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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 hBZh0x y  
    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! ( K-7z  
     
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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 7z&^i-l.  
    function z = zernfun(n,m,r,theta,nflag) w/0;N`YB  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. 4=ha$3h$  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N d/?0xLW  
    %   and angular frequency M, evaluated at positions (R,THETA) on the j1@PfKh  
    %   unit circle.  N is a vector of positive integers (including 0), and j;rxr1+w  
    %   M is a vector with the same number of elements as N.  Each element ~bjT,i  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) v@!r$jZ  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, 3A b_Z  
    %   and THETA is a vector of angles.  R and THETA must have the same Zvz}Z8jW  
    %   length.  The output Z is a matrix with one column for every (N,M) }Oy/F  
    %   pair, and one row for every (R,THETA) pair. F.R0c@&W  
    % na/,1iI<  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike w4&-9[@Y  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), (5^SL Y  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral 7mS_Cz+cB  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 28,HZaXhc  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized ;xE1#ZT  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ?rwHkPJ{*  
    % 6[1lK8o  
    %   The Zernike functions are an orthogonal basis on the unit circle. Bv=:F5hLG  
    %   They are used in disciplines such as astronomy, optics, and 8g 2'[ci$q  
    %   optometry to describe functions on a circular domain. kh*td(pfP9  
    % ]O68~+6  
    %   The following table lists the first 15 Zernike functions.  ~\+m o  
    % NEMC  
    %       n    m    Zernike function           Normalization \o!B:Vb<  
    %       -------------------------------------------------- V_Y2@4  
    %       0    0    1                                 1 YcuHYf5  
    %       1    1    r * cos(theta)                    2 E'_$?wWn5  
    %       1   -1    r * sin(theta)                    2 {B\lk:"X  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) 9O#?r82  
    %       2    0    (2*r^2 - 1)                    sqrt(3) !% yd'"6Dl  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) T+<OlXpL  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) & MfnH  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) |G>Lud  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) 6?jSe<4x  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) H Ff9^  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) ,Z]4`9c  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Q-S5("  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) ehYGw2  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) h`p9H2}0  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) xHdv?69,  
    %       -------------------------------------------------- qgLj^{  
    % TYr"yZ([  
    %   Example 1: X6c['Zrc  
    % y <21~g=  
    %       % Display the Zernike function Z(n=5,m=1) 3MFb\s&Fq  
    %       x = -1:0.01:1; +QVe -  
    %       [X,Y] = meshgrid(x,x); aruT eJF  
    %       [theta,r] = cart2pol(X,Y); * d[sja+  
    %       idx = r<=1; 8Ow0A  
    %       z = nan(size(X)); I!-5 #bxD  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); }>u<,  
    %       figure .1& F p  
    %       pcolor(x,x,z), shading interp e$@azi1  
    %       axis square, colorbar mq~L1< f  
    %       title('Zernike function Z_5^1(r,\theta)') ,;wc$-Z!8  
    % ~w9ZSSb4  
    %   Example 2: {VrjDj+Xy  
    % -nrfu)G  
    %       % Display the first 10 Zernike functions ('.r_F  
    %       x = -1:0.01:1; @#5PPXp  
    %       [X,Y] = meshgrid(x,x); T8rf+B/.L  
    %       [theta,r] = cart2pol(X,Y); @=1kr ^i  
    %       idx = r<=1; 86\B|!   
    %       z = nan(size(X)); Kzd)Z fnD0  
    %       n = [0  1  1  2  2  2  3  3  3  3]; q+-Bl  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; x?B8b-*  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; Z}'"c9oB  
    %       y = zernfun(n,m,r(idx),theta(idx));  =:-x;  
    %       figure('Units','normalized') &-0 eWwMW  
    %       for k = 1:10 HN tl>H  
    %           z(idx) = y(:,k); S7 Tem:/  
    %           subplot(4,7,Nplot(k)) D#,P-0+%  
    %           pcolor(x,x,z), shading interp w_!]_6%{b  
    %           set(gca,'XTick',[],'YTick',[]) +b]+5!  
