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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 正序阅读 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, Sa.nUj{M=  
    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, nDckT+eJ  
    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?* du#6  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? P$AHw;n[R  
    +@8, uL  
    (o{x*';i4  
    K~^o06 Y  
    <bhJ>  
    function z = zernfun(n,m,r,theta,nflag) 7hMh%d0d(_  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. lY,9bSF$  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ,1<6=vL  
    %   and angular frequency M, evaluated at positions (R,THETA) on the 4-'0# a  
    %   unit circle.  N is a vector of positive integers (including 0), and sMJa4P>O@  
    %   M is a vector with the same number of elements as N.  Each element "av/a   
    %   k of M must be a positive integer, with possible values M(k) = -N(k) ,5t_}d|3C=  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, o2]Np~`g,  
    %   and THETA is a vector of angles.  R and THETA must have the same -H_#et3&i  
    %   length.  The output Z is a matrix with one column for every (N,M) z [u!C/  
    %   pair, and one row for every (R,THETA) pair. X ) =-a  
    % |6Iw\YU  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike i \lr KA  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), @&Yl'&pn-R  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral n36@&q+B&  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, P^lRJB<$Q  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized m#6p=E  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. Xfg?\j/  
    % XC/M:2$  
    %   The Zernike functions are an orthogonal basis on the unit circle. !l .^]|  
    %   They are used in disciplines such as astronomy, optics, and 7s:cg  
    %   optometry to describe functions on a circular domain. OMYbCy^  
    % }J\7IsM&  
    %   The following table lists the first 15 Zernike functions. ^tjM1uaZ5(  
    % ^QHgc_oDm  
    %       n    m    Zernike function           Normalization = 4'r+2[  
    %       -------------------------------------------------- +f_3JL$  
    %       0    0    1                                 1 H6 $pA^  
    %       1    1    r * cos(theta)                    2 irB}h!@  
    %       1   -1    r * sin(theta)                    2 0PUSCka'6  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) vsI|HxpyC,  
    %       2    0    (2*r^2 - 1)                    sqrt(3) 0\"]XYOH  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) 5g9K|-  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) iy_3#x5>  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) YsTF10  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) ._'.F'd  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) brF) %x`  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) "Vwk&~B%  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ah!RQ2hDrV  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) HXqG;Fds(  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Q b5vyV `  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) H}1XK|K3#H  
    %       -------------------------------------------------- N{!@M_C^%R  
    %  /zir$  
    %   Example 1: c+1<3)Q<  
    % :pP l|"  
    %       % Display the Zernike function Z(n=5,m=1) = o1&.v2j  
    %       x = -1:0.01:1; *zX^Sg-[  
    %       [X,Y] = meshgrid(x,x); dFnu&u"  
    %       [theta,r] = cart2pol(X,Y); Nb>C5TjR  
    %       idx = r<=1; 5VLC\QgK^  
    %       z = nan(size(X)); dJ{'b '#  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); U owbk:  
    %       figure XJ7mvLM;  
    %       pcolor(x,x,z), shading interp ITU6Eq  
    %       axis square, colorbar $*+UX   
    %       title('Zernike function Z_5^1(r,\theta)') @iYr<>iDZ  
    % K7 tSSX<N  
    %   Example 2: R&L^+?  
    % "/\- ?YJjw  
    %       % Display the first 10 Zernike functions \!r,>P   
    %       x = -1:0.01:1; _w9 :([_  
    %       [X,Y] = meshgrid(x,x); 0VI[6t@  
    %       [theta,r] = cart2pol(X,Y); a  ,<u  
    %       idx = r<=1; lhFv2.qR  
    %       z = nan(size(X)); j sw0"d(  
    %       n = [0  1  1  2  2  2  3  3  3  3]; %h=cwT6  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; nrz2f7d$  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; sYyya:ykxT  
    %       y = zernfun(n,m,r(idx),theta(idx)); >=L<3W1  
    %       figure('Units','normalized') H3BMN}K~  
    %       for k = 1:10 t^<ki?*  
    %           z(idx) = y(:,k); 7{u1ynt   
    %           subplot(4,7,Nplot(k)) |%Ssb;M  
    %           pcolor(x,x,z), shading interp D{, b|4  
    %           set(gca,'XTick',[],'YTick',[]) /2]=.bLwz  
    %           axis square X&|y|  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) V#d8fRm  
    %       end 77zDHq=  
    % o~p%ODH  
    %   See also ZERNPOL, ZERNFUN2. @-jI<g  
    8$6^S{M3  
    1n+JHXR\  
    %   Paul Fricker 11/13/2006 "@+r|x  
    P&8QKX3 j^  
    ?h|w7/9  
    XZ1<sm8t."  
    @:G#[>nKe  
    % Check and prepare the inputs: K<D=QweOon  
    % ----------------------------- 9]*hP](  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Zd[6-/-:  
        error('zernfun:NMvectors','N and M must be vectors.') aQ.mvuMa7'  
    end bl`vT3  
    )R9QJSe  
    c *]6>50  
    if length(n)~=length(m) ;,jms~ik  
        error('zernfun:NMlength','N and M must be the same length.') 4qLH3I[Y  
    end ){,v&[  
    p#fV|2'  
    sRf?JyB  
    n = n(:); pe7R1{2Q_s  
    m = m(:); G' a{;3  
    if any(mod(n-m,2)) AU/L_hg  
        error('zernfun:NMmultiplesof2', ... }SF<. A  
              'All N and M must differ by multiples of 2 (including 0).') 3/?{= {  
    end jMB&(r  
    zD}2Zh]  
    Umt?COc  
    if any(m>n) t"L-9kCM  
        error('zernfun:MlessthanN', ... , aQ{  
              'Each M must be less than or equal to its corresponding N.') j`$d W H/2  
    end >$iQDVh!  
