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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 I+/fX0-Lib  
    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! Nj{;  
     
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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 v]#[bqB.b  
    function z = zernfun(n,m,r,theta,nflag) n~ZZX={a  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. qERJEyU?  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N n#sK31;yb  
    %   and angular frequency M, evaluated at positions (R,THETA) on the =7[}:haB{  
    %   unit circle.  N is a vector of positive integers (including 0), and cRE6/qrXGg  
    %   M is a vector with the same number of elements as N.  Each element S9[Y1qH>K  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) NA$%Up  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, R$`%<Y3)  
    %   and THETA is a vector of angles.  R and THETA must have the same &eb8k2S  
    %   length.  The output Z is a matrix with one column for every (N,M) `A#0If  
    %   pair, and one row for every (R,THETA) pair. %,S{9q  
    % vSR5F9  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike {Ve3EYYm  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), yqH9*&KH{  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral UW1i%u k  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, RL[F 9g  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized ~14|y|\/  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. y&/bp<Z  
    % <zm:J4&>T  
    %   The Zernike functions are an orthogonal basis on the unit circle. qHvU4v  
    %   They are used in disciplines such as astronomy, optics, and cG&@PO]+.  
    %   optometry to describe functions on a circular domain. z<%dWz  
    % G#ELQ/Q  
    %   The following table lists the first 15 Zernike functions. !ST7@D  
    % (*kKfg4Wj  
    %       n    m    Zernike function           Normalization G'`^U}9V\  
    %       -------------------------------------------------- 7yjun|Lt}X  
    %       0    0    1                                 1 4C )sjk?m  
    %       1    1    r * cos(theta)                    2 8@b`a]lgrd  
    %       1   -1    r * sin(theta)                    2 hiv {A9a?  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) iRx`Nx<@  
    %       2    0    (2*r^2 - 1)                    sqrt(3) ttls.~DG  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) -3 Sb%V\  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) &DjA?0`J  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) U2LD_-HZ  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) ;GKL[ tI"  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) O{\%{XrW  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) FzykC  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) vz)R84   
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) ?op;#/Q(  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) W)'*Dcd  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) e.^?hwl  
    %       -------------------------------------------------- #^yOW^  
    % =[zP  
    %   Example 1: WX]O1Y  
    % e tL?UF$  
    %       % Display the Zernike function Z(n=5,m=1) p+5J  
    %       x = -1:0.01:1; vvs2:87zvJ  
    %       [X,Y] = meshgrid(x,x); $j8CF3d.6  
    %       [theta,r] = cart2pol(X,Y); 5<e{)$C  
    %       idx = r<=1; YWJ$Pp  
    %       z = nan(size(X)); @^DVA}*b)  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); a4 7e  
    %       figure 22;B:  
    %       pcolor(x,x,z), shading interp [LQOP3f  
    %       axis square, colorbar ;Qi!~VsP;  
    %       title('Zernike function Z_5^1(r,\theta)') cucmn*o?  
    % ?JTTl;  
    %   Example 2: 1GIBqs~-  
    % 2h#.:!/SMw  
    %       % Display the first 10 Zernike functions \B:k|Pw6~  
    %       x = -1:0.01:1; &,3s2,1U(  
    %       [X,Y] = meshgrid(x,x); mU  
    %       [theta,r] = cart2pol(X,Y); m`):= ^nC  
    %       idx = r<=1; oRJ!TAbD  
    %       z = nan(size(X)); 'Z:wEt!  