    %           axis square w oSI 2i  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}'])  $VCWc#  
    %       end x GHS  
    % WSW,}tFp"  
    %   See also ZERNPOL, ZERNFUN2. 4h[^!up.7  
     /P/S0  
    %   Paul Fricker 11/13/2006 "~lGSWcU  
    G}b LWA  
    *Q8d &$ ^  
    % Check and prepare the inputs: yXx}'=&!0  
    % ----------------------------- t~0}Emgp<(  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) _ %HyXd  
        error('zernfun:NMvectors','N and M must be vectors.') CL$mK5u  
    end `)W}4itm  
    Dab1^H!KT  
    if length(n)~=length(m) JUlV$b.)J  
        error('zernfun:NMlength','N and M must be the same length.') .Lk2S "+  
    end .{1MM8 Q  
    v&EHp{8Qd  
    n = n(:); @:s|X  
    m = m(:); _YH)E^If  
    if any(mod(n-m,2)) YrR}55V,  
        error('zernfun:NMmultiplesof2', ... m{bw(+r  
              'All N and M must differ by multiples of 2 (including 0).') b)A$lP%`  
    end IRZ?'Im  
    S/ Y1NH  
    if any(m>n) VlVd"jW  
        error('zernfun:MlessthanN', ... 5"[Qs|VjA6  
              'Each M must be less than or equal to its corresponding N.') TY=BP!s  
    end m*BtD-{  
    }>w;(R  
    if any( r>1 | r<0 ) *HwTq[y  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') ;q&>cnLDR  
    end *p.P/w@1  
    hNV" {V3`{  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) vTD`Ja#h  
        error('zernfun:RTHvector','R and THETA must be vectors.') Xa2QtJq  
    end [Uezi1I  
    ]~z2s;J{/  
    r = r(:); wL2d.$?TEg  
    theta = theta(:); > @ulvHL  
    length_r = length(r); I hvL2 zB  
    if length_r~=length(theta) L44-: 3  
        error('zernfun:RTHlength', ... iaq0\d.[7  
              'The number of R- and THETA-values must be equal.') $o`N%]  
    end u8*Uia*vwH  
    (d[)U<  
    % Check normalization: pbivddi2  
    % -------------------- h{]l?6`  
    if nargin==5 && ischar(nflag) AO9F.A<T5  
        isnorm = strcmpi(nflag,'norm'); i8nCTW  
        if ~isnorm %/H  
            error('zernfun:normalization','Unrecognized normalization flag.') HzM^Zn57%  
        end w*ig[{ I  
    else 5w`v 3o  
        isnorm = false; tWi@_Rlx;  
    end #Vanw!  
    r}P{opn$t  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% .h^."+TJ  
    % Compute the Zernike Polynomials -\j}le6;c  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% =F ZvtcCa  
    9[|Ql  
    % Determine the required powers of r: nVoPTr  
    % ----------------------------------- Z-b^{uP  
    m_abs = abs(m); L^ VG?J  
    rpowers = []; Xb42R1  
    for j = 1:length(n) -lyT8qZ:(  
        rpowers = [rpowers m_abs(j):2:n(j)]; pd,5.d  
    end R\+p`n$  
    rpowers = unique(rpowers); <UG}P \N  
    2_0OSbFv'P  
    % Pre-compute the values of r raised to the required powers, z@$7T: H>  
    % and compile them in a matrix: g!<@6\RB  
    % ----------------------------- j3~:\H  
    if rpowers(1)==0 Tc@r#!.m  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 0vUX^<  
        rpowern = cat(2,rpowern{:}); _ 9Tv*@  
        rpowern = [ones(length_r,1) rpowern]; QdG_zK>|e  
    else K!k,]90Ko  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); r9@W8](\  
        rpowern = cat(2,rpowern{:}); }7vX4{Yn  
    end 9xC,i )  
    Ud:v3"1  
    % Compute the values of the polynomials: APuG8 <R,  
    % -------------------------------------- 8(D>ws$  
    y = zeros(length_r,length(n)); \Btv76*,  
    for j = 1:length(n) eQno]$-\  
        s = 0:(n(j)-m_abs(j))/2; kVQKP  U  
        pows = n(j):-2:m_abs(j); ;]MHU/  
        for k = length(s):-1:1 w:& m_z#M  
            p = (1-2*mod(s(k),2))* ... +is;$ 1rq  
                       prod(2:(n(j)-s(k)))/              ... wa W2$9O  
                       prod(2:s(k))/                     ... :=^JHE{  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ^!1mChf  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); AU$W=Z*  
            idx = (pows(k)==rpowers); I1 j-Q8  
            y(:,j) = y(:,j) + p*rpowern(:,idx); #Z}\;a{vZ  
        end %K /=7  
         k#ED#']N  
        if isnorm 3IZ^!J  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); t&wtw  
        end veAGUE %3  
    end ~ DVAk|fc  
    % END: Compute the Zernike Polynomials qp^O\>c  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% (J][(=s;a  
