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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 g7\MFertR^  
    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! A|jmp~@K)+  
     
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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 k&JB,d-mJ%  
    function z = zernfun(n,m,r,theta,nflag) [K5#4k  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. A=N &(k  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N n,,hE_  
    %   and angular frequency M, evaluated at positions (R,THETA) on the ;i;2cq  
    %   unit circle.  N is a vector of positive integers (including 0), and ?WVp,vP  
    %   M is a vector with the same number of elements as N.  Each element wl^7.IR  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) mBAI";L3  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, fRcs@yZnS  
    %   and THETA is a vector of angles.  R and THETA must have the same $*k(h|XfwW  
    %   length.  The output Z is a matrix with one column for every (N,M) dSdP]50M  
    %   pair, and one row for every (R,THETA) pair. v@xbur\L  
    % _1>Xk_  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike +, IMN)?;z  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 3bWYRW  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral -'!K("  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 3y# U|&]{  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized yW =I*f  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. !sTOo  
    % vk:k~   
    %   The Zernike functions are an orthogonal basis on the unit circle. OV~]-5gau  
    %   They are used in disciplines such as astronomy, optics, and N}|<P[LW  
    %   optometry to describe functions on a circular domain. rofGD9f   
    % A'zXbp:%  
    %   The following table lists the first 15 Zernike functions. pxGDzU  
    % OuZPgN  
    %       n    m    Zernike function           Normalization S]"U(JmW\  
    %       -------------------------------------------------- k vu SE  
    %       0    0    1                                 1 \Fh#CI  
    %       1    1    r * cos(theta)                    2 ce&Q}_  
    %       1   -1    r * sin(theta)                    2 R>C^duos.  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) o[A y2"e?  
    %       2    0    (2*r^2 - 1)                    sqrt(3) z~m{'O`  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) l* ap$1'  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) tz^2?wO  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) nO\c4#ce  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) <<SUIY@X  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) $~;h}I  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) NMy+=GZu^  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) xj!G9x<!  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) uY_vX\;67z  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) M+|J;caX  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) Nn/f*GDvK  
    %       -------------------------------------------------- yIq. m=  
    % .#OD=wkN0  
    %   Example 1: m)1+D"z  
    % mVs<XnA47  
    %       % Display the Zernike function Z(n=5,m=1) ,N1I\f  
    %       x = -1:0.01:1; ! ^ DQX=1  
    %       [X,Y] = meshgrid(x,x); xHpB/P~  
    %       [theta,r] = cart2pol(X,Y); ahUc ;S:v#  
    %       idx = r<=1; <i$ud&D  
    %       z = nan(size(X)); qlU"v)Mx  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); {CaTu5\  
    %       figure SDbR(oV  
    %       pcolor(x,x,z), shading interp [Yyb)Qf  
    %       axis square, colorbar \RF{ITV$kD  
    %       title('Zernike function Z_5^1(r,\theta)') Lu.C+zgQ  
    % AE@N:a  
    %   Example 2: uD0<|At/  
    % dI%#cf1  
    %       % Display the first 10 Zernike functions w9aLTLv-  
    %       x = -1:0.01:1; |y%M";MI  
    %       [X,Y] = meshgrid(x,x); #,5v#| u|7  
    %       [theta,r] = cart2pol(X,Y); dR GgiQO  
    %       idx = r<=1; oro^'#ki  
    %       z = nan(size(X)); s[n*fV']A  
    %       n = [0  1  1  2  2  2  3  3  3  3]; 2FxrjA  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; DX b=Ku  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; X4$86  
    %       y = zernfun(n,m,r(idx),theta(idx)); ?l/+*/AR;  
    %       figure('Units','normalized') (/[wM>q:r  
    %       for k = 1:10 O/ ih9,  
    %           z(idx) = y(:,k); tj1M1s|a  
    %           subplot(4,7,Nplot(k)) gLzQM3{X9  
    %           pcolor(x,x,z), shading interp N]dsGvX  
    %           set(gca,'XTick',[],'YTick',[]) W }  
    %           axis square 3$n O@rOS  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 6mml96(  
    %       end EG#mNpxE  
    % JU`5K}H<  
    %   See also ZERNPOL, ZERNFUN2. \\(3gB.Gd  
    x@Ze%$'  
    %   Paul Fricker 11/13/2006 $gPR3*0  
    wgcKeTD9  
    q_b,3Tp  
    % Check and prepare the inputs: n:P++^ j  
    % ----------------------------- 9k*1_  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) qZB}}pM#  
        error('zernfun:NMvectors','N and M must be vectors.') ><DXT nt'x  
    end 1=Y pNXX  
    TD^w|U.  
