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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, )XFMlSx)  
    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, 5VfP@{  
    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? gTT-7  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? =*pu+o,?  
    %?seX+ne  
    Qk= w ,`  
    hwJ.M4  
    M6>l%[  
    function z = zernfun(n,m,r,theta,nflag)  2B#WWb  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. -kO=pYP*O  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 4'M#m|V  
    %   and angular frequency M, evaluated at positions (R,THETA) on the 7">.{ @S  
    %   unit circle.  N is a vector of positive integers (including 0), and O`eNuQSv  
    %   M is a vector with the same number of elements as N.  Each element 1EN5ZN,  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) | zf||ju  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, pR $c<p  
    %   and THETA is a vector of angles.  R and THETA must have the same 1D$k:|pP~  
    %   length.  The output Z is a matrix with one column for every (N,M) _v\QuI6  
    %   pair, and one row for every (R,THETA) pair. Z(s} #-  
    % Q]\x O/  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike pw,.*N3P  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), Aq-v3$XL  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral shD$,! k  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1,  EpiagCS  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized xg8<b  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ZISR]xay  
    % 5HB4B <2  
    %   The Zernike functions are an orthogonal basis on the unit circle. NJ~'`{3v  
    %   They are used in disciplines such as astronomy, optics, and uo0(W3Q *  
    %   optometry to describe functions on a circular domain. oq|K:<l  
    % ` H"5nQRV  
    %   The following table lists the first 15 Zernike functions. Y9Pb  
    % Y\rKw!u_!  
    %       n    m    Zernike function           Normalization |:AjQ&PM)  
    %       -------------------------------------------------- :c\NBKHv*  
    %       0    0    1                                 1 $]_=B Jyu  
    %       1    1    r * cos(theta)                    2 ]2<g"zo0  
    %       1   -1    r * sin(theta)                    2 )}EwEM  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) 7M4iBk4I  
    %       2    0    (2*r^2 - 1)                    sqrt(3) 90q*V%cS  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) \"Np'$4eu  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) OSBE5  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) + 7Z%N9  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) Tb}b*d3  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) V{8mx70  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) v K$W)(Z  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) d"V^^I)yx&  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) u`ZnxD>  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10)  z\ \MLyS  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) %T&kK2d;  
    %       -------------------------------------------------- H;v*/~zl  
    % % $J^dF_0  
    %   Example 1: Dx8^V%b  
    % 4"GY0) Q  
    %       % Display the Zernike function Z(n=5,m=1) D=3NI  
    %       x = -1:0.01:1; MQI6e".  
    %       [X,Y] = meshgrid(x,x); F:n7yey  
    %       [theta,r] = cart2pol(X,Y); 0_ ;-QAd  
    %       idx = r<=1; dfNNCPu]+  
    %       z = nan(size(X)); CzwnmSv{.  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); $+Xohtt  
    %       figure ?&[`=ZVn  
    %       pcolor(x,x,z), shading interp Ts.6 1Rx  
    %       axis square, colorbar H#f FU  
    %       title('Zernike function Z_5^1(r,\theta)') n|8fdiK#}  
    % 5y.kOe4vH  
    %   Example 2: ZN. #g_  
    % +] FdgmK:  
    %       % Display the first 10 Zernike functions um[.r,++  
    %       x = -1:0.01:1; Hi )n]OE  
    %       [X,Y] = meshgrid(x,x); WXJ%bH  
    %       [theta,r] = cart2pol(X,Y); W&* 0F~  
    %       idx = r<=1; z+;+c$X  
    %       z = nan(size(X)); >1W)J3  
    %       n = [0  1  1  2  2  2  3  3  3  3]; f- _~rQ  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; LnLuWr<;}  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; C}7Sh6  
    %       y = zernfun(n,m,r(idx),theta(idx)); b8Y-!] F  
    %       figure('Units','normalized') Qax=_[r  
    %       for k = 1:10 0DGXMO$;  
    %           z(idx) = y(:,k); :X+7}!Wlo  
    %           subplot(4,7,Nplot(k)) _/hWzj=q  
    %           pcolor(x,x,z), shading interp )!3sB{ H  
    %           set(gca,'XTick',[],'YTick',[]) 'v?Z~"w=  
    %           axis square <5=^s%H  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) : x W.(^(d  
    %       end .|!Kv+yD  
    % w?Y;pc}1B  
    %   See also ZERNPOL, ZERNFUN2. dtJ?J<m}  
    >Ka}v:E  
    K;Fy&p^d  
    %   Paul Fricker 11/13/2006 G8j$&1`:  
    T$!. :v  
     {ZB7,\  
    bIR7g(PJ.b  
    ca5Ir<mL  
    % Check and prepare the inputs: !ouJ3Jn   
    % ----------------------------- ht)J#Di  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Ub3^Js!b%  
        error('zernfun:NMvectors','N and M must be vectors.') uvi+#4~G  
    end ApR>b%  
    .O@T#0&=_  
    4 1q|R[js!  