    F>)u<f,C  
    % Compute the Zernike functions: ^$24231^  
    % ------------------------------ MMD4b}p  
    idx_pos = m>0; E:(flW=  
    idx_neg = m<0; ;_,=  
    U/m6% )Yx(  
    z = y; 2md1GWyP  
    if any(idx_pos) 1-1x,U7w  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); \q(RqD  
    end WL7R.!P  
    if any(idx_neg) D&/(Avx.  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); d /jO~+jP  
    end q*\ #H C  
    1[a;2x A~  
    % EOF zernfun
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) >0=`3X|Y7  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. PtCwr)B,  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated +n%uIv  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive G7DEavtr  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, T/V8&'^i  
    %   and THETA is a vector of angles.  R and THETA must have the same * 'WzIk2  
    %   length.  The output Z is a matrix with one column for every P-value, #qGfo)  
    %   and one row for every (R,THETA) pair. Bl2y~fCA  
    % x^qmYX$'1b  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike KY<>S/  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2)  Y5 $5qQ  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) 3 ~0Z.!O  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 |Ma"B4  
    %   for all p. Pq>r|/~_  
    % PCH&eTKN  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 ~&[Wqn@MZ  
    %   Zernike functions (order N<=7).  In some disciplines it is DmM<Kkg.J  
    %   traditional to label the first 36 functions using a single mode DKAqQ?fS  
    %   number P instead of separate numbers for the order N and azimuthal znw\Dn?g  
    %   frequency M. .dt7b4.kd  
    % b{=2#J-  
    %   Example: E'?yI' ~=  
    % (GJ)FWen0"  
    %       % Display the first 16 Zernike functions h!?7I=p~#  
    %       x = -1:0.01:1; $(H%|Oyn  
    %       [X,Y] = meshgrid(x,x); w7$*J:{  
    %       [theta,r] = cart2pol(X,Y); d_BECx <\  
    %       idx = r<=1; |G~LJsXW!v  
    %       p = 0:15; %g{m12  
    %       z = nan(size(X)); ^\Nsx)Y;  
    %       y = zernfun2(p,r(idx),theta(idx)); r`pg`ChHv  
    %       figure('Units','normalized') aB;syl{  
    %       for k = 1:length(p) `$\g8Mo  
    %           z(idx) = y(:,k); XMd-r8yYr  
    %           subplot(4,4,k) acz8 H 0cS  
    %           pcolor(x,x,z), shading interp O6NgI2[O  
    %           set(gca,'XTick',[],'YTick',[])  ~Jrtm7  
    %           axis square xAw$bJj~s  
    %           title(['Z_{' num2str(p(k)) '}']) 47ra`*  
    %       end U+F?b\  
    % Sg(fZ' -  
    %   See also ZERNPOL, ZERNFUN. Xi^3o  
    S H2|xn  
    %   Paul Fricker 11/13/2006 ;|6kFBGC"+  
    L}*:,&Y/  
    j-8v$ 0'  
    % Check and prepare the inputs: dR<sBYo  
    % ----------------------------- wN\%b}pp  
    if min(size(p))~=1 2`tdH|Z`  
        error('zernfun2:Pvector','Input P must be vector.') lG/M%i  
    end 2F[smUL  
    @,F8gv*  
    if any(p)>35 9>\P]:  
        error('zernfun2:P36', ... "Kx2k>ym  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... @.;] $N&J  
               '(P = 0 to 35).']) y.AVH`_u  
    end !'o5X]s  
    0)`{]&  
    % Get the order and frequency corresonding to the function number: [`nY /g:  
    % ---------------------------------------------------------------- o4,fwPkB  
    p = p(:); YjN2 ,Xi  
    n = ceil((-3+sqrt(9+8*p))/2); wYQTG*&h  
    m = 2*p - n.*(n+2); s+&Ts|c#  
    rVgz+'rFD[  
    % Pass the inputs to the function ZERNFUN: x%ju(B>  
    % ---------------------------------------- _-eF &D  
    switch nargin ~G|un}g=  
        case 3 99w;Q 2k  
            z = zernfun(n,m,r,theta); eL3HX _2(  
        case 4 -.-j e"E  
            z = zernfun(n,m,r,theta,nflag); YUU|!A8x  
        otherwise AuuZWd  
            error('zernfun2:nargin','Incorrect number of inputs.') P0NGjS|Z{  
    end 4A^=4"BCV  
    f_a.BTtNO  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) 4+%;eY.A  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. eAKK uML  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of vH/<!jtI  
    %   order N and frequency M, evaluated at R.  N is a vector of {* S8n09v  