    %       n = [0  1  1  2  2  2  3  3  3  3]; o4OB xHKy  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; 2(x| %  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; w^=(:`  
    %       y = zernfun(n,m,r(idx),theta(idx)); f$9|qfW'$  
    %       figure('Units','normalized') *B \ @L  
    %       for k = 1:10 3,`M\#z%K  
    %           z(idx) = y(:,k); TvS<;0~K  
    %           subplot(4,7,Nplot(k)) >56fa6=3@  
    %           pcolor(x,x,z), shading interp wt;`_}g  
    %           set(gca,'XTick',[],'YTick',[]) q`.=/O'  
    %           axis square d[5v A/8O  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) mq:WBSsV  
    %       end '9zKaL  
    % ~kj96w4eAR  
    %   See also ZERNPOL, ZERNFUN2. {:b~^yW  
    /Oi(5?Jn  
    %   Paul Fricker 11/13/2006 ; yE.R[I  
    Ihr[44#  
    wnK6jMjkSf  
    % Check and prepare the inputs: "FhC"}N  
    % ----------------------------- z@o6[g/*Q  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) *M*WjEOA  
        error('zernfun:NMvectors','N and M must be vectors.') F6{/iF  
    end ,grx'to(X  
    Q+wO\TtE  
    if length(n)~=length(m) J] w3iYK  
        error('zernfun:NMlength','N and M must be the same length.') T8)X?>CIW  
    end mdQe)>  
    a7uL {*ZR  
    n = n(:); `IJ)'$pn  
    m = m(:); ya5HAs  
    if any(mod(n-m,2)) Yk)fBPHr  
        error('zernfun:NMmultiplesof2', ... MxUbx+_N  
              'All N and M must differ by multiples of 2 (including 0).') yPe9KN_  
    end 2{Dnfl'k  
    BOR$R}q  
    if any(m>n) ;DhAw1  
        error('zernfun:MlessthanN', ... B0A y  
              'Each M must be less than or equal to its corresponding N.') fAz4>_4  
    end E.sZjo1  
    cH$( *k9%M  
    if any( r>1 | r<0 ) #H<}xC2  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') J]zhwM  
    end e=p_qhBt  
    tZm`(2S  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) zDEgC  
        error('zernfun:RTHvector','R and THETA must be vectors.') \ykA7Y%  
    end n7Bv~?DM  
    isy[RAP<  
    r = r(:); Gc*=n*@^K  
    theta = theta(:); !fd>wvJ,:  
    length_r = length(r); Y2tBFeWY  
    if length_r~=length(theta) p:$kX9mT&  
        error('zernfun:RTHlength', ... #8 ^b]  
              'The number of R- and THETA-values must be equal.') <gGO  
    end ?b'(39fj  
    f*88k='\W  
    % Check normalization: z_'!?K{  
    % -------------------- [{R>'~  
    if nargin==5 && ischar(nflag) 5} <OB-9  
        isnorm = strcmpi(nflag,'norm'); =8TBkxG  
        if ~isnorm k%\y,b*  
            error('zernfun:normalization','Unrecognized normalization flag.') J%B/(v`  
        end JUj.:n2e  
    else ^!i4d))  
        isnorm = false; i `p1e5$  
    end BB9eQ: xO  
      )*6  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% g:xg ~H2  
    % Compute the Zernike Polynomials 5-k gGOt  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0%;| B  
    *?C8,;=2r  
    % Determine the required powers of r: .@EzHe ^W  
    % ----------------------------------- |+JO]J#bc  
    m_abs = abs(m); J7oj@Or9  
    rpowers = []; Zn40NKYc  
    for j = 1:length(n) F7w\ctUP  
        rpowers = [rpowers m_abs(j):2:n(j)]; n9 FA` e  
    end ^_ V0irv  
    rpowers = unique(rpowers); WBJn1  
    H^`J(J+  
    % Pre-compute the values of r raised to the required powers, U(x$&um(l  
    % and compile them in a matrix: Wd(|w8J{a  
    % ----------------------------- 8 $H\b &u  
    if rpowers(1)==0 [+CFQf>  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 3D5adI<aq"  
        rpowern = cat(2,rpowern{:}); bA$ElKT  
        rpowern = [ones(length_r,1) rpowern]; tn _\E/Q  
    else =B'Yx  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); Q%!xw(  
        rpowern = cat(2,rpowern{:}); s!yD%zO  
    end  5T9[a  
    9-&@Y  
    % Compute the values of the polynomials: W>Pcj EI  
    % -------------------------------------- F3$8l[O_  
    y = zeros(length_r,length(n)); K.&6c,P]  
    for j = 1:length(n) 'Z,7{U1P  
        s = 0:(n(j)-m_abs(j))/2; `*yOc6i]  
        pows = n(j):-2:m_abs(j); yLnTIE3)  
        for k = length(s):-1:1 g2}aEfp!H  
            p = (1-2*mod(s(k),2))* ... WLh!L='{BK  
                       prod(2:(n(j)-s(k)))/              ... 8@rF~^-_  
                       prod(2:s(k))/                     ... 3m21n7F4*  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ){u# (sW  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); FEopNDy@y  
            idx = (pows(k)==rpowers); -`Zk`s|!  
