = cart2pol(X,Y); 0Gj/yra9MO  
    %       idx = r<=1; ,eTdQI;   
    %       z = nan(size(X)); `yq) y>_  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); @[joM*U  
    %       figure NI"Zocp  
    %       pcolor(x,x,z), shading interp xj33g6S  
    %       axis square, colorbar ommW  
    %       title('Zernike function Z_5^1(r,\theta)') *DcIC]ao[  
    % 8m H6?,@6  
    %   Example 2: sRLjKi2D  
    % ~*1Z1aZ  
    %       % Display the first 10 Zernike functions y}FG5'5$13  
    %       x = -1:0.01:1; Ho}*Bn~ic  
    %       [X,Y] = meshgrid(x,x); GR(m+%Vw!  
    %       [theta,r] = cart2pol(X,Y); {{gd}g  
    %       idx = r<=1;  %o/@0.w  
    %       z = nan(size(X)); #k<l5x`  
    %       n = [0  1  1  2  2  2  3  3  3  3]; 1c/<2xO~  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; n[y=DdiKGS  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; aPe*@py3T  
    %       y = zernfun(n,m,r(idx),theta(idx)); p-"wY?q  
    %       figure('Units','normalized') ~{g/  
    %       for k = 1:10 I;AS.y  
    %           z(idx) = y(:,k); Y,mo}X<>  
    %           subplot(4,7,Nplot(k)) (=rDt93J  
    %           pcolor(x,x,z), shading interp )( YJ6l  
    %           set(gca,'XTick',[],'YTick',[]) ph)=:*A6&  
    %           axis square kL s{B  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}'])  W;yg{y   
    %       end 9(X~  
    % T__@hfT  
    %   See also ZERNPOL, ZERNFUN2. ZH=Bm^  
    - A}$5/  
    6}@T^?  
    %   Paul Fricker 11/13/2006  S\ZCZ0  
    W@GU;Nr  
    q~18JB4WPJ  
    ,F!-17_vt  
    )2Q0NbDn  
    % Check and prepare the inputs: b+RU <qR  
    % ----------------------------- ]ml'd  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) /QlzWson  
        error('zernfun:NMvectors','N and M must be vectors.') ?3LV$S)U  
    end ~y Dl & S  
    i+Ne.h  
    ^eoW+OxH  
    if length(n)~=length(m) .2P3 !KCL  
        error('zernfun:NMlength','N and M must be the same length.') zB7 ^L^Y  
    end ?=?*W7  
    oQ Vm)Bn'R  
    3 ?gfDJfE  
    n = n(:); !}`[s2ji  
    m = m(:); $rjm MSxi  
    if any(mod(n-m,2)) 9l[C&0w#\  
        error('zernfun:NMmultiplesof2', ... tT A  
              'All N and M must differ by multiples of 2 (including 0).') va(6?"9  
    end \/wk!mWV@  
    M_?B*QZJI  
    ~y 2joStx  
    if any(m>n) x `%x f  
        error('zernfun:MlessthanN', ... hOqNZ66{  
              'Each M must be less than or equal to its corresponding N.') 0|hOoO]?q&  
    end $Zi {1w  
    F_}y[Yn^  
    IAmMO[9H  
    if any( r>1 | r<0 ) e'v_eD T^  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') !t)uRJ   
    end X)TZ  S  
    fA V.Mj-  
    EN>a^B+!  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) su60j^e*  
        error('zernfun:RTHvector','R and THETA must be vectors.') j9%vw.3b  
    end rJp9ut'FEz  
    ]RVme^=  
    ]G! APE  
    r = r(:); E_z,%aD[  
    theta = theta(:); d.>O`.Mu)}  
    length_r = length(r); za.^vwkBk2  
    if length_r~=length(theta) &` "uKO]  
        error('zernfun:RTHlength', ... \u/=?b  
              'The number of R- and THETA-values must be equal.') H!y-o'Z  
    end % 5!Y#$:{o  
    Q."rE"}<  
    )Ps<u-V  
    % Check normalization: =*?XZA)c  
    % -------------------- o}D7 $6  
    if nargin==5 && ischar(nflag) v9D[| 4  
        isnorm = strcmpi(nflag,'norm'); od vUU#l  
        if ~isnorm Z 2uU'T  
            error('zernfun:normalization','Unrecognized normalization flag.') 2Y}A9Veb  
        end TU| 0I  
    else o:%;AOcl  
        isnorm = false; @nj`T{*.  
