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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, on=I*?+R  
    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, w`?Rd  
    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? &D[pX|!  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? ]XAJ|[]sj*  
    yXdJ5Me(T  
    49("$!  
    ,%a7sk<5k  
    xn)eb#r  
    function z = zernfun(n,m,r,theta,nflag) O^AF+c\n  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. qXQ/M]  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 1p[Z`m*9  
    %   and angular frequency M, evaluated at positions (R,THETA) on the V>2mz c  
    %   unit circle.  N is a vector of positive integers (including 0), and U =G^w L  
    %   M is a vector with the same number of elements as N.  Each element .PhH|jrCW^  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) Lk-%I?  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, eyiGe1^C  
    %   and THETA is a vector of angles.  R and THETA must have the same / W,K% s]  
    %   length.  The output Z is a matrix with one column for every (N,M) >nnjL rI  
    %   pair, and one row for every (R,THETA) pair. P(Fd|).j$  
    % u?>]C6$  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ?5oeyBA@  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 5"]t{-PD  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral g+-=/Ge  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, v GT#BS%  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized UW%.G  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. )38M~/ ^l  
    % -p:X]Ov  
    %   The Zernike functions are an orthogonal basis on the unit circle. #''q :^EQ  
    %   They are used in disciplines such as astronomy, optics, and L,XWX8  
    %   optometry to describe functions on a circular domain. j9=QOq  
    % <$\En[u0  
    %   The following table lists the first 15 Zernike functions. +cw;a]o^>  
    % JBsHr%!i  
    %       n    m    Zernike function           Normalization mu(EmAoenQ  
    %       -------------------------------------------------- o~*5FN}%+l  
    %       0    0    1                                 1 {[&_)AW6m%  
    %       1    1    r * cos(theta)                    2 Z{|U!tn  
    %       1   -1    r * sin(theta)                    2 BK_x5mGu3  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) cN{-&\ 6L  
    %       2    0    (2*r^2 - 1)                    sqrt(3) *1Lkde@|{  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) ]/p)XHKo  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) G(puC4 "&  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) ?Q< o-o;B  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) 3']yjj(gHr  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) !U@?Va~Zn  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) r# }`{C;+5  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) T|h/n\fx)a  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) S'I{'jP5  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) {ER%r'(4Z  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) 8qEK6-  
    %       -------------------------------------------------- jZm57{C#*?  
    % j]#-DIL  
    %   Example 1: kW#{[,7r  
    % #l(cBM9sz  
    %       % Display the Zernike function Z(n=5,m=1) rSYzrVc  
    %       x = -1:0.01:1; u= |hRTD=  
    %       [X,Y] = meshgrid(x,x); 4DL;/Z:  
    %       [theta,r] = cart2pol(X,Y); S=^a''bg  
    %       idx = r<=1; LN8V&'>  
    %       z = nan(size(X)); b ;Vy=f  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); 4No!`O-!&  
    %       figure '~^3 =[Z  
    %       pcolor(x,x,z), shading interp w/KCu W<  
    %       axis square, colorbar 8q6b3q:c  
    %       title('Zernike function Z_5^1(r,\theta)') fR>(b?C  
    % [8k7-}[  
    %   Example 2: TB]B l.  
    % kpM5/=f/@  
    %       % Display the first 10 Zernike functions m,e @bJ-  
    %       x = -1:0.01:1; GRanR'xG  
    %       [X,Y] = meshgrid(x,x); %5=XszS  
    %       [theta,r] = cart2pol(X,Y); \(lt [=  
    %       idx = r<=1; $lj1924?^  
    %       z = nan(size(X)); 2EubMG  
    %       n = [0  1  1  2  2  2  3  3  3  3]; 4s<*rKm~  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; vG'JMzAm  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; ndkV(#wQS  
    %       y = zernfun(n,m,r(idx),theta(idx)); t(4%l4i;X  
    %       figure('Units','normalized') U!"+~d)  
    %       for k = 1:10 silTL_$  
    %           z(idx) = y(:,k); P5+FZzQ  
    %           subplot(4,7,Nplot(k)) Q?GmSeUi  
    %           pcolor(x,x,z), shading interp 9w -t9X>X  
    %           set(gca,'XTick',[],'YTick',[]) cS98%@DR  
    %           axis square 6#+&_ #9  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) Rx$5#K!%M  
    %       end 7Q<xC  
    % E%M~:JuKd?  
    %   See also ZERNPOL, ZERNFUN2. I$4GM  
    */Oq$3QGsV  
    y ?FKou'  
    %   Paul Fricker 11/13/2006 3A_7R-sQ  
    R qS2Qo]  
    s4 o-*1R*`  
    8>TDrpT}  
    =GpO }t">  
    % Check and prepare the inputs: }bG|(Wp9  
    % ----------------------------- @Z.s:FV[  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) (m[]A&u  
        error('zernfun:NMvectors','N and M must be vectors.') Ed3 *fY  
    end +Io[o6*  
    hlxZq  
    7FMg6z8~  
    if length(n)~=length(m) 3F ;+ D  
        error('zernfun:NMlength','N and M must be the same length.') -r_/b  
    end sgDlT=c'  
    ?d1H]f<M  
    Oslbt8)U6  
    n = n(:); xBhfC!AK}  
    m = m(:); 2G8f4vsC[  
    if any(mod(n-m,2)) *<2+tI  
        error('zernfun:NMmultiplesof2', ... ^$aj,*Aj~  
              'All N and M must differ by multiples of 2 (including 0).') DCv~^  
    end =<I90j~)  
    9g#L"T=  
    p]uwGWDI  
    if any(m>n) 2H8,&lY.p  
        error('zernfun:MlessthanN', ... ]3<k>?  