    BF(.^oh"n0  
    p:8&&v~I  
    if any( r>1 | r<0 ) x $ oId{;  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') >`!Lh`n7_  
    end h oL"K  
    pz@wbu=($4  
    kc&MO`2 W\  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) f6-OR]R5  
        error('zernfun:RTHvector','R and THETA must be vectors.') y72=d?]W  
    end HOrD20  
    is [p7-  
    982n G-"  
    r = r(:); ;`MKi5g  
    theta = theta(:); %nkP?gn"a  
    length_r = length(r); Yr9!</;T  
    if length_r~=length(theta) WTP~MJ#C  
        error('zernfun:RTHlength', ... BK16~Wl  
              'The number of R- and THETA-values must be equal.') E[N3`"  
    end XrD@q  
    xsIuPL#_  
    o?l9$"\sqb  
    % Check normalization: BVk&TGa;[$  
    % -------------------- S>s{t=AY~  
    if nargin==5 && ischar(nflag) %uWq)D4r  
        isnorm = strcmpi(nflag,'norm'); P7`sJ("#  
        if ~isnorm %qf ?_2v  
            error('zernfun:normalization','Unrecognized normalization flag.') b _#r_`  
        end &6mXsx$  
    else ndU<,{r  
        isnorm = false;  "0( _  
    end +vh 4I  
    ;"77? )  
    D[ -Gzqh  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% > R5<D'cEN  
    % Compute the Zernike Polynomials _:0  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% `78:TU~5S  
    #nOS7Q#uW  
    R8U?s/*  
    % Determine the required powers of r: fxKhe[;  
    % ----------------------------------- ^YLk&A)X  
    m_abs = abs(m); wZ_k]{J  
    rpowers = []; -U"h3Ye^  
    for j = 1:length(n) zJ2dPp~u  
        rpowers = [rpowers m_abs(j):2:n(j)]; Rt^~db  
    end !C$bOhc  
    rpowers = unique(rpowers); AQH\ ;L  
    Y "RjMyQh  
    9+o`/lk1  
    % Pre-compute the values of r raised to the required powers, ogrh"  
    % and compile them in a matrix: Fuuy_+p@G  
    % ----------------------------- gLyE,1Z}u  
    if rpowers(1)==0 O*8 .kqlgt  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); B]o5 HA<k  
        rpowern = cat(2,rpowern{:}); GYq.!d@O  
        rpowern = [ones(length_r,1) rpowern]; k15B5  
    else )@O80uOFh  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); uGxh}'&  
        rpowern = cat(2,rpowern{:}); u\9t+wi}<  
    end im6Rx=}E{  
    vl{G;[6  
    1D6F WYV8  
    % Compute the values of the polynomials: .(7 end<  
    % -------------------------------------- *)6:yn  
    y = zeros(length_r,length(n)); LR5X=&k  
    for j = 1:length(n) O"D0+BK79e  
        s = 0:(n(j)-m_abs(j))/2; hrRkam !y  
        pows = n(j):-2:m_abs(j); AP8YY8,  
        for k = length(s):-1:1 P'dH*}H  
            p = (1-2*mod(s(k),2))* ... |H LU5=Y  
                       prod(2:(n(j)-s(k)))/              ... PSM~10l,  
                       prod(2:s(k))/                     ... Kn !n}GtR  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... X<OOgC  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); $T)EJe  
            idx = (pows(k)==rpowers); E9]/sFA-]  
            y(:,j) = y(:,j) + p*rpowern(:,idx); |NsrO8H   
        end /R2K3E#  
         0KQDw  
        if isnorm tocZO  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); sSM^net0  
        end _|!FhZ  
    end e B$ S d  
    % END: Compute the Zernike Polynomials |N{?LKR %  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% nsRZy0@$t  
    @w H+,]xE  
    :j}]nS  
    % Compute the Zernike functions: `H:5D5]  
    % ------------------------------ Z;nbnRz  
    idx_pos = m>0; `H_.<``>  
    idx_neg = m<0; [M7&  
    ] X9e|  
    uEK9  
    z = y; s C/5N  
    if any(idx_pos) ? x*Ve2+]  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); "o=*f/M  
    end Y'75DE<BC  
    if any(idx_neg) Vh.9/$xQ  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Z`?Z1SBt  
    end 80p?qe  
    rW~hFSrV[o  
    $[p<}o/6v]  
    % EOF zernfun 9q ##)  
     
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    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
    离线jssylttc
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    只看该作者 4楼 发表于: 2012-05-14
    顶顶·········
    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)  3GH@|id  
    2"31k2H[  
    DDE还是手动输入的呢? q*^Y8s~3I  
    X8F@U ^@  
    zygo和zemax的zernike系数,类型对应好就没问题了吧
    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。
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