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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 zg"<N  
    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! Dd :Qotu  
     
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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 gT?:zd=;  
    function z = zernfun(n,m,r,theta,nflag) k0Rd:DxO  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. !S$LRm\ '  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N Jvgx+{Xu  
    %   and angular frequency M, evaluated at positions (R,THETA) on the DTH;d-Z  
    %   unit circle.  N is a vector of positive integers (including 0), and 7CWz)LT  
    %   M is a vector with the same number of elements as N.  Each element <$qe2Ft Uq  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) 'MVE5  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, -Uh3A\#(  
    %   and THETA is a vector of angles.  R and THETA must have the same r[ni{ &  
    %   length.  The output Z is a matrix with one column for every (N,M) ]>B>.s  
    %   pair, and one row for every (R,THETA) pair. :bNqK0[rS  
    % ..)O/g.  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike *@^9 ]$*$  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ViKN|W >T  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral 6Q"fRXM   
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ?:H4Xd7  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized *S%~0=  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ~M _ @_  
    % `O/1aW1  
    %   The Zernike functions are an orthogonal basis on the unit circle. #{-B`FAQ  
    %   They are used in disciplines such as astronomy, optics, and ckykRqk}  
    %   optometry to describe functions on a circular domain. bbddbRj;  
    % @Fvp~]jCb  
    %   The following table lists the first 15 Zernike functions. k[#<=G_=/E  
    % pMndyuoJl  
    %       n    m    Zernike function           Normalization {DlQTgP  
    %       -------------------------------------------------- THEpW{.E  
    %       0    0    1                                 1 /Ps/m!  
    %       1    1    r * cos(theta)                    2 -Ri/I4Xj  
    %       1   -1    r * sin(theta)                    2 g3B%}!|  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) Rr A9@95+  
    %       2    0    (2*r^2 - 1)                    sqrt(3) AWo\u!j  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) ~XU%_Hz  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) L6<.>\^Z"  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) 8~* |muN.e  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) "Tt5cqUQoY  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) 57@6O-t-  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) s3<gq x-&r  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) GO4IAUA  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) vJI]ZnL{  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) #$n >+ lc  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) tx`gXtO$  
    %       -------------------------------------------------- [/E|n[Bx  
    % FWC\(f  
    %   Example 1: F)K&a  
    % ^jh c(ZW"  
    %       % Display the Zernike function Z(n=5,m=1) U</Vcz  
    %       x = -1:0.01:1; S,0h &A9  
    %       [X,Y] = meshgrid(x,x); ? $$Xg3w_#  
    %       [theta,r] = cart2pol(X,Y); )@(IhU )  
    %       idx = r<=1; W=G8l%  
    %       z = nan(size(X)); }jdMo83  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); W?TvdeBx  
    %       figure 1#tFO  
    %       pcolor(x,x,z), shading interp 2 Sgv  
    %       axis square, colorbar D*0[7:NSO  
    %       title('Zernike function Z_5^1(r,\theta)') db*yA@2Lg  
    % 8f`r!/j  
    %   Example 2: s$g3__|Y  
    % ^ruz-N^Y!  
    %       % Display the first 10 Zernike functions W79Sz}):  
    %       x = -1:0.01:1; LS:^K  
    %       [X,Y] = meshgrid(x,x); Wr+/ 9  
    %       [theta,r] = cart2pol(X,Y); SL[EOz#  
    %       idx = r<=1; 9z#z9|hj)3  
    %       z = nan(size(X)); @oKW$\  
    %       n = [0  1  1  2  2  2  3  3  3  3]; GHWt3K:*w  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; W*;r}!ro  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; A'6-E{  
    %       y = zernfun(n,m,r(idx),theta(idx)); (l+0*o,(  
    %       figure('Units','normalized') QtHK`f>4#n  
    %       for k = 1:10 &v)/mc7D  
    %           z(idx) = y(:,k); .+) AeGh  
    %           subplot(4,7,Nplot(k)) zFi)R }Ot  
    %           pcolor(x,x,z), shading interp (&i c3/-  
    %           set(gca,'XTick',[],'YTick',[]) X<sM4dwxE  
    %           axis square FFtB#  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 6w `.'5  
    %       end 7TtDI=f  
    % ]y9u5H^  
    %   See also ZERNPOL, ZERNFUN2. `T,^os#6  
    W"!{f  
    %   Paul Fricker 11/13/2006 JA09 o(  
    &|fPskpy  
    }D]y -BbA.  
