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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 j^/<:e c.  
    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! n<EIu  
     
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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 A;xH{vo{  
    function z = zernfun(n,m,r,theta,nflag) {Y+e|B0  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. Zz|et206  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N rJ4A9d3:  
    %   and angular frequency M, evaluated at positions (R,THETA) on the 4fL>Ou[YuX  
    %   unit circle.  N is a vector of positive integers (including 0), and 6[Mu3.T  
    %   M is a vector with the same number of elements as N.  Each element u~t%GIg  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) FLumI-se!  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, lE bV)&'  
    %   and THETA is a vector of angles.  R and THETA must have the same 'cN3Vv k  
    %   length.  The output Z is a matrix with one column for every (N,M) nwJub$5  
    %   pair, and one row for every (R,THETA) pair. ],<pZ1V;  
    % lA,[&  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike F{&0(6^p!  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), /z1-4:^`A[  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral %B>>J%  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, uq!d8{IMu  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized Urm(A9|N  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. /u'V>=D;f  
    % =b3<}]  
    %   The Zernike functions are an orthogonal basis on the unit circle. fQfd1=4  
    %   They are used in disciplines such as astronomy, optics, and a{I(Qh!}  
    %   optometry to describe functions on a circular domain. "f!H[F1~  
    % w`Cs,  
    %   The following table lists the first 15 Zernike functions. UnTvot6~  
    % )"bP]t^_  
    %       n    m    Zernike function           Normalization +o 6"Z)  
    %       -------------------------------------------------- I~M@v59C  
    %       0    0    1                                 1 s9b+uUt%  
    %       1    1    r * cos(theta)                    2 `g'9)Xf4KT  
    %       1   -1    r * sin(theta)                    2 C%P"\>5@  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) F^DDN7AKH  
    %       2    0    (2*r^2 - 1)                    sqrt(3) %&$s0=+  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) ];{CNDAL2  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) > 8!9  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) Qv;^nj{\qV  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) d r=h;[Q'  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) ' '|R$9\@  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) /n9,XD&)  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) H3 |x  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) y(6*)~Dh  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) )K 0rPnYV  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) kSqMI'89  
    %       -------------------------------------------------- ?Hf8<C}3  
    % <QRRD*\  
    %   Example 1: <`=(Ui$fD  
    % u85Uy yN  
    %       % Display the Zernike function Z(n=5,m=1) ^' b[#DG>F  
    %       x = -1:0.01:1; Cbr>\;sc2Z  
    %       [X,Y] = meshgrid(x,x); ,6T3:qkkvF  
    %       [theta,r] = cart2pol(X,Y); Ei\tn`I&  
    %       idx = r<=1; !-|{B3"6  
    %       z = nan(size(X)); >~* w  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); ,uhOf! |  
    %       figure -*0U&]T  
    %       pcolor(x,x,z), shading interp .5YW >PV  
    %       axis square, colorbar ujoJ6UOG  
    %       title('Zernike function Z_5^1(r,\theta)') v?#W/].C+  
    % ~i9'9PHX@  
    %   Example 2: /-C6I:  
    % sxn^1|O;m  
    %       % Display the first 10 Zernike functions l%xjCuuhU  
    %       x = -1:0.01:1; _*dUH5  
    %       [X,Y] = meshgrid(x,x); A:Kit_A  
    %       [theta,r] = cart2pol(X,Y); {$qLMx';  
    %       idx = r<=1; A}(Q^|6  
    %       z = nan(size(X)); `Ct fe8  
    %       n = [0  1  1  2  2  2  3  3  3  3];  :)Z.!  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; 5| bc*iqU  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; 6n Hyd<o  
    %       y = zernfun(n,m,r(idx),theta(idx)); drf?7%v  
    %       figure('Units','normalized') 5>"X?U}He  
    %       for k = 1:10 Hyz:i)2  
    %           z(idx) = y(:,k); #bdSH)V  
    %           subplot(4,7,Nplot(k)) 0X4%Ccs  
    %           pcolor(x,x,z), shading interp (GDW9:  
    %           set(gca,'XTick',[],'YTick',[]) r2xIbZ  
    %           axis square V.kRV{43  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) LHgEb9\Q  
    %       end ~"#[<d  
    % }E](NvCq  
    %   See also ZERNPOL, ZERNFUN2. Kv>P+I'|r  
    e"ur+7  
    %   Paul Fricker 11/13/2006 )_Wo6l)i  
     `\#J&N  
    <aGfQg|554  
    % Check and prepare the inputs: -}Q^A_xK  
    % ----------------------------- B|6_4ry0U  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 5AO' IhpL  
        error('zernfun:NMvectors','N and M must be vectors.') wYLJEuS|  
    end vSG$ 2g=  
    f" g-Hbl5  
    if length(n)~=length(m) ,5HC &@  
        error('zernfun:NMlength','N and M must be the same length.') u:s[6T0  
    end d{G*1l(X  
    M*lCoJ  
    n = n(:); MWron_xg  
    m = m(:); FJ(}@U}57  
    if any(mod(n-m,2)) k_hs g6Ur.  
