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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, Z_mQpt|y  
    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, FJB B@<>:  
    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? #VsS C1  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? VFKFO9  
    z&\N^tBv  
    CtZOIx.;|  
    )!;20Po  
    #X0Xc2}{f  
    function z = zernfun(n,m,r,theta,nflag) X2{Aa T*M  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. q9c-UQB(!  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N #q5tG\gnM  
    %   and angular frequency M, evaluated at positions (R,THETA) on the Zh_3ydMD1  
    %   unit circle.  N is a vector of positive integers (including 0), and u(8dsg R  
    %   M is a vector with the same number of elements as N.  Each element t+M'05-U2  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) _>_ "cKS  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, 55=YM'5]  
    %   and THETA is a vector of angles.  R and THETA must have the same QQD7NN>  
    %   length.  The output Z is a matrix with one column for every (N,M) 9<o*aFgCa  
    %   pair, and one row for every (R,THETA) pair. -?-yeJP2  
    % iu2O/l# r  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike .vd*~U"  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ]l`V#Rd  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral +^%)QH>9   
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1,  )|W6Z  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized En4!-pWHQ  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. G/_xn5XDD  
    % g'2}Y5m$`  
    %   The Zernike functions are an orthogonal basis on the unit circle. +o35${  
    %   They are used in disciplines such as astronomy, optics, and -fN5-AC  
    %   optometry to describe functions on a circular domain. a%`L+b5-$  
    % !vuun |  
    %   The following table lists the first 15 Zernike functions. fS p  
    % ""IPaNHQ  
    %       n    m    Zernike function           Normalization qC q?`0&#  
    %       -------------------------------------------------- 2iC BF-,  
    %       0    0    1                                 1 I1JL`\;4  
    %       1    1    r * cos(theta)                    2 ,rOh*ebF  
    %       1   -1    r * sin(theta)                    2 l~[ K.p&  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) W{1l?Wo  
    %       2    0    (2*r^2 - 1)                    sqrt(3) =%|f-x  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) ~*`wRiUhis  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) $QwzL/a  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) j$4lyDfD  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) !j3Xzn9  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) "V5_B^Gzb]  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) JURg=r]LI  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ZgmK~iJ  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) Q |hBGH9:B  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) b#n  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) Z% ]LZ/O8  
    %       -------------------------------------------------- c?6(mU\x  
    % R<^E?FI   
    %   Example 1: QBA{*@ A-  
    % +e#(p<  
    %       % Display the Zernike function Z(n=5,m=1) Wxgs66   
    %       x = -1:0.01:1; Equ%6x  
    %       [X,Y] = meshgrid(x,x); &SPIu,  
    %       [theta,r] = cart2pol(X,Y); ]qx!51S  
    %       idx = r<=1; ony;U#^T  
    %       z = nan(size(X)); g_eR&kuh  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); hTPvt  
    %       figure "$q"Kilj%  
    %       pcolor(x,x,z), shading interp Z/;hbbG  
    %       axis square, colorbar "&\(:#L  
    %       title('Zernike function Z_5^1(r,\theta)') ~/Y8wxg  
    % )iZhE"?z  
    %   Example 2: S+?*l4QK  
    % |T-Y tuy8  
    %       % Display the first 10 Zernike functions )ri'W <l  
    %       x = -1:0.01:1; |aj]]l[@S  
    %       [X,Y] = meshgrid(x,x); cca]@Ox]  
    %       [theta,r] = cart2pol(X,Y); 7w\!3pv  
    %       idx = r<=1; 9h<iw\ $'  
    %       z = nan(size(X)); Z*(OcQ-  
    %       n = [0  1  1  2  2  2  3  3  3  3]; 0:x+;R<P*w  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; ANR611-a  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; Ko "JH=<  
    %       y = zernfun(n,m,r(idx),theta(idx)); 6!){-IV  
    %       figure('Units','normalized') I,V'J|=j  
    %       for k = 1:10 k1LbWR1%wB  
    %           z(idx) = y(:,k); uL^X$8K;(  
    %           subplot(4,7,Nplot(k)) R5 EC/@  
    %           pcolor(x,x,z), shading interp y }h2  
    %           set(gca,'XTick',[],'YTick',[]) 1M[|9nWUC  
    %           axis square r)<n)eXeD  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) .SBN^fq  
    %       end f Qw|SW  
    % }@53*h i(  
    %   See also ZERNPOL, ZERNFUN2. >_X(rar0  
    }-&#vP~I  
    ~\zIb/ #  
    %   Paul Fricker 11/13/2006 'NnmLM(oh  
    @d|]BqQ4jh  
    l_fERp#y  
    0IsnG?"  