    % Check and prepare the inputs: y9Pw'4R  
    % ----------------------------- |mQC-=6t;Y  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) sK@]|9ciQ  
        error('zernfun:NMvectors','N and M must be vectors.') :z-?L0C=0  
    end 0" F\ V  
    MK.TBv  
    if length(n)~=length(m) b5)1\ANq  
        error('zernfun:NMlength','N and M must be the same length.') SFjRSMi  
    end >H5_,A}f  
    3Yf~5csY  
    n = n(:); PDpuHHB  
    m = m(:); zeshM8=  
    if any(mod(n-m,2)) 5SEGV|%  
        error('zernfun:NMmultiplesof2', ... 8I~*9MUp  
              'All N and M must differ by multiples of 2 (including 0).') B{K_?ae!  
    end 6!@p$ pm)a  
    ]+5Y\~I  
    if any(m>n) G0u H6x?  
        error('zernfun:MlessthanN', ... [(; .D  
              'Each M must be less than or equal to its corresponding N.') T"DG$R,Aj  
    end |RH^|2:x9Q  
    *7{{z%5Pu  
    if any( r>1 | r<0 ) !{F\ \D/  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') XnKf<|j6k  
    end " 1h~P,  
    &,QBJx<#  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) qzWnl[3  
        error('zernfun:RTHvector','R and THETA must be vectors.') \I7&F82e  
    end I@kMM12>c  
    _D{{C  
    r = r(:); 4}t$Lf_  
    theta = theta(:); &hE k m  
    length_r = length(r); r*c x_**  
    if length_r~=length(theta) s( :N>K5*  
        error('zernfun:RTHlength', ... =)f.Yf|A*  
              'The number of R- and THETA-values must be equal.') nTE\EZ+=2  
    end v2ab84 C*  
    je74As[  
    % Check normalization: ^YB3$:@$U  
    % -------------------- 8w ]'U  
    if nargin==5 && ischar(nflag) ?NxaJ^  
        isnorm = strcmpi(nflag,'norm'); %~\I*v04  
        if ~isnorm 6RfS_  
            error('zernfun:normalization','Unrecognized normalization flag.') CN6b 982&  
        end V8 G.KA "  
    else g6h=Q3@  
        isnorm = false; M1eM^m8U  
    end gMPvzBpP  
    ynn>d  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% z J V>;  
    % Compute the Zernike Polynomials )%q )!x  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [M?&JA_$}  
    l M a||  
    % Determine the required powers of r: UAds$ 9  
    % ----------------------------------- o;v_vCLO  
    m_abs = abs(m); 2 U3WH.o  
    rpowers = []; #;\tgUQ  
    for j = 1:length(n) SpM Hq_MLM  
        rpowers = [rpowers m_abs(j):2:n(j)]; 0BN=>]V~j7  
    end >Ft:&N9L{  
    rpowers = unique(rpowers); $*7AG  
    'kekJ.wJ;  
    % Pre-compute the values of r raised to the required powers, 8p]Krs:  
    % and compile them in a matrix: }q)dXFL=I#  
    % ----------------------------- #VuiY  
    if rpowers(1)==0 X1="1{8H  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); i+|/V&#3[  
        rpowern = cat(2,rpowern{:}); <$8e;:#:  
        rpowern = [ones(length_r,1) rpowern]; w"!zLB&9[  
    else (X|lK.W y  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); qbo W<W<H1  
        rpowern = cat(2,rpowern{:}); 30QQnMH3  
    end 2*Mu"v,  
    N lB%Qu  
    % Compute the values of the polynomials: |E FbT>  
    % -------------------------------------- 5xc-MkIRL  
    y = zeros(length_r,length(n)); )z7+%nTO  
    for j = 1:length(n) lsOZ%p%fV  
        s = 0:(n(j)-m_abs(j))/2; b$}@0  
        pows = n(j):-2:m_abs(j); -l$-\(,M`#  
        for k = length(s):-1:1 #+;0=6+SM  
            p = (1-2*mod(s(k),2))* ... %M-B"#OB7  
                       prod(2:(n(j)-s(k)))/              ... Lu~M=Fh  
                       prod(2:s(k))/                     ... [4HOWM>\  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... xilA`uw`1  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); B3yp2tncj  
            idx = (pows(k)==rpowers); BoXGoFn  
            y(:,j) = y(:,j) + p*rpowern(:,idx); 6zJ>n~&(  
        end Nk shJ2  
         rY(^6[!  
