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    [求助]ansys分析后面型数据如何进行zernike多项式拟合? [复制链接]

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    离线niuhelen
     
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    只看楼主 倒序阅读 楼主  发表于: 2011-03-12
    小弟不是学光学的,所以想请各位大侠指点啊!谢谢啦 LL3| U  
    就是我用ansys计算出了镜面的面型的数据,怎样可以得到zernike多项式系数,然后用zemax各阶得到像差!谢谢啦! ,:}VbQ:3I  
     
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    离线phility
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    只看该作者 1楼 发表于: 2011-03-12
    可以用matlab编程,用zernike多项式进行波面拟合,求出zernike多项式的系数,拟合的算法有很多种,最简单的是最小二乘法,你可以查下相关资料,挺简单的
    离线phility
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    只看该作者 2楼 发表于: 2011-03-12
    泽尼克多项式的前9项对应象差的
    离线niuhelen
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    只看该作者 3楼 发表于: 2011-03-12
    回 2楼(phility) 的帖子
    非常感谢啊,我手上也有zernike多项式的拟合的源程序,也不知道对不对,不怎么会有 `$;%%/tx  
    function z = zernfun(n,m,r,theta,nflag)  ,8p-EH  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. Q;2k bVWY  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N @DR?^ qp  
    %   and angular frequency M, evaluated at positions (R,THETA) on the zq^eL=%:  
    %   unit circle.  N is a vector of positive integers (including 0), and N':d T  
    %   M is a vector with the same number of elements as N.  Each element ?y*yl  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) ug`Jn&x!  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, ~|wh/]{b9  
    %   and THETA is a vector of angles.  R and THETA must have the same .a]av   
    %   length.  The output Z is a matrix with one column for every (N,M) 8`b_,(\N  
    %   pair, and one row for every (R,THETA) pair. ;ahI}}  
    % $>l65)(E\  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike HFj@NRE6  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi),  #|l#  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral PsS8b  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 98l-  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized ^zS|O]Tx  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. (TGG?V  
    % VelX+|w  
    %   The Zernike functions are an orthogonal basis on the unit circle. #5IfF~* i  
    %   They are used in disciplines such as astronomy, optics, and  D z>7.'3  
    %   optometry to describe functions on a circular domain. ,n{ |d33  
    % M059"X="  
    %   The following table lists the first 15 Zernike functions. hKK"D:?PRs  
    % 2I~a{:O  
    %       n    m    Zernike function           Normalization iJ`v3PP  
    %       -------------------------------------------------- yD&UH_ 1g  
    %       0    0    1                                 1 Y5Z<uD  
    %       1    1    r * cos(theta)                    2 ?)c9!hR  
    %       1   -1    r * sin(theta)                    2 xOpCybmc  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) V0$:t^^  
    %       2    0    (2*r^2 - 1)                    sqrt(3) XM*%n8q7#N  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) a: OuDjFp  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) O:O +Q!58  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) bcprhb  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) |S VL%agZ  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) ApAHa]Ccp  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) <NX6m|DD  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) e~BUAz  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) %MUwd@,  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ji|tc9#6  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) 3HmJixy  
    %       -------------------------------------------------- }#f~"-O  
    % .3 T#:Hl  
    %   Example 1: GCA?sFwo>  
    % 6/thhP3`-  
    %       % Display the Zernike function Z(n=5,m=1) V\o& {7!  
    %       x = -1:0.01:1; wTY8={p]  
    %       [X,Y] = meshgrid(x,x); &!FWo@  
    %       [theta,r] = cart2pol(X,Y); iYxpIqWw  
    %       idx = r<=1; HOAgRhzE  
    %       z = nan(size(X)); {B lM<  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); a)Ca:p  
    %       figure 4m$Xjj`vE  
    %       pcolor(x,x,z), shading interp 3DO ^vV  
    %       axis square, colorbar 9"~,ha7S$  
    %       title('Zernike function Z_5^1(r,\theta)') zc#aQ.  
