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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]5H  
    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! ^}RCoE  
     
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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 d z|or9&  
    function z = zernfun(n,m,r,theta,nflag) {$oj.V 4  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. VG5i{1  0  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N _T60;ZI+^  
    %   and angular frequency M, evaluated at positions (R,THETA) on the )+#` CIv  
    %   unit circle.  N is a vector of positive integers (including 0), and H8=N@l  
    %   M is a vector with the same number of elements as N.  Each element /l3V3B7  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) .e#w)K  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, "69s) ~  
    %   and THETA is a vector of angles.  R and THETA must have the same J4hL_iCQ  
    %   length.  The output Z is a matrix with one column for every (N,M) O 2V  
    %   pair, and one row for every (R,THETA) pair. !t"4!3  
    % . '6gZKXY  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 10Q ]67  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ZtNN<7  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral : 6jbt:  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, }{Pp]*I<A  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized 9X6h  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. G/E+L-N#`  
    % "Bkfoi  
    %   The Zernike functions are an orthogonal basis on the unit circle. 9 ql~q  
    %   They are used in disciplines such as astronomy, optics, and <)Dj9' _J  
    %   optometry to describe functions on a circular domain. }RF(CwZr(  
    % \  #F  
    %   The following table lists the first 15 Zernike functions. HZE#Ab*L  
    % : $1?i)  
    %       n    m    Zernike function           Normalization G[PtkPSJ  
    %       -------------------------------------------------- #\{l"-  
    %       0    0    1                                 1 H*n-_{h"t  
    %       1    1    r * cos(theta)                    2 =jN.1}  
    %       1   -1    r * sin(theta)                    2 .^`{1%  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) `v!urE/gg%  
    %       2    0    (2*r^2 - 1)                    sqrt(3) yZY\MB/  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) iQ67l\{R  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) e+7"/icK  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) [>I<#_^~  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) >NV @R&  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) k=$TGqQY?  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) q>_.[+6  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) !/b>sN}  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) BKCiIfkZ  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) b#%hY{$j  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) mthA4sz  
    %       -------------------------------------------------- ;+R&}[9,A)  
    % ?FZ HrA  
    %   Example 1:  tU5zF.%  
    % UW={[h{.|@  
    %       % Display the Zernike function Z(n=5,m=1) =ZznFVJ`={  
    %       x = -1:0.01:1; /KaZH R.  
    %       [X,Y] = meshgrid(x,x); :`#d:.@]o@  
    %       [theta,r] = cart2pol(X,Y); y-b%T|p9  
    %       idx = r<=1; VBlYvZ;$*  
    %       z = nan(size(X)); n+9=1Oo"  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); R_cA:3qc~  
    %       figure tKuwpT1Qc  
    %       pcolor(x,x,z), shading interp J1U/.`Oy  
    %       axis square, colorbar !?jrf] A@  
    %       title('Zernike function Z_5^1(r,\theta)') Dj?> <@  
    % }-{H  Y  
    %   Example 2: O/(`S<iip  
    % |3b^~?S  
    %       % Display the first 10 Zernike functions 3pROf#M  
    %       x = -1:0.01:1; &m7]v,&  
    %       [X,Y] = meshgrid(x,x); a5^] 20Fa  
    %       [theta,r] = cart2pol(X,Y); ~vhE|f  
    %       idx = r<=1;  %\#8{g  
