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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, vpdPW%B  
    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, :9x]5;ma  
    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? M0)0~#?.D  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? hgDFhbHtd6  
    @8aV*zjB  
    h -091N  
    S5Pn6'w  
    7zU~ X,  
    function z = zernfun(n,m,r,theta,nflag) d1t_o2  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. q&NXF (  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N E[zq<&P@  
    %   and angular frequency M, evaluated at positions (R,THETA) on the kVt/Hhd9  
    %   unit circle.  N is a vector of positive integers (including 0), and QGGBI Ku   
    %   M is a vector with the same number of elements as N.  Each element dNqj|Vu  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) ZZ :*c"b:  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, Fe$o*r,  
    %   and THETA is a vector of angles.  R and THETA must have the same 0(Z:QqpU$  
    %   length.  The output Z is a matrix with one column for every (N,M) /P46k4M1U  
    %   pair, and one row for every (R,THETA) pair. C8)s6  
    % `fJ;4$4  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike xdaq` ^Bbt  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), =JP Y{'VO  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral ]]}iSw'  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 'Ce?!U O  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized \'('HFr,  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. R*k;4*1u  
    % $/(``8li_  
    %   The Zernike functions are an orthogonal basis on the unit circle. Hv:~)h$  
    %   They are used in disciplines such as astronomy, optics, and )Wt&*WMFXl  
    %   optometry to describe functions on a circular domain. E(1G!uu<  
    % |DVFi2   
    %   The following table lists the first 15 Zernike functions. Ic&YiATj  
    % U%#Vz-r  
    %       n    m    Zernike function           Normalization -y3[\zNe  
    %       -------------------------------------------------- R6z *!W{  
    %       0    0    1                                 1 R `ob;>[Q  
    %       1    1    r * cos(theta)                    2 cf"!U+x  
    %       1   -1    r * sin(theta)                    2 3G^A^]h  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) 8-kR {9r  
    %       2    0    (2*r^2 - 1)                    sqrt(3) e85E+S%  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) )7P>Hj  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) < %<nh`D  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) q%]5/.J  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) #KHj.Vg  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) E0!0 uSg&  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) _o+OkvhU  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) N6S@e\*  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) !Zc#E,  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) -sDl[  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) GH3RRzp r  
    %       -------------------------------------------------- ka(3ONbG  
    % W&I:z-VH  
    %   Example 1: ,LLx&jS  
    % #BH]`A J  
    %       % Display the Zernike function Z(n=5,m=1) I?\P^f  
    %       x = -1:0.01:1; AxO.adQE%  
    %       [X,Y] = meshgrid(x,x); 2sEG# /Y=  
    %       [theta,r] = cart2pol(X,Y); !g|[A7<|  
    %       idx = r<=1; c3<H272\  
    %       z = nan(size(X)); Y$|KY/)H)  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx));  3(*vZ  
    %       figure m|]"e@SF2  
    %       pcolor(x,x,z), shading interp dV*9bDkM/  
    %       axis square, colorbar h*Mi/\  
    %       title('Zernike function Z_5^1(r,\theta)') (58r9WhS  
    % 3fYfj  
    %   Example 2: }h3[QUVf%  
    % mr]~(]B?r  
    %       % Display the first 10 Zernike functions c@j3L23B  
    %       x = -1:0.01:1; LJ z6)kz  
    %       [X,Y] = meshgrid(x,x); !# xi^I  
    %       [theta,r] = cart2pol(X,Y);  91fZ r  
    %       idx = r<=1; R.GDCGAL  
    %       z = nan(size(X)); E=,fdyj.  
    %       n = [0  1  1  2  2  2  3  3  3  3]; *N6sxFs  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; *W 04$N  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; mWLiXKnb  
    %       y = zernfun(n,m,r(idx),theta(idx)); g]?>6 %#rA  
    %       figure('Units','normalized') k@>(sXs  
    %       for k = 1:10 G%}k_vi&q  
    %           z(idx) = y(:,k); +*e Vi3  
    %           subplot(4,7,Nplot(k)) &*Kk> 4  
    %           pcolor(x,x,z), shading interp oXVx9dZ  
    %           set(gca,'XTick',[],'YTick',[]) |gT8QP  
    %           axis square 9El{>&Fs4  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ]&='E.f  
    %       end i0?/\@gd  
    % D7jbo[GgS  
    %   See also ZERNPOL, ZERNFUN2. }p8iq  
    I}}>M#  
    Cw5 B p9  
    %   Paul Fricker 11/13/2006 4:s,e<Tc4v  
    85A7YraL  
    7$R^u7DZ  
    ,~Lx7 5{  
    /(%!txSNEt  
    % Check and prepare the inputs: UdpuQzV<4`  
    % ----------------------------- 'Awd:Aed5  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ? Z2`f6;W4  
        error('zernfun:NMvectors','N and M must be vectors.') lpbcpB  
    end a`U/|[JM  
    = ^%*:iT  
    -V'Y^Df  
    if length(n)~=length(m) vnlHUQLO  
        error('zernfun:NMlength','N and M must be the same length.') eK\i={va  
    end %T}*DC$&S  
     |vBy=:  
    YlZ&4   
    n = n(:); fK^;?4  
    m = m(:); =W gzj|Kr  
    if any(mod(n-m,2)) hSj@<#b>F  
        error('zernfun:NMmultiplesof2', ... S++jwP  
              'All N and M must differ by multiples of 2 (including 0).') owA.P-4  
    end $+U 6c~^^  
    +h1X-K:I  
    -AB0uMot  
    if any(m>n) tU.~7f#+A  
        error('zernfun:MlessthanN', ... m:9|5W  
              'Each M must be less than or equal to its corresponding N.') Y7')~C`up^  
    end 4S* X=1  
    8 9maN  
    ]r\!Z <<(  
    if any( r>1 | r<0 ) 3/,}&SX  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') m mH xPd  
    end $OzVo&P;  
    jK{qw  
    M>{*PHze0  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 4(`U]dNcs  
        error('zernfun:RTHvector','R and THETA must be vectors.') jq_ i&~S  
    end 2r@9|}La  
    M'pIAm1p  
    P<LmCY m  
    r = r(:); 59X XmVg  
    theta = theta(:); vm =d?*cR  
    length_r = length(r); wZ_"@j<  
    if length_r~=length(theta) LMLrH.  