            y(:,j) = y(:,j) + p*rpowern(:,idx); k%-UW%  
        end 3BLH d<  
         =z<sx2#*  
        if isnorm #a9R3-aP  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); eYjF"Aq  
        end RLLL=?W@  
    end (r'NB  
    % END: Compute the Zernike Polynomials N>P" $  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% p&`I#6{  
    H.l WHM+H4  
    % Compute the Zernike functions: nSZp,?^  
    % ------------------------------ [{T/2IGq  
    idx_pos = m>0; ~j!|(a7  
    idx_neg = m<0; IsFL"Vx  
    QZO<'q`L  
    z = y; L+lye Ir'  
    if any(idx_pos) K&=6DvfR  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); M3GFKWQI,`  
    end $SniQ  
    if any(idx_neg) i !SN"SY  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ^;\6ju2  
    end rXe+#`m2  
    4`r-*Lx  
    % 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) @Hp=xC9V  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. Ye  >+  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated J+hifO  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive (1Jc-`  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, . ve a[  
    %   and THETA is a vector of angles.  R and THETA must have the same BT5~MYBl  
    %   length.  The output Z is a matrix with one column for every P-value, |B),N f|a  
    %   and one row for every (R,THETA) pair. $')Uie<!8  
    % 5Ak>/QF9  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike &23t/`   
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) *NI hYg6  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) "+4r4  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 6" . v6  
    %   for all p. 9v}vCg  
    % -$D#u  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 <bBgevL+_K  
    %   Zernike functions (order N<=7).  In some disciplines it is ;, u7)  
    %   traditional to label the first 36 functions using a single mode $I\lJ8  
    %   number P instead of separate numbers for the order N and azimuthal DJRr  
    %   frequency M. `{J(S'a`  
    % t;[?Q\  
    %   Example: (i^<er q  
    % &+pp;1ls  
    %       % Display the first 16 Zernike functions `S=4cSH(  
    %       x = -1:0.01:1; 7)Bizlf  
    %       [X,Y] = meshgrid(x,x); Yp9%u9tNq  
    %       [theta,r] = cart2pol(X,Y); 7{ QjE  
    %       idx = r<=1; ogE|8`Tq^  
    %       p = 0:15; t~]tw  
    %       z = nan(size(X)); -/6Ms%O  
    %       y = zernfun2(p,r(idx),theta(idx)); (R{z3[/u&  
    %       figure('Units','normalized') NUX2{8gs  
    %       for k = 1:length(p) <d3N2  
    %           z(idx) = y(:,k); 9 J~KM=p  
    %           subplot(4,4,k) HwZ@T &_4  
    %           pcolor(x,x,z), shading interp %0eVm   
    %           set(gca,'XTick',[],'YTick',[]) KWT[b?  
    %           axis square }cI _$  
    %           title(['Z_{' num2str(p(k)) '}']) 6 Zv~c(   
    %       end YoRD9M~iG~  
    % D?? \H\  
    %   See also ZERNPOL, ZERNFUN. f1/i f:~6  
    +}!FP3KgT  
    %   Paul Fricker 11/13/2006 C6}`qD  
    d0 yZ9-t  
    0]t7(P"F6  
    % Check and prepare the inputs: K9euNa  
    % ----------------------------- +WFa4NZ  
    if min(size(p))~=1 n'Z5rXg  
        error('zernfun2:Pvector','Input P must be vector.') )'t&LWS~  
    end P;l D ri  
    =:v5` :  
    if any(p)>35 C]%}L%,  
        error('zernfun2:P36', ... $PKUcT0N9  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... hc5iIJ]  
               '(P = 0 to 35).']) G!m;J8#m(  
    end *Y9'tHI  
    L)/^%/!  
    % Get the order and frequency corresonding to the function number: >WW5;7$  
    % ---------------------------------------------------------------- 83YQ c  
    p = p(:); [5jXYqD=vj  
    n = ceil((-3+sqrt(9+8*p))/2); g 2&P  
    m = 2*p - n.*(n+2); hvU\l`m  
    Qx}hiv/  
    % Pass the inputs to the function ZERNFUN: &+F}$8,  
    % ---------------------------------------- u1i ?L'  
    switch nargin ,zH\&D$>u  
        case 3 's6hCs&|NV  
            z = zernfun(n,m,r,theta); _^u^@.Q'i<  
        case 4 Y^J/jA0\B  
            z = zernfun(n,m,r,theta,nflag); W&Gt^5  
        otherwise dRnO5 7+{  
            error('zernfun2:nargin','Incorrect number of inputs.') \jThbCb  
    end 91j.%#[v'  
    !3Me 6&$O  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) YO^iEI.  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. |F^h >^ x  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of GjvTYg~  
    %   order N and frequency M, evaluated at R.  N is a vector of LS4|$X4H`!  