    % o@0p  
    %   Example 2: 6o/!H  
    % 2f$6}m'Ad  
    %       % Display the first 10 Zernike functions G+xdh  
    %       x = -1:0.01:1; o}K!p %5_  
    %       [X,Y] = meshgrid(x,x); [6Gb@jG  
    %       [theta,r] = cart2pol(X,Y); U#!f^@&AB  
    %       idx = r<=1; ,] ,dOIOwn  
    %       z = nan(size(X)); #!X4\+)  
    %       n = [0  1  1  2  2  2  3  3  3  3]; nXOJ  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; 6>Szxkz  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; wtw  
    %       y = zernfun(n,m,r(idx),theta(idx)); I=I'O?w  
    %       figure('Units','normalized') r/vRaOg>X  
    %       for k = 1:10 r8E)GBH-|  
    %           z(idx) = y(:,k); 5b2_{6t  
    %           subplot(4,7,Nplot(k)) L.@o  
    %           pcolor(x,x,z), shading interp 7 a}qnk %  
    %           set(gca,'XTick',[],'YTick',[]) -?$Hr\  
    %           axis square jQs"8[=s  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) !4f0VQI  
    %       end _*O^|QbM  
    % HsGyNkr?r  
    %   See also ZERNPOL, ZERNFUN2. ]dKLzW:l  
    &u'$q  
    %   Paul Fricker 11/13/2006 CcHf1 _CI  
    gOA  
    T~rPpi&  
    % Check and prepare the inputs: C"P40VQoo  
    % ----------------------------- M6P`~emX2  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) v} $KlT  
        error('zernfun:NMvectors','N and M must be vectors.') f|f9[h'  
    end *3A[C-1~.  
    lklMdsIdj  
    if length(n)~=length(m) ,5_Hen=PI  
        error('zernfun:NMlength','N and M must be the same length.') O!D0 hW4  
    end o7*z@R"  
    #FBq8iJ  
    n = n(:); .(0'l@#fT  
    m = m(:); sacaL4[_<  
    if any(mod(n-m,2)) ^Z{W1uYi  
        error('zernfun:NMmultiplesof2', ... 8)D5loS  
              'All N and M must differ by multiples of 2 (including 0).') 9o]h}Xc  
    end x05yU  
    p<2A4="&  
    if any(m>n) =~i~SG/f  
        error('zernfun:MlessthanN', ... y-TS?5Dr]  
              'Each M must be less than or equal to its corresponding N.') 32r2<QrX  
    end ESl-k2  
    h98_6Dw(]  
    if any( r>1 | r<0 ) ,3t('SE  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') vPn(~d_  
    end 5m`@ 4%)zp  
    .&AS-">Z  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) <303PPX^6  
        error('zernfun:RTHvector','R and THETA must be vectors.') J3oj}M*  
    end uj_ OWre  
    Efm37Kv5l  
    r = r(:); a3wTcp "r  
    theta = theta(:); ZLBv\VQ  
    length_r = length(r); 06 kjJ4  
    if length_r~=length(theta) .~+I"V{y F  
        error('zernfun:RTHlength', ... Rl7V~dUY  
              'The number of R- and THETA-values must be equal.') ik@g;>pQD  
    end u.t(78N  
    "(6]K}k@  
    % Check normalization: >bia FK>t  
    % -------------------- J 00%,Ju_  
    if nargin==5 && ischar(nflag) =rV*iLy  
        isnorm = strcmpi(nflag,'norm'); xD}ha  
        if ~isnorm f-N:  
            error('zernfun:normalization','Unrecognized normalization flag.') QfuKpcT &  
        end NJG-~ w  
    else X&1R6 O  
        isnorm = false; }xx[=t=nUf  
    end 9Z,vpTE  
    #:{Bd8PS  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% pm+_s]s,  
    % Compute the Zernike Polynomials b]v.jgD  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% }|rnyYA  
     o *2TH2  
    % Determine the required powers of r: }JXAG/<  
    % ----------------------------------- bDa(@QJ-  
    m_abs = abs(m);  7(;M  
    rpowers = []; X'4g\)*  
    for j = 1:length(n) 8Yr_$5R  
        rpowers = [rpowers m_abs(j):2:n(j)]; J G xuB*}  