    if length(n)~=length(m) N#&/d nV  
        error('zernfun:NMlength','N and M must be the same length.') g+pj1ycw/  
    end slH3c:j\  
    2 e9lk$  
    n = n(:); u d$*/ )/  
    m = m(:); @E !`:/k  
    if any(mod(n-m,2)) &<$YR~g5j$  
        error('zernfun:NMmultiplesof2', ... 3cB=9Y{<  
              'All N and M must differ by multiples of 2 (including 0).') e"^n^_9  
    end w(cl,W/w  
    bPMkBm  
    if any(m>n) %$ ^ eY'-'  
        error('zernfun:MlessthanN', ... VI(2/**  
              'Each M must be less than or equal to its corresponding N.') LQDU8[-  
    end 9 lH00n+'  
    +<$b6^>!$  
    if any( r>1 | r<0 ) `Qxdb1>mjY  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') Nu4PY@m]C  
    end )9~-^V0A^>  
    z$b'y;k  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) +et)!2N  
        error('zernfun:RTHvector','R and THETA must be vectors.') Xd@_:ds  
    end 9^2l<4^Z  
    `CqF&b  
    r = r(:); v?<Tkw ^F  
    theta = theta(:); 5hg ^K^ZZ  
    length_r = length(r); R$M>[Kjn  
    if length_r~=length(theta) qt,;Yxx#^  
        error('zernfun:RTHlength', ... }:xj%?ki  
              'The number of R- and THETA-values must be equal.') q 7aH=dhw  
    end 2|:x_rcj  
    %WO4uOi:@  
    % Check normalization: DEN (pA\  
    % -------------------- g?> V4WF  
    if nargin==5 && ischar(nflag) 5o2vj8::  
        isnorm = strcmpi(nflag,'norm'); a F5=k: k  
        if ~isnorm O]-s(8Oo3  
            error('zernfun:normalization','Unrecognized normalization flag.') WX}pBmU  
        end DUlvlQW  
    else ;Vlt4,s)  
        isnorm = false; y#?AW`|  
    end $I4:g.gKpG  
    vfpK|=[7o  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% h:~ 8WV|  
    % Compute the Zernike Polynomials Mx_O'D  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ?8TIPz J  
    :Lh`Q"a  
    % Determine the required powers of r: -;W`0 k^  
    % ----------------------------------- ;T\'|[bY   
    m_abs = abs(m); qN@0k>11?  
    rpowers = []; L3|~ i&k  
    for j = 1:length(n) [;,Xp/  
        rpowers = [rpowers m_abs(j):2:n(j)]; V m]u-R`{  
    end zTb,h  
    rpowers = unique(rpowers); bY!1t}ALh  
    |>!tqgq  
    % Pre-compute the values of r raised to the required powers,  mm9xO%  
    % and compile them in a matrix: @78%6KZ`i  
    % ----------------------------- 0.!!rq,  
    if rpowers(1)==0 Eq/oq\(/6  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); hVf;{p &  
        rpowern = cat(2,rpowern{:}); D{G~7P\.  
        rpowern = [ones(length_r,1) rpowern]; @; 0t+  
    else VB&`g<  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); o8!uvl}:9  
        rpowern = cat(2,rpowern{:}); 7>!Rg~M  
    end LY1dEZ-)A  
    R~!md  
    % Compute the values of the polynomials: b5t:" >wC  
    % -------------------------------------- CCp&+LRvR  
    y = zeros(length_r,length(n)); _h0hl]rf  
    for j = 1:length(n) Rr"D)|Y;C(  
        s = 0:(n(j)-m_abs(j))/2; N5jJ,iz  
        pows = n(j):-2:m_abs(j); G * '1[Bu  
        for k = length(s):-1:1 #{x4s?   
            p = (1-2*mod(s(k),2))* ... vD3j(d  
                       prod(2:(n(j)-s(k)))/              ... u7}C):@H  
                       prod(2:s(k))/                     ... /@feY?glc  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ~%d*#Yxq  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); mz?1J4rt  
            idx = (pows(k)==rpowers); @8"cT-  
            y(:,j) = y(:,j) + p*rpowern(:,idx); -I*NS6  
        end Wj"GS!5  
         e%EE|  
        if isnorm 3w$Ib}7   
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); $i;_yTht  
        end ) ={ H  
    end ,Uu#41ZOKL  
    % END: Compute the Zernike Polynomials /6yH ,{(a  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% >@uFye$  
    = @n`5g  
    % Compute the Zernike functions: FC }r~syqA  
    % ------------------------------ (ioJ G-2u  
    idx_pos = m>0; _&}z+(Ug  
    idx_neg = m<0; mt*/%>@7R  
    WYY&MHp  
    z = y; U~s-'-C /  
    if any(idx_pos) {bMOT*X=A  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); yR4++yk  
    end o6c>sh  
    if any(idx_neg) 0p[-M`D  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); IfzZ\x .  