    %   positive integers (including 0), and M is a vector with the -z$&lP]  
    %   same number of elements as N.  Each element k of M must be a 0I@Cx {$  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) JPfE`NZ  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is 8 |iMD1  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix 0vm}[a4+i;  
    %   with one column for every (N,M) pair, and one row for every X-=J7G`\h#  
    %   element in R. QHuh=7u)  
    % ^L;k  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- v"a.%" oN8  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is _ 0Ced&i  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to oc3}L^aD  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 3teanU`  
    %   for all [n,m]. =C.WM*='  
    % L=WB'*N  
    %   The radial Zernike polynomials are the radial portion of the koAM",5D  
    %   Zernike functions, which are an orthogonal basis on the unit fnm:Wa|,%|  
    %   circle.  The series representation of the radial Zernike LQrm/)4bF5  
    %   polynomials is '+{dr\nJ  
    % E)7ODRVbl  
    %          (n-m)/2 :},/ D*v  
    %            __ rCa2$#Z  
    %    m      \       s                                          n-2s k|c=O6GO  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r S0<m><|kl  
    %    n      s=0 Z6vm!#\  
    % 4C{3>BE  
    %   The following table shows the first 12 polynomials. ^9C9[$Q  
    % Y2,\WKa  
    %       n    m    Zernike polynomial    Normalization ep3iI77/  
    %       --------------------------------------------- L7lRh=D  
    %       0    0    1                        sqrt(2) f:-dw6a=s  
    %       1    1    r                           2 P7iU_CgyW  
    %       2    0    2*r^2 - 1                sqrt(6) JKsdPW<?  
    %       2    2    r^2                      sqrt(6) Ly$s0.!  
    %       3    1    3*r^3 - 2*r              sqrt(8) {? dW-  
    %       3    3    r^3                      sqrt(8) rX<gcntv  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) sB,>4*Zd  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) zsx12b^w  
    %       4    4    r^4                      sqrt(10) *jF VYg  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) g6. =(je  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) aab?hR  
    %       5    5    r^5                      sqrt(12) 0w_2E  
    %       --------------------------------------------- Kc:} Ky  
    % D< 4!7*9%  
    %   Example: H}$hk  
    % Hf'yRKACj  
    %       % Display three example Zernike radial polynomials dIR6dI   
    %       r = 0:0.01:1; MXxE)"G*a  
    %       n = [3 2 5]; -)Y?1w  
    %       m = [1 2 1]; *I}_B\kY  
    %       z = zernpol(n,m,r); F& 'HZX  
    %       figure O<x53MN^  
    %       plot(r,z) UT9=S21  
    %       grid on KrFV4J[  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') XTZI !  
    % *0^t;A+  
    %   See also ZERNFUN, ZERNFUN2. '\2lWR]ndd  
    2.K"+%  
    % A note on the algorithm. D=fB&7%@  
    % ------------------------ :-f"+v  
    % The radial Zernike polynomials are computed using the series r]=3aebR.  
    % representation shown in the Help section above. For many special zq5_&AeW  
    % functions, direct evaluation using the series representation can Lz VvUVk  
    % produce poor numerical results (floating point errors), because ,QpDz{8  
    % the summation often involves computing small differences between GZN@MK*co  
    % large successive terms in the series. (In such cases, the functions pP'-}%  
    % are often evaluated using alternative methods such as recurrence lE54RX}e4  
    % relations: see the Legendre functions, for example). For the Zernike A/U tf0{3"  
    % polynomials, however, this problem does not arise, because the ~%Ws"1  
    % polynomials are evaluated over the finite domain r = (0,1), and YSuw V)Y  
    % because the coefficients for a given polynomial are generally all bz~-uHC  
    % of similar magnitude. QsmG(1=  
    % iDO~G($C  
    % ZERNPOL has been written using a vectorized implementation: multiple DOXRU5uP3  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] Oed&B  
    % values can be passed as inputs) for a vector of points R.  To achieve XU0"f!23x  