    %       z = nan(size(X)); u~:y\/Y6  
    %       n = [0  1  1  2  2  2  3  3  3  3]; FX&~\kmV'j  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; &|1<v<I5  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28];  qA7>vi%  
    %       y = zernfun(n,m,r(idx),theta(idx)); & ywPuTt  
    %       figure('Units','normalized') Ta0|+IYk<  
    %       for k = 1:10 ,-LwtePJ0  
    %           z(idx) = y(:,k); (,\+tr8r8  
    %           subplot(4,7,Nplot(k))  DPxM'7  
    %           pcolor(x,x,z), shading interp Xl{P8L  
    %           set(gca,'XTick',[],'YTick',[]) UhWNl]Z  
    %           axis square ZQsJL\x[UK  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) -Cpl?Io`r5  
    %       end x+:UN'"r  
    % \)904W5R  
    %   See also ZERNPOL, ZERNFUN2. IPKbMlV#d  
    9&2O 9Nz6  
    %   Paul Fricker 11/13/2006 wssRA?9<  
    U$.@]F4&  
    T*Exs|N2P-  
    % Check and prepare the inputs: n nEgx;Nl0  
    % ----------------------------- P )"m0Lu<  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) /SR*W5#s  
        error('zernfun:NMvectors','N and M must be vectors.') dq6m>;`  
    end 3S@7]Pg  
    6<SAa#@ey  
    if length(n)~=length(m) xh,qNnGGi  
        error('zernfun:NMlength','N and M must be the same length.') [PM 2\#K  
    end }OR@~V{Gj  
    )[6U^j4  
    n = n(:); J?1 uKR  
    m = m(:); A RuA<vQ  
    if any(mod(n-m,2)) P6`u._mX  
        error('zernfun:NMmultiplesof2', ... bHYy}weZ  
              'All N and M must differ by multiples of 2 (including 0).') 4jM Fr,  
    end rQs)O<jl  
    dr}`H,X"3  
    if any(m>n) {hjhL: pg  
        error('zernfun:MlessthanN', ... {SPq$B_VR  
              'Each M must be less than or equal to its corresponding N.') n1t*sk/J  
    end G@\1E+Ip  
    %6,SKg p  
    if any( r>1 | r<0 ) +F` S>U  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') #aJ(m&  
    end faX#**r  
    .Iw AK/QS  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Ecefi pG  
        error('zernfun:RTHvector','R and THETA must be vectors.') @Zu5VpJ  
    end Qh3YJ=X&  
    gQg"j)  
    r = r(:); K~{$oD7!  
    theta = theta(:); ~d4 )/y  
    length_r = length(r); )gIKH{JYL  
    if length_r~=length(theta) Q7\w+ANf0  
        error('zernfun:RTHlength', ... wLH>:yKUU  
              'The number of R- and THETA-values must be equal.') A*2jENgci  
    end ]EBxl=C}D  
    )JLdO*H  
    % Check normalization: XGWSdPJLr  
    % -------------------- kQSy+q  
    if nargin==5 && ischar(nflag) mt{nm[D!Xp  
        isnorm = strcmpi(nflag,'norm'); KIf dafRL  
        if ~isnorm w^|*m/h|@u  
            error('zernfun:normalization','Unrecognized normalization flag.') /GN<\_o=q  
        end - q1?? u  
    else Tod&&T'UW  
        isnorm = false; 4N_R:B-V u  
    end HGs $*  
    85:=4N%  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  I<mV+ex  
    % Compute the Zernike Polynomials TH&U j1  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% n u[ML  
    L-WT]&n_  
    % Determine the required powers of r: m@2QnA[ 4  
    % ----------------------------------- KNvZm;Q6  
    m_abs = abs(m); Uw. `7b>B  
    rpowers = []; =JEv,ZGT3  
    for j = 1:length(n) mb TEp*H  
        rpowers = [rpowers m_abs(j):2:n(j)]; ]I dk:et  
    end ]Ji.Zk  
    rpowers = unique(rpowers); iDp)FQ$  
    /sx&=[ D  
    % Pre-compute the values of r raised to the required powers, wr/"yQA]  
    % and compile them in a matrix: |O|V-f{l  
    % ----------------------------- x.!V^HQSN  
    if rpowers(1)==0 {0wIR_dGX  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); Z, Yb&b  
        rpowern = cat(2,rpowern{:}); {j?FNOJn  
        rpowern = [ones(length_r,1) rpowern]; $oID(P  
    else %~H-)_d20  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ?W?c 1>  
        rpowern = cat(2,rpowern{:}); (ylTp]~mR-  
    end p Z|V 3  
    W.f/pu  
    % Compute the values of the polynomials: 30#s aGV  
    % -------------------------------------- mZS >O_E  
    y = zeros(length_r,length(n)); Eex~xiiV  
    for j = 1:length(n) %+W{iu[|  
