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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 n"c[,k+R`U  
    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! eceP0x  
     
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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 4V)kx[j  
    function z = zernfun(n,m,r,theta,nflag) .SU8)T  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. K0|FY=#2y  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N "*e$aTZB\  
    %   and angular frequency M, evaluated at positions (R,THETA) on the kTOzSiq  
    %   unit circle.  N is a vector of positive integers (including 0), and YYBDRR"  
    %   M is a vector with the same number of elements as N.  Each element I-]?"Q7Jz  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) dO! kk"qn  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, s+$ Q}|?u  
    %   and THETA is a vector of angles.  R and THETA must have the same 6]WAUK%h  
    %   length.  The output Z is a matrix with one column for every (N,M) Q{>+ft U  
    %   pair, and one row for every (R,THETA) pair. KQ!8ks]  
    % y.mda:$~=  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike [}E='m}u9+  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), U ]H#MiC!  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral hF~n)oQ  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, FXG]LoP  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized H)kwQRfu  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. Fo5FNNiID  
    % &[?\k>  
    %   The Zernike functions are an orthogonal basis on the unit circle. 823Y\x~>  
    %   They are used in disciplines such as astronomy, optics, and O:;w3u7;u  
    %   optometry to describe functions on a circular domain. ;u_X)  
    % J?"B%B5c  
    %   The following table lists the first 15 Zernike functions. )l C)@H}  
    % %S960  
    %       n    m    Zernike function           Normalization ohGJ1  
    %       -------------------------------------------------- 6_GhO@lOG  
    %       0    0    1                                 1 > PRFWO  
    %       1    1    r * cos(theta)                    2 V1N3iI  
    %       1   -1    r * sin(theta)                    2 vxBgGl  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) Q%`@0#"]Sv  
    %       2    0    (2*r^2 - 1)                    sqrt(3) @e.C"@G  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) _YhES-Ff  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) w e//|fA<  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) ].w4$OJ?  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) y@S$^jk.  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) %D{6[8  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) 'x#~'v*  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) yW=::=  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) zZPO&akB"  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) UmP/h@8  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) %v M-mbX  
    %       -------------------------------------------------- 5uGq%(24  
    % ?=sDM& '  
    %   Example 1: S6DKREO  
    % L\J;J%fz.  
    %       % Display the Zernike function Z(n=5,m=1) iHM%iUV  
    %       x = -1:0.01:1; D0-3eV -  
    %       [X,Y] = meshgrid(x,x); zFfr. g;L  
    %       [theta,r] = cart2pol(X,Y); AlaW=leTe  
    %       idx = r<=1; ]m3HF&  
    %       z = nan(size(X)); oWT3apGO  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); IVY]EkEG~  
    %       figure Qz1E 2yJ  
    %       pcolor(x,x,z), shading interp q 'yva  
    %       axis square, colorbar W aRw05r  
    %       title('Zernike function Z_5^1(r,\theta)') Vx u0F]%  
    % 6P l<'3&  
    %   Example 2: B6DYZ+7A  
    % W:2( .?  
    %       % Display the first 10 Zernike functions 6@5+m 0`u3  
    %       x = -1:0.01:1; `Y$4 H,8L  
    %       [X,Y] = meshgrid(x,x); *Hn8)x}E  
    %       [theta,r] = cart2pol(X,Y); & '`g#N  
    %       idx = r<=1; b{&)6M)zo  
    %       z = nan(size(X)); p?OoC  
    %       n = [0  1  1  2  2  2  3  3  3  3]; By!o3}~g  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; -`h)$&,  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; jvL[ JI,b  
    %       y = zernfun(n,m,r(idx),theta(idx)); Ax7[;|2  
    %       figure('Units','normalized') <)H9V-5aZ  
    %       for k = 1:10 v@L;x [Q  
    %           z(idx) = y(:,k); p8O2Z? \  
    %           subplot(4,7,Nplot(k)) \!ZTL1b8t  
    %           pcolor(x,x,z), shading interp Q Z  
    %           set(gca,'XTick',[],'YTick',[]) ! n@KU!&k  
    %           axis square 83_h J  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) E{`fF8]K  
    %       end AQvudx)@"  
    % ]h+j)J}[A  
    %   See also ZERNPOL, ZERNFUN2. +I|vzz`ZVr  
    O<?R)NH-P  
    %   Paul Fricker 11/13/2006 R&k<AZ  
    cdT7 @  
    YjKxb9  
    % Check and prepare the inputs: ;N0XFjdR  
    % ----------------------------- ^hM4j{|&M  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 7R\<inCQ  
        error('zernfun:NMvectors','N and M must be vectors.') $%#!bV  
    end fIU#M]Xx  
    aX'*pK/-  
    if length(n)~=length(m) uy$e?{Jf  
        error('zernfun:NMlength','N and M must be the same length.') p_%Rt"!  