    end YN1P9j#0d  
    rpowers = unique(rpowers); - Dm/7Sxd`  
    Hmt} @  
    % Pre-compute the values of r raised to the required powers, :yN;_bC!b%  
    % and compile them in a matrix: Y_3 {\g|x  
    % ----------------------------- 12\h| S~  
    if rpowers(1)==0 S) /(~  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); SomA`y+ERn  
        rpowern = cat(2,rpowern{:}); ^YddVp  
        rpowern = [ones(length_r,1) rpowern]; Y27x;U  
    else -4|\,=j  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); C zKU;~D=B  
        rpowern = cat(2,rpowern{:}); gVM9*3LH6  
    end c"w}<8  
    eRkvNI  
    % Compute the values of the polynomials: ]iewukB4  
    % -------------------------------------- c:0nOP  
    y = zeros(length_r,length(n)); 5;wA7@  
    for j = 1:length(n) +H5= zf2  
        s = 0:(n(j)-m_abs(j))/2; 1b:3'E.#w  
        pows = n(j):-2:m_abs(j); MA\"JAP/  
        for k = length(s):-1:1 ~y.{WuUD  
            p = (1-2*mod(s(k),2))* ... 5mwtlC':l?  
                       prod(2:(n(j)-s(k)))/              ... vd Fy}#X  
                       prod(2:s(k))/                     ... R}MdBE  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... .4c*  _$  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); 1 "'t5?XW  
            idx = (pows(k)==rpowers); GAONgz|ZI  
            y(:,j) = y(:,j) + p*rpowern(:,idx); 3n;UXYJ%  
        end gs)wQgJ[  
         {&,9Zy]"S  
        if isnorm iR;Sd >)  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); &kKopJH  
        end X{A|{u=  
    end P;o6rQf  
    % END: Compute the Zernike Polynomials SoZ$1$o2  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |QwX  
    Z?k4Kb  
    % Compute the Zernike functions: $]IX11.m  
    % ------------------------------ vzl+0"  
    idx_pos = m>0; %n-:mSus  
    idx_neg = m<0; s`W\`w}  
    $\kqh$")  
    z = y; U4]>8L  
    if any(idx_pos) KE3/sw0  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 5$o]D  
    end }oH A@o5  
    if any(idx_neg) {3@lvoDT  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 4.)hCb  
    end d;` bX+K  
    ?bwF$Ku  
    % EOF zernfun
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    function z = zernfun2(p,r,theta,nflag) ZS51QB  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. }HB)%C50.  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated o:E+c_^q`  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive ~e,k71  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, Qhlgu!  
    %   and THETA is a vector of angles.  R and THETA must have the same JBa( O- T  
    %   length.  The output Z is a matrix with one column for every P-value, =KfV;.&  
    %   and one row for every (R,THETA) pair. '"C$E922  
    % 5 _X|U*+5  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike |0 #J=am  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) LX{[9   
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) k fER  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 =fmM=@!$<  
    %   for all p. dKyJ.p   
    % t}LV[bj1u  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 s'\PU1{  
    %   Zernike functions (order N<=7).  In some disciplines it is *B"p:F7J|  
    %   traditional to label the first 36 functions using a single mode v;.7-9c*  
    %   number P instead of separate numbers for the order N and azimuthal s)Bl1\Q  
    %   frequency M. jt|e?1:vF  
    % EVc Ees  
    %   Example: gf/$M[H!   
