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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 M!G/5:VZ  
    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! h6Vd<sV\tf  
     
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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 "jq F  
    function z = zernfun(n,m,r,theta,nflag) Kn+B):OY+  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. K`R  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N )q+;+J`>  
    %   and angular frequency M, evaluated at positions (R,THETA) on the )=h+5Z>E1  
    %   unit circle.  N is a vector of positive integers (including 0), and e58tf3  
    %   M is a vector with the same number of elements as N.  Each element h>NuQo*  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) -A8CW9|mk  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, n\scOM)3  
    %   and THETA is a vector of angles.  R and THETA must have the same k1^&;}/f:  
    %   length.  The output Z is a matrix with one column for every (N,M) 9&4z4@on  
    %   pair, and one row for every (R,THETA) pair. $8_b[~%2  
    % p-8x>dmP(  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike q-0( Wx9|  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), )J|~'{z:  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral ~EhM"go  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 'k$j^ |r>  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized /;1h-Rc>  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. sr$JFMTO11  
    % ^paM{'J\\)  
    %   The Zernike functions are an orthogonal basis on the unit circle. (d5kD#.N  
    %   They are used in disciplines such as astronomy, optics, and "IdN*K  
    %   optometry to describe functions on a circular domain. XuW>GT/  
    % {Ve_u  
    %   The following table lists the first 15 Zernike functions. X04JQLhy"  
    % /`B:F5r  
    %       n    m    Zernike function           Normalization LT '2446  
    %       -------------------------------------------------- , rc %#eF  
    %       0    0    1                                 1 Pu|3_3^  
    %       1    1    r * cos(theta)                    2 G C3G=DTt  
    %       1   -1    r * sin(theta)                    2 &p^8zEs  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) TqXB2`7Ri  
    %       2    0    (2*r^2 - 1)                    sqrt(3) Oc?]L&ap  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) B}p{$g!  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) n|I5ylt  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) eN ]9=Y~-K  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) K@@[N17/8  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) 39,7N2uY  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) nJo6;_MI!  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) w97%5[-T  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) DlbNW& V  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) D j@7vM%_  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) _ye74$#  
    %       -------------------------------------------------- 0h~7"qUF@  
    % +/~;y{G..z  
    %   Example 1:  [E|%  
    % iyf vcKO  
    %       % Display the Zernike function Z(n=5,m=1) <MBpV^Y}  
    %       x = -1:0.01:1; 6{PlclI !  
    %       [X,Y] = meshgrid(x,x); p{4nWeH?B  
    %       [theta,r] = cart2pol(X,Y); YeCS`IXm  
    %       idx = r<=1; vTU*6)  
    %       z = nan(size(X)); loFApBD=$^  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); 1|Z!8:&pj  
    %       figure ,O/ t6'  
    %       pcolor(x,x,z), shading interp "_T8Km008  
    %       axis square, colorbar i"o %Gc  
    %       title('Zernike function Z_5^1(r,\theta)') &C=[D_h  
    % [oh0 )wzB  
    %   Example 2: i_6 Y6  
    % f& >[$zh  
    %       % Display the first 10 Zernike functions hV]]%zwR+  
    %       x = -1:0.01:1; g/6>>p`J  
    %       [X,Y] = meshgrid(x,x); xF8^#J6>  
    %       [theta,r] = cart2pol(X,Y); gG6j>%y  
    %       idx = r<=1; &!5S'J %  
    %       z = nan(size(X)); m3E`kW |  
    %       n = [0  1  1  2  2  2  3  3  3  3]; hMvLx>q3)  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; (^).$g5Hg  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; <*5 5d2  
    %       y = zernfun(n,m,r(idx),theta(idx)); 7$u}uv`j  
    %       figure('Units','normalized') Zw<\^1  
    %       for k = 1:10 #?EmC]N7  
    %           z(idx) = y(:,k); %^CoWbU  
    %           subplot(4,7,Nplot(k)) XIJW$CY  
    %           pcolor(x,x,z), shading interp 9( "<NB0y  
    %           set(gca,'XTick',[],'YTick',[]) RO+N>Wkt  
    %           axis square J}'a|a@bk  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ^TF71u o  
    %       end /f0*NNSat-  
    % I=G-(L/&  
    %   See also ZERNPOL, ZERNFUN2. hY S}PE  
    r0 fxEYze&  
    %   Paul Fricker 11/13/2006 E\Et,l#|LY  
    AeN$AqQd/  
    c Y(2}Ay  
    % Check and prepare the inputs: KJ;;825?  