              'Each M must be less than or equal to its corresponding N.') tWYKW3~]  
    end o'@VDGS`  
    Mg]q^T.a  
    ZYoWz(  
    if any( r>1 | r<0 ) i{w<4E3  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') yz!j9pJ  
    end  Hq h  
    bZk7)b;1o  
    Y!9'Wf/^  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Hd6g0  
        error('zernfun:RTHvector','R and THETA must be vectors.') NaC^q*>9  
    end vW`{BWd  
    wn[q?|1  
    XCO{}wU)>  
    r = r(:); pC0l}hnUg  
    theta = theta(:); dI<s)!  
    length_r = length(r); 7vRJQe)  
    if length_r~=length(theta) :e:jILQ[  
        error('zernfun:RTHlength', ... MV5'&" ,oB  
              'The number of R- and THETA-values must be equal.') PZ~uHX_d>  
    end !']=7It{  
    M@S6V7  
    *4Cq,o`o>  
    % Check normalization: O:3pp8  
    % -------------------- ;JMd(\+-  
    if nargin==5 && ischar(nflag) KFBo1^9N  
        isnorm = strcmpi(nflag,'norm'); Af5O;v\  
        if ~isnorm QIVpO /@  
            error('zernfun:normalization','Unrecognized normalization flag.') ,x}p1EZ  
        end L)JpMf0  
    else TOV531   
        isnorm = false; k.>*!l0  
    end P]-d (N}/H  
    1 ry:Z2  
    ^Humy DD6  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% dIe-z7x  
    % Compute the Zernike Polynomials RG|]Kt8  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% l2KR=& SX/  
    ]Qe;+p9vU  
    /|Za[  
    % Determine the required powers of r: &yv%"BPV  
    % ----------------------------------- u^SXg dj  
    m_abs = abs(m); sY!PXD0Q  
    rpowers = []; g,U~3#   
    for j = 1:length(n) I&qT3/SVI  
        rpowers = [rpowers m_abs(j):2:n(j)]; JX(JZ/8B^  
    end q05_5  
    rpowers = unique(rpowers); fD#|C~:=  
    &mDKpYrB  
    7. 9n  
    % Pre-compute the values of r raised to the required powers, :-7`Lfi@%  
    % and compile them in a matrix: iPX6 r4-  
    % ----------------------------- :\x53-&hO4  
    if rpowers(1)==0 &=5  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); Gd1%6}<~  
        rpowern = cat(2,rpowern{:}); qlmz@kTb  
        rpowern = [ones(length_r,1) rpowern]; Fyoy)y*  
    else 6T0E'kv S  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); vU LlAQG  
        rpowern = cat(2,rpowern{:}); "knSc0 ,u  
    end gP1~N^hke]  
    8=OK8UaU  
    E6|!G  
    % Compute the values of the polynomials: %m1k^  
    % -------------------------------------- kVE% "  
    y = zeros(length_r,length(n)); C#[YDcp4  
    for j = 1:length(n) {C Qo}@.7  
        s = 0:(n(j)-m_abs(j))/2; cZT;VmC  
        pows = n(j):-2:m_abs(j); #z 3tSnmp  
        for k = length(s):-1:1 |rkj$s,  
            p = (1-2*mod(s(k),2))* ... x&7% U  
                       prod(2:(n(j)-s(k)))/              ... gsd9QW  
                       prod(2:s(k))/                     ... j7=I!<w V  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... VQV7W  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); F;Ms6 "K  
            idx = (pows(k)==rpowers); -~ytk=  
            y(:,j) = y(:,j) + p*rpowern(:,idx); -q\5)nY  
        end QPjmIO  
         ?#idmb}(  
        if isnorm N r5 aU6]  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); :D6"h[7  
        end _,(]T&j #2  
    end ^l;nBD#nJ  
    % END: Compute the Zernike Polynomials K[Bq,nPo  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% iD,iv  
    cMOvM0f  
    3>qUYxG8  
    % Compute the Zernike functions: R?!xO-^t  
    % ------------------------------ FU/yJy  
    idx_pos = m>0; \)859x&(  
    idx_neg = m<0; L+2!Sc,>  
    0o2o]{rM{2  
    GCCmUR9d  
    z = y; tyFhp:ZB  
    if any(idx_pos) |4//%Ll/  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); {^gb S  
    end itb0dF1G  
    if any(idx_neg) Z)Y--`*  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ]^MOFzSz~  
    end {?m;DY v  
    Dv?'(.z  
    Z#YkAQHv5  
    % EOF zernfun ?F'gh4  
     
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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)  td4*+)'FY  
    $/,qw   
    DDE还是手动输入的呢? :Oo  
    ,^O**k9F  
    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