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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, G{<wXxq%  
    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, 8EQ;+V  
    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? DN+iS  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? 5|Uub ,  
    W cnYD)  
    QJ QQ-  
    iV%% VR8b  
    iJcl0)|  
    function z = zernfun(n,m,r,theta,nflag) Q{RHW@_/  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. m@~HHwj  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N }-!$KR]:s  
    %   and angular frequency M, evaluated at positions (R,THETA) on the HO' HkVA  
    %   unit circle.  N is a vector of positive integers (including 0), and z&eJ?wb  
    %   M is a vector with the same number of elements as N.  Each element j_Fr3BWS  
    %   k of M must be a positive integer, with possible values M(k) = -N(k)  W* YfyM  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, DZvpt%q  
    %   and THETA is a vector of angles.  R and THETA must have the same Jv5G:M5+~  
    %   length.  The output Z is a matrix with one column for every (N,M) t]V)3Ww  
    %   pair, and one row for every (R,THETA) pair. 7Sokn?~i  
    % $>+-=XMVB  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike z-,'W`  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), &{8 "- dw  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral E:7vm@+  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ]HRE-g  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized 0]T ;{  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. R,(^fM  
    % dK=BH=S2?X  
    %   The Zernike functions are an orthogonal basis on the unit circle. Z|)~2[Roa  
    %   They are used in disciplines such as astronomy, optics, and oY{*X6:6<  
    %   optometry to describe functions on a circular domain. =%bc;ZUu  
    % ,y^By_1wS  
    %   The following table lists the first 15 Zernike functions. {T$;BoR#O  
    % $.`(2  
    %       n    m    Zernike function           Normalization sQR;!-j  
    %       -------------------------------------------------- bw@tA7Y  
    %       0    0    1                                 1 ?p`}6s Q}  
    %       1    1    r * cos(theta)                    2 ?Hy++  
    %       1   -1    r * sin(theta)                    2  d(k`Yk8  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) :D(:( `A=  
    %       2    0    (2*r^2 - 1)                    sqrt(3) c$p1Sovw  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) OuX/BMG  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) 0DN:{dJz  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) luV%_[F  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8)  -"<eq0  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) -WEiY  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) <>-UPRw qI  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ,TL~];J'  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) K^Xg^9  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) U9Y'eP.2  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) 1cUC>_%?  
    %       -------------------------------------------------- n 6oVx 5/  
    % p/@z4TCNX  
    %   Example 1: O'(qeN<^w  
    % b\}`L"  
    %       % Display the Zernike function Z(n=5,m=1) E#T'=f[r~  
    %       x = -1:0.01:1; i`E]gJ$  
    %       [X,Y] = meshgrid(x,x); 9) wjVk  
    %       [theta,r] = cart2pol(X,Y); 3n X7$$X  
    %       idx = r<=1; a29mVmi>  
    %       z = nan(size(X)); guBOR 0x`  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); fE7Kv_N-%  
    %       figure Yzd-1Jvk  
    %       pcolor(x,x,z), shading interp zm"&8/l  
    %       axis square, colorbar N#|c2n+  
    %       title('Zernike function Z_5^1(r,\theta)') IN_GL18^MV  
    % 1`b?nX  
    %   Example 2: wp$SO^?-  
    % e;Q~P]x  
    %       % Display the first 10 Zernike functions Rb#?c+&#  
    %       x = -1:0.01:1; NmK%k jCx  
    %       [X,Y] = meshgrid(x,x); N$pO] p  
    %       [theta,r] = cart2pol(X,Y); 6Bs_" P[  
    %       idx = r<=1; WpRi+NC}ln  
    %       z = nan(size(X)); KPKby?qQ^  
    %       n = [0  1  1  2  2  2  3  3  3  3]; !iITX,'8  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; UGl}=hwKkG  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28];  c,x2   
    %       y = zernfun(n,m,r(idx),theta(idx)); Jg^tr>I~  
    %       figure('Units','normalized') 8iq~ha$]|  
    %       for k = 1:10 r/8,4:rh  
    %           z(idx) = y(:,k); OG0ro(|dI  
    %           subplot(4,7,Nplot(k)) ^fH]Rlx  
    %           pcolor(x,x,z), shading interp (gz|6N  
    %           set(gca,'XTick',[],'YTick',[]) * _U z**M  
    %           axis square ,v(G2`Z  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) obUh+9K  
    %       end fyT!/  
    % <PXA`]x~  
    %   See also ZERNPOL, ZERNFUN2. N/]TZu~k z  
    y=-d*E  
    7M5HIK6_  
    %   Paul Fricker 11/13/2006 c~@I1M  
    +STT(bMn  
    8&H1w9NrX_  
    iQ~cG[6  
    G| ^tqI  
    % Check and prepare the inputs: ("wPkm^  
    % ----------------------------- 9NKZE?5P|D  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ]oZ$,2#;~  
        error('zernfun:NMvectors','N and M must be vectors.') 2qw~hWX  
    end 2L ~U^  
    ;z!~-ByzL  
    n6 )  
    if length(n)~=length(m) HA"LU;5>2J  
        error('zernfun:NMlength','N and M must be the same length.') =v1s@5 ;~  
    end $O7>E!uVD  
    >L)Xyq  
    1,BtOzuRo  
    n = n(:); Z3"f7l6  
    m = m(:); [BmondOx  
    if any(mod(n-m,2)) w ~Es,@  
        error('zernfun:NMmultiplesof2', ... }4\>q$8'  
              'All N and M must differ by multiples of 2 (including 0).') #>[+6y]U!  