        s = 0:(n(j)-m_abs(j))/2; \O3m9,a   
        pows = n(j):-2:m_abs(j); [I,Z2G,Jb  
        for k = length(s):-1:1 O>b C2;+s  
            p = (1-2*mod(s(k),2))* ... 7hD>As7`/  
                       prod(2:(n(j)-s(k)))/              ... 2 /\r)$ 2i  
                       prod(2:s(k))/                     ... dk#k bG;  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... s^G.]%iU  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); |}s*E_/[  
            idx = (pows(k)==rpowers); 'j8:vq^d  
            y(:,j) = y(:,j) + p*rpowern(:,idx); w7.V6S$Ga  
        end C\Wmq [  
         EPI4!3]  
        if isnorm 9iIhte.  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); m<T%Rb4?@  
        end %op**@4/t\  
    end 1y@i}<9F  
    % END: Compute the Zernike Polynomials ,i?nWlh+  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% sk<3`x+  
    p?%y82E  
    % Compute the Zernike functions:  ul6]!Iy  
    % ------------------------------ .LnGL]/  
    idx_pos = m>0; F3[T.sf  
    idx_neg = m<0; In"ZIKaC  
    i4Q@K,$  
    z = y; KEo ,m  
    if any(idx_pos) ` xEx^P^7  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); O_ muD\  
    end e\`&p  
    if any(idx_neg) ed{ -/l~j  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); f M :]&  
    end >-RQ]?^  
    4<w.8rR:A  
    % 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) e9Wa<i 8  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. hlvK5Z   
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated +5g_KS  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive @muRxi  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, n:I,PS0H<  
    %   and THETA is a vector of angles.  R and THETA must have the same z>1Pz(  
    %   length.  The output Z is a matrix with one column for every P-value, Gt8M&S-;  
    %   and one row for every (R,THETA) pair. >NGj =L<  
    % U*rcd-@  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike D# 9m\o_  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) > ym,{EHK  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) P[G)sA_"  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 0I-9nuw,^;  
    %   for all p. 6##_%PO<m  
    % #X+JHl  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 n$A9_cHF7  
    %   Zernike functions (order N<=7).  In some disciplines it is T#T*Zw"+  
    %   traditional to label the first 36 functions using a single mode Di,^%  
    %   number P instead of separate numbers for the order N and azimuthal pTth}JM>  
    %   frequency M. hIYNhZv  
    % v|)4ocFK  
    %   Example: "=HA Y  
    % @(EAq<5{  
    %       % Display the first 16 Zernike functions ,i ^9 |Oeq  
    %       x = -1:0.01:1; =g7x' kN  
    %       [X,Y] = meshgrid(x,x); W]$w@.oW[  
    %       [theta,r] = cart2pol(X,Y); 1 fp?  
    %       idx = r<=1; //up5R_nx  
    %       p = 0:15; :I.mGH!^  
    %       z = nan(size(X)); Co9^OF-k  
    %       y = zernfun2(p,r(idx),theta(idx)); P1. [  
    %       figure('Units','normalized') \i>?q   
    %       for k = 1:length(p) CImWd.W9~  
    %           z(idx) = y(:,k); ].avItg  
    %           subplot(4,4,k) np|Sy;:  
    %           pcolor(x,x,z), shading interp ]? c B:}  
    %           set(gca,'XTick',[],'YTick',[]) ; }I:\P  
    %           axis square '&P%C" 5  
    %           title(['Z_{' num2str(p(k)) '}']) ?> 9/#Nv  
    %       end + )AG*  
    % &Q/W~)~  
    %   See also ZERNPOL, ZERNFUN. ^`i#$  
    LRxZcxmy  
    %   Paul Fricker 11/13/2006 MVpGWTH@F  
    X;+sUj8  
    xJpA0_xfG  
    % Check and prepare the inputs: B6+khuG(  
    % ----------------------------- B B{$&Oh  
    if min(size(p))~=1 ~f2z]JLr:  
        error('zernfun2:Pvector','Input P must be vector.') V5@:#BIs  
    end ZuzEg*lb  
    RXMISt3+{y  
    if any(p)>35 Gm&Za,4%4  
        error('zernfun2:P36', ... #Qw0&kM7I  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... {S]}.7`l9(  
               '(P = 0 to 35).']) @(w@e\Bq  
    end +%z> H"J.  
