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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 E3f9<hm   
    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! "c,!vc4  
     
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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 PrSkHxm  
    function z = zernfun(n,m,r,theta,nflag) 5V@&o`!=h  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. 9afh[3qm  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N QrC/ssf}  
    %   and angular frequency M, evaluated at positions (R,THETA) on the VNj@5s  
    %   unit circle.  N is a vector of positive integers (including 0), and ,H39V+Y*  
    %   M is a vector with the same number of elements as N.  Each element XsUUJuCG  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) ],[)uTZc  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, 9P.(^SD][z  
    %   and THETA is a vector of angles.  R and THETA must have the same J>%t<xYf4  
    %   length.  The output Z is a matrix with one column for every (N,M) d0 -~| `5  
    %   pair, and one row for every (R,THETA) pair. M3(k'q7&:  
    % 6Y7H|>g)  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike C),7- ?  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), M4?8xuC  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral Jq .L:>x  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, `G?qY8  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized qS.)UaA  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. w!`Umll2  
    % Z^# ]#f  
    %   The Zernike functions are an orthogonal basis on the unit circle. +.@c{5J<  
    %   They are used in disciplines such as astronomy, optics, and "K?Q  
    %   optometry to describe functions on a circular domain. TvQ^DZbe  
    % .N"~zOV<#  
    %   The following table lists the first 15 Zernike functions. K\&o2lo]  
    % Q\9K2=4  
    %       n    m    Zernike function           Normalization |s=`w8p  
    %       -------------------------------------------------- vv.PF~:  
    %       0    0    1                                 1 f^9&WT  
    %       1    1    r * cos(theta)                    2 Rri`dmH   
    %       1   -1    r * sin(theta)                    2 Hm9<fQuM  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) 8!zb F<W9  
    %       2    0    (2*r^2 - 1)                    sqrt(3) G{b:i8}l  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) >]&X ^V%Q#  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) S&?7K-F>_o  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) kdcQw7G  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) `#6x=24  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) 9y^/GwUQ  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) "8(U\KaX  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) SRL-Z&M  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) Wx]d $_  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) q*8lnk  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) >85zQ 1aL  
    %       -------------------------------------------------- wsnK3tM7-  
    % @6&JR<g*t  
    %   Example 1: ;&f1vi4  
    % {'R)4hL  
    %       % Display the Zernike function Z(n=5,m=1) rWQY?K@  
    %       x = -1:0.01:1; }1QF+C f  
    %       [X,Y] = meshgrid(x,x); Fr5 Xp  
    %       [theta,r] = cart2pol(X,Y); "!L kp2\  
    %       idx = r<=1; saW!9HQj  
    %       z = nan(size(X)); S " pI  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); "C74  
    %       figure {1~T]5  
    %       pcolor(x,x,z), shading interp : ejJV 6.  
    %       axis square, colorbar uLVBM]Qj  
    %       title('Zernike function Z_5^1(r,\theta)') };S0 G!  
    % n29(!10Px  
    %   Example 2: #a,9B-X  
    % kMxjS^fr  
    %       % Display the first 10 Zernike functions vV^dm)?  
    %       x = -1:0.01:1; C;qMw-*F  
    %       [X,Y] = meshgrid(x,x); yA;W/I4  
    %       [theta,r] = cart2pol(X,Y); }htPTOy5  
    %       idx = r<=1; Ty+I8e]{  
    %       z = nan(size(X)); &88oB6$D^q  
    %       n = [0  1  1  2  2  2  3  3  3  3]; KQmZ#W%2m  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; IOEM[zhb$  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; Z8&' f,  
    %       y = zernfun(n,m,r(idx),theta(idx)); 3?E}t*/  
    %       figure('Units','normalized') A';QuWdT  