    end h?fv:^vSi  
    H#G'q_uHH  
    ?\.P  
    if any(m>n) 4LKOBiEM  
        error('zernfun:MlessthanN', ... znX2W0V  
              'Each M must be less than or equal to its corresponding N.') 4e1Zyi!  
    end %;9wToyK>  
    %q(n'^#Z.y  
    Qq^>7OU>Co  
    if any( r>1 | r<0 ) 866n{lyL  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') M {_`X  
    end : !J!l u  
    e>y"V; Mj  
    ZN',=&;n'  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) T<?JL.8g_  
        error('zernfun:RTHvector','R and THETA must be vectors.') !dStl:B  
    end $UgM7V$  
    WZ;f3 "  
    Jc:*X4-'  
    r = r(:); VI[ikNpX  
    theta = theta(:); ?,TON5Fl-  
    length_r = length(r); Yc+ /="&z  
    if length_r~=length(theta) _D[vMr[  
        error('zernfun:RTHlength', ... / IAK'/  
              'The number of R- and THETA-values must be equal.') eB^:+h#A_  
    end =A GsW  
    |'b=xeH.^<  
    f \[Z`D  
    % Check normalization: s0qA8`Yu  
    % -------------------- Of SYOL7o  
    if nargin==5 && ischar(nflag) "PLZZL$+  
        isnorm = strcmpi(nflag,'norm'); p8Ts5n  
        if ~isnorm $yI!YX&  
            error('zernfun:normalization','Unrecognized normalization flag.') E;9SsA  
        end qbFzA i  
    else z9u"?vdA  
        isnorm = false; J'.U+XU  
    end zf4@:GM`  
    VLkK6W.u  
    e(,sFhR  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% owJPEx  
    % Compute the Zernike Polynomials *GTCVxu  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% aKhI|%5kA  
    X+ h|sy  
    DU|0#z=*t5  
    % Determine the required powers of r: iK s/8n  
    % ----------------------------------- 9^c\$"2B  
    m_abs = abs(m); VD<W  
    rpowers = []; N?ky2wG  
    for j = 1:length(n) G<Z|NT  
        rpowers = [rpowers m_abs(j):2:n(j)]; xmT(yv,  
    end w*f.Fu(su  
    rpowers = unique(rpowers); YJ_LD6PL9  
    :(!il?  
    5kofO  
    % Pre-compute the values of r raised to the required powers, e{>X2UNW  
    % and compile them in a matrix: qR--lvO  
    % ----------------------------- qWfG@hn  
    if rpowers(1)==0 ?s dVd  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); BI3Q~ADV  
        rpowern = cat(2,rpowern{:}); &zynfj#o  
        rpowern = [ones(length_r,1) rpowern]; gV9 1=Pj  
    else W]4Gs;  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); HUfH/x3zj]  
        rpowern = cat(2,rpowern{:}); CZS{^6Ye  
    end l+*^P'0u  
    !gWV4vC  
    w<lHY=z E  
    % Compute the values of the polynomials: [2a*TI  
    % -------------------------------------- @K7#}7,t  
    y = zeros(length_r,length(n)); q1;}~}W;z4  
    for j = 1:length(n) 0-oR { {  
        s = 0:(n(j)-m_abs(j))/2; I;S[Ft8d  
        pows = n(j):-2:m_abs(j); QyuSle  
        for k = length(s):-1:1 $21+6  
            p = (1-2*mod(s(k),2))* ... X@*$3z#Z  
                       prod(2:(n(j)-s(k)))/              ... S ])Ap'E  
                       prod(2:s(k))/                     ... k^}8=,j}  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... pE[ul  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); b?hdWQSW7  
            idx = (pows(k)==rpowers); y<.0+YL-e+  
            y(:,j) = y(:,j) + p*rpowern(:,idx); zZ3,e L  
        end lUJ/ nG0l  
         6'3@/.  
        if isnorm G,FYj'<!7,  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); R+ lwOVX  
        end T@;z o8:  
    end Y4sf 2w  
    % END: Compute the Zernike Polynomials h3$.` >l  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% t|jX%s=  
    iov55jT~l@  
    rDX_$,3L  
    % Compute the Zernike functions: yQ?N*'}$  
    % ------------------------------ ,Drd s"H  
    idx_pos = m>0; 9[N+x2q  
    idx_neg = m<0; K'+GK S7.  
    }#ZQ\[  
    gk>-h,>"  
    z = y; Uc/MPCqZ  
    if any(idx_pos) lpQsmd#  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ^a4y+!  
    end WBFG_])  
    if any(idx_neg) rR@ t5  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); s PYG?P(l  
    end (Hb i+IHV  
    j(F&*aH78  
    aL$m  
    % EOF zernfun $`W .9  
     
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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)  tj/X 7|  
    _1jd{? kt  
    DDE还是手动输入的呢?  U, _nEx  
    ? RL[#d+y  
    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