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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 5a(<%Q <"  
    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! eO{@@?/y  
     
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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 Ea4 * o  
    function z = zernfun(n,m,r,theta,nflag) 4p x_ZD#J  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. -]QguZE  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N k6J\Kkk(  
    %   and angular frequency M, evaluated at positions (R,THETA) on the y#bK,}  
    %   unit circle.  N is a vector of positive integers (including 0), and {{E jMBg{  
    %   M is a vector with the same number of elements as N.  Each element 3G&0Ciet  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) ?48AY6  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, "=ElCaP}  
    %   and THETA is a vector of angles.  R and THETA must have the same l7Y8b`  
    %   length.  The output Z is a matrix with one column for every (N,M) t{=i=K 3  
    %   pair, and one row for every (R,THETA) pair. O3+)qb!X  
    % P/`m3aSzX.  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike c `ud;lI  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), fqX~xp  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral &9@gm--b:  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, !u%9;>T7  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized a hwy_\  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. &GU@8  
    % (0g7-Ci  
    %   The Zernike functions are an orthogonal basis on the unit circle. K>LpN')d  
    %   They are used in disciplines such as astronomy, optics, and ^vn\4  
    %   optometry to describe functions on a circular domain. ?C~X@sq  
    % nF j-<!  
    %   The following table lists the first 15 Zernike functions. mo*'"/  
    % *pK lA&_  
    %       n    m    Zernike function           Normalization ?k::tNv0  
    %       -------------------------------------------------- T\cR2ZT~  
    %       0    0    1                                 1 TC@bL<1  
    %       1    1    r * cos(theta)                    2 wlL8X7+:  
    %       1   -1    r * sin(theta)                    2 S"{GlRpd  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) N Z)b:~a  
    %       2    0    (2*r^2 - 1)                    sqrt(3) dQ`:8S K  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) [%t3[p<)O  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) u6p5:oJj,  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) )"F5lOA6  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) wH#-mu#Yl<  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) " SLvUzO>q  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) nIR*_<ow  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) iz$FcA]  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) a(5y>HF  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) v#-E~;C cC  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) $mD>r x  
    %       -------------------------------------------------- 2pjW,I!`  
    % m'SmN{(t  
    %   Example 1: QS5H >5M)  
    % \.kTe<.:_  
    %       % Display the Zernike function Z(n=5,m=1) pY, O_ t$  
    %       x = -1:0.01:1; 2H3(HZv  
    %       [X,Y] = meshgrid(x,x); +!Q!m 3/I  
    %       [theta,r] = cart2pol(X,Y); Gxo# !  
    %       idx = r<=1;  A<2I!  
    %       z = nan(size(X)); 2DUr7r M  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); ;hkro$  
    %       figure Ogd8!'\  
    %       pcolor(x,x,z), shading interp l`G(O$ct  
    %       axis square, colorbar X:R%1+&*  
    %       title('Zernike function Z_5^1(r,\theta)') u\:rY)V  
    % *$JB`=Q  
    %   Example 2: pK<%<dIc  
    % ^g-Fg>&M  
    %       % Display the first 10 Zernike functions T7^ulG1'  
    %       x = -1:0.01:1; D9,e3.?p  
    %       [X,Y] = meshgrid(x,x); K q/~T7Ru  
    %       [theta,r] = cart2pol(X,Y); _IC,9bbg  
    %       idx = r<=1; ([[)Ub$U  
    %       z = nan(size(X)); !8we8)7  
    %       n = [0  1  1  2  2  2  3  3  3  3]; 8g.AT@ ,Q  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; Is<x31R  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; ;x,+*%  
    %       y = zernfun(n,m,r(idx),theta(idx)); 0GS{F8f~,  
    %       figure('Units','normalized') 692Rw}/  
    %       for k = 1:10 vJ~4D*(]l  
    %           z(idx) = y(:,k); 2ve lH;  
    %           subplot(4,7,Nplot(k)) \y[Bu^tk  
    %           pcolor(x,x,z), shading interp uXjoGcW  
    %           set(gca,'XTick',[],'YTick',[]) T=^jCH &  
    %           axis square L7s>su|c(  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) :m]/u( /N  
    %       end 2]2{&bu  
    % LjSLg[i  
    %   See also ZERNPOL, ZERNFUN2. qo)Q}0  
    _yiR h:  
    %   Paul Fricker 11/13/2006 ht2 f-EKf{  
    Qk+=znJ  
    j.<:00<  
    % Check and prepare the inputs: (D0C#<4P  
    % ----------------------------- w'!ECm>*`  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) u82h6s<'W  
        error('zernfun:NMvectors','N and M must be vectors.') f3/SO+Me}  
    end o<Xc,mP  
    W+8BQ- 2  
    if length(n)~=length(m) xOwNCh  
        error('zernfun:NMlength','N and M must be the same length.') lr-12-D%-  
    end TNyK@~#m  
    qUifw @  
    n = n(:); fL(':W&n-  
    m = m(:); v&p,Clt-2  
    if any(mod(n-m,2)) P#w}3^  
        error('zernfun:NMmultiplesof2', ... &7$,<9.  
