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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, {Q}F.0Q  
    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, =C2KHNc  
    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? P8(hHuO  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? 31~nay15  
    Gp{,v  
    (3"N~\9m  
    $nb.[si\  
    o_1N "o%  
    function z = zernfun(n,m,r,theta,nflag) Mj{w/'  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. W=#AfPi$&  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ?-zuy US  
    %   and angular frequency M, evaluated at positions (R,THETA) on the $J^fpXO  
    %   unit circle.  N is a vector of positive integers (including 0), and 9Ta0Li  
    %   M is a vector with the same number of elements as N.  Each element DXo]O}VF  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) ^)wKS]BQ..  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, `BQv;NtP  
    %   and THETA is a vector of angles.  R and THETA must have the same <PVwf`W.  
    %   length.  The output Z is a matrix with one column for every (N,M) ae2Q^yLA  
    %   pair, and one row for every (R,THETA) pair. $~S~pvT  
    % kU+|QBA@  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ?=ffv]v|  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ?G5,}%  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral {#:31)P  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, {zWR)o .=  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized vQ L$.A3>  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 8PzGUn;\  
    % a}uYv:  
    %   The Zernike functions are an orthogonal basis on the unit circle. 0$]iRE;O]  
    %   They are used in disciplines such as astronomy, optics, and r\d(*q3B  
    %   optometry to describe functions on a circular domain. m`l9d4p w?  
    % *5 +GJWKN  
    %   The following table lists the first 15 Zernike functions. A#6zI NK#B  
    % {vGJ}q?Sd"  
    %       n    m    Zernike function           Normalization {9yf0n  
    %       -------------------------------------------------- ~_-]> SI  
    %       0    0    1                                 1 (c>g7d<>n  
    %       1    1    r * cos(theta)                    2 qa-FLUkIk!  
    %       1   -1    r * sin(theta)                    2 R0}1:1}$Sn  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) K Ax=C}9  
    %       2    0    (2*r^2 - 1)                    sqrt(3) ni&|;"Nt-  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) 0|RofL&o  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) d)e mTXB(  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) ~\mh\a&  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) ."H;bfcL_  
    %       3    3    r^3 * sin(3*theta)             sqrt(8)  ]'`E  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) {BmqUoZrC  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) `XhH{*Q"X  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) [Q0V5P~Q'  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ^3TNj  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) a8f#q]TyQ  
    %       -------------------------------------------------- U fyhd  
    % 1!KROes4  
    %   Example 1: \4L ur  
    % HMCLJ/  
    %       % Display the Zernike function Z(n=5,m=1) X58U>4a  
    %       x = -1:0.01:1; ? Bpnnwx  
    %       [X,Y] = meshgrid(x,x); Vw1>d+<~-)  
    %       [theta,r] = cart2pol(X,Y); n&njSj/  
    %       idx = r<=1; )Cl>%9  
    %       z = nan(size(X)); O|V0WiY<  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); _Xt/U>N  
    %       figure `UTPX'Vz  
    %       pcolor(x,x,z), shading interp mUa#sTm  
    %       axis square, colorbar &h0LWPl  
    %       title('Zernike function Z_5^1(r,\theta)') T@tsM|pI  
    % 4AS%^&ah  
    %   Example 2: l!f_ +lv  
    % +Yc^w5 !(  
    %       % Display the first 10 Zernike functions B;r_[^  
    %       x = -1:0.01:1; J5G<Y*q  
    %       [X,Y] = meshgrid(x,x); 68XJ`/d  
    %       [theta,r] = cart2pol(X,Y); :$$~$P  
    %       idx = r<=1; x ;|HT  
    %       z = nan(size(X)); `Pvi+:6\Y  
    %       n = [0  1  1  2  2  2  3  3  3  3]; &KjMw:l  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; -K'UXoU1  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; %dzt'uz  
    %       y = zernfun(n,m,r(idx),theta(idx)); [UA*We 1  
    %       figure('Units','normalized') P |t yyjO  
    %       for k = 1:10 ) 2Ei<  
    %           z(idx) = y(:,k); 509T?\r  
    %           subplot(4,7,Nplot(k)) ?$|tT\SFV  
    %           pcolor(x,x,z), shading interp 2y - QH  
    %           set(gca,'XTick',[],'YTick',[]) )Ka-vX)D@  
    %           axis square 1.du#w  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) )8A.Wg4S;c  
    %       end ga;nM#/  
    % 9;+&}:IVS  
    %   See also ZERNPOL, ZERNFUN2. ZAr6RRv ^  
    Ddr.6`VJ  
    zR<{z  
    %   Paul Fricker 11/13/2006 .dU91> ~Ov  
    ~JT`q: l-q  
    #yochxF_  
    Cw,a)XB  
    <D.E .^Y  
    % Check and prepare the inputs: ^3dc#5]Xf  
    % ----------------------------- 1eD#-tzV  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) AkQ(V  
        error('zernfun:NMvectors','N and M must be vectors.') M{J>yN  
    end rRRh-%.RU  
    m^QoB  
    U4"^NLAq  
    if length(n)~=length(m) G`!,>n 3  
        error('zernfun:NMlength','N and M must be the same length.') VZi1b0k1.  