    % ----------------------------- L|H:&|F  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) VHihC]ks,  
        error('zernfun:NMvectors','N and M must be vectors.') mxfmK +'_  
    end KrDG  
    H%z9VJ*!0  
    if length(n)~=length(m) BL^\"Xh$|  
        error('zernfun:NMlength','N and M must be the same length.') /l&$B  
    end sOUQd-!"  
    VW/ICX~"d  
    n = n(:); @n Oj6b  
    m = m(:); ;bhD:$NB X  
    if any(mod(n-m,2)) E6zSMl5b  
        error('zernfun:NMmultiplesof2', ... 7`_`V&3s  
              'All N and M must differ by multiples of 2 (including 0).') J70r`   
    end o3OtG#g2  
    X&14;lu%p  
    if any(m>n) JRYCM}C]  
        error('zernfun:MlessthanN', ... 6I!B>V#U+  
              'Each M must be less than or equal to its corresponding N.') -b`O"Ck*  
    end bc}BQ|Q  
    @&xWd{8'  
    if any( r>1 | r<0 ) ]u<8j r  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') =d M'n}@U  
    end k ^(RSu<  
    `+0K~k|DC  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) z<u*I@;  
        error('zernfun:RTHvector','R and THETA must be vectors.') s5cY>  
    end >Xv Fg  
    _my!YS5n  
    r = r(:); sf5F$  
    theta = theta(:); &6q67  
    length_r = length(r); 0wnC"2GUX  
    if length_r~=length(theta) ;:A/WU.^  
        error('zernfun:RTHlength', ... |\3X7)^8D  
              'The number of R- and THETA-values must be equal.') fIwG9cR  
    end &~{0@/  
    }u?DK,R  
    % Check normalization: wMvAm%}+  
    % -------------------- V3v/h V:  
    if nargin==5 && ischar(nflag) >%1mx\y^  
        isnorm = strcmpi(nflag,'norm'); wx[Y2lUh6  
        if ~isnorm Zv&<r+<g  
            error('zernfun:normalization','Unrecognized normalization flag.') 6TkV+\  
        end _A]=45cn~  
    else gO%o A} !i  
        isnorm = false; Or<OmxJg  
    end 9[ ,+4&wX7  
     O3~7  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% dk>qTY+j5  
    % Compute the Zernike Polynomials - xKa-3  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% wT@{=s,  
    Bh ,GQHJ  
    % Determine the required powers of r: '<-F3  
    % ----------------------------------- PSREQK@}E  
    m_abs = abs(m); caD)'FSES  
    rpowers = []; 9AP."RV  
    for j = 1:length(n) U#>K(  
        rpowers = [rpowers m_abs(j):2:n(j)]; UR<a7j"@2  
    end Pe?=M[u2  
    rpowers = unique(rpowers); wzf%~ats  
    .{,fb  
    % Pre-compute the values of r raised to the required powers, ZWXA%u7V  
    % and compile them in a matrix: iOXsj  
    % ----------------------------- /ht-]Js$G  
    if rpowers(1)==0 '1vm]+oM  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); N.F //n  
        rpowern = cat(2,rpowern{:}); ?M9?GodbP.  