              'All N and M must differ by multiples of 2 (including 0).') ;RNM   
    end f-vZ2+HP  
    8$2l^  
    if any(m>n) $dlnmNP+  
        error('zernfun:MlessthanN', ... \_lG#p|  
              'Each M must be less than or equal to its corresponding N.') 7bA4P*  
    end >]:N?[Y_~}  
    $?[1#%  
    if any( r>1 | r<0 ) TTeAa  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') X!,#'&p&  
    end 30A`\+^f  
    =$^Wkau  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 0|.7Kz^  
        error('zernfun:RTHvector','R and THETA must be vectors.') Aqa6R+c  
    end A)hq0FPp  
    C$$"{FfgU"  
    r = r(:); ,:v.L}+Z  
    theta = theta(:); 0 $n8b/%.  
    length_r = length(r); T r SN00  
    if length_r~=length(theta) Zx}N Fcn  
        error('zernfun:RTHlength', ... 9=Y,["br$_  
              'The number of R- and THETA-values must be equal.') (:_%kmu  
    end jHs<s`#h  
    @o}1n?w  
    % Check normalization: aEcktg6h  
    % -------------------- +CsI,Uf4*  
    if nargin==5 && ischar(nflag) ,"P5D&,_  
        isnorm = strcmpi(nflag,'norm'); *Gg1h@&  
        if ~isnorm KU1+<OCh  
            error('zernfun:normalization','Unrecognized normalization flag.') zkjPLeX  
        end a9yIV5_N  
    else GCcwEl!K^  
        isnorm = false; ?R|fS*e2EB  
    end X)`(nj  
    |HaU3E*R  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 4 ?9soc  
    % Compute the Zernike Polynomials *4(/t$)pEl  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ^/_\etV  
    r!{w93rPX  
    % Determine the required powers of r: 9F2w.(m  
    % ----------------------------------- X@6zI-Y %  
    m_abs = abs(m); {toyQ)C7  
    rpowers = []; el <<D  
    for j = 1:length(n) /2g)Z!&+L  
        rpowers = [rpowers m_abs(j):2:n(j)]; Ft3N#!ubl  
    end tb-OKZq  
    rpowers = unique(rpowers); Q3B'-BZe  
    j[$B\H  
    % Pre-compute the values of r raised to the required powers, Z:\;R{D  
    % and compile them in a matrix: ^>,< *p  
    % ----------------------------- .nj?;).  
    if rpowers(1)==0 y)?W-5zL  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); kWZ/ej  
        rpowern = cat(2,rpowern{:}); * kX3sG$8  
        rpowern = [ones(length_r,1) rpowern]; GN htnB  
    else <.PPs:{8#  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 7h9fQ&y  
        rpowern = cat(2,rpowern{:}); ) xfc-Q  
    end ]C}u- B746  
    E3CiZ4=5  
    % Compute the values of the polynomials: xG*lV|<7>  
    % -------------------------------------- W (=Wg|cr  
    y = zeros(length_r,length(n)); 6e$sA (a=i  
    for j = 1:length(n) uPv;y!Lsa@  
        s = 0:(n(j)-m_abs(j))/2; 3b g4#c  
        pows = n(j):-2:m_abs(j); 'k-u9  
        for k = length(s):-1:1 !wLH&X$XT  
            p = (1-2*mod(s(k),2))* ... mV:RmA  
                       prod(2:(n(j)-s(k)))/              ... 7]M,yIwc  
                       prod(2:s(k))/                     ... <F#*:Re_y  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... Sy+]SeF&  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); <B ]i80.  
