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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 zi3v, Kq  
    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! ]s\vc:cc?  
     
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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 > ^zNKgSQ  
    function z = zernfun(n,m,r,theta,nflag) pZopdEFDK|  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. BJb,  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 3N- '{c6]U  
    %   and angular frequency M, evaluated at positions (R,THETA) on the q4w]9b/  
    %   unit circle.  N is a vector of positive integers (including 0), and iKV|~7nwO  
    %   M is a vector with the same number of elements as N.  Each element `ovMfL.u  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) "qF/7`e[  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, du$M  
    %   and THETA is a vector of angles.  R and THETA must have the same H`fJ< So?  
    %   length.  The output Z is a matrix with one column for every (N,M) F nXm;k,9*  
    %   pair, and one row for every (R,THETA) pair. L&)e}"  
    % !J<Xel {  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike RV_I&HD!  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), K mH))LIv  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral E;s_=j1f  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, &z40l['4bz  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized .=Oww  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. Z8FgxR  
    % Nv.  
    %   The Zernike functions are an orthogonal basis on the unit circle. P?f${ t+  
    %   They are used in disciplines such as astronomy, optics, and :%J;[bS+  
    %   optometry to describe functions on a circular domain. xok T  
    % aReJ@  
    %   The following table lists the first 15 Zernike functions. He'VqUw_  
    % Z81;Y=(  
    %       n    m    Zernike function           Normalization )Cj1VjAg  
    %       -------------------------------------------------- T=u"y;&L  
    %       0    0    1                                 1 ?xH{7)dO  
    %       1    1    r * cos(theta)                    2 4V4S5V  
    %       1   -1    r * sin(theta)                    2 yOQae m^O  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) rf|Nu3AJ  
    %       2    0    (2*r^2 - 1)                    sqrt(3) ^gx~{9`RR  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) {+_p?8X  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) ^ '|y^t  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) ]58~b%s  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) r'#!w3*Cy  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) ,)*[Xa_n  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) jQm~F` z  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) aV|V C $  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) OYt_i'Q  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) \RR` F .7  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) K/Yeh<_&  
    %       -------------------------------------------------- Z3c\}HLY  
    % -hW>1s<  
    %   Example 1: (0Br`%!F  
    % s<#BxN  
    %       % Display the Zernike function Z(n=5,m=1) 3e^0W_>6  
    %       x = -1:0.01:1;  rn( drG  
    %       [X,Y] = meshgrid(x,x); H!7?#tRU  
    %       [theta,r] = cart2pol(X,Y); *,CJ 3< >  
    %       idx = r<=1; #z&R9$  
    %       z = nan(size(X)); }JST(d&  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); :Bt,.uN C  
    %       figure eL"'-d+]  
    %       pcolor(x,x,z), shading interp WO9vOS>  
    %       axis square, colorbar z(Uz<*h8  
    %       title('Zernike function Z_5^1(r,\theta)') mMllen  
    % GqsV 6kH  
    %   Example 2: 8g)$%Fy+N  
    % d2i ?FT>  
    %       % Display the first 10 Zernike functions e8dZR3JL  
    %       x = -1:0.01:1; $mKExW  
    %       [X,Y] = meshgrid(x,x); ;}f {o^]'  
    %       [theta,r] = cart2pol(X,Y); 5<`83; R9  
    %       idx = r<=1; hy;V~J#  
    %       z = nan(size(X)); eDP&W$s#  
    %       n = [0  1  1  2  2  2  3  3  3  3]; +U J~/XV  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; uwI"V|g%a&  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; tzd !r7  
    %       y = zernfun(n,m,r(idx),theta(idx)); C.#Ha-@uz  
    %       figure('Units','normalized') H'udxPF  
    %       for k = 1:10 $eT[`r  
    %           z(idx) = y(:,k); 6l2O>V  
    %           subplot(4,7,Nplot(k)) l3^'bp6HQ  
    %           pcolor(x,x,z), shading interp 8$]SvfX  
    %           set(gca,'XTick',[],'YTick',[]) x?B`p"ifS  
    %           axis square q:M'|5P  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) %hBwc#^  
    %       end n(#yGzq  
    % w/ZP. B  
    %   See also ZERNPOL, ZERNFUN2. b|k^   
    zQ,M795@EA  
    %   Paul Fricker 11/13/2006 "{E%Y*  
    9eHqOmz  
    E A55!  