        rpowern = [ones(length_r,1) rpowern]; g(QT"O!dY  
    else 9:JQ*O$  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); :J;&Z{  
        rpowern = cat(2,rpowern{:}); SbK6o:[  
    end J7FzOwd1h  
    6xiCTs0@  
    % Compute the values of the polynomials: D*}_L   
    % -------------------------------------- ;GV~MH-F  
    y = zeros(length_r,length(n)); Mem1X rBH  
    for j = 1:length(n) J u"K"  
        s = 0:(n(j)-m_abs(j))/2; T%O2=h\} E  
        pows = n(j):-2:m_abs(j); 2#)z%K6T  
        for k = length(s):-1:1 gn)>(MG  
            p = (1-2*mod(s(k),2))* ... 6Ij'z9nJw  
                       prod(2:(n(j)-s(k)))/              ... E'+?7ZGWj  
                       prod(2:s(k))/                     ... 'LMMo4o3  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 2F#DJN#  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); 8fWk C<f}  
            idx = (pows(k)==rpowers); 'dJ(x  
            y(:,j) = y(:,j) + p*rpowern(:,idx); *o6hDhg  
        end  LOi/+;>  
         \'.|7{Xu  
        if isnorm }s@vN8C  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); QE4TvnhK  
        end P-?R\(QYtR  
    end <~ }NxY\5  
    % END: Compute the Zernike Polynomials ypLt6(1j%  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% OJT%?P%@{  
    K9QC$b9(  
    % Compute the Zernike functions: t{Wu5<F:  
    % ------------------------------ 75W@B}dZd  
    idx_pos = m>0; LO@o`JF  
    idx_neg = m<0; j]'ybpMT"  
    '7JM/AcC#K  
    z = y; A@ZsL  
    if any(idx_pos) 'cPE7uNT  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 5bo')^xa  
    end :wY(</H  
    if any(idx_neg) IN?6~O p  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); BPd *@l  
    end "5@\"L  
    ub9,Wd"^  
    % EOF zernfun
    离线niuhelen
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) pM}n)Q!{3"  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. ^\g?uH6k U  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated 5 51p* B2  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive \[L|  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, j5!pS xOC  
    %   and THETA is a vector of angles.  R and THETA must have the same ~; vt{pk  
    %   length.  The output Z is a matrix with one column for every P-value, kE854Ej  
    %   and one row for every (R,THETA) pair. !|~yf3  
    % k-I U}|Xz  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike =3|5=ZU034  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) #Q/xQ`+|.  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) YQ`88 z  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 ^_t7{z%sA[  
    %   for all p. r#NR3_@9  
    % B3W2?5p  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 2#X>^LH  
    %   Zernike functions (order N<=7).  In some disciplines it is @me ( pnD  
    %   traditional to label the first 36 functions using a single mode .#LvvAeh  
    %   number P instead of separate numbers for the order N and azimuthal 4VP$, |a  
    %   frequency M. 7_`_iymR  
    % C:p`  
    %   Example: ]SUW"5L-  
    % s&M#]8x;x  
    %       % Display the first 16 Zernike functions juB/?'$~  
    %       x = -1:0.01:1; _-z;  
    %       [X,Y] = meshgrid(x,x); ,U.|+i{  
    %       [theta,r] = cart2pol(X,Y); 5`)[FCQ  
    %       idx = r<=1; T/ P   
    %       p = 0:15; nU/x,W[}  
    %       z = nan(size(X)); 7T?T0x3>  
    %       y = zernfun2(p,r(idx),theta(idx)); /X;! F>  
    %       figure('Units','normalized') \L"0Pmt[  
    %       for k = 1:length(p) +HEL^  
    %           z(idx) = y(:,k);  Sr+ &  
    %           subplot(4,4,k) Id{Ix(O  
    %           pcolor(x,x,z), shading interp Dv}VmC""  
    %           set(gca,'XTick',[],'YTick',[]) tS[%C)  
    %           axis square z'}z4^35,  
    %           title(['Z_{' num2str(p(k)) '}']) )<`/Aaie  
    %       end : 2A\X' @  
    % O +Xu ?W]  
    %   See also ZERNPOL, ZERNFUN. @6V kNe9  
    H(DI /"N  
    %   Paul Fricker 11/13/2006 H={5>;8G  