    end =At)?A9[  
    ^_!2-QY.~  
    % 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) -@?4Tfl  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. L]#b =Y  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated NkY7Hg0  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive [[' (,,r  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, 9 gWqs'  
    %   and THETA is a vector of angles.  R and THETA must have the same 0W!S.]^1  
    %   length.  The output Z is a matrix with one column for every P-value, erdA ?  
    %   and one row for every (R,THETA) pair. hL}AgY@  
    % #kRt\Fzq  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike uE-|]QQo  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) 84f^==Y  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) $J!WuOz4^i  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 B)=)@h[f  
    %   for all p. #?klVK&e/  
    % l }WvO]  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 )UN_,'H/V  
    %   Zernike functions (order N<=7).  In some disciplines it is +!eh\.u|]  
    %   traditional to label the first 36 functions using a single mode UB3hC`N\  
    %   number P instead of separate numbers for the order N and azimuthal `IH*~d]  
    %   frequency M. {I]>!V0j!  
    % 0^mCj<g  
    %   Example: C1po]Ott*  
    % E<r<ObeRv`  
    %       % Display the first 16 Zernike functions zr^"zcfz&  
    %       x = -1:0.01:1; )8 %lZ {  
    %       [X,Y] = meshgrid(x,x); %hN7K  
    %       [theta,r] = cart2pol(X,Y); rB}2F*eT  
    %       idx = r<=1; }Wz[ox9b  
    %       p = 0:15; IZOO>-g'f  
    %       z = nan(size(X)); ?#Y1E~N  
    %       y = zernfun2(p,r(idx),theta(idx)); NQIbav^5  
    %       figure('Units','normalized') yt1dYF0Xq  
    %       for k = 1:length(p) *IIuGtS  
    %           z(idx) = y(:,k); ~en'E  
    %           subplot(4,4,k) cd]def[d  
    %           pcolor(x,x,z), shading interp ^|ul3_'?  
    %           set(gca,'XTick',[],'YTick',[]) !MEA@^$#  
    %           axis square k293 wS  
    %           title(['Z_{' num2str(p(k)) '}']) !;E{D  
    %       end Dgm"1+  
    % O[15x H,  
    %   See also ZERNPOL, ZERNFUN. v{oHC4  
    !m6=Us  
    %   Paul Fricker 11/13/2006 I;+>@Cn(g<  
    |DLmMsS4  
    1!&m1  
    % Check and prepare the inputs: F`M`c%  
    % ----------------------------- g^[BnP)I  
    if min(size(p))~=1 )Ag{S[yZ  
        error('zernfun2:Pvector','Input P must be vector.') >N-l2?rE  
    end "J>8ZUP  
    H' %#71  
    if any(p)>35 A+ LX37B  
        error('zernfun2:P36', ... MTAq} 8  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... ]+d> ;$O  
               '(P = 0 to 35).']) wPxtQv  
    end CSlPrx2\  
    /TY=ig1z  
    % Get the order and frequency corresonding to the function number: m(CAXq-t  
    % ---------------------------------------------------------------- BjV;/<bt  
    p = p(:); v%;Ny ab6$  
    n = ceil((-3+sqrt(9+8*p))/2); [ ]42$5eof  
    m = 2*p - n.*(n+2); e,Sxu[2  
    Bj* M W  
    % Pass the inputs to the function ZERNFUN: 5v"Sv  
    % ---------------------------------------- ~RRS{\,  
    switch nargin mO&zE;/[  
        case 3 CtE <9?  
            z = zernfun(n,m,r,theta); C^7M>i  
        case 4 )}1S `*J/O  
            z = zernfun(n,m,r,theta,nflag); cXbQ  
        otherwise yeIc Q%  
            error('zernfun2:nargin','Incorrect number of inputs.') *}=z^;_oq  
    end MR/gLm(8(  
    8SoTABHV  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) ?wREX[Tqs  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. }NKnV3G/Z  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of ~2[mZias  
    %   order N and frequency M, evaluated at R.  N is a vector of Y?\PU{ O  
    %   positive integers (including 0), and M is a vector with the -YY@[5x?u  
    %   same number of elements as N.  Each element k of M must be a gmN$}Gy}  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) nx@,oC4  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is ?Lbn R~/J  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix Dr5AJ`y9A  
    %   with one column for every (N,M) pair, and one row for every > *soc!#Y  
    %   element in R. zo:NE0 0  
    % 3u[5T|D'  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- |f2 bb  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is U\i7'9w]3  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to "mc ]^ O  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 lsKQZ@LN`  
    %   for all [n,m]. remRm Y?  