    % Check and prepare the inputs: PE6,9i0ee  
    % ----------------------------- { g[kn^|  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) vs+aUT C\  
        error('zernfun:NMvectors','N and M must be vectors.') 9pj6`5Zn@6  
    end <>$CYTb  
    4zhh **]B  
    if length(n)~=length(m) jPz1W4pk  
        error('zernfun:NMlength','N and M must be the same length.') p ]jLs|tat  
    end . 4RU'9M  
    ^fO9oPM|  
    n = n(:); j~.tyxOq#  
    m = m(:); o-&0_Zq_  
    if any(mod(n-m,2)) *v(Q-FW  
        error('zernfun:NMmultiplesof2', ... l44QB8 9  
              'All N and M must differ by multiples of 2 (including 0).') rrEf<A}  
    end o[eZ"}~  
    HbPn<x^7  
    if any(m>n) ADOA&r[  
        error('zernfun:MlessthanN', ... u' kG(<0Y  
              'Each M must be less than or equal to its corresponding N.') %zY5'$v `  
    end \v=@'  
    Crj7n/mp]s  
    if any( r>1 | r<0 ) GNuIcy  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') +3XaAk  
    end =a+  } 6  
    9* 3;v;F  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) JJg;X :p  
        error('zernfun:RTHvector','R and THETA must be vectors.') b?,%M^9\`  
    end ^jRX6  
    3HcduJntl  
    r = r(:); #ucb  
    theta = theta(:); \I}EWI  
    length_r = length(r); 9(!AKKrr;  
    if length_r~=length(theta) NySa%7@CD  
        error('zernfun:RTHlength', ... \JR^uJ{Y  
              'The number of R- and THETA-values must be equal.') [742s]j  
    end ]o=ON95ja  
    P{n*X  
    % Check normalization: umnQ$y 0  
    % -------------------- 'xnI N u  
    if nargin==5 && ischar(nflag) v{"yrC  
        isnorm = strcmpi(nflag,'norm'); q=`n3+N_H~  
        if ~isnorm YjL'GmL<  
            error('zernfun:normalization','Unrecognized normalization flag.') 2,g4yXws5  
        end h*1T3U$  
    else W)T'?b'.  
        isnorm = false; /uR/,R++  
    end H=~7g3  
    eGpKoq7a  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Z42EnJ  
    % Compute the Zernike Polynomials )'RaMo` 4  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [ "3s  
    IqepR >5t  
    % Determine the required powers of r: #XqCz>Z  
    % ----------------------------------- :IJ<Mmb  
    m_abs = abs(m); U~?mW,iRL  
    rpowers = []; o%;ly  
    for j = 1:length(n) ,3-^EfccW  
        rpowers = [rpowers m_abs(j):2:n(j)]; K*,,j\Q.  
    end KDGrX[L:6  
    rpowers = unique(rpowers); uHmvHA~/c8  
    q`L )^In"  
    % Pre-compute the values of r raised to the required powers, o_k)x3I?  
    % and compile them in a matrix: |sFd5X  
    % ----------------------------- ns\I Y<Yo  
    if rpowers(1)==0 /)K;XtcN  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); EN/t5d  
        rpowern = cat(2,rpowern{:}); IDos4nM27]  
        rpowern = [ones(length_r,1) rpowern]; 's5rl  
    else < Mu`,Kv*  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); Jn| i!  
        rpowern = cat(2,rpowern{:}); vV 9vB3K5?  