    N3ccn  
    {4r }jH  
    % Check and prepare the inputs: __i))2  
    % ----------------------------- s[%@3bY!7  
    if min(size(p))~=1 G&@_,y|  
        error('zernfun2:Pvector','Input P must be vector.') K<p)-q  
    end 1 Ay.^f  
    {; 3a^K  
    if any(p)>35 ?|{tWR,Vb  
        error('zernfun2:P36', ... M-A{{q   
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... CJt(c,!z  
               '(P = 0 to 35).']) =~'y'K]  
    end &G_XgQsg{  
    "LM[WcDX  
    % Get the order and frequency corresonding to the function number: iz2;xa*  
    % ---------------------------------------------------------------- L DdgI  
    p = p(:); ;M5]XCP k  
    n = ceil((-3+sqrt(9+8*p))/2); "(yw(/  
    m = 2*p - n.*(n+2); wj\kx\+  
    \iAs  
    % Pass the inputs to the function ZERNFUN: s}?QA cC  
    % ---------------------------------------- 0>yu Bgh  
    switch nargin V-lp';bD  
        case 3 .`^wRpa2M  
            z = zernfun(n,m,r,theta); DYTC2  
        case 4 ,p6o "-  
            z = zernfun(n,m,r,theta,nflag); u(g9-O  
        otherwise \'+P5,  
            error('zernfun2:nargin','Incorrect number of inputs.') |t_SN,)dd  
    end )@6iQ  
    ,R=)^Gh{  
    % EOF zernfun2
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    function z = zernpol(n,m,r,nflag) E]ZIm  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. Im@Yx^gc   
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of z*`nfTw l  
    %   order N and frequency M, evaluated at R.  N is a vector of uk)D2.eS,  
    %   positive integers (including 0), and M is a vector with the A3Y}|7QA  
    %   same number of elements as N.  Each element k of M must be a @Wd1+Yky  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) kjj?X|Un  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is 6IKi*}  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix v+ "9&  
    %   with one column for every (N,M) pair, and one row for every PGA `R  
    %   element in R. #fxdZm,  
    % ,<fs+oi  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- e4 -7&8N+  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is +kT o$_Wkz  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to aV G4D f  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 x_#'6H\1ga  
    %   for all [n,m]. %R?#Y1Tq;  
    % zJG=9C?  
    %   The radial Zernike polynomials are the radial portion of the CwsC)]{/o  
    %   Zernike functions, which are an orthogonal basis on the unit >RXDuCVi  
    %   circle.  The series representation of the radial Zernike 8:jakOeT  
    %   polynomials is p~dj-w  
    % YH{FTVOt{C  
    %          (n-m)/2 cT# R B7  
    %            __ TFtD>q X  
    %    m      \       s                                          n-2s TTz_w-68  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r U9h@1:  
    %    n      s=0 3| 0OW Jk  
    % JvM:xy9  
    %   The following table shows the first 12 polynomials. ;b^"b{  
    % @!%HEs!# #  
    %       n    m    Zernike polynomial    Normalization {\ J%i|u  
    %       --------------------------------------------- e%KCcU  
    %       0    0    1                        sqrt(2) + EM^  
    %       1    1    r                           2 dA$qzQ  
    %       2    0    2*r^2 - 1                sqrt(6) D*!9K8<o  
    %       2    2    r^2                      sqrt(6) 7DIFJJE'  
    %       3    1    3*r^3 - 2*r              sqrt(8) =VF%Z[Gm  
    %       3    3    r^3                      sqrt(8) =xM:8 hm  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) 9/6=[)  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) 8Oo16LPD  
    %       4    4    r^4                      sqrt(10) >\Ml \CyL  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) 2w>yW]  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) "SU O2-Gj  
    %       5    5    r^5                      sqrt(12) dU#} Tk  
    %       --------------------------------------------- R<e ~Cb-  