    end ;0dH@b  
    ';3>rv_  
    tg\Nm7I  
    n = n(:); uVqc:Q"  
    m = m(:); Fqeqn[,  
    if any(mod(n-m,2)) t{] 6GlW  
        error('zernfun:NMmultiplesof2', ... -s 0SQe{!_  
              'All N and M must differ by multiples of 2 (including 0).') z:-{Y2F  
    end g=\(%zfsxr  
    dHY@V> D'-  
    6>WkisxG  
    if any(m>n) B&_:20^y~  
        error('zernfun:MlessthanN', ... mfj{_fR3  
              'Each M must be less than or equal to its corresponding N.') ~!({U nt+'  
    end BbX$R`f  
    uU)t_W&-J  
    t\/H.Hb  
    if any( r>1 | r<0 ) ? X8`+`nh  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') >&.N_,*  
    end "q?(rx;  
    `:iMGq ZN  
    j EbmW*   
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) %`bs<ZWT  
        error('zernfun:RTHvector','R and THETA must be vectors.') |B (,53  
    end NuO@N r  
    12 )  
    =#2%[kGq  
    r = r(:); tV=Qt[|@  
    theta = theta(:); >J9Qr#=H2  
    length_r = length(r); ,O:4[M!$w  
    if length_r~=length(theta) a0ms9%Y;Q[  
        error('zernfun:RTHlength', ... ]4t1dVD  
              'The number of R- and THETA-values must be equal.') >7WT4l)7!b  
    end L[zTT\a  
    OFo hyy(  
    !S<p"   
    % Check normalization: ) P7oL.)  
    % -------------------- QO$18MBcc  
    if nargin==5 && ischar(nflag) .B^ tEBGVD  
        isnorm = strcmpi(nflag,'norm'); mg*iW55g  
        if ~isnorm /[Nkk)8-  
            error('zernfun:normalization','Unrecognized normalization flag.') |~76dxU  
        end s1OSuSL>  
    else Nn_b  
        isnorm = false; w%wVB/(  
    end 3x(Y+ ymP  
    VN-0hw/A  
    f:8!@,I  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% c*owP  
    % Compute the Zernike Polynomials R UCUEo63  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% lGet)/w;c  
    "U9e)a0v  
    G$)q% b;Lz  
    % Determine the required powers of r: 5#z7Hj&w  
    % ----------------------------------- ,M]W_\N~E  
    m_abs = abs(m); ^E, #}cW  
    rpowers = []; fm#7}Y  
    for j = 1:length(n) fhk(<KZvJ  
        rpowers = [rpowers m_abs(j):2:n(j)]; `_&vvJPn@!  
    end s|WcJV  
    rpowers = unique(rpowers); )l*3^kwL{U  
    )[99SM   
    5bZ0}^FYF  
    % Pre-compute the values of r raised to the required powers, 7yG%E  
    % and compile them in a matrix: B|syb!g  
    % ----------------------------- #x;d+Q@  
    if rpowers(1)==0 C^?/9\  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); -Nr*na^H9#  
        rpowern = cat(2,rpowern{:}); 2n"-~'3\  
        rpowern = [ones(length_r,1) rpowern]; nF-l4=  
    else <&+0  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); dRs\e(H'  
        rpowern = cat(2,rpowern{:}); af[dkuv  
    end  v?d`fd  
    "SuG6!k3  
    ga'G)d3oS  
    % Compute the values of the polynomials: bz1`f>%l  
    % -------------------------------------- ,A#gF_8  
    y = zeros(length_r,length(n)); 0{!-h  
    for j = 1:length(n) L{ej<0yr  
        s = 0:(n(j)-m_abs(j))/2; Yl1l$[A$  
        pows = n(j):-2:m_abs(j); ~Y1nU-  
        for k = length(s):-1:1 4U$M0 =  
            p = (1-2*mod(s(k),2))* ... 4<EC50@.  
                       prod(2:(n(j)-s(k)))/              ... zl, Vj%d  
                       prod(2:s(k))/                     ... 0W 1bZPM  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ]:#W$9,WL  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); X&Ospl@H  
            idx = (pows(k)==rpowers); aYtW!+#  
            y(:,j) = y(:,j) + p*rpowern(:,idx); %U[H`E  
        end )eX{a/Be  
         2L.6!THG  
        if isnorm uxX 3wY;M  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); PTQN.[bBh  
        end !(S.7#-r  
    end `/G9*tIR8g  
    % END: Compute the Zernike Polynomials xNJ*TA[+  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% tI0D{Xrc  
    dF&@q,  
    "-HWw?rx/  
    % Compute the Zernike functions: T7Y+ WfYh  
    % ------------------------------ oB<!U%BN  
    idx_pos = m>0; H.Z<T{y;  
    idx_neg = m<0; X2 <fS~m  
    l?X)]1  
    )|]*"yf:E  
    z = y; |*~SR.[`  
    if any(idx_pos) eS%8WmCV9<  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); HbCcROl(  
    end M"Y ,kA|+  
    if any(idx_neg) h5n@SE>G  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); n "I{aJ]K  
    end MHCwjo"  
    ^C2SLLgeJ  
    y?iW^>|?L=  
    % EOF zernfun f?QP(+M5.  
     
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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)  CTQJ=R"  
     =tc!"{  
    DDE还是手动输入的呢? wB"`lY   
    %0%Tp  
    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