            idx = (pows(k)==rpowers); /%ODJ1M  
            y(:,j) = y(:,j) + p*rpowern(:,idx); } #\;np  
        end U}RS*7`  
         WuP([8  
        if isnorm e' /  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 0@sr NuW  
        end t_dw}I   
    end 9Sx<tj_4P{  
    % END: Compute the Zernike Polynomials rj2r#{[  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% #q~3c;ec  
    9I`Mm}v@  
    % Compute the Zernike functions: xY\*L:TwW  
    % ------------------------------ E]u'MX  
    idx_pos = m>0; gCk y(4  
    idx_neg = m<0; dbMu6Bm\G  
    !_XU^A>  
    z = y; F9u:8;\@`  
    if any(idx_pos) zulf%aaL  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ;G%wc!  
    end 9z| >roNe  
    if any(idx_neg) {0A[v}X ~  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); D_yY0rRM  
    end /+<%,c$n  
    :]u}x Dv3  
    % 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)WS}&  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. w}U5dM`  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated (v'lb!j^#  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive H%01&u  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, vHI"C %  
    %   and THETA is a vector of angles.  R and THETA must have the same d5sGkR`(  
    %   length.  The output Z is a matrix with one column for every P-value, !0. 5  
    %   and one row for every (R,THETA) pair. ?(,5eg  
    % $@u^Jt, ?  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike f1]AfH#  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) XY+aunLf  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) N}l]Ilm$34  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 xPfnyAo?%z  
    %   for all p. l9ifUh e  
    % n6%jhv9H  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 3f:1D=f  
    %   Zernike functions (order N<=7).  In some disciplines it is ];b!*Z  
    %   traditional to label the first 36 functions using a single mode t0asW5f  
    %   number P instead of separate numbers for the order N and azimuthal <SC|A|  
    %   frequency M. F'5d\v  
    % 7>2j=Y_Kp  
    %   Example: MR)KLM0  
    % $ vw}p.  
    %       % Display the first 16 Zernike functions XJxs4a1[t  
    %       x = -1:0.01:1; /_k hFw  
    %       [X,Y] = meshgrid(x,x); /[0 /8f6  
    %       [theta,r] = cart2pol(X,Y); !(ux.T0  
    %       idx = r<=1; rt@-Pw!B  
    %       p = 0:15; @rK>yPhf  
    %       z = nan(size(X)); "mP*}VF  
    %       y = zernfun2(p,r(idx),theta(idx)); e}Af"LI  
    %       figure('Units','normalized') Pu%>j'A  
    %       for k = 1:length(p) $MJDB  
    %           z(idx) = y(:,k); @Pb 1QLiz  
    %           subplot(4,4,k) mk`cyN>m  
    %           pcolor(x,x,z), shading interp P{i8  
    %           set(gca,'XTick',[],'YTick',[]) L$u&~"z-  
    %           axis square 7]. IT(  
    %           title(['Z_{' num2str(p(k)) '}']) MY<!\4/  
    %       end }'.Sn{OWf  
    % -{:Lx E  
    %   See also ZERNPOL, ZERNFUN. _[OEE<(  
    6dS1\Y  
    %   Paul Fricker 11/13/2006 }PIGj}F/  
    w7t"&=pF7  
    ?9r,Y;,H  
    % Check and prepare the inputs: 3~3(G[w  
    % ----------------------------- &v9PT!R~  
    if min(size(p))~=1 m/F(h-?  
        error('zernfun2:Pvector','Input P must be vector.') #L4Kwy  
    end 9uq+Ve>  
    /DG`Hg  
    if any(p)>35 Tk#&Ux{ZJ  
        error('zernfun2:P36', ... '3_B1iAv  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... mKFHT  
               '(P = 0 to 35).']) =3v 1]7 X  
    end hPXVPLm7I  
    p|9ECdU>;  
    % Get the order and frequency corresonding to the function number: vzrD"  
    % ---------------------------------------------------------------- :qSi>KCGh  
    p = p(:); ^Lv ^W  
    n = ceil((-3+sqrt(9+8*p))/2); 4FHX#`  
    m = 2*p - n.*(n+2); D)~nAkVq  
    ) Q  
    % Pass the inputs to the function ZERNFUN: > %cWTC  
    % ---------------------------------------- WWs[]zr  
    switch nargin 6!Ji>h.Ak  
        case 3 .' 2gJ"?,  
            z = zernfun(n,m,r,theta); ;!Z7-OZX  
        case 4 - 2na::<K  
            z = zernfun(n,m,r,theta,nflag); NF\^'W@N  
        otherwise |XdrO  
            error('zernfun2:nargin','Incorrect number of inputs.') & Dl'*|  
    end cLko  
    ^xNe Eb  
    % 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) 2#A9D.- h  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. @P5@ &G  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of 7(USp#"  
    %   order N and frequency M, evaluated at R.  N is a vector of {2*l :'  
    %   positive integers (including 0), and M is a vector with the Eno2<<  
    %   same number of elements as N.  Each element k of M must be a Q2pboZ86  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) u{nWjqrM*5  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is XoQk'7"f  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix Vh9s.=*P@  