    % nIGElt]  
    %   The radial Zernike polynomials are the radial portion of the IGF25-7B  
    %   Zernike functions, which are an orthogonal basis on the unit C sCH :>  
    %   circle.  The series representation of the radial Zernike uLSuY}K0  
    %   polynomials is ftD(ed  
    % 5=/j  
    %          (n-m)/2 <aQ5chf7  
    %            __  1t }  
    %    m      \       s                                          n-2s *vOk21z77d  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r f7:}t+d  
    %    n      s=0 ##nC@h@  
    % RKy!=#;17  
    %   The following table shows the first 12 polynomials. qm< mw"]  
    % C/MQY:X4  
    %       n    m    Zernike polynomial    Normalization Xo3@-D_c!c  
    %       --------------------------------------------- rDv`E^\  
    %       0    0    1                        sqrt(2) >DR$}{IV  
    %       1    1    r                           2 aUq 2$lw1  
    %       2    0    2*r^2 - 1                sqrt(6) +P<#6<gR  
    %       2    2    r^2                      sqrt(6) vQy+^deW  
    %       3    1    3*r^3 - 2*r              sqrt(8) e?+&2zMq  
    %       3    3    r^3                      sqrt(8) 009Q#[A  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) C4)m4r%  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) }J $\<ZT  
    %       4    4    r^4                      sqrt(10) mI5!rrRD|  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) p\DSFB  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) RcOfesW o  
    %       5    5    r^5                      sqrt(12) +!)_[ zo  
    %       --------------------------------------------- vF)eo"_s*  
    % [F 24xC+  
    %   Example: 6N[X:F 3`,  
    % -Eoq#ULvR  
    %       % Display three example Zernike radial polynomials WWT1_&0  
    %       r = 0:0.01:1; U04&z 91"  
    %       n = [3 2 5]; tgY/8& $M  
    %       m = [1 2 1]; X'j9l4Ph7  
    %       z = zernpol(n,m,r); o8~<t]Ejw  
    %       figure D B(!*6#?  
    %       plot(r,z) myF/_o&Ty  
    %       grid on 6eb~Z6n&?  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest')  %e(DPX  
    % 5,?^SK|'x  
    %   See also ZERNFUN, ZERNFUN2. Q9i[?=F:z  
    q4Mv2SPT  
    % A note on the algorithm. ij?Ww'p9>  
    % ------------------------ W&v|-#7=6  
    % The radial Zernike polynomials are computed using the series s7,D}Zz  
    % representation shown in the Help section above. For many special 3~?m?vj|Y  
    % functions, direct evaluation using the series representation can *!ecb1U5  
    % produce poor numerical results (floating point errors), because ZE9.r`  
    % the summation often involves computing small differences between V=<AI.Z:w  
    % large successive terms in the series. (In such cases, the functions Y]DC; ,  
    % are often evaluated using alternative methods such as recurrence q@1xYz:J  
    % relations: see the Legendre functions, for example). For the Zernike S|F:[(WaM  
    % polynomials, however, this problem does not arise, because the <==6fc>s  
    % polynomials are evaluated over the finite domain r = (0,1), and Cv [1HO<  
    % because the coefficients for a given polynomial are generally all A=wG};%_  
    % of similar magnitude. y-{?0mLq  
    % ZS[Ut  
    % ZERNPOL has been written using a vectorized implementation: multiple HSVl$66  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] ~_ u3_d.  