        error('zernfun:NMmultiplesof2', ... Dt\rMSjZ9  
              'All N and M must differ by multiples of 2 (including 0).') uQ'Izdm  
    end s'yT}XQ;r  
    ;7w4BJcq']  
    if any(m>n) ,f?+QV\T.  
        error('zernfun:MlessthanN', ... LyA}Nd]pyq  
              'Each M must be less than or equal to its corresponding N.') i*ErxWzu  
    end G[M{TS3&Ds  
    B~t[Gy  
    if any( r>1 | r<0 ) d\A!5/LG  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') ;<)<4N"  
    end Rt2<F-gY  
    "`&1"*  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) :o+&>z  
        error('zernfun:RTHvector','R and THETA must be vectors.') TW5Pt{X= f  
    end ]3bXJE  
    ym.:I@b?6  
    r = r(:); (,!G$~Sy  
    theta = theta(:); #Qnl,lf  
    length_r = length(r); , UiA?7k  
    if length_r~=length(theta) 3}9c0%}F  
        error('zernfun:RTHlength', ... [/IN820t  
              'The number of R- and THETA-values must be equal.') ?A`8c R=)I  
    end l0-zu6i w  
    5svM3  #  
    % Check normalization: IFfB3{J  
    % -------------------- 8JbN&C  
    if nargin==5 && ischar(nflag) 3C7}V{?  
        isnorm = strcmpi(nflag,'norm'); }{( J *T  
        if ~isnorm <Gkmk?x`A  
            error('zernfun:normalization','Unrecognized normalization flag.') mKN#dmw6  
        end T5b*Ia  
    else !au%D?w  
        isnorm = false; i]M:ntB"  
    end L>.* ^]  
    C])b 3tM,7  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% i_M0P12  
    % Compute the Zernike Polynomials %6eQ;Rp*  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% _m2p>(N|  
    QdtGFY4f,  
    % Determine the required powers of r: dAkJ5\=*  
    % ----------------------------------- 1}Mdo&:t  
    m_abs = abs(m); C6ry]R@  
    rpowers = []; QssU\@ / Q  
    for j = 1:length(n) FhVoN}  
        rpowers = [rpowers m_abs(j):2:n(j)]; .qGfLvx%  
    end h}knn3"S  
    rpowers = unique(rpowers); g6p:1;Evf  
    T>qI,BEY  
    % Pre-compute the values of r raised to the required powers, ^a{cK  
    % and compile them in a matrix: L j>HZS$F  
    % ----------------------------- |E5\_Z  
    if rpowers(1)==0 t`oH7)nut  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); i^2-PKPg{  
        rpowern = cat(2,rpowern{:}); yHIZpU|(j  
        rpowern = [ones(length_r,1) rpowern]; Wd<|DmSy  
    else fNnX{Wq  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); AaX][2y8  
        rpowern = cat(2,rpowern{:}); W&`{3L  
    end c|KN@)A  
    >3&Oe  
    % Compute the values of the polynomials: s !XJ   
    % -------------------------------------- F\IJim-Rh  
    y = zeros(length_r,length(n)); (`me}8  
    for j = 1:length(n) ~us1Df0bp  
        s = 0:(n(j)-m_abs(j))/2; yZcnky  
        pows = n(j):-2:m_abs(j); 3Eu;_u_  
        for k = length(s):-1:1 lJIcU RI4  
            p = (1-2*mod(s(k),2))* ... U+2U#v=<  
                       prod(2:(n(j)-s(k)))/              ... o~J~-$T{  
                       prod(2:s(k))/                     ... [,86||^  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... @r=v*hu  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); <2,NWn.  
            idx = (pows(k)==rpowers); +u\kTn  
            y(:,j) = y(:,j) + p*rpowern(:,idx); w+W! dM  
        end QuWW a|g^.  