    % this vectorization most efficiently, the algorithm in ZERNPOL } V4"-;P  
    % involves pre-determining all the powers p of R that are required to V,uhBMT#  
    % compute the outputs, and then compiling the {R^p} into a single 9tS& $-  
    % matrix.  This avoids any redundant computation of the R^p, and |jV4]7Luq  
    % minimizes the sizes of certain intermediate variables. RU `TzD  
    % J<_&f_K0]  
    %   Paul Fricker 11/13/2006 q\[31$i$  
    ^}8_tZs8\  
    ?.A6HrAPB  
    % Check and prepare the inputs: IBVP4&}x$  
    % ----------------------------- 0nAeeVz|  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) tS2lex%  
        error('zernpol:NMvectors','N and M must be vectors.') lb1(1 |#  
    end -X4`,0y%{O  
    @D"1}CW  
    if length(n)~=length(m) e_6 i896  
        error('zernpol:NMlength','N and M must be the same length.') gWS4 9*O  
    end Smk]G))o{  
    O)5-6lm  
    n = n(:); &V( LeSI  
    m = m(:); AmSJ!mTd8o  
    length_n = length(n); 7k3":2 :  
    RpLm'~N'  
    if any(mod(n-m,2)) >[xQUf,p  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') TF^]^XS'  
    end m$J'nA  
    73xI8  
    if any(m<0) 7<.f&1MgI  
        error('zernpol:Mpositive','All M must be positive.') n.l p ena  
    end oS_p/$F,  
    dl{3fldb  
    if any(m>n) g6W.Gl"5\w  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') sCb?TyN'n  
    end & 8&WY1cU  
    !9)*.9[8  
    if any( r>1 | r<0 ) !#iP)"O  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') QW2% Gv:  
    end ^U_jeAuk8[  
    # |UrHK;  
    if ~any(size(r)==1) r9vC&pWZ  
        error('zernpol:Rvector','R must be a vector.') y6j TT%  
    end E$G "R =  
    Pq4sv`q)S  
    r = r(:); xD lC]loi7  
    length_r = length(r); {{DW P-v4  
    hrAI@.Bo  
    if nargin==4 eB]ZnJ2^=  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); mU&J,C  
        if ~isnorm rWvJ{-%  
            error('zernpol:normalization','Unrecognized normalization flag.') A`r&"i OKA  
        end f:utw T  
    else Ta[}k/zW  
        isnorm = false; YT:5J%"  
    end vRY4N{v(<  
    U&eLj"XZ  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 4*dT|NU  
    % Compute the Zernike Polynomials  03#_ (  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% pI^n("|  
    7I.[1V`  
    % Determine the required powers of r: /n_HUY  
    % ----------------------------------- gh 0\9;h  
    rpowers = []; L|H{;r'  
    for j = 1:length(n) ]jYl:41yI  
        rpowers = [rpowers m(j):2:n(j)]; '",5Bu#C  
    end HxM-VK '  
    rpowers = unique(rpowers); `H|g~7KD&  
    L'6zs:i  
    % Pre-compute the values of r raised to the required powers, 9%dNktt  
    % and compile them in a matrix: #e0+;kBh  
    % ----------------------------- [,e_2<   
    if rpowers(1)==0 !kz\ {  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); S n+Yi  
        rpowern = cat(2,rpowern{:}); kR_[p._  
        rpowern = [ones(length_r,1) rpowern]; ~p 1y+  
    else M>^IQ  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); lubS{3<  
        rpowern = cat(2,rpowern{:}); ~\_E%NR yA  
    end  6$Dbeb  
    l-npz)EM  
    % Compute the values of the polynomials: & 3a+6!L[  
    % -------------------------------------- %$}iM<  
    z = zeros(length_r,length_n); C^~iz in  
    for j = 1:length_n BdYh:  
        s = 0:(n(j)-m(j))/2; O|/tRkDMP{  
        pows = n(j):-2:m(j); bC{~/ JP  
        for k = length(s):-1:1 xSf3Ir(,  
            p = (1-2*mod(s(k),2))* ... {!4%Z9G  
                       prod(2:(n(j)-s(k)))/          ... I[|I\tW  
                       prod(2:s(k))/                 ... 2,fB$5+  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... :`|,a (  
                       prod(2:((n(j)+m(j))/2-s(k))); aG ,uF  
            idx = (pows(k)==rpowers); .6hH}BM  
            z(:,j) = z(:,j) + p*rpowern(:,idx); ^m7PXY  
        end )Qc$UI8L  
         ?\yo~=N^  
        if isnorm x{- caOH  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); c2U>89LlZ  
        end r3-3*_  
    end ~DD/\V  
    OwEz( pj@  
    % 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)  v Q"s  
    hF"g 91P  
    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。 T:dm0iau  
    cmhN(==  
    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)