    % /mLOh2 T  
    %       % Display the first 16 Zernike functions Xq`|'6]/  
    %       x = -1:0.01:1; uM"G)$I\  
    %       [X,Y] = meshgrid(x,x);  y/t{*a  
    %       [theta,r] = cart2pol(X,Y); FHpS?htRy  
    %       idx = r<=1; j'Ry.8}  
    %       p = 0:15; ceN*wkGyB  
    %       z = nan(size(X)); S;#S3?G  
    %       y = zernfun2(p,r(idx),theta(idx)); hES_JbX}]  
    %       figure('Units','normalized') 7PG&G5  
    %       for k = 1:length(p) #({0HFSC:j  
    %           z(idx) = y(:,k); ((i%h^tGa;  
    %           subplot(4,4,k) @]r,cPx0Y  
    %           pcolor(x,x,z), shading interp X`kTbIZ|  
    %           set(gca,'XTick',[],'YTick',[]) ZMO7 o 1"  
    %           axis square b#;%TbDF  
    %           title(['Z_{' num2str(p(k)) '}']) r\J"|{)e  
    %       end 5~&9/ ALk5  
    % ;Z]i$Vi_r  
    %   See also ZERNPOL, ZERNFUN. *?'nA{a)E  
    7b7~D +b  
    %   Paul Fricker 11/13/2006 WW33ZJ  
    -a:+ h\K  
    v'`VyXetl  
    % Check and prepare the inputs: },9Hq~TA  
    % ----------------------------- \9Nd"E[B  
    if min(size(p))~=1 eSvS<\p  
        error('zernfun2:Pvector','Input P must be vector.') dg[ &5D1Q  
    end aO :wedfl  
    Le#>uWM  
    if any(p)>35 6Y4sv5G  
        error('zernfun2:P36', ... D:`b61sWi_  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... ~,[<R  
               '(P = 0 to 35).']) f9FJ:?  
    end O_%X>Q9  
    Ne7HPSWiOP  
    % Get the order and frequency corresonding to the function number: jWHv9XtW  
    % ---------------------------------------------------------------- 3^m0 k E  
    p = p(:); _*\:UBZx6  
    n = ceil((-3+sqrt(9+8*p))/2); M*M,Z  
    m = 2*p - n.*(n+2); i("ok  
    ' S%?&4  
    % Pass the inputs to the function ZERNFUN: K q;X(&Z  
    % ---------------------------------------- DC?U +  
    switch nargin T,z 7U2O  
        case 3 AE`z~L,  
            z = zernfun(n,m,r,theta); , y%!s27  
        case 4 _a?c,<A  
            z = zernfun(n,m,r,theta,nflag);  )l 0\TF  
        otherwise [n%=2*1p  
            error('zernfun2:nargin','Incorrect number of inputs.') xgsEJE  
    end fmqHWu*wG  
    ZDHm@,d  
    % EOF zernfun2
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    function z = zernpol(n,m,r,nflag) bH 6i1c8  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. 2G=prS`s  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of jtS-nQ|  
    %   order N and frequency M, evaluated at R.  N is a vector of -^C^3pms  
    %   positive integers (including 0), and M is a vector with the {lv@V*_Y0  
    %   same number of elements as N.  Each element k of M must be a V)|]w[(Y  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) "{TVd>9_  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is @\ udaZc  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix o03Y w)*  
    %   with one column for every (N,M) pair, and one row for every /6Bm <k%  
    %   element in R. 42E%&DF  
    % CEQs}bz  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- b!lS=zIN  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is '!\t!@I$  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to sVT:1 kI  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 4%refqWK  
    %   for all [n,m]. 4$~A%JN3  
    % c&> S  
    %   The radial Zernike polynomials are the radial portion of the :v$][jZ2  
    %   Zernike functions, which are an orthogonal basis on the unit W}6OMAbsE;  
    %   circle.  The series representation of the radial Zernike qDlh6W?}k  
    %   polynomials is $p(  
    % G;jX@XqZ  
    %          (n-m)/2 +f){x9 :  
    %            __ "`6pF8k  
    %    m      \       s                                          n-2s 4,g[g#g<q  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r :OEovk(`  
    %    n      s=0 =lb5 #  
    % a_z1S Z2[  
    %   The following table shows the first 12 polynomials. \+iZdZD  
    % z^,P2kqK_  
    %       n    m    Zernike polynomial    Normalization %]"eN{Uvn  
    %       --------------------------------------------- lGhhH _  
    %       0    0    1                        sqrt(2) Rz03he  
    %       1    1    r                           2 $j(laD#AR  
    %       2    0    2*r^2 - 1                sqrt(6) . DrGr:UW  
    %       2    2    r^2                      sqrt(6) h/s8".\  
    %       3    1    3*r^3 - 2*r              sqrt(8) 8wH1x .  