         }Md5a%s<  
        if isnorm 5[5|_H+0  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); cf`g.9pjlx  
        end {;-wXzv`  
    end iPeW;=-2Wk  
    % END: Compute the Zernike Polynomials }eq*dr1`  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% X4I+  
    K);)$8K  
    % Compute the Zernike functions: G%FLt[  
    % ------------------------------ i2&I<:  
    idx_pos = m>0; Ehw2o-s^  
    idx_neg = m<0; "HwSW4a]  
    -.!+i8d>  
    z = y; J_`a}ox  
    if any(idx_pos) )~2~q7  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); M,we9];N  
    end  up==g  
    if any(idx_neg) [ @ASAhV^+  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); V7(-<})8  
    end Or3GrZ!H  
    -50Qy[0."  
    % EOF zernfun
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) G<M9 6V  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. F!7\Za,  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated des.TSZ  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive C'.^2s#e8  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, M.3ULt8  
    %   and THETA is a vector of angles.  R and THETA must have the same Dt:NBN  
    %   length.  The output Z is a matrix with one column for every P-value, 0`KR8# A@  
    %   and one row for every (R,THETA) pair. d.xT8l}sS  
    % 7Q0vwKC8>  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike T%]@R4z#q  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) Zdy{e|-Zn  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) l8H8c &  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 ^I:f4RWo  
    %   for all p. r)|6H"n#]S  
    % ;Z.sK-NJ4  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 j.kv!;Rj=  
    %   Zernike functions (order N<=7).  In some disciplines it is w JF(&P  
    %   traditional to label the first 36 functions using a single mode jp880}  
    %   number P instead of separate numbers for the order N and azimuthal k@P?,r  
    %   frequency M. M4)Y%EPc  
    % b ,e"x48q  
    %   Example: p`)Mk<`dYD  
    % K^e4w`F|  
    %       % Display the first 16 Zernike functions .2V?G]u  
    %       x = -1:0.01:1; pmc)$3u  
    %       [X,Y] = meshgrid(x,x); V='A;gs  
    %       [theta,r] = cart2pol(X,Y); GJIZu&C  
    %       idx = r<=1; 3R<VpN){  
    %       p = 0:15; FBeo@  
    %       z = nan(size(X)); 6%Pvh- ~_  
    %       y = zernfun2(p,r(idx),theta(idx)); !CUM*<iV  
    %       figure('Units','normalized') sL], @z8<k  
    %       for k = 1:length(p) nhy:5eSK  
    %           z(idx) = y(:,k); -,q qQf  
    %           subplot(4,4,k) VQ;'SY:`  
    %           pcolor(x,x,z), shading interp WI1DL&*B@<  
    %           set(gca,'XTick',[],'YTick',[]) [L=M=;{4  
    %           axis square nQ@<[KNd  
    %           title(['Z_{' num2str(p(k)) '}']) Yy0U2N [i  
    %       end U;#G $  
    % "2ZuI; w  
    %   See also ZERNPOL, ZERNFUN. R>hL.+l.  
    yG2rAG_ G&  
    %   Paul Fricker 11/13/2006 -_BX\iP{  
    VE)) `?  
    49=L9:  
    % Check and prepare the inputs: rN'8,CV  
    % ----------------------------- ?{%"v\w  
    if min(size(p))~=1 0IyT(1hS  
        error('zernfun2:Pvector','Input P must be vector.') e)$a;6  
    end %wco)2  
    umN4|X  
    if any(p)>35 3LZvlcLb  
        error('zernfun2:P36', ... X*M2 O%g`L  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... 9^E!2CJ  
               '(P = 0 to 35).']) 45H9pY w  
    end 5DJ!:QY!  