    %   with one column for every (N,M) pair, and one row for every #`)-$vUv^f  
    %   element in R. `k%#0E*H  
    % Qufv@.'AY  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- S9#N%{8P  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is 3pjYY$'  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to BRzWZq%r3  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 qg:I+"u  
    %   for all [n,m]. Y~SlipY_  
    % ${6'  
    %   The radial Zernike polynomials are the radial portion of the ;)pV[3[  
    %   Zernike functions, which are an orthogonal basis on the unit }' 0Xz9/ l  
    %   circle.  The series representation of the radial Zernike ~Q 9)Q  
    %   polynomials is XoiYtx53  
    % $AhX@|?z  
    %          (n-m)/2 k^\ &.63(  
    %            __ $yqq.#1  
    %    m      \       s                                          n-2s QuRg(K%:  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r ` +UMZc  
    %    n      s=0 :#htOsP  
    % Vd>.fb\U2  
    %   The following table shows the first 12 polynomials. eo52X &I  
    % p[cL# fBz  
    %       n    m    Zernike polynomial    Normalization yrNc[kS/  
    %       --------------------------------------------- zXU{p\;)\  
    %       0    0    1                        sqrt(2) ;fME4Sp  
    %       1    1    r                           2 s1$#G!'  
    %       2    0    2*r^2 - 1                sqrt(6) YaT+BRh?  
    %       2    2    r^2                      sqrt(6) <$2zr4  
    %       3    1    3*r^3 - 2*r              sqrt(8) zd4y5/aoS  
    %       3    3    r^3                      sqrt(8) Z?aR9OTP  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) 6|qvo+%  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) $#W6z:  
    %       4    4    r^4                      sqrt(10) =\7p0cq&*  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) CWsv#XOg]  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) g*.(! !  
    %       5    5    r^5                      sqrt(12) _rVX_   
    %       --------------------------------------------- R eu J=|F  
    % D % ,yA  
    %   Example: ?JTyNg4<  
    % Y]Vc}-a(h  
    %       % Display three example Zernike radial polynomials J15T!_AW<  
    %       r = 0:0.01:1; 8"UG&wLT  
    %       n = [3 2 5]; xxcDd_z  
    %       m = [1 2 1]; Vy~$%H94  
    %       z = zernpol(n,m,r); 5(`GF|  
    %       figure +p6\R;_E  
    %       plot(r,z) `0sk2fn  
    %       grid on 7[0k5-  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') rh&Eu qE%  
    % Z9~Wlt'?  
    %   See also ZERNFUN, ZERNFUN2. )nxIxr0d-  
    2e@\6l,!^  
    % A note on the algorithm. (w B[ ]O$@  
    % ------------------------ eaP$/U D?  
    % The radial Zernike polynomials are computed using the series <X&:tZ #/  
    % representation shown in the Help section above. For many special Fe< t@W  
    % functions, direct evaluation using the series representation can =,G(1#  
    % produce poor numerical results (floating point errors), because u$p|hd d  
    % the summation often involves computing small differences between ^O*hs%eO%  
    % large successive terms in the series. (In such cases, the functions # h|< >  
    % are often evaluated using alternative methods such as recurrence Ayt!a+J  
    % relations: see the Legendre functions, for example). For the Zernike :.df(1(RL  
    % polynomials, however, this problem does not arise, because the x#mk[SV  
    % polynomials are evaluated over the finite domain r = (0,1), and Vho0f<`E  
    % because the coefficients for a given polynomial are generally all {5-{f=Rk  
    % of similar magnitude. yS3s5C{C  
    % *sVxjZvV  
    % ZERNPOL has been written using a vectorized implementation: multiple h l'k_<a*  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] ~0Q72  
    % values can be passed as inputs) for a vector of points R.  To achieve 6WM_V9Tidq  
    % this vectorization most efficiently, the algorithm in ZERNPOL 7N=VVD~!b  
    % involves pre-determining all the powers p of R that are required to .cog9H'  
    % compute the outputs, and then compiling the {R^p} into a single }"H900WE|  
    % matrix.  This avoids any redundant computation of the R^p, and o e"ShhT  
    % minimizes the sizes of certain intermediate variables. @?? 6)C  
    % Mg/2 w  
    %   Paul Fricker 11/13/2006 y36aoKH  
    ofCP>Z-  
    ur7a%NH  
    % Check and prepare the inputs: x:lf=D lA  
    % ----------------------------- RE$-{i  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) (oEA)yc|  
        error('zernpol:NMvectors','N and M must be vectors.') Te#[+B?  