    %       3    3    r^3                      sqrt(8) v#^_|  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) eeVzOq(  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) i;l0)q  
    %       4    4    r^4                      sqrt(10) s#BSZP  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) OCN:{  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) +T8h jOkC  
    %       5    5    r^5                      sqrt(12) mb GL)NI  
    %       --------------------------------------------- r-e-2y7  
    % '/U%-/@  
    %   Example: # A#,]XP  
    % KFhnv`a.0  
    %       % Display three example Zernike radial polynomials 5>\Lk>rI  
    %       r = 0:0.01:1; +*`>7m<^  
    %       n = [3 2 5]; &iTTal.6  
    %       m = [1 2 1]; boeIO\2}P0  
    %       z = zernpol(n,m,r); Q$^)z_jai  
    %       figure E0t%]?1  
    %       plot(r,z) `p#u9M>  
    %       grid on Yc`PK =!l  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') oAt{ #v  
    % tq.g4X ;_  
    %   See also ZERNFUN, ZERNFUN2. O5v~wLx9e  
    MA+{7 [  
    % A note on the algorithm. 19]O;  
    % ------------------------ rD!UP1Nb  
    % The radial Zernike polynomials are computed using the series @W.0YU0|J  
    % representation shown in the Help section above. For many special W<\*5oB%H  
    % functions, direct evaluation using the series representation can /4>|6l=  
    % produce poor numerical results (floating point errors), because (.~,I+Cz'  
    % the summation often involves computing small differences between LZ4Z]!V  
    % large successive terms in the series. (In such cases, the functions Uqd2{fji=#  
    % are often evaluated using alternative methods such as recurrence M?v`C>j  
    % relations: see the Legendre functions, for example). For the Zernike 5E!Wp[^  
    % polynomials, however, this problem does not arise, because the zgPUW z X=  
    % polynomials are evaluated over the finite domain r = (0,1), and -Gj."ks  
    % because the coefficients for a given polynomial are generally all {C'9?4&  
    % of similar magnitude. jRBKy8?[C  
    % *@E&O^%cO  
    % ZERNPOL has been written using a vectorized implementation: multiple ,R*YI  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] 4"et4Y7  
    % values can be passed as inputs) for a vector of points R.  To achieve 9xRor<  
    % this vectorization most efficiently, the algorithm in ZERNPOL \Lz4ZZjSY  
    % involves pre-determining all the powers p of R that are required to |IZFWZd  
    % compute the outputs, and then compiling the {R^p} into a single #eY?6Kjn  
    % matrix.  This avoids any redundant computation of the R^p, and }kF*I@:g  
    % minimizes the sizes of certain intermediate variables. -&0HAtc  
    % 55V&[>|K5  
    %   Paul Fricker 11/13/2006 !=p^@N7  
    CuA A)Bj  
    yIf>8ed]#  
    % Check and prepare the inputs: >U6 2vX"  
    % ----------------------------- V_P,~!  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ` 7jdV  
        error('zernpol:NMvectors','N and M must be vectors.') FQBAt0  
    end </li<1  
    aMO+ y91Y(  
    if length(n)~=length(m) NaC}KI`  
        error('zernpol:NMlength','N and M must be the same length.') ]cP$aixd  
    end ZJ'FZ8Sx  
    \heQVWRl  
    n = n(:); @YI- @  
    m = m(:); G0Wv=tX|  
    length_n = length(n); KF f6um  
    A08{]E#v>  
    if any(mod(n-m,2)) q/3 )yG6s  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') 8]A`WDO3  
    end Pi'[d7o  
    P3+?gW'  
    if any(m<0) xf 4`+[  
        error('zernpol:Mpositive','All M must be positive.') o0FVVSl  
    end JAS!eF  
    0 ChdFf7  
    if any(m>n) ?T7ndXX  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.')  &DX  
    end l^4!  
