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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, =hlu, By  
    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, U]$3NIe  
    这个,跟我们用zygo干涉仪直接拟合出的36项zernike系数,有何关系呢? M*uG`Eo&  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? GjG3aqP&!  
    <ZdNPcT<s  
    K@hUif|([  
    x~^nlnKVf  
    0&~u0B{  
    function z = zernfun(n,m,r,theta,nflag) '& :"/4@)  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. CB1u_E_  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 5w9<_W0d  
    %   and angular frequency M, evaluated at positions (R,THETA) on the }5U f`pM8  
    %   unit circle.  N is a vector of positive integers (including 0), and JH#?}L/0Fe  
    %   M is a vector with the same number of elements as N.  Each element kMXl {  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) Zv93cv  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, j&5Xjl>4  
    %   and THETA is a vector of angles.  R and THETA must have the same l"8YIsir  
    %   length.  The output Z is a matrix with one column for every (N,M) Mr(3]EfgO  
    %   pair, and one row for every (R,THETA) pair. 2T9Z{v  
    % 1L7,x @w  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike X;tk\Ixd  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), _{%H*PxTn=  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral K(2s%  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, -%|I  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized RwWQ$Eb_s  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. Qt 2hb  
    % kF .b)  
    %   The Zernike functions are an orthogonal basis on the unit circle. ZxQP,Ys_Y  
    %   They are used in disciplines such as astronomy, optics, and 7O#>N}|  
    %   optometry to describe functions on a circular domain. %#~Wk|8} Q  
    % <5%We(3  
    %   The following table lists the first 15 Zernike functions. uip]K{/A!e  
    % 9m{rQ P/  
    %       n    m    Zernike function           Normalization 6~LpBlb  
    %       -------------------------------------------------- yM@cml6Ox  
    %       0    0    1                                 1 X*Zv,Wm  
    %       1    1    r * cos(theta)                    2 75f.^4/%  
    %       1   -1    r * sin(theta)                    2 AP%h!b5v  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) clNP9{  
    %       2    0    (2*r^2 - 1)                    sqrt(3) ?|\Lm3%J  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) om6R/K  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) dQ]j r.  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) 7Z_iQ1  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) &3V4~L1aEg  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) +8M{y D9#  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) ojri~erJE?  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) gN%R-e0  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) f&'md  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 65v'/m!ys  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) #A!0KN;GC2  
    %       -------------------------------------------------- G)Y!aX  
    % 566EMy|  
    %   Example 1: O9Aooe4W=  
    % x& S>Mr  
    %       % Display the Zernike function Z(n=5,m=1) n^K]R}S  
    %       x = -1:0.01:1; i{2KMa{K  
    %       [X,Y] = meshgrid(x,x); _ sd?l  
    %       [theta,r] = cart2pol(X,Y); xlAaIo)T  
    %       idx = r<=1; }O/Nn0,  
    %       z = nan(size(X)); #~b9H05D  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); ) =[Tgh  
    %       figure  ~$B ,K]  
    %       pcolor(x,x,z), shading interp ryN-d%t?  
    %       axis square, colorbar UWHC]V?  
    %       title('Zernike function Z_5^1(r,\theta)') |@RO&F  
    % <OUAppH  
    %   Example 2: 4/b#$o<I?  
    % 2r,fF<WQ  
    %       % Display the first 10 Zernike functions TR|; /yJ  
    %       x = -1:0.01:1; e(Ve rd:c  
    %       [X,Y] = meshgrid(x,x); #qWEyb2UZ  
    %       [theta,r] = cart2pol(X,Y); qF ?S[Z;  
    %       idx = r<=1; (_* a4xGF  
    %       z = nan(size(X)); dx^3(#B  
    %       n = [0  1  1  2  2  2  3  3  3  3]; ;1KhUf;&F  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; (w*$~p  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; ="`y<J P  
    %       y = zernfun(n,m,r(idx),theta(idx)); ]zO]*d=m  
    %       figure('Units','normalized') ep5aBrN]"  
    %       for k = 1:10 ,Gfnf%H\8>  
    %           z(idx) = y(:,k); x {rt\OT  
    %           subplot(4,7,Nplot(k)) 04s N 4C  
    %           pcolor(x,x,z), shading interp \ys3&<;b  
    %           set(gca,'XTick',[],'YTick',[]) MmX42;Pw  
    %           axis square 2}NfR8 N  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) #xmUND`@  
    %       end  m ]\L1&  
    % bnlL-]]9z  
    %   See also ZERNPOL, ZERNFUN2. ` F)Iv:;y,  
    IAfYlS#<yD  
    z, OMR`W  
    %   Paul Fricker 11/13/2006 ZrTq)BZ  
    HV}NT~  
    <C&UD j  
    yH Cc@`1.  