        if isnorm ,IG?(CK|  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); ^/jALA9!  
        end K[i|OZWu  
    end Z"a]AsG/Q#  
    % END: Compute the Zernike Polynomials H_7X%TvXb  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ~[ x}  
    k&3'[&$I*,  
    % Compute the Zernike functions: Sv03="&  
    % ------------------------------ M-NY&@Nj  
    idx_pos = m>0; )-d &XN7  
    idx_neg = m<0; N2`u ]*"0  
    M2y"M,k4  
    z = y; ZTP&*+d  
    if any(idx_pos) \:91BQP c  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); fAV=O%^  
    end c>e~$b8  
    if any(idx_neg) =j!Ruy1  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); /,2${$c!  
    end f m'Qif q^  
    x0x/2re  
    % 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) rWMG_eP:  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. jaI mO  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated ]o\y(!  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive JOJ? .H&su  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, edD"jq)J  
    %   and THETA is a vector of angles.  R and THETA must have the same -g]g  
    %   length.  The output Z is a matrix with one column for every P-value, M/mUY  
    %   and one row for every (R,THETA) pair. 0`dMT>&I  
    % B?)=d,E  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike GwaU7[6  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) F,-S&d  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) ghd*EXrF H  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 &r Lg/UEV-  
    %   for all p. *eo<5YUHt  
    % jPf*qe>U  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 -w:F8k ~  
    %   Zernike functions (order N<=7).  In some disciplines it is s8]9OG3g  
    %   traditional to label the first 36 functions using a single mode < l%3P6|  
    %   number P instead of separate numbers for the order N and azimuthal aD:vNX  
    %   frequency M. -] .Y";  
    % ^#VyIF3q  
    %   Example: ^N5BJ'[F:  
    % ,pz^8NJAI  
    %       % Display the first 16 Zernike functions + B#3!  
    %       x = -1:0.01:1; )m Uc !TP  
    %       [X,Y] = meshgrid(x,x); :5`BhFAd  
    %       [theta,r] = cart2pol(X,Y); A+lP]Oy0S  
    %       idx = r<=1; 4^0L2BVcv  
    %       p = 0:15; R1DXi  
    %       z = nan(size(X)); &[hq !v  
    %       y = zernfun2(p,r(idx),theta(idx)); R~],5_|  
    %       figure('Units','normalized') duKR;5:  
    %       for k = 1:length(p) t3)nG8> )  
    %           z(idx) = y(:,k); '<C I^5^  
    %           subplot(4,4,k) HV??B :  
    %           pcolor(x,x,z), shading interp jK^'s6i#  
    %           set(gca,'XTick',[],'YTick',[]) yjbqby7  
    %           axis square \HB4ikl  
    %           title(['Z_{' num2str(p(k)) '}']) |*im$[g=-  
    %       end ^p0BeSRiy;  
    % 4#z@B1Jx  
    %   See also ZERNPOL, ZERNFUN. :>.~"uWo{  
    /f9jLY +  
    %   Paul Fricker 11/13/2006 ^< ,Np+  
    I4Ys ,n  
    z Lw=*  
    % Check and prepare the inputs: Ny>tJ~I  
    % ----------------------------- MT,LO<.  