    % values can be passed as inputs) for a vector of points R.  To achieve [_: GQ  
    % this vectorization most efficiently, the algorithm in ZERNPOL Nh\o39=  
    % involves pre-determining all the powers p of R that are required to L_o/fTz4  
    % compute the outputs, and then compiling the {R^p} into a single ""*g\  
    % matrix.  This avoids any redundant computation of the R^p, and BZ(I]:oDL  
    % minimizes the sizes of certain intermediate variables. z\$(@:{A  
    % )iFXa<5h  
    %   Paul Fricker 11/13/2006 a'A<'(yv  
    6 [?5hmc"w  
    3,n"d-  
    % Check and prepare the inputs: MG~bDM4  
    % ----------------------------- ';v1AX}5q  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) BN~ndWRK  
        error('zernpol:NMvectors','N and M must be vectors.') 4W}mPeEeV  
    end .y lvJ$  
    $9@AwS@Uu  
    if length(n)~=length(m) P3nBxw"  
        error('zernpol:NMlength','N and M must be the same length.') s+(@UUl  
    end Jt0U`_  
    Wb[k2V  
    n = n(:); L|B! ]}  
    m = m(:); lB.n5G  
    length_n = length(n); "Q{ l])N  
    h1l%\3ZH  
    if any(mod(n-m,2)) z#1"0Ks&P  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') `jVRabZ0  
    end 2[qoqd(  
    a$Hq<~46  
    if any(m<0) cL][sI  
        error('zernpol:Mpositive','All M must be positive.') #jd.i  
    end |>Fz:b d  
    D c;k)z=  
    if any(m>n) +bT[lJ2O>G  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') tG ZMIG_  
    end uP(t+}dQ+3  
    $HtGB]  
    if any( r>1 | r<0 ) gA% A})  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') fI1 9p Q  
    end ZCV i ZWo  
    p_X{'=SQ1  
    if ~any(size(r)==1) Y B,c=Wx  
        error('zernpol:Rvector','R must be a vector.') aOS,%J^ ?  
    end y#th&YC_b  
    noM=8C&U  
    r = r(:); E@yo/S  
    length_r = length(r); r-yUWIr S  
    *,IK4F6>:  
    if nargin==4 v5@M 34  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); ;FW <%  
        if ~isnorm -*i_8`  
            error('zernpol:normalization','Unrecognized normalization flag.') (m6V)y  
        end o8|qT)O@U  
    else ifu!6_b.  
        isnorm = false; dfKGO$}V  
    end vbd)L$$20+  
    ;8dffsyq  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% >^GV #z  
    % Compute the Zernike Polynomials V)l:fUm2  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% JgA{1@h  
    w%8y5v5  
    % Determine the required powers of r: @0]WMI9B"B  
    % ----------------------------------- d4*SfzB  
    rpowers = []; ir"t@"Y;o  
    for j = 1:length(n) l#%7BGwzY  
        rpowers = [rpowers m(j):2:n(j)]; &1R#!|h1W  
    end O"Nr$bS(Y  
    rpowers = unique(rpowers); C#^y{q  
    tfZ@4%'  
    % Pre-compute the values of r raised to the required powers, M=lU`Sm  
    % and compile them in a matrix: \$*CXjh3G  
    % ----------------------------- =QGmJ3  
    if rpowers(1)==0 ^L)TfI_n  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); GBT|1c'i  
        rpowern = cat(2,rpowern{:}); ; +(VO  
        rpowern = [ones(length_r,1) rpowern]; FO%pdLs,  
    else 8Znr1=1   
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); &)gc{(4$  
        rpowern = cat(2,rpowern{:}); 6 /5,n0  
    end T^(W _S  
    Dmr*Lh~  
    % Compute the values of the polynomials: 0<a|=kZ  
    % -------------------------------------- BV:Ca34&  
    z = zeros(length_r,length_n); `[g$EXX  
    for j = 1:length_n kfZ`|w@q  
        s = 0:(n(j)-m(j))/2; IGQBTdPUa  
        pows = n(j):-2:m(j); -Dx3*ZhP  
        for k = length(s):-1:1 K) fKL   
            p = (1-2*mod(s(k),2))* ... 4=/jh:h  
                       prod(2:(n(j)-s(k)))/          ... ZTV)D  
                       prod(2:s(k))/                 ... l`ZL^uT  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... A|S)cr8z  
                       prod(2:((n(j)+m(j))/2-s(k))); T3@2e0u )  
            idx = (pows(k)==rpowers); $!MP0f\q g  
            z(:,j) = z(:,j) + p*rpowern(:,idx); Qn.dL@W  
        end wbrOL(q.m  
         ]oVP_ &E  
        if isnorm 6QCV i  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); j1%8r*Jj  
        end fmj}NV&ma  
    end =eUKpYI  
    &:&l+  
    % 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)  N<<wg{QO  
    i`l;k~rP  
    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。 n-M6~   
    9'H:pb2  
    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)