    end Zotv]P2k  
    >[=`{B  
    if length(n)~=length(m) 7z&u92dJI  
        error('zernpol:NMlength','N and M must be the same length.') (@ sKE  
    end uB5o Ghu-  
    1bs95Fh9Q  
    n = n(:); -bfd><bs  
    m = m(:); -;z\BW5 y  
    length_n = length(n); _nq n|  
    R.;59s  
    if any(mod(n-m,2)) )6WU&0>AU8  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') Big-)7?  
    end ^o:5B%}#[  
    t$iU|^'uV  
    if any(m<0) M - TK  
        error('zernpol:Mpositive','All M must be positive.') CP^^ct-C  
    end Wfy+7$14M  
    3I(H.u  
    if any(m>n) c[q3O**  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') wE2?/wb  
    end Ta$<#wb  
    -+/|  
    if any( r>1 | r<0 ) g'E^@1{  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') 2Y{9Df  
    end 5S bSz!s`$  
    i.&16AY  
    if ~any(size(r)==1) E;0"1 P|S  
        error('zernpol:Rvector','R must be a vector.') C?k4<B7V  
    end 7lu;lAAP  
    u}_q'=<\  
    r = r(:); a8TE  
    length_r = length(r); [MG:Ym).2`  
    4V5*6O9(u  
    if nargin==4 q9^r2OO  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); 3kg+*]tLx  
        if ~isnorm 3{z }[@N  
            error('zernpol:normalization','Unrecognized normalization flag.') 3:S"!F  
        end mi?Fy0\  
    else bfgLU.1I  
        isnorm = false; 1>1!oml1E  
    end ?jnEHn  
    ;H.r6  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% de[_T%A  
    % Compute the Zernike Polynomials w:Vs$,  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% O"GzeEY7  
    WJWhx4Hk  
    % Determine the required powers of r: Lm/^ 8V+  
    % ----------------------------------- Ff30%  
    rpowers = []; zi'?FM[f)  
    for j = 1:length(n) 0vEa]ljS  
        rpowers = [rpowers m(j):2:n(j)]; j*nCIxF  
    end }Na*jr0y9{  
    rpowers = unique(rpowers); 3:RZ@~u=  
    E#OKeMK  
    % Pre-compute the values of r raised to the required powers, (RF6K6~  
    % and compile them in a matrix: }T6jQ:?@  
    % ----------------------------- .KV?;{~q@  
    if rpowers(1)==0 @<+(40`*  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); bbnAmZ   
        rpowern = cat(2,rpowern{:}); 1'=brc YR  
        rpowern = [ones(length_r,1) rpowern]; y g7z?AZ  
    else v,s]:9f`\>  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); %pdfGM 9g  
        rpowern = cat(2,rpowern{:}); SVJ3!1B,  
    end LQ._?35r  
    k5 8lmuU  
    % Compute the values of the polynomials: 3e"G.0vJ  
    % -------------------------------------- Xc.~6nYp  
    z = zeros(length_r,length_n); 6Dch+*4*@  
    for j = 1:length_n %PQC9{hUy$  
        s = 0:(n(j)-m(j))/2; /#HY-b  
        pows = n(j):-2:m(j); "dt3peH  
        for k = length(s):-1:1 /3fo=7G6  
            p = (1-2*mod(s(k),2))* ... nt7ui*k  
                       prod(2:(n(j)-s(k)))/          ... DF#Ob( 1  
                       prod(2:s(k))/                 ... +r '  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... ?tBEB5  
                       prod(2:((n(j)+m(j))/2-s(k))); .w`8_v&Y  
            idx = (pows(k)==rpowers); 7G xNI  
            z(:,j) = z(:,j) + p*rpowern(:,idx); dM)fr  
        end Yq.Omr!  
         nBQG.3  
        if isnorm ~ohW9Z1  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); A ="h}9ok  
        end >q}EZC  
    end E~24b0<7  
    St?vd+(>  
    % 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)  zVFz}kJa  
    i!@L`h!rw  
    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。 Tl6%z9rY@  
    t 5g@t0$  
    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)