    ,"D1!0  
    % Check and prepare the inputs: |A2.W8`o  
    % ----------------------------- '@$?A>.cj  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) F?UL0Q|uv  
        error('zernfun:NMvectors','N and M must be vectors.') oR+Fn}mG  
    end p'H5yg3h  
    7l?=$q>k"  
    ?}W:DGudZ  
    if length(n)~=length(m) .8"o&%$`V  
        error('zernfun:NMlength','N and M must be the same length.') (k[<>$hL*  
    end `p!.K9r7   
    h.67] U7m  
    G^+0</Q  
    n = n(:); wtZe\ h  
    m = m(:); U<*dDE~z  
    if any(mod(n-m,2)) iB\d `NUf  
        error('zernfun:NMmultiplesof2', ... l!oU9  
              'All N and M must differ by multiples of 2 (including 0).') =%a.C(0&G  
    end w'UP#vT5&  
    9Vp$A$7M  
    o:?IT/>  
    if any(m>n) 46mu,v  
        error('zernfun:MlessthanN', ... zP5HTEz  
              'Each M must be less than or equal to its corresponding N.') &=f%(,+  
    end UOa{J|k>h  
    77)C`]0(  
    jildiT[s  
    if any( r>1 | r<0 ) b7\nCRY  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') Sna7r~ j  
    end d~.#KS  
    poM VB{U  
    U7{, *  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) RlpW)\{j?  
        error('zernfun:RTHvector','R and THETA must be vectors.') %cBJ haR{(  
    end wt-)5f'{  
    r1,RloyZS  
    &ed.%:  
    r = r(:); {~Phc 2z  
    theta = theta(:); J H6\;G6  
    length_r = length(r); $[IuEdc/  
    if length_r~=length(theta) IuRKj8J)o  
        error('zernfun:RTHlength', ... e\\ I,  
              'The number of R- and THETA-values must be equal.') dD#A.C,Rz  
    end w@hm>6j  
     M*%iMz  
    qF>}"m  
    % Check normalization: Cfa?LgSz  
    % -------------------- ,;UVQwY  
    if nargin==5 && ischar(nflag) 1;SWfKU?.  
        isnorm = strcmpi(nflag,'norm'); N'TL &]  
        if ~isnorm d6wsT\S  
            error('zernfun:normalization','Unrecognized normalization flag.') d'PjO-"g  
        end Zpg$:Rr  
    else uQrD}%GI  
        isnorm = false; xaejG/'iK  
    end e;,D!  
    `NrxoU=  
    o ;.j_  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% CmM K\R.  
    % Compute the Zernike Polynomials )~rN{W<s`H  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% kAU[lPt*R  
    =! /S |  
    TN(1oJ:  
    % Determine the required powers of r: MH1??vW  
    % ----------------------------------- .#P'NF(5#  
    m_abs = abs(m); {73Z$w1%  
    rpowers = []; @MTm8E6au  
    for j = 1:length(n)  #\Lt0  
        rpowers = [rpowers m_abs(j):2:n(j)];   "Qm  
    end GoZr[=d  
    rpowers = unique(rpowers); B_nim[72  
    5^*I]5t8  
    "4|D"|wI)  
    % Pre-compute the values of r raised to the required powers, o|$l+TC  
    % and compile them in a matrix: j$siCsF  
    % ----------------------------- *JUP~/Nr  
    if rpowers(1)==0 < OCy  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); b v~"_)C  
        rpowern = cat(2,rpowern{:}); cd#@"&r  
        rpowern = [ones(length_r,1) rpowern]; vEk jd#  
    else DhYQ>Gv8U  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); UOi8>;k`  
        rpowern = cat(2,rpowern{:}); Z-.`JkKd8  
    end K#kU6/  
    gYw4YP0Gz  
    `~F=  
    % Compute the values of the polynomials: +g>)Bur  
    % -------------------------------------- a)/!ifJ;  
    y = zeros(length_r,length(n)); 0ERA(=w5  
    for j = 1:length(n) =E,^ +`M  
        s = 0:(n(j)-m_abs(j))/2; 5L'X3g  
        pows = n(j):-2:m_abs(j); Z`Rrv$M!  
        for k = length(s):-1:1 p<: bP w  
            p = (1-2*mod(s(k),2))* ... 7X*$Fu<  
                       prod(2:(n(j)-s(k)))/              ... :>c33X}  
                       prod(2:s(k))/                     ... \ym3YwP4/:  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... := C-P7  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); K1Snag  
            idx = (pows(k)==rpowers); _?]bd-E  
            y(:,j) = y(:,j) + p*rpowern(:,idx); 8XIG<Nc  
        end ;*Ldnj;B  
         fucG 9B  
        if isnorm EIPNR:6t  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); ]yiwdQ  
        end | U )  
    end w3WBgH  
    % END: Compute the Zernike Polynomials 7j| ^ZuI+  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ZxF RE#y~2  
    t2uX+1F  
    $aDAD4mmm  
    % Compute the Zernike functions: )!z<q}i5  
    % ------------------------------ V{+'(<SV  
    idx_pos = m>0; V(3^ev/  
    idx_neg = m<0; F9IrbLS9c  
    DybuLB$f  
    wVX]"o  
    z = y; <[Oe.0SGu  
    if any(idx_pos) H]As2$[  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); !tJQ75Hwv  
    end 1fUg  
    if any(idx_neg) (?4m0Sn>#h  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 0}H7Xdkp  
    end WR,MqM20  
    |C"(K-do  
    (d mLEt  
    % EOF zernfun &y_Ya%Z3*e  
     
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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)  =odKi"-6  
    oTJ^WePZQ  
    DDE还是手动输入的呢? WJJmM*>JW  
    h_HPmh5  
    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