    end T2azHo7  
    Qhc; Zl  
    % Compute the values of the polynomials: olxxs(  
    % -------------------------------------- gCG #?f  
    y = zeros(length_r,length(n)); Kj3Gm>B<y  
    for j = 1:length(n) QT%vrXzz  
        s = 0:(n(j)-m_abs(j))/2; 6H  U*,  
        pows = n(j):-2:m_abs(j); TKGaGMx6@  
        for k = length(s):-1:1 >35w"a7S  
            p = (1-2*mod(s(k),2))* ... I''n1v?N  
                       prod(2:(n(j)-s(k)))/              ... <pHm=q/U  
                       prod(2:s(k))/                     ... eu_ZsseZ  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... VEI ct{  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); f#GMJ mCQs  
            idx = (pows(k)==rpowers); ?r8hl.Z>  
            y(:,j) = y(:,j) + p*rpowern(:,idx); $2i@@#g8  
        end (&v|,.c^)1  
         sb8bCEm- \  
        if isnorm > 3(,s^  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 5%fWX'mS  
        end GU@#\3  
    end yx4pQL7  
    % END: Compute the Zernike Polynomials N#e9w3Rli  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% hqjjd-S0  
    e?+-~]0  
    % Compute the Zernike functions: n9J{f"`m  
    % ------------------------------ i+~BVb  
    idx_pos = m>0; Tt{z_gU6  
    idx_neg = m<0; 0}`-vOLd-  
    EleJ$ `/  
    z = y; Dg0rVV6c  
    if any(idx_pos) W  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); P\6:euI  
    end 0wV9Trp  
    if any(idx_neg) <)(W7#Ks  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); [Eu) ~J*  
    end 5n}<V-yJ*m  
    BQg3+w:>  
    % 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) L2EQ 9i'[  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. 9oO~UP!ag  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated Z/?{{}H+  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive {xov8 M  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, (Y7zaAG]  
    %   and THETA is a vector of angles.  R and THETA must have the same /PTRe5-7  
    %   length.  The output Z is a matrix with one column for every P-value, X"_ ^^d-  
    %   and one row for every (R,THETA) pair. { ,c*OR  
    % r#)1/`h  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike !Pnjr T  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) dr=Q9%  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) 5~IdWwG*w  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 `kQosQV  
    %   for all p. +D]raU  
    % KHHYk>FR  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 !\BZ_guz  
    %   Zernike functions (order N<=7).  In some disciplines it is t4v'X}7q]  
    %   traditional to label the first 36 functions using a single mode VlA]A,P}i  
    %   number P instead of separate numbers for the order N and azimuthal JOt(r}gU  
    %   frequency M. 6V JudNA  
    % R3|4|JlGR  
    %   Example: JRFUNy1+e1  
    % 3s%ND7!/  
    %       % Display the first 16 Zernike functions O=}Rp 1  
    %       x = -1:0.01:1; 8t*sp-cy|  
    %       [X,Y] = meshgrid(x,x); J({D~  
    %       [theta,r] = cart2pol(X,Y); I(pq3_9$  
    %       idx = r<=1; Qm[s"pM  
    %       p = 0:15; eeCG#NFY5  
    %       z = nan(size(X)); Q9O_>mZy  
    %       y = zernfun2(p,r(idx),theta(idx)); Z*mbhod  
    %       figure('Units','normalized') -X$EE$:  
    %       for k = 1:length(p) |w{Qwf!2  
    %           z(idx) = y(:,k); S}%z0g<  
    %           subplot(4,4,k) @g5y_G{SP  
    %           pcolor(x,x,z), shading interp j`RG Moq  
    %           set(gca,'XTick',[],'YTick',[]) yFDeY PZP  
    %           axis square .aY $-Y<  
    %           title(['Z_{' num2str(p(k)) '}']) T Q4L~8  
    %       end ?=&S?p)-<  
    % Uz!3){E  
    %   See also ZERNPOL, ZERNFUN. <O'U-. Gc  
    IV. })8  
    %   Paul Fricker 11/13/2006 3_XLx{["'  
    13 #ff  
    # lvt4a"P"  
    % Check and prepare the inputs: ~>=.^  
    % ----------------------------- 65~E<)UJ  
    if min(size(p))~=1 V2T% tn;rp  
        error('zernfun2:Pvector','Input P must be vector.') e c]kt'  
    end ~CnnN[g(_  
    =+j3E<w  
    if any(p)>35 >bZ-mX)j\0  
        error('zernfun2:P36', ... $-1ajSVJ  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... j%nN*ms  
               '(P = 0 to 35).']) ZJBb% d1;  
    end ^)>( <6  
    66eJp-5e8  
    % Get the order and frequency corresonding to the function number: t7F0[E'=5\  
    % ---------------------------------------------------------------- \Yr*x7!  
    p = p(:); VmPh''Z%-  
    n = ceil((-3+sqrt(9+8*p))/2); T@yQOD7  
    m = 2*p - n.*(n+2); zG ='U  
    4DCh+|r  
    % Pass the inputs to the function ZERNFUN: ;Y*K!iFWH  
    % ---------------------------------------- mk1R~4v  
    switch nargin LsERcjwwK  
        case 3 S^p b9~  
            z = zernfun(n,m,r,theta); 3i!a\N4 K  
        case 4 hTn"/|_SW  
            z = zernfun(n,m,r,theta,nflag); c 1F^Gj!8  
        otherwise 6Oy:5Ps8a  
            error('zernfun2:nargin','Incorrect number of inputs.') :zKW[sF  
    end @r*GGI!  