    kM l+yli3c  
    % Get the order and frequency corresonding to the function number: tn\yI!a  
    % ---------------------------------------------------------------- Pjf"CW+A  
    p = p(:); G6Axs1a  
    n = ceil((-3+sqrt(9+8*p))/2); P-_6wfg,;>  
    m = 2*p - n.*(n+2); sPpH*,(  
    *uRBzO}  
    % Pass the inputs to the function ZERNFUN: ]"As1"  
    % ---------------------------------------- [-1^-bb  
    switch nargin dmtr*pM_  
        case 3 (*9$`!wS  
            z = zernfun(n,m,r,theta); biD$qg  
        case 4 T3.&R#1M8-  
            z = zernfun(n,m,r,theta,nflag); S&5&];Ag  
        otherwise FBX'.\@`  
            error('zernfun2:nargin','Incorrect number of inputs.') aH(J,XY  
    end f1RWP@iar  
    wD}l$ & +  
    % 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) 2eS~/Pq5=i  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. mfn,Gjt3O  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of \A#41  
    %   order N and frequency M, evaluated at R.  N is a vector of WM$ MPs  
    %   positive integers (including 0), and M is a vector with the 2DDtu[}  
    %   same number of elements as N.  Each element k of M must be a T@B/xAq5!  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) U[-o> W#  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is vzAaxk%  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix E?f-wQF  
    %   with one column for every (N,M) pair, and one row for every q'F+OQb1  
    %   element in R. Y;M|D'y+  
    % N7zft  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- yjX9oxhtL  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is ZgcMv,=  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to h 0Q5-EA  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 '3tCH)s  
    %   for all [n,m]. ibk6|pp  
    % K#d`Hyx  
    %   The radial Zernike polynomials are the radial portion of the O"9\5(w  
    %   Zernike functions, which are an orthogonal basis on the unit >z>!Luw  
    %   circle.  The series representation of the radial Zernike CAWNDl4  
    %   polynomials is %JBz5G  
    % ;7V%#-  
    %          (n-m)/2 `5.'_3  
    %            __ _4So{~Gf1  
    %    m      \       s                                          n-2s '@KEi%-^>  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r %)W2H^  
    %    n      s=0 '[:D$q;  
    % n5NsmVW\x  
    %   The following table shows the first 12 polynomials. xGg )Y#  
    % {rw|#Z>A  
    %       n    m    Zernike polynomial    Normalization j{A y\n(  
    %       --------------------------------------------- azp):*f("  
    %       0    0    1                        sqrt(2) 'G4ICtHQ  
    %       1    1    r                           2 }<SQ  
    %       2    0    2*r^2 - 1                sqrt(6) @o _}g !9=  
    %       2    2    r^2                      sqrt(6) LckK\`mh  
    %       3    1    3*r^3 - 2*r              sqrt(8) }2.`N%[  
    %       3    3    r^3                      sqrt(8) osAd1<EIC  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10)  }q`S$P;  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) nRS}}6Q  
    %       4    4    r^4                      sqrt(10) Jhhb7uU+  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) 3yF,ak {Sl  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) l<LI7Z]A  
    %       5    5    r^5                      sqrt(12)  "Og7rl  
    %       --------------------------------------------- E A1?)|}n  
    % .j0$J\:i  
    %   Example: P@Oo$ o  
    % IY\5@PVZ  
    %       % Display three example Zernike radial polynomials *C*U5~Zq7:  
    %       r = 0:0.01:1; x2\qXN/R  
    %       n = [3 2 5]; />pI8 g<  
    %       m = [1 2 1]; 3$>1FoSk  
    %       z = zernpol(n,m,r); q"8e a/  
    %       figure k"zv~`i'  
    %       plot(r,z) c9u`!'g`i  
    %       grid on >W+%8e  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') c9 _ rmz8  
    % XJ| <?   
    %   See also ZERNFUN, ZERNFUN2. 9F;>W ET  
    k)=s>&hl  
    % A note on the algorithm. BG]#o| KW  
    % ------------------------ "_NN3lD)X  
    % The radial Zernike polynomials are computed using the series C1n>M}b  
    % representation shown in the Help section above. For many special ~-Qw.EdC  
    % functions, direct evaluation using the series representation can A[{yCn`tM  
    % produce poor numerical results (floating point errors), because 'yEHI  
    % the summation often involves computing small differences between #gs`#6 ,'  
    % large successive terms in the series. (In such cases, the functions D.u{~  
    % are often evaluated using alternative methods such as recurrence eJX9_6m-  
    % relations: see the Legendre functions, for example). For the Zernike `e&Suyf4B  
    % polynomials, however, this problem does not arise, because the <=/hi l  
    % polynomials are evaluated over the finite domain r = (0,1), and ,<P vovg_  
    % because the coefficients for a given polynomial are generally all _8UU'1d  
    % of similar magnitude. G<J?"oQbRT  
    % `mJ6K&t$<  
    % ZERNPOL has been written using a vectorized implementation: multiple H40p86@M  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] V6X 0^g  
    % values can be passed as inputs) for a vector of points R.  To achieve .?sx&2R2  
    % this vectorization most efficiently, the algorithm in ZERNPOL mNTzUoZF'@  
    % involves pre-determining all the powers p of R that are required to qqY"*uJ'  
    % compute the outputs, and then compiling the {R^p} into a single Wt-GjxGi  
    % matrix.  This avoids any redundant computation of the R^p, and ^k">A:E2  
    % minimizes the sizes of certain intermediate variables. Y]2A&0  
    % N<VJ(20y  
    %   Paul Fricker 11/13/2006 ?NsW|w_  
    })Vi  
    xY(*.T9K  
    % Check and prepare the inputs: 7[XRd9a5(  
    % -----------------------------  d{3QP5  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) &B1WtW  
        error('zernpol:NMvectors','N and M must be vectors.') 9qzHS~l  
    end 0 /U{p,r6`  
    \Uq(Zga4)  
    if length(n)~=length(m) &}B|"s[  
        error('zernpol:NMlength','N and M must be the same length.') [waIi3Dv\  
    end "@0]G<H  
    m&&m,6``P  
    n = n(:); . 3T3E X|G  
    m = m(:); hhc,uJ">!  