    tA^CuJR  
    % Get the order and frequency corresonding to the function number: T0N6k acl  
    % ---------------------------------------------------------------- KG GJ\r6  
    p = p(:); O-!Q~;3][  
    n = ceil((-3+sqrt(9+8*p))/2); 3Xm> 3  
    m = 2*p - n.*(n+2); 1[!7xA0j  
    kAs=5_?I  
    % Pass the inputs to the function ZERNFUN: O*yA50Cn  
    % ---------------------------------------- ?8FJMFv;4%  
    switch nargin vs@u*4.Ut<  
        case 3 <Qt9MO`a  
            z = zernfun(n,m,r,theta); "sT)<Wc  
        case 4 [WI'oy  
            z = zernfun(n,m,r,theta,nflag); :Sn4Pg `Q  
        otherwise +zK?1llt  
            error('zernfun2:nargin','Incorrect number of inputs.') yIg^iZD  
    end h-\Ov{~  
    <j1r6.E)  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) ;]k\F  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. i(kx'ua?  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of jhJ<JDJ?`  
    %   order N and frequency M, evaluated at R.  N is a vector of ,y@WFRsx  
    %   positive integers (including 0), and M is a vector with the J> "qeR /  
    %   same number of elements as N.  Each element k of M must be a #Z,@yJ2wl  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) S VypR LVB  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is o#>Mf464I  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix JvNd'u)Z<  
    %   with one column for every (N,M) pair, and one row for every ~[=d{M!$W  
    %   element in R. < <F  
    % E{}J-_oS45  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- =Y*@8=V  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is ;.^! 7j  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to 9@}5FoX"  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 0sH~H[ap  
    %   for all [n,m]. y6Ez.$M  
    % gLg.mV1<  
    %   The radial Zernike polynomials are the radial portion of the Q(O0z3b  
    %   Zernike functions, which are an orthogonal basis on the unit dnV&U%fO  
    %   circle.  The series representation of the radial Zernike }2S)CL=  
    %   polynomials is wc-v]$DW  
    % ^=8/Iw  
    %          (n-m)/2 .hUlI3z9  
    %            __ CR;E*I${  
    %    m      \       s                                          n-2s Ti7 @{7>  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r 9W, %[  
    %    n      s=0 ) I(9qt>Y  
    % JJ'f\f9  
    %   The following table shows the first 12 polynomials. 9|Ylv:sR  
    % 3Pp+>{2_?  
    %       n    m    Zernike polynomial    Normalization brG!TJ   
    %       --------------------------------------------- #m;o)KkH$r  
    %       0    0    1                        sqrt(2) CH q5KB98+  
    %       1    1    r                           2 7U-}Y  
    %       2    0    2*r^2 - 1                sqrt(6) p'6XF{  
    %       2    2    r^2                      sqrt(6) 6:AEg  
    %       3    1    3*r^3 - 2*r              sqrt(8) 5z\,]  
    %       3    3    r^3                      sqrt(8) bdfs'udt9  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) lnK  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) Jfo'iNOu  
    %       4    4    r^4                      sqrt(10) X|D-[|P  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) j-VwY/X  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) #,97 ]  
    %       5    5    r^5                      sqrt(12) FM(EOsWk  
    %       --------------------------------------------- @/:7G.  
    % |Y?<58[!)  
    %   Example: TL)7X.1'L  
    % {:3:GdM6  
    %       % Display three example Zernike radial polynomials U| ?68B3  
    %       r = 0:0.01:1; y4$$*oai&  
    %       n = [3 2 5]; 5g O9 <  
    %       m = [1 2 1]; _+ Sf+ta  
    %       z = zernpol(n,m,r); \3"B$Sp|=  
    %       figure mSvSdKKKlI  
    %       plot(r,z) G!wb|-4<$  
    %       grid on &5XEjY>@  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') >P:U9 b  
    % (h= ]Ox  
    %   See also ZERNFUN, ZERNFUN2. ?k($Tc&Q  
    :RxMZwa=  
    % A note on the algorithm. ^2{6W6=  
    % ------------------------ U_;="y  
    % The radial Zernike polynomials are computed using the series Gt _tL%  
    % representation shown in the Help section above. For many special E'98JZ5ga  
    % functions, direct evaluation using the series representation can "K$c9Z8  
    % produce poor numerical results (floating point errors), because hc3tzB  
    % the summation often involves computing small differences between T>7N "C  
    % large successive terms in the series. (In such cases, the functions ofw&? Sk0  
    % are often evaluated using alternative methods such as recurrence !uO@4]:Y  
    % relations: see the Legendre functions, for example). For the Zernike &:u3-:$:9  
    % polynomials, however, this problem does not arise, because the v*FbvrY  
    % polynomials are evaluated over the finite domain r = (0,1), and D~Ef%!&  
    % because the coefficients for a given polynomial are generally all W7gY$\1<&  
    % of similar magnitude. &/-MUKN  
    % e6mm;@F>  
    % ZERNPOL has been written using a vectorized implementation: multiple .tppCy  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] r:$*pC&{  
    % values can be passed as inputs) for a vector of points R.  To achieve nnvS.s`O  
    % this vectorization most efficiently, the algorithm in ZERNPOL B3D}'<  
    % involves pre-determining all the powers p of R that are required to ;\6@s3  
    % compute the outputs, and then compiling the {R^p} into a single G-|c%g!ejf  
    % matrix.  This avoids any redundant computation of the R^p, and <SQR";  
    % minimizes the sizes of certain intermediate variables. i*$~uuY  
    % (6NDY5h~=n  
    %   Paul Fricker 11/13/2006 68(^*  
    '/t9#I@G\  
    aXG|IN5 *m  
    % Check and prepare the inputs: L N.:>,  
    % ----------------------------- =:xX~,qmv  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) HY1K(T  
        error('zernpol:NMvectors','N and M must be vectors.') []aw;\7}Y  
    end _+nk3-yQw  
    ]z8/S!?  