    o WcBQ|   
    if any( r>1 | r<0 ) ]7 2wv#-  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') {f[X)  
    end mVEHVz $  
    (db4.G+0  
    if ~any(size(r)==1) :'K%&e?7s  
        error('zernpol:Rvector','R must be a vector.') $#7~  
    end t'DYT"3  
    ;`}b .S =n  
    r = r(:); !/6KQdF  
    length_r = length(r); >o8N@`@VK-  
    lPOcX'3\  
    if nargin==4 Nh+ZSV4WJ:  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); 3PRK.vf  
        if ~isnorm IWP[?U=  
            error('zernpol:normalization','Unrecognized normalization flag.') '`/w%OEVC5  
        end  &&sCaNb  
    else ?%wM8?  
        isnorm = false; ZE"Z_E;r  
    end TptXH?  
    FX:'38-fk  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% QP1 bm]QYA  
    % Compute the Zernike Polynomials V8IEfU  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% U(u$5  
    r^$WX@ t&  
    % Determine the required powers of r: Bw8&Amxx:  
    % ----------------------------------- @DK;i_i  
    rpowers = []; 7 J+cs^2  
    for j = 1:length(n) 0\Ga&Q0-(O  
        rpowers = [rpowers m(j):2:n(j)]; riY[p,  
    end )ZQML0}P;  
    rpowers = unique(rpowers); q?2kD"%$  
    Kk<MS$Ov  
    % Pre-compute the values of r raised to the required powers, ]q|^?C  
    % and compile them in a matrix: dT4e[4l  
    % ----------------------------- Hpq?I-g<^  
    if rpowers(1)==0 Rln JlY/  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); u|uPvbM  
        rpowern = cat(2,rpowern{:}); @T 8$/  
        rpowern = [ones(length_r,1) rpowern]; .m \y6  
    else N77EM  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); E1V;eoK.D  
        rpowern = cat(2,rpowern{:}); Q2HULz{  
    end bsgrg  
    P},d`4Ty@  
    % Compute the values of the polynomials: H_+F~P5RC  
    % -------------------------------------- q'4qSu  
    z = zeros(length_r,length_n); (b4;c=<[{  
    for j = 1:length_n 7l|D!`BS  
        s = 0:(n(j)-m(j))/2; >5+]~[S  
        pows = n(j):-2:m(j); U2)y fhI  
        for k = length(s):-1:1 $jtXN E?  
            p = (1-2*mod(s(k),2))* ... "8YXFg  
                       prod(2:(n(j)-s(k)))/          ... 7n*[r*$  
                       prod(2:s(k))/                 ... sPUn"7  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... +/Q ?<*[  
                       prod(2:((n(j)+m(j))/2-s(k))); -+w^"RBV  
            idx = (pows(k)==rpowers); hY-;Vh0J  
            z(:,j) = z(:,j) + p*rpowern(:,idx); /pOK4"  
        end 5Sfz0  
         o6~9.~_e  
        if isnorm X__>r ?oJ  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); H&3i[D!p  
        end k6PHyt`3'  
    end ~[d|:]  
    \ 4r?=5v*  
    % EOF zernpol
    离线niuhelen
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    只看该作者 6楼 发表于: 2011-03-12
    这三个文件,我不知道该怎样把我的面型节点的坐标及轴向位移用起来,还烦请指点一下啊,谢谢啦!
    离线li_xin_feng
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    只看该作者 7楼 发表于: 2012-09-28
    我也正在找啊
    离线guapiqlh
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    只看该作者 8楼 发表于: 2014-03-04
    我也一直想了解这个多项式的应用,还没用过呢
    离线phoenixzqy
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    只看该作者 9楼 发表于: 2014-04-22
    回 guapiqlh 的帖子
    guapiqlh:我也一直想了解这个多项式的应用,还没用过呢 (2014-03-04 11:35)  DeAi'"&  
    kYVn4Wq  
    数值分析方法看一下就行了。其实就是正交多项式的应用。zernike也只不过是正交多项式的一种。 bHi0N@W!vG  
    @"\j]ZEnY  
    07年就写过这方面的计算程序了。
    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)