    end +jGUp\h%9;  
    y+.(E-g  
    ;UQ&yj%x  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 'Te'wh=Y  
    % Compute the Zernike Polynomials 2Aq+:ud)P  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% lzz68cT  
    5)4?i p  
    rsK b9G  
    % Determine the required powers of r: rqM_#[Y?  
    % ----------------------------------- yAJrdY"  
    m_abs = abs(m); QSo48OFs  
    rpowers = []; I\82_t8  
    for j = 1:length(n) cc3+ Wx_  
        rpowers = [rpowers m_abs(j):2:n(j)]; 4d-"kx3X  
    end ;Ac!"_N?7  
    rpowers = unique(rpowers); ub{Yg5{3S\  
    W$R@Klz  
    AIwp2Fz  
    % Pre-compute the values of r raised to the required powers, I"jub kI=Z  
    % and compile them in a matrix: Wc/B_F?2  
    % ----------------------------- I\6^]pi,  
    if rpowers(1)==0 ;uU 8$  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 38RyUHL=  
        rpowern = cat(2,rpowern{:}); XCO;t_%  
        rpowern = [ones(length_r,1) rpowern]; VC NQ}h[D  
    else 74~ %4  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ,Ct1)%   
        rpowern = cat(2,rpowern{:}); Oyjhc<6  
    end rdm&YM`J  
    yu'@gg(  
    _ Av_jw`m  
    % Compute the values of the polynomials: u,Cf4H*xS  
    % -------------------------------------- 14Jkr)N  
    y = zeros(length_r,length(n)); &m@DK>  
    for j = 1:length(n) @(e/Y/  
        s = 0:(n(j)-m_abs(j))/2; $,7Yo nc  
        pows = n(j):-2:m_abs(j); %y\  
        for k = length(s):-1:1 j1$s^-9  
            p = (1-2*mod(s(k),2))* ... %t,Fxj4F  
                       prod(2:(n(j)-s(k)))/              ... :W1B"T<  
                       prod(2:s(k))/                     ... WS ^%< h#  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... =&?BPhJE  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); !JwR[X\f  
            idx = (pows(k)==rpowers); * @'N/W/8  
            y(:,j) = y(:,j) + p*rpowern(:,idx); jL#`CD  
        end ygTc Y  
         b,RQ" {  
        if isnorm E$E #c8I:  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); )ocr.wU@  
        end Eg#WR&Uq"  
    end Fpy-? U  
    % END: Compute the Zernike Polynomials )Es|EPCx!  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% JE/Kf<  
    4Y}{?]>pu  
    4#w Z#}  
    % Compute the Zernike functions:  i(n BXV{  
    % ------------------------------ @7,k0H9Moa  
    idx_pos = m>0; MJI`1*(  
    idx_neg = m<0; 2lRE+_qz  
    ~~3 BV,  
    7F wo t&  
    z = y; 6^"Spf]  
    if any(idx_pos) xIa8Ac  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 45tQ$jr`1  
    end pu6@X7W"  
    if any(idx_neg) 6{TUs>~  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); UB|}+WA3  
    end di]TS9&9  
    L+2<J,   
    y^hCO:`l3  
    % EOF zernfun p;9"0rj,z  
     
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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)  4-m6e$p;  
    7@ \:l~{  
    DDE还是手动输入的呢? h8dFW"cpC  
    Bmt^*;WY+  
    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