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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, ]2m=lt1  
    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, B0b|+5WhR  
    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? T3oFgzoO  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? CbM~\6 R  
    U>*@VOgB  
    e">&B]#}  
    0x~+=GUN  
    8i] S[$Fc  
    function z = zernfun(n,m,r,theta,nflag) Vwp>:'Pu  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. ppIXS(  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 1oO(;--u_  
    %   and angular frequency M, evaluated at positions (R,THETA) on the @xdtl{5G  
    %   unit circle.  N is a vector of positive integers (including 0), and  dHx4yFS  
    %   M is a vector with the same number of elements as N.  Each element x} =,'Ko}3  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) @Dsw.@/  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, O:GPuVb\  
    %   and THETA is a vector of angles.  R and THETA must have the same Ag0 6M U  
    %   length.  The output Z is a matrix with one column for every (N,M) eg*aVb  
    %   pair, and one row for every (R,THETA) pair. O<p=&=TD7  
    % DtBvfYO8)>  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ).jQ+XE'>  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 00;SK!+$  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral &w^9#L  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, spP[S"gI  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized &,{ >b[  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. r jn:E  
    % g0B-<>E  
    %   The Zernike functions are an orthogonal basis on the unit circle. UUz{Qm%  
    %   They are used in disciplines such as astronomy, optics, and  Me z&@{  
    %   optometry to describe functions on a circular domain. D,..gsg  
    % !j7mY9x+  
    %   The following table lists the first 15 Zernike functions. ugN%8N  
    % . h)VR 5?j  
    %       n    m    Zernike function           Normalization )kjQ W&)g  
    %       -------------------------------------------------- " TCJT390  
    %       0    0    1                                 1 uM 'n4oH  
    %       1    1    r * cos(theta)                    2 v @M6D}  
    %       1   -1    r * sin(theta)                    2 J1(SL~e],  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) }f;TG:6  
    %       2    0    (2*r^2 - 1)                    sqrt(3) =C$"e4%Be  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) =k d-rIBc  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) O6$,J1 2l  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) nnhI]#,a{  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) uDG>m7(}/h  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) b' ^<0c  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) =g6~2p=H  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) zK~_e\m  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) b&E"r*i|  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) l@w\ Vxr  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) PSAEW.L  
    %       -------------------------------------------------- T] H 'l  
    % k {{eyC  
    %   Example 1: /kr|}`# Z  
    % m~=VUhPd  
    %       % Display the Zernike function Z(n=5,m=1) 'S}3lsIE  
    %       x = -1:0.01:1; vt"bB  
    %       [X,Y] = meshgrid(x,x); ~b *|V  
    %       [theta,r] = cart2pol(X,Y); q}jh>`d  
    %       idx = r<=1; fif'ptK  
    %       z = nan(size(X)); 7?g({]  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); ]srL>29_b  
    %       figure CEkf0%YJ  
    %       pcolor(x,x,z), shading interp Q& d;UVp  
    %       axis square, colorbar }t(5n$go6  
    %       title('Zernike function Z_5^1(r,\theta)') !b0A %1W;  
    % 8@;R2]Q  
    %   Example 2: |Z>}#R!,P  
    % WllQM,h  
    %       % Display the first 10 Zernike functions ,^1 #Uz8  
    %       x = -1:0.01:1; 4VF]t X?o  
    %       [X,Y] = meshgrid(x,x); 1)}hzA  
    %       [theta,r] = cart2pol(X,Y); 8rJf2zL  
    %       idx = r<=1; 4j+M<g  
    %       z = nan(size(X)); Qg1kF^=  
    %       n = [0  1  1  2  2  2  3  3  3  3]; bly `m p8#  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; sw1gpkX  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; =j w?*  
    %       y = zernfun(n,m,r(idx),theta(idx)); .+8#&Uy  
    %       figure('Units','normalized') !RLXB$@`  
    %       for k = 1:10 TRgj`FG  
    %           z(idx) = y(:,k); _W Hi<,-  
    %           subplot(4,7,Nplot(k)) sjLm-pn3  
    %           pcolor(x,x,z), shading interp p;zT #%  
    %           set(gca,'XTick',[],'YTick',[]) (O:&RAkk7  
    %           axis square v8\_6}*I  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) HYdt3GtJ?  