    % >?GCH(eW%  
    %   Example: B#[.c$  
    % {'[S.r`  
    %       % Display three example Zernike radial polynomials n!%'%%o2v  
    %       r = 0:0.01:1; <xr\1VjA  
    %       n = [3 2 5]; /xkF9   
    %       m = [1 2 1]; .a `ojT  
    %       z = zernpol(n,m,r); B^"1V{M  
    %       figure U )J/so)  
    %       plot(r,z) NSQ#\:3:S  
    %       grid on KNqs=:i  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') Wh1'?#  
    % 6_O3/   
    %   See also ZERNFUN, ZERNFUN2.  Yk yB  
    7/6%92T/B  
    % A note on the algorithm. +%LR1+/%b  
    % ------------------------ g)^g_4  
    % The radial Zernike polynomials are computed using the series N_f>5uv  
    % representation shown in the Help section above. For many special D' oy% 1Q}  
    % functions, direct evaluation using the series representation can Y]H,rO  
    % produce poor numerical results (floating point errors), because ]xN)>A2  
    % the summation often involves computing small differences between $;B0x  
    % large successive terms in the series. (In such cases, the functions _#T bO fu  
    % are often evaluated using alternative methods such as recurrence a,KqTQB  
    % relations: see the Legendre functions, for example). For the Zernike vABUUAo!Jr  
    % polynomials, however, this problem does not arise, because the &v<Am%!N  
    % polynomials are evaluated over the finite domain r = (0,1), and 8U7X/L  
    % because the coefficients for a given polynomial are generally all @;h$!w<  
    % of similar magnitude. h"On9  
    % #]lK!:  
    % ZERNPOL has been written using a vectorized implementation: multiple XJZS}Z7h  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] (GbZt{.  
    % values can be passed as inputs) for a vector of points R.  To achieve *&~wl(+O=  
    % this vectorization most efficiently, the algorithm in ZERNPOL oE'Flc.  
    % involves pre-determining all the powers p of R that are required to "&1h<>  
    % compute the outputs, and then compiling the {R^p} into a single 2t`d. s=  
    % matrix.  This avoids any redundant computation of the R^p, and Fle pM*  
    % minimizes the sizes of certain intermediate variables. Jn)DZv8?  
    % m-XS_5x\  
    %   Paul Fricker 11/13/2006 zOA2chy4  
    `E-cf7%  
    X"O^4MnvI  
    % Check and prepare the inputs: L~ e{Vv8UR  
    % ----------------------------- Fps:6~gD  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) )1o<}7  
        error('zernpol:NMvectors','N and M must be vectors.') XcM.<Dn3  
    end |J^$3RX  
    ^6FU]  
    if length(n)~=length(m) Y@Ti2bI`v  
        error('zernpol:NMlength','N and M must be the same length.') C,nU.0  
    end n+:}p D  
    *#Hw6N0#   
    n = n(:); |ZJ<N\\h-  
    m = m(:); 8 ;o*c6+  
    length_n = length(n); 1eMz"@ Q9  
    `rZS\A  
    if any(mod(n-m,2)) (K3eb  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') G~KYFNHr  
    end nbdjk1E`~  
    l|5;&(Y+s  
    if any(m<0) Rg~F[j$N  
        error('zernpol:Mpositive','All M must be positive.') )@a_|q@V  
    end ZA.i\ ;2  
    1cY,)Z%l #  
    if any(m>n) &~#y-o"  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') #Hi$squJ  
    end N Ah^2X  
    X^9eCj;c  
    if any( r>1 | r<0 ) eGQ -Ht,N  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') WC`x^HI  
    end [WW ~SOJe  
    od RtJ[   
    if ~any(size(r)==1) pW1(1M)[%Z  
        error('zernpol:Rvector','R must be a vector.') (V9 ;  
    end DBk]2W|i  
    J3,m{%EtNM  
    r = r(:); s|d"2w6t  
    length_r = length(r); ! ,&{1p  
    E>Lgf&R#W  
    if nargin==4 ,@0D_&JAl  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); XG}9) fT  
        if ~isnorm !FHm.E_>  
            error('zernpol:normalization','Unrecognized normalization flag.') %)p?&_  
        end :FX|9h  
    else p~f=0K  
        isnorm = false; aYws{Vii  
    end ?yy,3:  
    #MAXH7[  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ?OF9{$m3?  