    %       for k = 1:10 ~<r i97)  
    %           z(idx) = y(:,k); >Ko[Xb-8^_  
    %           subplot(4,7,Nplot(k)) P!<[U!<hH  
    %           pcolor(x,x,z), shading interp ngyY  
    %           set(gca,'XTick',[],'YTick',[]) "2hh-L7ql  
    %           axis square rK|*hcy  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) Ceg!w#8Z,  
    %       end +>YfRqz:KB  
    % lhPGE_\  
    %   See also ZERNPOL, ZERNFUN2. 5 9 -!6;T  
    '^}+Fv<O  
    %   Paul Fricker 11/13/2006 (3%t+aqq  
    P))^vUt~  
    Jqfm@Y  
    % Check and prepare the inputs: &u8z5pls8  
    % ----------------------------- )#[|hb=o  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) flnVYQe  
        error('zernfun:NMvectors','N and M must be vectors.') rFu ez$  
    end -=5)NH t  
    U Cb02h  
    if length(n)~=length(m) /[pqI0sf<A  
        error('zernfun:NMlength','N and M must be the same length.') =NDOS{($  
    end 5H !y46z  
    hh"-w3+  
    n = n(:); rt!r2dq"  
    m = m(:); !%S4 n  
    if any(mod(n-m,2)) 2\@Z5m3B  
        error('zernfun:NMmultiplesof2', ... D>kD1B1  
              'All N and M must differ by multiples of 2 (including 0).') {o|k.zy  
    end "H+,E_&(  
    e7k%6'@  
    if any(m>n) *g$i5!yM'  
        error('zernfun:MlessthanN', ... `W5-.Tv  
              'Each M must be less than or equal to its corresponding N.') O\Eqr?%L)  
    end wNDbHR  
    @d&H]5  
    if any( r>1 | r<0 ) vsMmCd)7U  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') n=!uNu7  
    end GyC)EFd  
    2wlKBSON  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ,8VU&?`<}  
        error('zernfun:RTHvector','R and THETA must be vectors.') <nzN$"%  
    end 6 /Y1 wu  
    G|4^_`-  
    r = r(:); 4Z5#F]OA7  
    theta = theta(:); H3O@9YU  
    length_r = length(r); ht6244:  
    if length_r~=length(theta) aC^$*qN-)  
        error('zernfun:RTHlength', ... reiU%C  
              'The number of R- and THETA-values must be equal.') U A-7nb  
    end ..qd,9H  
    u, kU$  
    % Check normalization: J;QUPpH Z  
    % -------------------- Pe ~c  
    if nargin==5 && ischar(nflag) l-O$m  
        isnorm = strcmpi(nflag,'norm'); ls|LCQPx  
        if ~isnorm 6X_\Ve  
            error('zernfun:normalization','Unrecognized normalization flag.') :b /J\  
        end 2qU&l|>  
    else zx%X~U   
        isnorm = false; X0$@Ik  
    end = r4!V>  
    4s.]M>Yb  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% j8$Zv%Ca%  
    % Compute the Zernike Polynomials Poy^RpnX  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Mr3-q  
    ")UwkF  
    % Determine the required powers of r: q]c5MlJXF  
    % ----------------------------------- C"eXs#A  
    m_abs = abs(m); -$-8W  
    rpowers = []; h*l&RR:i  
    for j = 1:length(n) 6|;Uq'  
        rpowers = [rpowers m_abs(j):2:n(j)]; Q!'qC*Gyfn  
    end ^DAu5|--R  
    rpowers = unique(rpowers); /@YCA}|/  
    wEEn?  
    % Pre-compute the values of r raised to the required powers, C/@LZ OEL  
    % and compile them in a matrix: cxyM\@QB3  
    % ----------------------------- ?S[Y:<R{:  
    if rpowers(1)==0 %J7UP4  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); \:_3i\2p  
        rpowern = cat(2,rpowern{:}); ERz;H!pU8  
        rpowern = [ones(length_r,1) rpowern]; 7+,vTsCd  
    else  xvm5   
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); W`)<vGn=Y  
        rpowern = cat(2,rpowern{:}); eph)=F$  
    end FfFak@H  
    2}WDw>V  
    % Compute the values of the polynomials: pbBoy+.>  
    % -------------------------------------- $P {K2"Oc  
    y = zeros(length_r,length(n)); T0QvnIaP  
    for j = 1:length(n) 7&%^>PU7  
        s = 0:(n(j)-m_abs(j))/2; ff2d @P,!  
        pows = n(j):-2:m_abs(j); ;)hw%Z]Jj$  
        for k = length(s):-1:1 Dd $qQ  
            p = (1-2*mod(s(k),2))* ... h#.N3o  
                       prod(2:(n(j)-s(k)))/              ... nWYCh7  
                       prod(2:s(k))/                     ... |%7cdMC  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... '\7G@g?UZ  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); U~f4e7x*O  
            idx = (pows(k)==rpowers); !!,0'c  
            y(:,j) = y(:,j) + p*rpowern(:,idx); L'A)6^d@S  
        end dF@)M  
         > s EjR!  
        if isnorm -j2 (R?a  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); u'5`[U -!  