    %   positive integers (including 0), and M is a vector with the eFDhJ  
    %   same number of elements as N.  Each element k of M must be a .J:04t1  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) y+xw`gR:  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is V^fSrW]  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix >^}nk04  
    %   with one column for every (N,M) pair, and one row for every gI8r SmH  
    %   element in R. ,,1H#;j  
    % "8MG[$Y  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- d'ddxT$GG  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is %"tLs%"7=P  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to 'PPVM@)fU  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 \l!^6G|c  
    %   for all [n,m]. 59K%bz5t  
    % 1,@-y#V_  
    %   The radial Zernike polynomials are the radial portion of the *V+6409m  
    %   Zernike functions, which are an orthogonal basis on the unit jO xH' 1I  
    %   circle.  The series representation of the radial Zernike YI05?J}  
    %   polynomials is YT7,=k_  
    % Sh'>5z2  
    %          (n-m)/2 C@+"d3  
    %            __ <[{Ty+  
    %    m      \       s                                          n-2s L4Ep7=  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r jR8~EI+  
    %    n      s=0 wq &|V  
    % lF-;h{   
    %   The following table shows the first 12 polynomials. !z@QoD  
    % o('W2Bs-o  
    %       n    m    Zernike polynomial    Normalization 8B ZTHlUB  
    %       --------------------------------------------- rl-r8?H}  
    %       0    0    1                        sqrt(2) U 7mA~t2E  
    %       1    1    r                           2 MgJiJ0y  
    %       2    0    2*r^2 - 1                sqrt(6) |jB]5ciT  
    %       2    2    r^2                      sqrt(6) bg.f';C  
    %       3    1    3*r^3 - 2*r              sqrt(8) tI50z khaB  
    %       3    3    r^3                      sqrt(8) 1Xy{&Ut\  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) :NB|r  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) hKq <e%oVH  
    %       4    4    r^4                      sqrt(10) q~*3Bk~  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) 9y=$ |"<(  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) Mazjn?f  
    %       5    5    r^5                      sqrt(12) BLx tS  
    %       --------------------------------------------- Z9-HQ5>  
    % zTi %j$o  
    %   Example: W[S4s/)mg  
    % tBEZ4 W>67  
    %       % Display three example Zernike radial polynomials >XzCHtEP  
    %       r = 0:0.01:1; 0Yz &aH  
    %       n = [3 2 5]; O8BxXa@5  
    %       m = [1 2 1]; 4^H(p  
    %       z = zernpol(n,m,r); ca>6r`  
    %       figure fdr.'aMf%  
    %       plot(r,z) `P;fD/I  
    %       grid on A#9@OWV5f  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') ydp?%RB3w  
    % I#(D.\P  
    %   See also ZERNFUN, ZERNFUN2. h%e}4U@X  
    a<+Qw'  
    % A note on the algorithm. _%WJ7~>  
    % ------------------------ 4>]^1J7Wz  
    % The radial Zernike polynomials are computed using the series ;)ff Gg>  
    % representation shown in the Help section above. For many special ;u;YfOr  
    % functions, direct evaluation using the series representation can |a@$KF$  
    % produce poor numerical results (floating point errors), because s=`1wkh0  
    % the summation often involves computing small differences between gE8=#%1<  
    % large successive terms in the series. (In such cases, the functions ahgm*Cpc  
    % are often evaluated using alternative methods such as recurrence }>:v  
    % relations: see the Legendre functions, for example). For the Zernike >T3HkOT  
    % polynomials, however, this problem does not arise, because the X$,#OR  
    % polynomials are evaluated over the finite domain r = (0,1), and /7Z0|Zw]  
    % because the coefficients for a given polynomial are generally all [~$Ji&Dd  
    % of similar magnitude. M ,.++W\  
    % ]/;0  
    % ZERNPOL has been written using a vectorized implementation: multiple hg7`jE&2  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] f:L%th  
    % values can be passed as inputs) for a vector of points R.  To achieve 42:~oKiQ$"  
    % this vectorization most efficiently, the algorithm in ZERNPOL vPuPSE%M  