    % Compute the Zernike Polynomials ,WQg.neOA  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 55Ag<\7  
    j-TRa,4bN  
    % Determine the required powers of r: h"t\x}8qq  
    % ----------------------------------- +wxDK A_  
    rpowers = []; Am"e%|:  
    for j = 1:length(n) JUd Q Q  
        rpowers = [rpowers m(j):2:n(j)]; @23~)uiZa  
    end 5Sx.'o$  
    rpowers = unique(rpowers); 'e:(61_  
    MkX=34oc^  
    % Pre-compute the values of r raised to the required powers, !8OgaMngzF  
    % and compile them in a matrix: &AP`k  
    % ----------------------------- J+r:7NvZ  
    if rpowers(1)==0 tZan1C%p>  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); R(p3* t&n  
        rpowern = cat(2,rpowern{:}); M<'He.n  
        rpowern = [ones(length_r,1) rpowern]; RJOW#e :  
    else 5%RiM|+  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); {B[ }}wX$  
        rpowern = cat(2,rpowern{:}); ubUVxYD?  
    end yZkS   
    f[wxt n'r  
    % Compute the values of the polynomials: VMH^jCFp  
    % -------------------------------------- ,)ZI&BL5  
    z = zeros(length_r,length_n); /_</m?&.U&  
    for j = 1:length_n do:IkjU~  
        s = 0:(n(j)-m(j))/2; }No8to  
        pows = n(j):-2:m(j); p((a(Q/  
        for k = length(s):-1:1 fi  [4F  
            p = (1-2*mod(s(k),2))* ... `T3B  
                       prod(2:(n(j)-s(k)))/          ... SVBo0wvz-  
                       prod(2:s(k))/                 ... -{A*`.[v  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... {BzE  
                       prod(2:((n(j)+m(j))/2-s(k))); xQkvK=~$  
            idx = (pows(k)==rpowers); <.hutU*1  
            z(:,j) = z(:,j) + p*rpowern(:,idx); _ o.j({S  
        end |dhKeg_  
         v n4z C  
        if isnorm DB3qf>@?  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); %"^8$A?>,k  
        end "{[\VsX|c  
    end ?OS0.  
    T{)_vQ  
    % EOF zernpol
    离线niuhelen
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    只看该作者 6楼 发表于: 2011-03-12
    这三个文件,我不知道该怎样把我的面型节点的坐标及轴向位移用起来,还烦请指点一下啊,谢谢啦!
    离线li_xin_feng
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    只看该作者 7楼 发表于: 2012-09-28
    我也正在找啊
    离线guapiqlh
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    只看该作者 8楼 发表于: 2014-03-04
    我也一直想了解这个多项式的应用,还没用过呢
    离线phoenixzqy
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    只看该作者 9楼 发表于: 2014-04-22
    回 guapiqlh 的帖子
    guapiqlh:我也一直想了解这个多项式的应用,还没用过呢 (2014-03-04 11:35)  &R?`QB2/  
    ZnRT$ l O  
    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。 -fk;Qq3O  
    p]EugLEmG  
    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)