    end e*NnVys  
    ?CPahU  
    n = n(:); }19\.z&J  
    m = m(:); iqWQ!r^  
    if any(mod(n-m,2)) ]N?kG`[  
        error('zernfun:NMmultiplesof2', ... ?Z/V~,  
              'All N and M must differ by multiples of 2 (including 0).') hz@bW2S.  
    end !Wnb|=j  
    vA8nvoi  
    if any(m>n) 8<Av@9 *}  
        error('zernfun:MlessthanN', ... j A%u 5V  
              'Each M must be less than or equal to its corresponding N.') 2c*GuF9(0  
    end E:nF$#<'N  
    s.C_Zf~3  
    if any( r>1 | r<0 ) X l5 A 'h  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') 8{sGNCvU  
    end u^  ~W+  
    EaN6^S=  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 83#mB:^R  
        error('zernfun:RTHvector','R and THETA must be vectors.') 4H&+dR I"  
    end ?6WY:Zec@  
    [{,1=AB  
    r = r(:); ~Mxvq9vaD  
    theta = theta(:); wb l&  
    length_r = length(r); $ddCTS^  
    if length_r~=length(theta) *$g-:ILRuZ  
        error('zernfun:RTHlength', ... Y$@?.)tY  
              'The number of R- and THETA-values must be equal.') "4{r6[dn  
    end S"H2 7  
    <RL]  
    % Check normalization: Q*Pq{]0K  
    % -------------------- ]c'A%:f<  
    if nargin==5 && ischar(nflag) 4Fr  
        isnorm = strcmpi(nflag,'norm'); /j.9$H'y  
        if ~isnorm Q^")jPd  
            error('zernfun:normalization','Unrecognized normalization flag.') S)@j6(HC4  
        end |yPu!pfl  
    else SvF<p3  
        isnorm = false; jmZI7?<z  
    end a\*yZlXKs  
    =T7.~W  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% }N52$L0[  
    % Compute the Zernike Polynomials =rdV ]{Wc  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% .7X^YKR  
    X"%gQ.1|{j  
    % Determine the required powers of r: DN6Mo<H  
    % ----------------------------------- {+>-7 9b  
    m_abs = abs(m); f3y=Wxk[  
    rpowers = []; N"ST@/j.A  
    for j = 1:length(n) TB31- ()  
        rpowers = [rpowers m_abs(j):2:n(j)]; } 0y"F  
    end do'GlU oMC  
    rpowers = unique(rpowers); $[ *w"iQ  
    7b+6%fV  
    % Pre-compute the values of r raised to the required powers, O;3>sLgc  
    % and compile them in a matrix: k+*u/neh  
    % ----------------------------- a d\ot#V  
    if rpowers(1)==0 cFXp  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); xskz) kk  
        rpowern = cat(2,rpowern{:}); MF'JeM;H  
        rpowern = [ones(length_r,1) rpowern]; 5[0?g@aO  
    else v`T c}c '  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); E2-\]?\F(  
        rpowern = cat(2,rpowern{:}); &UFZS94@r  
    end CWKm(@"5  
    M"L=L5OH-  
    % Compute the values of the polynomials: !5!<C,U  
    % -------------------------------------- |Y.?_lC  
    y = zeros(length_r,length(n)); ;n;p@Uu[ b  
    for j = 1:length(n) );YDtGip J  
        s = 0:(n(j)-m_abs(j))/2; 0> \sQ,T  
        pows = n(j):-2:m_abs(j); yB!dp;gM{  
        for k = length(s):-1:1 ^<6[.)  
            p = (1-2*mod(s(k),2))* ... m]&SNz=  
                       prod(2:(n(j)-s(k)))/              ... 3XNCAb2  
                       prod(2:s(k))/                     ... N2o7%gJw  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... #\ErY3k6&  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); nJ;.Td  
            idx = (pows(k)==rpowers); @ N m@]q  
            y(:,j) = y(:,j) + p*rpowern(:,idx); # f\rt   
        end lEBLZ}}\  
         NHE18_v5  
        if isnorm _#8MkW#]~  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); J .<F"r>  
        end B)UZ`?>c  
    end \b>] 8Un"  
    % END: Compute the Zernike Polynomials ! d gNtI@  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% CvdN"k  
     L"aeG  
    % Compute the Zernike functions: Ho]su?  