        end c z'5iK  
    end a \5FAkI  
    % END: Compute the Zernike Polynomials Ao.\  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% vZAv_8S)  
    B(5c9DI`  
    % Compute the Zernike functions: 8*a), 3aK  
    % ------------------------------ @w9{5D4  
    idx_pos = m>0; 4ne5=YY *  
    idx_neg = m<0; '+y_\  
    fw-\|fP  
    z = y; vT{kL  
    if any(idx_pos) J%rP$O$  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); X&\d)/Y  
    end d IB }_L  
    if any(idx_neg) Snw3`|Y~<  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); =?Fkn4t  
    end ]!"S+gT*C  
    PX O!t]*  
    % 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) ^_5|BT@  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. k[|~NLB8  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated {,$rkwW  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive PRu&3BP  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, -yH,5vD  
    %   and THETA is a vector of angles.  R and THETA must have the same @tUoD>f  
    %   length.  The output Z is a matrix with one column for every P-value, n.6T OF  
    %   and one row for every (R,THETA) pair. B{6<;u)[  
    % T2'RATfG  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike db72W x0>  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) Tbbz'b;{  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) .8gl< vX  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 (U1]:tZ<.  
    %   for all p. P.\nLE J=  
    % (nSml,gU  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 $hrIO+  
    %   Zernike functions (order N<=7).  In some disciplines it is R=)55qu  
    %   traditional to label the first 36 functions using a single mode </zXA$m  
    %   number P instead of separate numbers for the order N and azimuthal r|<DqTc6l  
    %   frequency M. k)\gWPH  
    % (#\pQ51  
    %   Example: 48D?'lW %  
    % *<4Em{rZ5  
    %       % Display the first 16 Zernike functions w|Cx>8P8@  
    %       x = -1:0.01:1; A&5:ATQ/|  
    %       [X,Y] = meshgrid(x,x); .i"W8~<e  
    %       [theta,r] = cart2pol(X,Y); AC%JC+  
    %       idx = r<=1; 77 r(*.O|  
    %       p = 0:15; zU5Hb2a  
    %       z = nan(size(X)); O'*@ Ytn  
    %       y = zernfun2(p,r(idx),theta(idx)); 0Rn+`UnwB  
    %       figure('Units','normalized') "j<bA8$Vw  
    %       for k = 1:length(p) []kN16F  
    %           z(idx) = y(:,k); s Kicn5  
    %           subplot(4,4,k) U Z_'><++  
    %           pcolor(x,x,z), shading interp AasZuO_I  
    %           set(gca,'XTick',[],'YTick',[]) 1QqHF$S  
    %           axis square E;Y;r"  
    %           title(['Z_{' num2str(p(k)) '}']) =<z.mzqu5  
    %       end /s:fW+C  
    % \6I +K"  
    %   See also ZERNPOL, ZERNFUN. {MdLX.ycc)  
    &K+0xnUH  
    %   Paul Fricker 11/13/2006 _~'+Qe_o$5  
    <W)u{KS#TY  
    '_P\#7$!MV  
    % Check and prepare the inputs: U/{6% Qy  
    % ----------------------------- W;6vpPhg#!  
    if min(size(p))~=1 ?DV5y|}pj  
        error('zernfun2:Pvector','Input P must be vector.') Ucw yxX I  
    end Rf-[svA  
    =}!Mf'  
    if any(p)>35 tiPa6tQ  
        error('zernfun2:P36', ... e$)300 o  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... 9O.YOiW  
               '(P = 0 to 35).']) (@0O   
    end ,3i,P(?(  
    Ey[On^$  
    % Get the order and frequency corresonding to the function number: eJCjJ)  
    % ---------------------------------------------------------------- {LzH&qu  
    p = p(:); sW#JjtK  
    n = ceil((-3+sqrt(9+8*p))/2); K4Dp:2/K%  
    m = 2*p - n.*(n+2); UaG1c%7?X  
    P(k(m< 0  
    % Pass the inputs to the function ZERNFUN: \G@wp5  
    % ---------------------------------------- I751 t  
    switch nargin V%0I%\0Y  
        case 3 az;Q"V'6  
            z = zernfun(n,m,r,theta); bizTd  
        case 4 a&{X!:X  
            z = zernfun(n,m,r,theta,nflag); "t=hzn"~%  
        otherwise G2{O9  
            error('zernfun2:nargin','Incorrect number of inputs.') >O9o,o/6R  
    end t`'iU$:1f  
    5+Mdh`  
    % 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) AwUi+|7r])  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. |WryBzZ>on  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of T=a=B(  