    length_n = length(n); VuZuS6~#J  
    ;iL#7NG-R  
    if any(mod(n-m,2)) W.KDVE$}f  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') F{;((VboN  
    end TKmf+ZT*r  
    "J_9WUN  
    if any(m<0) M%P:n/j  
        error('zernpol:Mpositive','All M must be positive.') c4eBt))}V  
    end R$[vm6T?  
    $B5aje}i  
    if any(m>n) 6mxfLlZ  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') \\;jw[P0  
    end 1K50Z.o&@  
    ` 7V]y -  
    if any( r>1 | r<0 ) <}9lZEqY  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') S3 Xl  
    end ],Do6 @M-  
    Cj lk  
    if ~any(size(r)==1) Z o(rTCZX  
        error('zernpol:Rvector','R must be a vector.') jasy<IqT!{  
    end l}A93jSL  
    @Qt{jI !  
    r = r(:); 6q.Uhe_B  
    length_r = length(r); _ *Pf  
    i2SR{e8:GF  
    if nargin==4 u>a5GkG.  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); z[qDkL  
        if ~isnorm Yufc{M00  
            error('zernpol:normalization','Unrecognized normalization flag.') 59;KQ  
        end T%*D~=fQ'  
    else ":QZy8f9%  
        isnorm = false; tJ$_lk ~6q  
    end 07{)?1cod4  
    t!7-DF|N  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ~6LN6}~|.  
    % Compute the Zernike Polynomials N6i Q8P -  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% gT6jYQ  
    {9.|2%a  
    % Determine the required powers of r: lA8`l>I  
    % ----------------------------------- UH"%N)[  
    rpowers = []; CB}2j  
    for j = 1:length(n) [FR`Z=%  
        rpowers = [rpowers m(j):2:n(j)]; `*1p0~cu  
    end j3E7zRm] \  
    rpowers = unique(rpowers); 4ID5q~  
    Qj3EXb  
    % Pre-compute the values of r raised to the required powers, :& ."ttf=  
    % and compile them in a matrix: #Ki[$bS~6  
    % ----------------------------- ^SrJu:Q_  
    if rpowers(1)==0 =]0&i]z[.  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); !'*-$e  
        rpowern = cat(2,rpowern{:}); Zp=U W*g^  
        rpowern = [ones(length_r,1) rpowern]; 3AN/ H  
    else |Ds1  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); fVpMx4&F   
        rpowern = cat(2,rpowern{:}); k~1?VQ+?M  
    end aO4?m+  
    Qh\60f>0  
    % Compute the values of the polynomials: 6i3$CW  
    % -------------------------------------- \z(gqkc 6  
    z = zeros(length_r,length_n); 'm kLCS  
    for j = 1:length_n 1#+S+g@#  
        s = 0:(n(j)-m(j))/2; 40m-ch6Q  
        pows = n(j):-2:m(j); 5VU2[ \  
        for k = length(s):-1:1 Q*~]h;6\{d  
            p = (1-2*mod(s(k),2))* ... '?(% Zxw%&  
                       prod(2:(n(j)-s(k)))/          ... 1/J=uH  
                       prod(2:s(k))/                 ... t;\Y{`  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... }:)&u|d_  
                       prod(2:((n(j)+m(j))/2-s(k))); ER.}CM6{[  
            idx = (pows(k)==rpowers); FVJ GL  
            z(:,j) = z(:,j) + p*rpowern(:,idx); hM@>q&q_  
        end @b2aNS<T  
         A6(/;+n  
        if isnorm +TDw+  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); RPRBmb940  
        end P+/e2Y  
    end C1QA)E['V  
    JZyAXm%  
    % 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)  buHJB*?9  
    7F~X,Dk_  
    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<e2d7oN  
    V>3X\)qu  
    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)