    % ------------------------------ Zwx%7l;C  
    idx_pos = m>0; B-mowmJ3dg  
    idx_neg = m<0; (;,sc$H]  
    @(lh%@hO  
    z = y; .RL=xb|[  
    if any(idx_pos) G+m }MOQP7  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 2KZneS`  
    end nr3==21Om4  
    if any(idx_neg) moE2G?R  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); GtHivC  
    end lLIA w$  
    A=>u 1h69  
    % 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) Mb=" Te>|  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. `F6C-  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated M3Kfd  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive 8;X-)&R  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, WF"k[2  
    %   and THETA is a vector of angles.  R and THETA must have the same #fM'>$N  
    %   length.  The output Z is a matrix with one column for every P-value, )`}:8y?  
    %   and one row for every (R,THETA) pair. PI<vxjOK`  
    % I}Q2Vu<  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike  .wr>]yN  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) rM "l@3hP  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) \`"ht  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 B erwI 7!=  
    %   for all p. g=I})s:CTp  
    % .|=\z9_7S8  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 1 {)Q[#l  
    %   Zernike functions (order N<=7).  In some disciplines it is :Q q#Z  
    %   traditional to label the first 36 functions using a single mode w#J2 wS  
    %   number P instead of separate numbers for the order N and azimuthal O H7FkR  
    %   frequency M. 8XbT`y  
    % B-ESFATc  
    %   Example: xLn%hxm?,  
    % 9>$p  
    %       % Display the first 16 Zernike functions v8D C21pb  
    %       x = -1:0.01:1; /7LR;>Bj  
    %       [X,Y] = meshgrid(x,x); Q=:|R3U/  
    %       [theta,r] = cart2pol(X,Y); :H[6Lg\*  
    %       idx = r<=1; },[}$m %  
    %       p = 0:15; t:c.LFrF  
    %       z = nan(size(X)); a: K[ y  
    %       y = zernfun2(p,r(idx),theta(idx)); F5#YOck&,  
    %       figure('Units','normalized') 5(8@%6>ruj  
    %       for k = 1:length(p) ~_ a-E  
    %           z(idx) = y(:,k); 2BobH_ H  
    %           subplot(4,4,k) tI{_y  
    %           pcolor(x,x,z), shading interp bjS {(  
    %           set(gca,'XTick',[],'YTick',[])  LIdF 0  
    %           axis square |Ds=)S" K  
    %           title(['Z_{' num2str(p(k)) '}']) Qei" '~1a  
    %       end =qIyqbXz  
    % cGD(.=  
    %   See also ZERNPOL, ZERNFUN. UZ$/Ni  
    P }uOJVQ_  
    %   Paul Fricker 11/13/2006 S@sO;-^+  
    07$o;W@  
    {y;n:^  
    % Check and prepare the inputs: QdC<Sk!G  
    % ----------------------------- %07SFu#  
    if min(size(p))~=1 M@ZI\  
        error('zernfun2:Pvector','Input P must be vector.') X 8`Sf>  
    end Lh<).<S  
    9k=3u;$v  
    if any(p)>35 IIqUZJ  
        error('zernfun2:P36', ... abEmRJTmW  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... 1i ] ^{;]  
               '(P = 0 to 35).']) o? $.fhD   
    end K4);HJ|=  
    UY 2OZ& &  
    % Get the order and frequency corresonding to the function number: ;S*}WqP,  
    % ---------------------------------------------------------------- <^uBoKB/f  
    p = p(:); k$7Jj-+~  
    n = ceil((-3+sqrt(9+8*p))/2);  f V(J|  
    m = 2*p - n.*(n+2); IqGdfL6[(  
    r"R#@V\'1b  
    % Pass the inputs to the function ZERNFUN: OUXR  
    % ---------------------------------------- a@*\o+Su  
    switch nargin I`p;F!s  
        case 3 "wHFN>5B  
            z = zernfun(n,m,r,theta); @OHm#`~  
        case 4 }iuw5dik+  
            z = zernfun(n,m,r,theta,nflag); @ry_nKr9  
        otherwise ?F;8Pa/  
            error('zernfun2:nargin','Incorrect number of inputs.') PiYxk+N  
    end ofv)SCjd  
    = 9]~ yt  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) Ez=Olbk  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. ^a1^\X.~  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of d<N:[Y\4l  
    %   order N and frequency M, evaluated at R.  N is a vector of n=ux5M  
    %   positive integers (including 0), and M is a vector with the 8pgEix/M5o  
    %   same number of elements as N.  Each element k of M must be a {8%a5DiM  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) u-5{U-^_  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is b%/ 1$>_  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix > "=>3  
    %   with one column for every (N,M) pair, and one row for every g'qa}/X  
    %   element in R. H+Sz=tg5  
    % j^2wb+`  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- t1y4 7fX6  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is eHDN\QA 2  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to 5N&?KA-  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 ~La>?:g <+  
    %   for all [n,m]. "!%l/_p?  