    6X$\:>  
    % Check and prepare the inputs: iT1HbAT]  
    % ----------------------------- ">nFzg?Y  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 3>z+3!I z  
        error('zernfun:NMvectors','N and M must be vectors.') 0%3T'N%  
    end `?T8NK  
    T8vMBaU!qY  
    g$8a B{)  
    if length(n)~=length(m) n>%TIoY  
        error('zernfun:NMlength','N and M must be the same length.') |^GN<y^cn  
    end RP wP4Z  
    P?y{ 9H*  
    u#Uc6? E  
    n = n(:); >*}m .'u  
    m = m(:);  ur k@v  
    if any(mod(n-m,2)) tgu fU  
        error('zernfun:NMmultiplesof2', ... [wJl]i  
              'All N and M must differ by multiples of 2 (including 0).') TJs@V>,  
    end ?QzN\f Y;  
    C}]rx{xC  
     TJb&f<  
    if any(m>n) iEMIzaR  
        error('zernfun:MlessthanN', ... td2bL4  
              'Each M must be less than or equal to its corresponding N.') 2V*<J:;wb  
    end l" H/PB<.  
    79U 7<]-!  
    m RtE~~p  
    if any( r>1 | r<0 ) oc&yz>%q  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') Pu!%sGjD  
    end 55`cNZ  
    R=HcSRTkA  
    GZ"J6/0-|  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) m*TJ@gI*t  
        error('zernfun:RTHvector','R and THETA must be vectors.') )W![TIp  
    end Pe7% 9  
    ?s: 2~Qlu  
    s ,GGO3^  
    r = r(:); H3o Um1  
    theta = theta(:); =[^_x+x hE  
    length_r = length(r); fkr; a`<W  
    if length_r~=length(theta) FDRpK 5cw  
        error('zernfun:RTHlength', ... {7o|*M  
              'The number of R- and THETA-values must be equal.') dp>LhTLc  
    end Jm G)=$,  
    +JL"Z4b@R}  
    n15lX,FI  
    % Check normalization: wB0ONH[  
    % -------------------- 1He'\/#  
    if nargin==5 && ischar(nflag) ehls:)F  
        isnorm = strcmpi(nflag,'norm'); -o0~xspF  
        if ~isnorm KRP)y{~o  
            error('zernfun:normalization','Unrecognized normalization flag.') uY=}w"Db  
        end "@E(}z'sM  
    else YPsuG -is  
        isnorm = false; 9#niMv9  
    end Y<-dd"\  
    1:lhZFZ  
    qnR{'d  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% _&6&sp<n  
    % Compute the Zernike Polynomials 68jq1Y Pv  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% tr\}lfK%  
    *HN0em  
    Ot_xeg;7  
    % Determine the required powers of r: g4*]R>f  
    % ----------------------------------- B^uQv|m  
    m_abs = abs(m); bi[gyl#  
    rpowers = []; hSD uByoi  
    for j = 1:length(n) n,NKJt  
        rpowers = [rpowers m_abs(j):2:n(j)]; iw^(3FcP@C  
    end |^E# cI  
    rpowers = unique(rpowers); A?*_14&  
    i<nUp1r(  
    ryp@<}A]!d  
    % Pre-compute the values of r raised to the required powers, E`SFr  
    % and compile them in a matrix: 9:tKRN_D  
    % ----------------------------- c B9`U4<  
    if rpowers(1)==0 }x1*4+Y1  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); HLcK d`$/  
        rpowern = cat(2,rpowern{:}); nzHsyL  
        rpowern = [ones(length_r,1) rpowern]; cC6W1K!  
    else _S;L| 1>S  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); u;*Wc9>sU  
        rpowern = cat(2,rpowern{:}); kiFTx &gf  
    end 0UvN ws  
    NPM}w!  
    ?5!>k^q  
    % Compute the values of the polynomials: |PED8K:rU  
    % -------------------------------------- %Ot^G%34  
    y = zeros(length_r,length(n)); ~Xg@,?Zr  
    for j = 1:length(n) S:GX!6>  
        s = 0:(n(j)-m_abs(j))/2; +;Jb)8  
        pows = n(j):-2:m_abs(j); I)Dd"I  
        for k = length(s):-1:1 VL/%D*  
            p = (1-2*mod(s(k),2))* ... \{G1d"n  
                       prod(2:(n(j)-s(k)))/              ... $z~sN  
                       prod(2:s(k))/                     ... K?q1I<94  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... GJY7vS^#  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); T.zU erbO  
            idx = (pows(k)==rpowers); \$Nx`d aFi  
            y(:,j) = y(:,j) + p*rpowern(:,idx); *@r)3  
        end |8b*BnS  
         1e>,QX  
        if isnorm MKYXYR  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); "1H?1"w~  
        end C{ {DZ*  
    end iQ9jt  
    % END: Compute the Zernike Polynomials Q2)z1'Wv  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% d aIt `}s  
    joh=0nk;D  
    mzz77i  
    % Compute the Zernike functions: sSC yjS'T  
    % ------------------------------ 2rq)U+   
    idx_pos = m>0; t/K<fy 6  
    idx_neg = m<0; Kd _tjWS  
    Brh<6Btl  
    f#a ~av9rC  
    z = y; dD3I.?DY  
    if any(idx_pos) XTD _q  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); "|S \J5-%  
    end 0.-2FHc9L  
    if any(idx_neg) 2 fX-J  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); H/p<lp  
    end 9Kw4K#IqQ  
    _W4i?Bde  
    8]Xwj].^C  
    % EOF zernfun O1Gd_wDC/i  
     
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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)  x:O?Fj  
    z)=D&\HX  
    DDE还是手动输入的呢? lz EF^6I  
    K0yTHX?(.  
    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