    if length(n)~=length(m) Q4L=]qc T  
        error('zernpol:NMlength','N and M must be the same length.') UN F\k1[  
    end >~]|o   
    IVZUB*wv)b  
    n = n(:); %3"3V1  
    m = m(:); K*2s-,b *  
    length_n = length(n); esE!i0%  
    %'_:#!9  
    if any(mod(n-m,2)) }9W[7V?  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') ?U[6X| 1  
    end , V,Q(!$F  
    upk+L^  
    if any(m<0) lY(_e#  
        error('zernpol:Mpositive','All M must be positive.') 27+faR  
    end RticGQy&5  
    uDkX{<_Xe  
    if any(m>n) qyFeq])  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') AXte&l=M  
    end _&U#*g  
    [KHlApL  
    if any( r>1 | r<0 ) reArXmU<u  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') 9}a$0H h  
    end 'J-a2oiM(  
    !OQ5AF$  
    if ~any(size(r)==1) !G\gqkSL  
        error('zernpol:Rvector','R must be a vector.') )8rF'pxI  
    end ?Js4 \X!uJ  
    .5!`wwVi  
    r = r(:); tP*GYWI48  
    length_r = length(r); VF";p^  
    z^.dYb7<  
    if nargin==4 FXn98UFY  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); _yR_u+5  
        if ~isnorm (n: A` ]  
            error('zernpol:normalization','Unrecognized normalization flag.') e1E_$oJP  
        end qm_m8   
    else @mxaZ5Vv}  
        isnorm = false; Vp~ cN  
    end O CIoY?a  
    \}W3\To_  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% pjTJZhT2I  
    % Compute the Zernike Polynomials (3D&GY!/  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ^2 H-_  
    60$;Q,]o  
    % Determine the required powers of r: !X$19"  
    % ----------------------------------- R) dP=W*  
    rpowers = [];  $RRX-  
    for j = 1:length(n) R"JXWw  
        rpowers = [rpowers m(j):2:n(j)]; CadIu x^  
    end 4r ~K`)/S'  
    rpowers = unique(rpowers); BY[7`@  
    bE mN tp^  
    % Pre-compute the values of r raised to the required powers, dR< d7  
    % and compile them in a matrix: 3)#Nc|  
    % ----------------------------- l4U*Lv>   
    if rpowers(1)==0 f~Pce||e  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 0L8fpGJ  
        rpowern = cat(2,rpowern{:}); ! }e75=x  
        rpowern = [ones(length_r,1) rpowern]; U*\K<fw   
    else 5Rs#{9YE  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); }0]uA|lH*  
        rpowern = cat(2,rpowern{:}); na~ FT[3 C  
    end /FC HF#yK  
    [a!AK kj  
    % Compute the values of the polynomials: UjoA$A!Od;  
    % -------------------------------------- ty#6%  
    z = zeros(length_r,length_n); X])iQyN  
    for j = 1:length_n v&/H6r#E.  
        s = 0:(n(j)-m(j))/2; 0&I*)Zt9x  
        pows = n(j):-2:m(j); PMbZv%.,-  
        for k = length(s):-1:1 /ILd|j(e  
            p = (1-2*mod(s(k),2))* ... *P7/ry^<F  
                       prod(2:(n(j)-s(k)))/          ... !1i-"rR  
                       prod(2:s(k))/                 ... }i^|.VZZ  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... +"BJjxG  
                       prod(2:((n(j)+m(j))/2-s(k))); 8UgogNR\  
            idx = (pows(k)==rpowers); 3T0-RP*  
            z(:,j) = z(:,j) + p*rpowern(:,idx); n-jPb064  
        end 4TPdq&';C:  
         m> P\}A^N  
        if isnorm C"**>OGe  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); !DSm[Z1  
        end ] L#c <0  
    end wf]?:'}  
    W]7<PL*u  
    % 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)  x 4LPrF1  
    Io IhQ  
    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。 `IBNBJy  
    (m Yi  
    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)