    if min(size(p))~=1 XTHy CK  
        error('zernfun2:Pvector','Input P must be vector.') ~(xIG  
    end uOqWMRsoi  
    ,?+rM ;  
    if any(p)>35 XQu~/{A=  
        error('zernfun2:P36', ... f.`noZN  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... lbv9 kk[  
               '(P = 0 to 35).']) szC~?]<YY  
    end _-*Lj;^V  
    $e;_N4d^  
    % Get the order and frequency corresonding to the function number: vnXa4\Vdy  
    % ---------------------------------------------------------------- aZYa<28?L%  
    p = p(:); "$? f&*  
    n = ceil((-3+sqrt(9+8*p))/2); Y" s1z<?  
    m = 2*p - n.*(n+2); vv  _I o  
    Y#_,Ig5.  
    % Pass the inputs to the function ZERNFUN: J3fcnI  
    % ----------------------------------------  t]Xdzy  
    switch nargin !aD/I%X  
        case 3 zLlu% Oc  
            z = zernfun(n,m,r,theta); FLO#!G  
        case 4 XQhBnam%  
            z = zernfun(n,m,r,theta,nflag); )DsC:cP  
        otherwise L{|V13?  
            error('zernfun2:nargin','Incorrect number of inputs.') > _1*/o JO  
    end <h2WM (n  
    Vt:]D?\3  
    % 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) wAxXK94#3  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. .N8AkQ(Ok  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of | eIN<RY5  
    %   order N and frequency M, evaluated at R.  N is a vector of mHo}, |  
    %   positive integers (including 0), and M is a vector with the ~#dNGWwG  
    %   same number of elements as N.  Each element k of M must be a *ta ``q  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) G}Cze Lw  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is sTO*  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix 4;_{*U-  
    %   with one column for every (N,M) pair, and one row for every 716JnG>  
    %   element in R. D[m;rcl  
    % \X5{>nNh  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- >v@R]9  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is [y| "iSD  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to -:`$8/A|  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 ywpk\  
    %   for all [n,m]. KFdTw{GlJ7  
    % C)`k{(-{  
    %   The radial Zernike polynomials are the radial portion of the c4oQ4  
    %   Zernike functions, which are an orthogonal basis on the unit gmy$_4+6o  
    %   circle.  The series representation of the radial Zernike u~\u8X3  
    %   polynomials is d6-a\]gF  
    % (,`ypD+3q  
    %          (n-m)/2  2&O!<C j  
    %            __ " 4#V$V  
    %    m      \       s                                          n-2s 1q<BYc+z  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r LY[XPV]t  
    %    n      s=0 zRdL-u%(#  
    % <\GP\G  
    %   The following table shows the first 12 polynomials. `3^ *K/K\  
    % D)XF@z;  
    %       n    m    Zernike polynomial    Normalization EA9`-xs|  
    %       --------------------------------------------- QWv+J a  
    %       0    0    1                        sqrt(2) bB'iK4  
    %       1    1    r                           2 @FKNB.>  
    %       2    0    2*r^2 - 1                sqrt(6) %geiJ z  
    %       2    2    r^2                      sqrt(6) ";yCo0*  
    %       3    1    3*r^3 - 2*r              sqrt(8) ^AK<]r<?L?  