        error('zernfun:RTHlength', ... UC.kI&A  
              'The number of R- and THETA-values must be equal.') JOwu_%  
    end D8WKy  
    qu;$I'Ul%  
    [|\#cVWs  
    % Check normalization: x+[ATZ([  
    % -------------------- >Udq{<]#r  
    if nargin==5 && ischar(nflag) PE?ICou  
        isnorm = strcmpi(nflag,'norm'); &<- S-e  
        if ~isnorm 5inCAPXz  
            error('zernfun:normalization','Unrecognized normalization flag.') )OK"H^}f  
        end 'oUTY *  
    else FRsp?i K)  
        isnorm = false; !Yz CK*av1  
    end n8i: /ypB  
    equi26jhr  
    jPn.w,=)27  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% <s$Jj><  
    % Compute the Zernike Polynomials vTC{  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% k+hl6$:Qj%  
    }-Jo9dNs  
    t~":'le`zr  
    % Determine the required powers of r: C)QKodI  
    % ----------------------------------- ;(Az   
    m_abs = abs(m); Ydyz-  
    rpowers = []; ;s+3 #Py  
    for j = 1:length(n) A f}o/g  
        rpowers = [rpowers m_abs(j):2:n(j)]; {4)d  
    end i9T<(sdK+  
    rpowers = unique(rpowers); z|zEsDh;  
    . "7-f]!  
    U U#tm  
    % Pre-compute the values of r raised to the required powers, LZQG.  
    % and compile them in a matrix: +x<OyjY5?]  
    % ----------------------------- ~(:0&w%e  
    if rpowers(1)==0 s|X_:3\x  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); PzustC|  
        rpowern = cat(2,rpowern{:}); p$` ^A  
        rpowern = [ones(length_r,1) rpowern]; G"".;}AV  
    else !>/J]/4>  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ja(ZJ[<`  
        rpowern = cat(2,rpowern{:}); j]aIJbi  
    end at1 oxmy  
    {XwDvLZ  
    4 []!Km  
    % Compute the values of the polynomials: <;cch6Z  
    % -------------------------------------- <&bBE"U4  
    y = zeros(length_r,length(n)); c{qTVi5e  
    for j = 1:length(n) O9N+<sU=X  
        s = 0:(n(j)-m_abs(j))/2; lI@Z)~  
        pows = n(j):-2:m_abs(j); }vg|05L  
        for k = length(s):-1:1 dux_v"Xl  
            p = (1-2*mod(s(k),2))* ... +]0hSpZ"p  
                       prod(2:(n(j)-s(k)))/              ... \tCK7sBn  
                       prod(2:s(k))/                     ... .')^4\  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... VFm)!'=I  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); !(3[z>  
            idx = (pows(k)==rpowers); Dj6^|R$z&  
            y(:,j) = y(:,j) + p*rpowern(:,idx); _qh \  
        end =5uhIU0O  
         12Fnv/[n'K  
        if isnorm k L4#  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); ngk:q5Tp  
        end @g*[}`8]y  
    end Y@qugQM>  
    % END: Compute the Zernike Polynomials 2EO9IxIf  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% u#Bj#y!  
    Ak$9\Sl  
    ;";>7k/}  
    % Compute the Zernike functions: 0T 0I<t  
    % ------------------------------ gADqIPu]  
    idx_pos = m>0; MJa` 4[/  
    idx_neg = m<0; o ,xy'  
    _ozg=n2(  
    x@:98P  
    z = y; tCGA3t  
    if any(idx_pos) jaMpi^C  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); %CgmZTz~<  
    end m}2hIhD9  
    if any(idx_neg) O"_QDl<ya  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); |:u5R%  
    end g;:3I\ L  
    TGjxy1A  
    $XKUw"%  
    % EOF zernfun S(rnVsW%Ki  
     
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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)  ay=f1<a  
    ck8Qs08  
    DDE还是手动输入的呢? ?;go5f+X  
    +ZRm1q   
    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