    % involves pre-determining all the powers p of R that are required to =8OPj cX.V  
    % compute the outputs, and then compiling the {R^p} into a single .Ajs0 T2  
    % matrix.  This avoids any redundant computation of the R^p, and Nt42v  
    % minimizes the sizes of certain intermediate variables. 2d+IROA  
    % kQlXcR  
    %   Paul Fricker 11/13/2006 U+sAEN_e k  
    GRz`fO  
    N>;"r]Rl"  
    % Check and prepare the inputs: nIVPh99  
    % ----------------------------- #p~tkQ:'1  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) W&`_cGoP  
        error('zernpol:NMvectors','N and M must be vectors.') l= 5kd.{  
    end ?}^e,.M0?s  
    ]dk44,EL  
    if length(n)~=length(m) 2GECcx53  
        error('zernpol:NMlength','N and M must be the same length.') _QCspPT' c  
    end Q%4>okj,  
    vW6Pf^yJ  
    n = n(:); *1iJa  
    m = m(:); @;x|+@r  
    length_n = length(n); ]5D?Sc#-  
    NwM=  
    if any(mod(n-m,2)) M`HXUA4  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') EOhUr=5~  
    end oPRvd_~  
    B]]_rl,  
    if any(m<0) 5o\yhYS:  
        error('zernpol:Mpositive','All M must be positive.') M$A"<5  
    end ^.jIus5  
    YhFB*D;  
    if any(m>n) VR5$[-E3  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') y]eH@:MJ;A  
    end <jU[&~p  
    VkFTIyt  
    if any( r>1 | r<0 ) &O5%6Sv3d  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') Q)yhpwrX  
    end +jzpB*@  
    Y ~RPspHW  
    if ~any(size(r)==1) H?ssV^k  
        error('zernpol:Rvector','R must be a vector.') MdT'xYomzQ  
    end uc~PKU?tO  
    N8:?Z#z  
    r = r(:); mzTF2K  
    length_r = length(r); P:t|'t  
    f33'2PYl  
    if nargin==4 t`03$&Cx7  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); u*`acmS>N  
        if ~isnorm Wfkm'BnV  
            error('zernpol:normalization','Unrecognized normalization flag.') yyxGVfr  
        end 1eI >Yy>}  
    else ^Qz8`1`;Z  
        isnorm = false; 'R8VCj  
    end NZYtA7  
    3(%hHM7DM  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% sxJKu  
    % Compute the Zernike Polynomials \\ M2_mT  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ?qYw9XQYL  
    g*\u8fpRq  
    % Determine the required powers of r: vG#|CO9  
    % ----------------------------------- wlBdA  
    rpowers = []; 2fTkHBhn&  
    for j = 1:length(n) ~ C6< 75  
        rpowers = [rpowers m(j):2:n(j)]; UZMo(rG.]{  
    end qO[6?q=c:  
    rpowers = unique(rpowers); dz &| 3o  
    yAR''>  
    % Pre-compute the values of r raised to the required powers, U*, 8 ,C  
    % and compile them in a matrix: B`<(qPD  
    % ----------------------------- =PAvPj&}e  
    if rpowers(1)==0 E*>tFw&[  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); Ki}PO`s  
        rpowern = cat(2,rpowern{:}); bP&o] ?dN  
        rpowern = [ones(length_r,1) rpowern]; =|S8.|r+  
    else :2Qm*Y&_$V  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); -% PUY(  
        rpowern = cat(2,rpowern{:}); kmNY ;b6Y$  
    end Y }'C'PR  
    Tz+2g&+  
    % Compute the values of the polynomials: ?2OT:/I,  
    % -------------------------------------- 4z Af|Je  
    z = zeros(length_r,length_n); "2+>!G RQ  
    for j = 1:length_n V\%;S  
        s = 0:(n(j)-m(j))/2; `da6}Vqj:  
        pows = n(j):-2:m(j); ?(>7v[=iT  
        for k = length(s):-1:1 !tv3.:eT  
            p = (1-2*mod(s(k),2))* ... 2:MB u5**  
                       prod(2:(n(j)-s(k)))/          ... YTQ|Hg6jO  
                       prod(2:s(k))/                 ... s ,\w00-:  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... X_+`7yCi"x  
                       prod(2:((n(j)+m(j))/2-s(k))); mJ<rzX  
            idx = (pows(k)==rpowers); NwD*EuPF:  
            z(:,j) = z(:,j) + p*rpowern(:,idx); Hx2UDHF  
        end oh%T4 $  
         !Qjpj KRy  
        if isnorm "v/^nH  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1));  kOETx  
        end 7'7bIaJk  
    end  Us k@{  
    U# Y ?'3:  
    % 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)  U2ecvq[T  
    @=K> uyB  
    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。 mz+>rc  
    TqKL(Qw E  
    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)