    %   order N and frequency M, evaluated at R.  N is a vector of \<0B1m  
    %   positive integers (including 0), and M is a vector with the iZ3W"Vd`b  
    %   same number of elements as N.  Each element k of M must be a UM*jKi2]"  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) vR6Bn  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is yqXH:757~  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix cV{%^0? D  
    %   with one column for every (N,M) pair, and one row for every _/*U2.xS  
    %   element in R. 3l<S}k@M)  
    % Z BUArIC  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- $/1c= Y@  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is U_l'3oPJw  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to ;!< Znw  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 D<i[LZd  
    %   for all [n,m]. BElJB&I  
    % /%aiEhL  
    %   The radial Zernike polynomials are the radial portion of the 5m:i6,4  
    %   Zernike functions, which are an orthogonal basis on the unit }{9&:!uA  
    %   circle.  The series representation of the radial Zernike [[~w0G~1  
    %   polynomials is Hy"x  
    % XNM a0  
    %          (n-m)/2 kU-t7'?4  
    %            __ Z4$cyL'$P  
    %    m      \       s                                          n-2s d1@%W;qX!  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r ;;$#)b  
    %    n      s=0 /y7M lU9  
    % if;71ZE  
    %   The following table shows the first 12 polynomials. I]"96'|N  
    % 9z;HsUv  
    %       n    m    Zernike polynomial    Normalization -V"22sR]  
    %       --------------------------------------------- '1fNBH2  
    %       0    0    1                        sqrt(2) t%zpNd2lk  
    %       1    1    r                           2 _sE#)@p  
    %       2    0    2*r^2 - 1                sqrt(6)  O|A_PyW  
    %       2    2    r^2                      sqrt(6) ]9=h%5Ji>  
    %       3    1    3*r^3 - 2*r              sqrt(8) LWsP ya  
    %       3    3    r^3                      sqrt(8) $P7iRM]  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) I-]>d;4.  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) Q(d9n8  
    %       4    4    r^4                      sqrt(10) iGDLZE+?  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) kL7#W9  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) ffXyc2o  
    %       5    5    r^5                      sqrt(12) {*ZY(6^  
    %       --------------------------------------------- Ogt]_  
    % 1QZ&Mj^^  
    %   Example: XS0xLt=  
    %  HBys  
    %       % Display three example Zernike radial polynomials V]c;^  
    %       r = 0:0.01:1; @\oz4^  
    %       n = [3 2 5]; cWGDee(  
    %       m = [1 2 1]; }),w1/#5u8  
    %       z = zernpol(n,m,r); b96%")  
    %       figure <D&)OxEn\  
    %       plot(r,z) &~UJf4b|A  
    %       grid on 3QSZ ZJ  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') rV({4cIe9R  
    % ]`g <w#  
    %   See also ZERNFUN, ZERNFUN2. 3Y)PU=  
    @cRZk`|1n  
    % A note on the algorithm. xR"M*%{@0  
    % ------------------------ +5.t. d  
    % The radial Zernike polynomials are computed using the series z|?R/Gf8  
    % representation shown in the Help section above. For many special qjJBcu_C'S  
    % functions, direct evaluation using the series representation can #>Y'sd5'A  
    % produce poor numerical results (floating point errors), because 7f<EoSK  
    % the summation often involves computing small differences between k+V6,V)my  
    % large successive terms in the series. (In such cases, the functions 8,O33qwH  
    % are often evaluated using alternative methods such as recurrence 4$/i%B#ad  
    % relations: see the Legendre functions, for example). For the Zernike 2#X4G~>#h  
    % polynomials, however, this problem does not arise, because the  $rXh0g  
    % polynomials are evaluated over the finite domain r = (0,1), and 7@.cOB`y@3  
    % because the coefficients for a given polynomial are generally all [ rNXQ` /  
    % of similar magnitude. ,U )"WLmY  
    % Bb&^ {7  
    % ZERNPOL has been written using a vectorized implementation: multiple FtufuL?JS  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] ,:Vm6u!  