    if length(n)~=length(m) ]U82A**n  
        error('zernfun:NMlength','N and M must be the same length.') C`Zz\DNG@  
    end (/JiOg^cw  
    )17CG*K1  
    c3lU  
    n = n(:); DY1UP (y  
    m = m(:); N 8 n`f  
    if any(mod(n-m,2)) 3,t3\`=  
        error('zernfun:NMmultiplesof2', ... 0F/o  
              'All N and M must differ by multiples of 2 (including 0).') O!#r2Y"?K1  
    end C8ek{o)%W  
    JYc;6p$<i  
    m5`<XwD9  
    if any(m>n) G*^4+^Vz?  
        error('zernfun:MlessthanN', ... >8PGyc*9  
              'Each M must be less than or equal to its corresponding N.') V^apDV\AV  
    end DxoW,G W  
    lk;4l Z  
    hT go  
    if any( r>1 | r<0 ) p?PK8GL  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') @Jr:+|v3B  
    end /fv;`?~d*  
    mQUI9  
    9vZ:oO  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) vY)5<z&  
        error('zernfun:RTHvector','R and THETA must be vectors.') m9M#)<@*  
    end :Y>FuE  
    wNl{,aH@  
    VUmf;~  
    r = r(:); {9B"'65o  
    theta = theta(:); &PZ&'N|P  
    length_r = length(r); 6 );8z!+  
    if length_r~=length(theta) iC2``[m"  
        error('zernfun:RTHlength', ... zi%Ql|zI~  
              'The number of R- and THETA-values must be equal.') H< 51dJn~  
    end e|> 5 R  
    Gu@n1/m@o  
    m55|&Ux|  
    % Check normalization: X)Zc*9XA  
    % -------------------- mUA!GzJ~u-  
    if nargin==5 && ischar(nflag) 4M*Z1  
        isnorm = strcmpi(nflag,'norm'); SFJ"(ey$  
        if ~isnorm VDT.L,9  
            error('zernfun:normalization','Unrecognized normalization flag.') C2 4"H|D  
        end ANWfRtiU#  
    else 18nT Iz_  
        isnorm = false; /:B2-4>Q!  
    end R`KlG/Tk  
    N1iP!m9Q  
    |tN:o= 6  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% X~g U$  
    % Compute the Zernike Polynomials WF] |-)vw  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% xB`j* %  
    }i._&x`):  
    g>E.Snj}  
    % Determine the required powers of r: oZ5 ,y+L4  
    % ----------------------------------- `NySTd)\  
    m_abs = abs(m); +N}yqgE  
    rpowers = []; %-fQ[@5  
    for j = 1:length(n) zt;aB>jz#  
        rpowers = [rpowers m_abs(j):2:n(j)]; ?[?;%Y  
    end 'C7$,H'  
    rpowers = unique(rpowers); P](/5KrK  
    l=UXikx  
    Z\r?>2  
    % Pre-compute the values of r raised to the required powers, b|pp}il  
    % and compile them in a matrix: 8'qq!WR~  
    % ----------------------------- ^u(-v/D9  
    if rpowers(1)==0 1 HY K& ',  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); %/oeV;D  
        rpowern = cat(2,rpowern{:}); i0n u5kD+d  
        rpowern = [ones(length_r,1) rpowern]; H S)$|m_  
    else nvB< pSm  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); smKp3_r  
        rpowern = cat(2,rpowern{:}); 8 qlQC.VA[  
    end &6e A.  
    yXQ 28A  
    `*WzHDv5p  
    % Compute the values of the polynomials: ]TVc 'G;  
    % -------------------------------------- #+&"m7 s  
    y = zeros(length_r,length(n));  oP~%7Jt  
    for j = 1:length(n) ~6=aoF5"3?  
        s = 0:(n(j)-m_abs(j))/2; ;Wgkf_3  
        pows = n(j):-2:m_abs(j); =%SH2kb  
        for k = length(s):-1:1 XTJA"y  
            p = (1-2*mod(s(k),2))* ... bgeJVI  
                       prod(2:(n(j)-s(k)))/              ... v]\T&w%9  
                       prod(2:s(k))/                     ... |G)P I`BH  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ` ZBOaN^if  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); ivg W[]  
            idx = (pows(k)==rpowers);  {b|V;/  
            y(:,j) = y(:,j) + p*rpowern(:,idx); O"}O~lZ[6T  
        end :}-VLp4b  
         &o]fBdn  
        if isnorm QtA@p  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); ?)gc;K  
        end [Lcy &+  
    end 2 ?F?C  
    % END: Compute the Zernike Polynomials [9d\WPLC  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% YpgO]\/w  
    (%'`t(<  
    e=+q*]>  
    % Compute the Zernike functions: _\<TjGtG  
    % ------------------------------ d ATAH}r&  
    idx_pos = m>0; 9* P-k.Bl  
    idx_neg = m<0; 5Y 7 %Z  
    W=y9mW|p/  
    M?5voV*  
    z = y; X4L@|"ZI  
    if any(idx_pos) M< H+$}[  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); b/_u\R ]-'  
    end 0v#p4@Z  
    if any(idx_neg) NtmmPJ|5  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); `_sKR,LhB  
    end F-XMy>9  
    ?69E_E  
    cd`P'GDF  
    % EOF zernfun XP[~ :+  
     
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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)  >wR)p\UEb  
    9JeT1\VvHY  
    DDE还是手动输入的呢? b?2 \j}  
    p9!jM\(  
    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