    % W1=H8 O  
    %   The radial Zernike polynomials are the radial portion of the 'ub@]ru|  
    %   Zernike functions, which are an orthogonal basis on the unit v-_e)m^  
    %   circle.  The series representation of the radial Zernike n#OB%@]<V  
    %   polynomials is '(L7;+E  
    % b`O'1r\Y;  
    %          (n-m)/2 /CG"]!2 "  
    %            __ )f<z% :I+Z  
    %    m      \       s                                          n-2s 4Ic*9t3  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r V /V9B2.$  
    %    n      s=0 ,>mrPtxN  
    % xx%j.zDI]  
    %   The following table shows the first 12 polynomials. SJ>vwmA4  
    % ^qD$z=z-  
    %       n    m    Zernike polynomial    Normalization ? '{SX9  
    %       --------------------------------------------- 8C9-_Ng`  
    %       0    0    1                        sqrt(2) (jl D+Y_  
    %       1    1    r                           2 OA"q[s  
    %       2    0    2*r^2 - 1                sqrt(6) h'&%>Q2  
    %       2    2    r^2                      sqrt(6) (S\[Y9  
    %       3    1    3*r^3 - 2*r              sqrt(8) l#Y,R 0  
    %       3    3    r^3                      sqrt(8) aH/ k Ua  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) X=fYWj[H,  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) Ks`J([(W&  
    %       4    4    r^4                      sqrt(10) [;) ,\\u,d  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) f5VLw`m}.8  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) jQ^|3#L\  
    %       5    5    r^5                      sqrt(12) ~;{; ,8!)  
    %       --------------------------------------------- WuUk9_ g  
    % iN8zo:&Z  
    %   Example: 'XP7" N47O  
    % V7fq4O^:  
    %       % Display three example Zernike radial polynomials fN^8{w/O  
    %       r = 0:0.01:1; Oso#+  
    %       n = [3 2 5]; !/i{l  
    %       m = [1 2 1]; h-<81"}j1  
    %       z = zernpol(n,m,r); G[I"8iS,  
    %       figure ++Ts  
    %       plot(r,z) c>:wd@w  
    %       grid on ZyPVy  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') 9u}Hmb  
    % rgtT~$S  
    %   See also ZERNFUN, ZERNFUN2. hxd`OG<gF  
    ex (.=X 1  
    % A note on the algorithm. peuZ&yK+"  
    % ------------------------ r 8rgY42  
    % The radial Zernike polynomials are computed using the series k(7&N0V%zz  
    % representation shown in the Help section above. For many special F {4bo$~>  
    % functions, direct evaluation using the series representation can tKx~1-  
    % produce poor numerical results (floating point errors), because MSqVlj  
    % the summation often involves computing small differences between 4`]^@"{  
    % large successive terms in the series. (In such cases, the functions O#~yKqB  
    % are often evaluated using alternative methods such as recurrence dkBIx$t  
    % relations: see the Legendre functions, for example). For the Zernike Tg)| or/ %  
    % polynomials, however, this problem does not arise, because the ][h%UrV  
    % polynomials are evaluated over the finite domain r = (0,1), and ^-Kf']hU  
    % because the coefficients for a given polynomial are generally all })8N5C+KU  
    % of similar magnitude. rt~d6|6  
    % Pz|>"'  
    % ZERNPOL has been written using a vectorized implementation: multiple /dQl)tL  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] QIvVcfM^  
    % values can be passed as inputs) for a vector of points R.  To achieve O{G?;H$  
    % this vectorization most efficiently, the algorithm in ZERNPOL 1&evG-#<:  
    % involves pre-determining all the powers p of R that are required to u9GQU  
    % compute the outputs, and then compiling the {R^p} into a single j9 4=hJVKi  
    % matrix.  This avoids any redundant computation of the R^p, and O/a4]r+_  
    % minimizes the sizes of certain intermediate variables. )E@.!Ut4o  
    % '(yAfL 9}  
    %   Paul Fricker 11/13/2006 lC("y' ::  
    wyj{zWRJp  
    (\hx` Yh=>  
    % Check and prepare the inputs: \lf;P?M^  
    % ----------------------------- 5Y'qaIFR  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) aweV#j(y  
        error('zernpol:NMvectors','N and M must be vectors.') 2 %@4]  
    end E=CsIK   
    #Z`q+@@ ]A  
    if length(n)~=length(m) ,+vy,<e&  
        error('zernpol:NMlength','N and M must be the same length.') m=A(NKZ   
    end m}aB?+i  
    kmsb hYM)  
    n = n(:); Agg<tM{yB  
    m = m(:); aS{n8P6vW  
    length_n = length(n); AJ? r,!)  