    % values can be passed as inputs) for a vector of points R.  To achieve I fO;S*Qt  
    % this vectorization most efficiently, the algorithm in ZERNPOL UgB'[@McS  
    % involves pre-determining all the powers p of R that are required to N3t0-6$_  
    % compute the outputs, and then compiling the {R^p} into a single e& ANp0|W  
    % matrix.  This avoids any redundant computation of the R^p, and d"G+8}.4  
    % minimizes the sizes of certain intermediate variables. y5= `ap  
    % 5_0(D;Q  
    %   Paul Fricker 11/13/2006 /$n ~lf  
    ~zm 7?_"@]  
    dk QaM@  
    % Check and prepare the inputs: _qvK*nE  
    % ----------------------------- A392=:N+Q  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) q0%  
        error('zernpol:NMvectors','N and M must be vectors.') S1n3(U:m  
    end c4e_6=Iv  
    L_Q#(in  
    if length(n)~=length(m) |vte=)%  
        error('zernpol:NMlength','N and M must be the same length.') Z') pf  
    end 9 7%0;a8  
    K.C> a:J  
    n = n(:); rzrl>9 h  
    m = m(:); 9[v1h,L  
    length_n = length(n); `=#01YX[0  
    oMcK`%ydm  
    if any(mod(n-m,2)) YL jHt\  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') QQk{\ PV  
    end rA0,`}8\  
    UX`]k{Mz  
    if any(m<0) y AF+bCXo  
        error('zernpol:Mpositive','All M must be positive.') )PkNWj6%y  
    end xy+QbD T  
    h9$ Fx  
    if any(m>n) PI A)d-Z  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') F Kc;W  
    end 2y0J`!/)  
    y`e4;*1  
    if any( r>1 | r<0 ) 3`hUo5K  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') z^o1GY  
    end n<Svw a}  
    |( (zTf  
    if ~any(size(r)==1) 8pM>Co!  
        error('zernpol:Rvector','R must be a vector.') Gx?+9C V  
    end QVZD/shq  
    d lH$yub  
    r = r(:); d {lP  
    length_r = length(r); RVtQ20e";r  
    a\kb^D=T  
    if nargin==4 PYB+FcR6?n  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); IWVlrGyM  
        if ~isnorm LW#M@  
            error('zernpol:normalization','Unrecognized normalization flag.') %v5R#14[n  
        end #L crI  
    else JGiKBm;  
        isnorm = false; y<W8Q<9  
    end hlvt$Jwq  
    F}Mhs17!|  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ,p{`pma  
    % Compute the Zernike Polynomials p\wJD1s  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% zHB_{(o7  
    ocwG7J\W  
    % Determine the required powers of r: sK$wN4k  
    % ----------------------------------- XXmE+aI  
    rpowers = []; s*M@%_A?  
    for j = 1:length(n) si1*Wt<3Bc  
        rpowers = [rpowers m(j):2:n(j)]; ?<X(]I.j  
    end |ifHSc.j<  
    rpowers = unique(rpowers); `U!y&Q$,  
    P#kGX(G9!  
    % Pre-compute the values of r raised to the required powers, BOlAm*tFt  
    % and compile them in a matrix: @mw "W{  
    % ----------------------------- (J$\-a7<f  
    if rpowers(1)==0 nR \'[~+  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); Mro4`GL  
        rpowern = cat(2,rpowern{:}); \`'KlF2  
        rpowern = [ones(length_r,1) rpowern]; NQTnhiM7$  
    else r'/;O  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 7&}P{<}o^  
        rpowern = cat(2,rpowern{:}); h4&;?T S  
    end c"YXxA J  
    -ML6d&cm  
    % Compute the values of the polynomials: 4z7G2  
    % -------------------------------------- \ v@({nB8  
    z = zeros(length_r,length_n); 9V1cdb~?"T  
    for j = 1:length_n ]*"s\ix  
        s = 0:(n(j)-m(j))/2; 1N`vCt]w  
        pows = n(j):-2:m(j); 2)iD4G`  
        for k = length(s):-1:1 TDK@)mP  
            p = (1-2*mod(s(k),2))* ... KM?1/KZ/~  
                       prod(2:(n(j)-s(k)))/          ... KV!<Oq  
                       prod(2:s(k))/                 ... _cJ[ FP1  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... `&7RMa4=  
                       prod(2:((n(j)+m(j))/2-s(k))); W-2i+g)  
            idx = (pows(k)==rpowers); Zp`T  
            z(:,j) = z(:,j) + p*rpowern(:,idx); )yee2(S  
        end 'aJgLws*w  
         PY\PUMF>  
        if isnorm -Q e~)7  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); ;uI~BV*3  
        end HP2wtN{Zs  
    end Pd=,$UQp  
    l?N`{ ,1^  
    % 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)  a;p6?kv  
    LXxl?D  
    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。 >4'21,q  
    -A@U0=o  
    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)