    G`0O5G:1  
    % 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) @9 S ::  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. $mGzJ4&  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of Z. gb'  
    %   order N and frequency M, evaluated at R.  N is a vector of .2@T|WD!Ah  
    %   positive integers (including 0), and M is a vector with the sX~E ~$_g  
    %   same number of elements as N.  Each element k of M must be a R|qrK  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) *g+ ZXB  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is > "F-1{  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix u/j\pDl.  
    %   with one column for every (N,M) pair, and one row for every HU?1>}4L  
    %   element in R. wL),/i&<  
    % .Bl:hk\  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- A2ye ^<-C.  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is j67a?0<C2U  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to !u]@Ru34  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 cOP'ql{"  
    %   for all [n,m]. 45.ks.  
    % F(`|-E"E;  
    %   The radial Zernike polynomials are the radial portion of the D*-  
    %   Zernike functions, which are an orthogonal basis on the unit S&UP;oc  
    %   circle.  The series representation of the radial Zernike ; j.d  
    %   polynomials is bDWL Hdu a  
    % vOlfyH>  
    %          (n-m)/2 V"4L=[le  
    %            __ jq)Bj#'7  
    %    m      \       s                                          n-2s y p{Dl  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r yy{YduI  
    %    n      s=0 q#AEu xI1  
    % 4Px|:7~wT8  
    %   The following table shows the first 12 polynomials. _=I1  
    % f u\j  
    %       n    m    Zernike polynomial    Normalization 7(o`>7x*  
    %       --------------------------------------------- +C!GV.q[  
    %       0    0    1                        sqrt(2) xbCR4upS  
    %       1    1    r                           2 e: Sd#H!  
    %       2    0    2*r^2 - 1                sqrt(6) ~2rQ80_  
    %       2    2    r^2                      sqrt(6) l3b=8yn.  
    %       3    1    3*r^3 - 2*r              sqrt(8) [6l0|Y  
    %       3    3    r^3                      sqrt(8) `:3nF'  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) G)s.~ T  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) _0Ea 3K  
    %       4    4    r^4                      sqrt(10) _$$.5?4  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) :|V650/  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) Tfh2>  
    %       5    5    r^5                      sqrt(12) K.QSt  
    %       --------------------------------------------- mF@7;dpr  
    % ( xooU 8d  
    %   Example: ++b[>};  
    % 9cB+ x`+Lu  
    %       % Display three example Zernike radial polynomials %7mGMa/  
    %       r = 0:0.01:1; H}GGUE&c*  
    %       n = [3 2 5]; nQV0I"f]?]  
    %       m = [1 2 1]; *yT>  
    %       z = zernpol(n,m,r); z**2-4 z  
    %       figure =At" Q6-O  
    %       plot(r,z) FLG"c690  
    %       grid on T=YVG@fm?  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') P x Q]$w  
    % 8'@5X-nD  
    %   See also ZERNFUN, ZERNFUN2. L<=Dl  
    %]7 6u7b/  
    % A note on the algorithm. toGd;2rl  
    % ------------------------ 2R,} j@  
    % The radial Zernike polynomials are computed using the series f$:Y'$Z1  
    % representation shown in the Help section above. For many special Y)/|C7~W  
    % functions, direct evaluation using the series representation can X>`03?L  
    % produce poor numerical results (floating point errors), because `Ns@W?  