    EZy)A$|  
    if any(mod(n-m,2)) gk[aM~p  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') nE&@Q  
    end (vPN5F  
    \y)rt )  
    if any(m<0) T/Gz94c  
        error('zernpol:Mpositive','All M must be positive.') ;R5`"`  
    end qL&[K>2z  
    8# >op6^  
    if any(m>n) H*QIB_  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') +xSHL|:b  
    end m!4ndO;0vh  
    djQH1^ (IU  
    if any( r>1 | r<0 ) *:YiimOY"  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') I<4Pur>"  
    end !M]uL&:  
    q J=~Y|(  
    if ~any(size(r)==1) ?(y*nD[a  
        error('zernpol:Rvector','R must be a vector.') 3n1;G8Nf  
    end C:* *;=.  
    ?m=N]!n  
    r = r(:); #`iB`|  
    length_r = length(r); @ ZwvBH  
    a| x.C6P e  
    if nargin==4 NP#w +Qw  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); !t %j?\f  
        if ~isnorm _AYK435>N  
            error('zernpol:normalization','Unrecognized normalization flag.') P*Uwg&Qz)  
        end ;| 5F[  
    else Ovt.!8  
        isnorm = false; M~#gRAUJ  
    end # E^1|:  
    y$F'(b| )  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ^q vbqfh  
    % Compute the Zernike Polynomials r CHl?J  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% } FlT%>Gw  
    [0[i5'K:  
    % Determine the required powers of r: GR.^glG?6  
    % ----------------------------------- | y# Jx  
    rpowers = []; vnt%XU,,Y  
    for j = 1:length(n) qu6D 5t  
        rpowers = [rpowers m(j):2:n(j)]; B6nX$T4zP  
    end vq0Tk bzs  
    rpowers = unique(rpowers); PbgP\JeX  
    6V:U (g  
    % Pre-compute the values of r raised to the required powers, r1m]HFN  
    % and compile them in a matrix: S6M}WR^,  
    % ----------------------------- )?n aN  
    if rpowers(1)==0 eIEeb,#i  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); E *6Cw l  
        rpowern = cat(2,rpowern{:}); H8zK$!  
        rpowern = [ones(length_r,1) rpowern]; IH&|Tcf\  
    else =/+-<px  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 4qh?,^Dq  
        rpowern = cat(2,rpowern{:}); ,~$p,ALwN7  
    end gNrjo=  
    ) 0W{]2  
    % Compute the values of the polynomials: 4Zddw0|2  
    % -------------------------------------- GL0L!="!  
    z = zeros(length_r,length_n); "]x'PI 4J  
    for j = 1:length_n JCzeXNY  
        s = 0:(n(j)-m(j))/2; #PW9:_BE  
        pows = n(j):-2:m(j); c(m<h+ 2VL  
        for k = length(s):-1:1 !bx;Ta.  
            p = (1-2*mod(s(k),2))* ... Y;Dp3v !  
                       prod(2:(n(j)-s(k)))/          ... =tn)}Y.<e  
                       prod(2:s(k))/                 ... rgv?gaQ>  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... J26 VnK  
                       prod(2:((n(j)+m(j))/2-s(k))); I)6+6pm  
            idx = (pows(k)==rpowers); a=1@*ID  
            z(:,j) = z(:,j) + p*rpowern(:,idx); M}-Rzc  
        end 2'\H\|  
         aQcleTb  
        if isnorm ]t,BMu=%  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); (9 GWbB?  
        end uc\Kg1{  
    end 7wqK>Y1a  
    PO^ij2eS  
    % 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)  YD6'#(  
    X+9>A.92  
    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。 b8UO,fY q  
    ~36!?&eA8  
    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)