    %       end k;Qm%B  
    % "kc%d'c(  
    %   See also ZERNPOL, ZERNFUN2. 8rBa}v9  
    Tsu\4 cL]  
    ;\13x][  
    %   Paul Fricker 11/13/2006 3# 0Nd"/0  
    OS(Ua  
    +sZY0(|K8  
    /55 3v;l<  
    >;M STHeW  
    % Check and prepare the inputs: @Z""|H"0  
    % ----------------------------- `]6W*^'PD  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) NeEV=+<-G  
        error('zernfun:NMvectors','N and M must be vectors.') lUnC+w#[  
    end ^Kl<<pUaV  
    iH)vLD  
    W^,p2  
    if length(n)~=length(m) h|z59h&X8G  
        error('zernfun:NMlength','N and M must be the same length.') P|fh4b4  
    end <gvgr4@^yR  
    %gqu7}'  
    (A_H[xP  
    n = n(:); XGnC8Be{4  
    m = m(:); 5}9rpN{y  
    if any(mod(n-m,2)) C?g*c  
        error('zernfun:NMmultiplesof2', ... >"]t4]GVf  
              'All N and M must differ by multiples of 2 (including 0).') [--] ?Dr  
    end C91'dM  
    rc{F17~vX  
    KAT^vbR  
    if any(m>n) 2mthUq9b*  
        error('zernfun:MlessthanN', ... ?[5_/0L,=  
              'Each M must be less than or equal to its corresponding N.') YpSK |(  
    end v~!_DD au  
    8Sf}z@~]  
    ,I f9w$(z  
    if any( r>1 | r<0 ) \S?;5LacZ  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') cn_KHz=  
    end J<iiA:&J  
    u69G #  
    Hg(nC*#/Q  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) dlV HyCW  
        error('zernfun:RTHvector','R and THETA must be vectors.') |JUAR{  
    end <;Td8T;  
    n3hlo@gYW  
    y<8o!=Tb5  
    r = r(:); j{%'A  
    theta = theta(:); .X4UDZQg  
    length_r = length(r); /-ewCCzZV  
    if length_r~=length(theta) b~rlh=(o#_  
        error('zernfun:RTHlength', ... Zr!CT5C5  
              'The number of R- and THETA-values must be equal.') > lK:~~1  
    end Ve\!:,(Y_  
    wqQrby<  
    !xC IvKW  
    % Check normalization: <qxqlEQT  
    % -------------------- S)@) @3  
    if nargin==5 && ischar(nflag) EhIa31>X  
        isnorm = strcmpi(nflag,'norm'); {*qz<U >  
        if ~isnorm M ~6k[ew  
            error('zernfun:normalization','Unrecognized normalization flag.') H#I%6k*\a  
        end HO8x:2m  
    else Oufdi3h  
        isnorm = false; rEs Gf+4  
    end S\11 8TpD  
    lJ4&kF=t  
    Jy#2 1  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0\Myhh~DLE  
    % Compute the Zernike Polynomials V7Mp<x%  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ddeH-Z  
    A|0\ct  
    cD4H@!=a  
    % Determine the required powers of r: l:"zYcp%  
    % ----------------------------------- ')v<MqBr  
    m_abs = abs(m); mr#XN&e  
    rpowers = []; a)M#O\i`  
    for j = 1:length(n) JiHk`e`  
        rpowers = [rpowers m_abs(j):2:n(j)]; pH!8vnoA  
    end 'sAs#  
    rpowers = unique(rpowers); P*8DM3':  
    *}NJ  
    ~]lVixr9  
    % Pre-compute the values of r raised to the required powers, y{uN+QS  
    % and compile them in a matrix: DWar3+u&0  
    % ----------------------------- 1ml{oqNj  
    if rpowers(1)==0 RVe UQ%  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 8G p%Q  
        rpowern = cat(2,rpowern{:}); ^U@E rc#d  
        rpowern = [ones(length_r,1) rpowern]; j[YO1q*  
    else 7S]akcT/  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); `Ot;KDz  
        rpowern = cat(2,rpowern{:}); T,Zfz9{n  
    end x4bj?=+  
    %'i`Chc^!;  
    i_qR&X  
    % Compute the values of the polynomials: 095Z Z20  
    % -------------------------------------- 1W2hd!J7C  
    y = zeros(length_r,length(n)); "G @(AE(  
    for j = 1:length(n) TYh_uox6  
        s = 0:(n(j)-m_abs(j))/2; B[6y2+6$0  
        pows = n(j):-2:m_abs(j); aJ}Cq k  
        for k = length(s):-1:1 H$n{|YO `  
            p = (1-2*mod(s(k),2))* ... JRl`evTS  
                       prod(2:(n(j)-s(k)))/              ... 3XomnL{  
                       prod(2:s(k))/                     ... h\qM5Qx+Q  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ...  MfNguh  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); !9JK95;  
            idx = (pows(k)==rpowers); -& \?Q_6  
            y(:,j) = y(:,j) + p*rpowern(:,idx); dKwY\)\  
        end _; ].  
         Yy:Q/zw o  
        if isnorm %?[H=v(b  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); x_TtS|   
        end L[FNr&  
    end kdHP v=/U  
    % END: Compute the Zernike Polynomials e^ygQ<6%  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% #4<Rs|K  
    !F&Ss|(}  
    AmmUoS\  
    % Compute the Zernike functions: (qM(~4|`  
    % ------------------------------ QX j4cg  
    idx_pos = m>0; .U:DuyT  
    idx_neg = m<0; ,5L[M&5  
    <MH| <hP  
    =9ISsI\Y6  
    z = y; )cX6o[oia  
    if any(idx_pos) qc-4;m o  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); \f7A j>  
    end :7+E fu  
    if any(idx_neg) h (`Erb  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); u.s-/ g  
    end hVAP )"5  
    KvrcO#-sL  
    ch%-Cg~%  
    % EOF zernfun ]7`)|PJ  
     
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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)  "#H@d+u  
    S~QL x  
    DDE还是手动输入的呢? =_%i5]89P  
    oaI|A^v  
    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