    %       3    3    r^3                      sqrt(8) -! Hn,93  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) -l<b|`s=w.  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) Ro$'|}(+A  
    %       4    4    r^4                      sqrt(10) W"+*%x  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) X[:Hp`_$  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) %mPIr4$Pg  
    %       5    5    r^5                      sqrt(12) )#z c$D^U  
    %       --------------------------------------------- = ;#?CAa:  
    % $ 5ZBNGr  
    %   Example: z=B*s!G  
    % Aa-L<wZVPt  
    %       % Display three example Zernike radial polynomials 5mUHk]W  
    %       r = 0:0.01:1; -hw^3Af  
    %       n = [3 2 5]; MW8GM}Ho[  
    %       m = [1 2 1]; 9 o6ig>C  
    %       z = zernpol(n,m,r); nS)U+q-x&o  
    %       figure JsI` #  
    %       plot(r,z) 6/Y3#d  
    %       grid on HtB>#`'  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') Z`<S_PPz  
    % Y%y=  
    %   See also ZERNFUN, ZERNFUN2. Ac}+U q  
    p'fq&a+  
    % A note on the algorithm. `"zXf-qeE  
    % ------------------------ +<7~yZ[Z8  
    % The radial Zernike polynomials are computed using the series yEIM58l  
    % representation shown in the Help section above. For many special ?U.+SQ  
    % functions, direct evaluation using the series representation can hAtf)  
    % produce poor numerical results (floating point errors), because 9HrT>{@  
    % the summation often involves computing small differences between FIhq>L.q4  
    % large successive terms in the series. (In such cases, the functions HpY-7QTPJ~  
    % are often evaluated using alternative methods such as recurrence S[(Tpk2_  
    % relations: see the Legendre functions, for example). For the Zernike U;u@\E@2  
    % polynomials, however, this problem does not arise, because the UZ7Zzc#g  
    % polynomials are evaluated over the finite domain r = (0,1), and Jt5\  
    % because the coefficients for a given polynomial are generally all $(B|$e^:(  
    % of similar magnitude. =V~p QbZ  
    % U8Z(=*Z3  
    % ZERNPOL has been written using a vectorized implementation: multiple N|-M|1w96  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] 3*x_S"h  
    % values can be passed as inputs) for a vector of points R.  To achieve &7_xr.c7  
    % this vectorization most efficiently, the algorithm in ZERNPOL K8M[xaI@  
    % involves pre-determining all the powers p of R that are required to 69ZGdN  
    % compute the outputs, and then compiling the {R^p} into a single %^tKt  
    % matrix.  This avoids any redundant computation of the R^p, and ]R[j ]E.  
    % minimizes the sizes of certain intermediate variables. a)w *  
    % 5<ZE.'O  
    %   Paul Fricker 11/13/2006 ))m\d*  
    y(/"DUx  
    v&Xsyb0CaM  
    % Check and prepare the inputs: y,'M3GGl  
    % ----------------------------- +*&bgGhT  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Z$ q{!aY  
        error('zernpol:NMvectors','N and M must be vectors.') h=h4`uA9  
    end 2GeJ\1k  
    & -L$B  
    if length(n)~=length(m) B8A-|S!,U  
        error('zernpol:NMlength','N and M must be the same length.') &$$KC?!w  
    end ZLm?8g6-  
    N?7MYP  
    n = n(:); HZ%2WM  
    m = m(:); e$kBpG"D  
    length_n = length(n); sZ,xbfZby  
    mQ(6ahD U  
    if any(mod(n-m,2)) @p` *MWU  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') *} @Y"y  
    end 5B6twn~[  
    b7-M'-Km0_  
    if any(m<0) 2OT RP4U  
        error('zernpol:Mpositive','All M must be positive.') -u+@5K;^Y  
    end jlaU3qXL  
    Xa o*h(Q@L  
    if any(m>n) b,C2(?hg  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') V+`gkWe/  
    end ZAATV+Z  
    -DAkVFsN  
    if any( r>1 | r<0 ) 0F48T<i  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') =Q+i(UGHi  
    end 0 PdeK'7  
    fv@mA--  
    if ~any(size(r)==1) {"kE u  
        error('zernpol:Rvector','R must be a vector.') ?XCFR t,ol  
    end @QOlo -u  
    LAs#g||M  
    r = r(:); |!t &ZpdD  
    length_r = length(r); A]<+Aq@{  
    v@,n]"  
    if nargin==4 2Xw=kwu  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); Q)]C~Q  
        if ~isnorm (U_`Q1Jo  
            error('zernpol:normalization','Unrecognized normalization flag.') {*yFTP"93  
        end JRgrg &#  
    else h D5NX  
        isnorm = false; da[=d*I.  