    % the summation often involves computing small differences between b%MZfaU  
    % large successive terms in the series. (In such cases, the functions [ S  
    % are often evaluated using alternative methods such as recurrence RdD>&D$I  
    % relations: see the Legendre functions, for example). For the Zernike 4r4 #u'Om  
    % polynomials, however, this problem does not arise, because the !D['}%  
    % polynomials are evaluated over the finite domain r = (0,1), and s.7=!JQ#]p  
    % because the coefficients for a given polynomial are generally all %C`P7&8m=O  
    % of similar magnitude. +0U=UV)U  
    % o#6QwbU25  
    % ZERNPOL has been written using a vectorized implementation: multiple z<9C-  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] BNJ0D  
    % values can be passed as inputs) for a vector of points R.  To achieve {E%c%zzQ  
    % this vectorization most efficiently, the algorithm in ZERNPOL  &ox  
    % involves pre-determining all the powers p of R that are required to |*JMPg?zI  
    % compute the outputs, and then compiling the {R^p} into a single !`N:.+DT  
    % matrix.  This avoids any redundant computation of the R^p, and 'd Be,@  
    % minimizes the sizes of certain intermediate variables. kiJ=C2'&  
    % S|| W  
    %   Paul Fricker 11/13/2006 vrb@::sy0T  
    rzHBop-8  
    h(yFr/  
    % Check and prepare the inputs: V~*>/2+  
    % ----------------------------- 6oLOA}q   
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ynM:]*~K  
        error('zernpol:NMvectors','N and M must be vectors.') VK*_p EV,}  
    end #z>I =gl  
    DgcS@N  
    if length(n)~=length(m) $\*Z   
        error('zernpol:NMlength','N and M must be the same length.') M`K]g&57hL  
    end ?7wcv$K5  
    =YVxQj  
    n = n(:); ]wDqdD y7S  
    m = m(:); tn Ufi8\ob  
    length_n = length(n); 'gor*-o:wu  
    X<IW5*   
    if any(mod(n-m,2)) Q/3*65  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') @"NP`#  
    end 'z,kxra|n  
    [- 92]  
    if any(m<0) -4+'(3qr  
        error('zernpol:Mpositive','All M must be positive.') QAx9W%  
    end :k ?`gm$  
    75p9_)>96  
    if any(m>n) sXEIC#rq  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') \?t8[N\_[(  
    end WqefH{PB  
    ,N:^4A  
    if any( r>1 | r<0 ) ,nL~?h-Zh  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') #!z-)[S.+  
    end :'%|LBc0  
    sE&nEc  
    if ~any(size(r)==1) > "rM\ Q  
        error('zernpol:Rvector','R must be a vector.') 1@{ov!YB]  
    end 7r?,wM  
    $!. [R}  
    r = r(:); k-3;3Mq  
    length_r = length(r); 9^g8VlQdT  
    BMO,eQcB  
    if nargin==4 MOB4t|  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); _ZavY<6  
        if ~isnorm &TRKd)wd  
            error('zernpol:normalization','Unrecognized normalization flag.') MkFWZ9c3  
        end 9;XbyA]  
    else :I7qw0?  
        isnorm = false; $:5h5Y#z  
    end XM!oN^  
    <w}i  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% xib}E[-l#  
    % Compute the Zernike Polynomials !]s=9(O  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% V^FM-bg%9  
     U%r{{Q1  
    % Determine the required powers of r: ='D%c^;O8'  
    % ----------------------------------- 037\LPO  
    rpowers = []; fhZwYx&t  
    for j = 1:length(n) L|APXy]>  
        rpowers = [rpowers m(j):2:n(j)]; W (TTsnnx  
    end 6 [XaIco=C  
    rpowers = unique(rpowers); &u|t{C#0  
    z5({A2q  
    % Pre-compute the values of r raised to the required powers, b/*QV0(  
    % and compile them in a matrix: An(gHi;1$  
    % ----------------------------- FEhBhv|m  
    if rpowers(1)==0 o7+<sL  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 1f^oW[w&  
        rpowern = cat(2,rpowern{:}); zx "EAF{  
        rpowern = [ones(length_r,1) rpowern]; *{5}m(5F  
    else +e>G V61  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); =3hJti9[  
        rpowern = cat(2,rpowern{:}); M6cybEk`  
    end 1 c"s+k]9  
    F ! v01]O  
    % Compute the values of the polynomials: N%:uOX8{  
    % -------------------------------------- o(v`  
    z = zeros(length_r,length_n); 7>7n|N  
    for j = 1:length_n IA1O]i S  
        s = 0:(n(j)-m(j))/2; xF) .S@  
        pows = n(j):-2:m(j); |af<2(d  
        for k = length(s):-1:1 :W&kl UU"  
            p = (1-2*mod(s(k),2))* ... tZ=|1lM  
                       prod(2:(n(j)-s(k)))/          ... OdyL j  
                       prod(2:s(k))/                 ... "o;%em*Bc  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... D.AiqO<z  
                       prod(2:((n(j)+m(j))/2-s(k))); eWYet2!Q  
            idx = (pows(k)==rpowers); #F .8x@  
            z(:,j) = z(:,j) + p*rpowern(:,idx); >Gg[J=7`  
        end Jp)PKS ![  
         T06w`'aL  
        if isnorm S WsD]rn  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); 7m8:odeF  
        end L}8 }Pns?&  
    end Q!~1Xc0S`p  
     H!y@.W{_  
    % 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)  |Z/ySAFM  
    q.X-2jjpx:  
    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。 J/*[wj  
    Z,&ywMm/G  
    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)