    end %<dvdIB  
    jpwR\"UJ  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% q8Jhs7fv  
    % Compute the Zernike Polynomials  ujin+;1  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% gm^j8  B  
    DUrfC[jpv  
    % Determine the required powers of r: Ga<Uvr%+  
    % ----------------------------------- =Ff _)k  
    rpowers = []; 4*0C_F@RX  
    for j = 1:length(n) r~[Bzw"c  
        rpowers = [rpowers m(j):2:n(j)]; 7];AB;0"  
    end WHF[l1  
    rpowers = unique(rpowers); Yamu"#  
    % _.kd"  
    % Pre-compute the values of r raised to the required powers, eW<|I  
    % and compile them in a matrix: 6 4,('+  
    % ----------------------------- \=1$$EDS9  
    if rpowers(1)==0 F>F2Yql&W  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); &u`]Zn   
        rpowern = cat(2,rpowern{:}); ?2(5 2?cJ  
        rpowern = [ones(length_r,1) rpowern]; 4 EE7gkM5  
    else ~:krJ[=  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); u+7S/9q8  
        rpowern = cat(2,rpowern{:}); 8(zE^W,[8"  
    end 8l.bT|#O  
    G+~f  
    % Compute the values of the polynomials: mAM:Q*a'  
    % -------------------------------------- Rs@>LA  
    z = zeros(length_r,length_n); V|{\8&  2  
    for j = 1:length_n jd.{J{o  
        s = 0:(n(j)-m(j))/2; ?W 6 :$  
        pows = n(j):-2:m(j); e S: 8Pn  
        for k = length(s):-1:1 H8x66}  
            p = (1-2*mod(s(k),2))* ... .vnQZ*6  
                       prod(2:(n(j)-s(k)))/          ... \<aR^Sj.  
                       prod(2:s(k))/                 ... P @Jo[J<  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... $ucDz f=o  
                       prod(2:((n(j)+m(j))/2-s(k))); wi/qI(O!  
            idx = (pows(k)==rpowers); 3<x1s2U  
            z(:,j) = z(:,j) + p*rpowern(:,idx); ;7>k[?'e  
        end x%'5 rnm|  
         <*Gd0 v%  
        if isnorm v]GQb  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); \1He9~6  
        end V8hmfV~=]P  
    end 9u;/l#?@T  
    [.Rdq]w6  
    % EOF zernpol
    离线niuhelen
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    只看该作者 6楼 发表于: 2011-03-12
    这三个文件,我不知道该怎样把我的面型节点的坐标及轴向位移用起来,还烦请指点一下啊,谢谢啦!
    离线li_xin_feng
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    只看该作者 7楼 发表于: 2012-09-28
    我也正在找啊
    离线guapiqlh
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    只看该作者 8楼 发表于: 2014-03-04
    我也一直想了解这个多项式的应用,还没用过呢
    离线phoenixzqy
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    只看该作者 9楼 发表于: 2014-04-22
    回 guapiqlh 的帖子
    guapiqlh:我也一直想了解这个多项式的应用,还没用过呢 (2014-03-04 11:35)  #hP>IU  
    2C1NDrS;}  
    数值分析方法看一下就行了。其实就是正交多项式的应用。zernike也只不过是正交多项式的一种。 -$X4RS  
    z4~p(tl  
    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)