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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 4xH/a1&p=  
    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! }|%1LL^pB  
     
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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 DY9]$h*y  
    function z = zernfun(n,m,r,theta,nflag) AYgXqmH~+  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. b>Y{,`E3  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N fGO\f;P  
    %   and angular frequency M, evaluated at positions (R,THETA) on the wapSpSt  
    %   unit circle.  N is a vector of positive integers (including 0), and 7 @ )  
    %   M is a vector with the same number of elements as N.  Each element wD=]U@t`,  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) Ml7 (<J  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, K)BQ0v.:[  
    %   and THETA is a vector of angles.  R and THETA must have the same *8WB($T}  
    %   length.  The output Z is a matrix with one column for every (N,M) u '7h(1@  
    %   pair, and one row for every (R,THETA) pair. ?oFd%|I  
    % ATl?./Tu  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike Y}1c>5{bE  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), xEp?|Q$  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral fEX=csZ86  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, o87kF!x  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized FO5a<6  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. aL( hWE  
    % -cM1]soT  
    %   The Zernike functions are an orthogonal basis on the unit circle. p,goYF??  
    %   They are used in disciplines such as astronomy, optics, and MDU#V  
    %   optometry to describe functions on a circular domain. &CQO+Yr$l  
    % 0Gc@AG{  
    %   The following table lists the first 15 Zernike functions. -}9^$}PR  
    % N,c!1: b  
    %       n    m    Zernike function           Normalization DK\XC%~m  
    %       -------------------------------------------------- /\c'kMAW!  
    %       0    0    1                                 1 t/\   
    %       1    1    r * cos(theta)                    2 H*'1bLzq  
    %       1   -1    r * sin(theta)                    2 \3$!)z  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) \&5V';  
    %       2    0    (2*r^2 - 1)                    sqrt(3) mK[Z#obc=  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) y %Q. (  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8)  ch8a  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) A^>@6d $2  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) MLu!8dgI  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) kFv*>>X`  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) ('tXv"fT  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) k*\Bl4g  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) -GA F>  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 6 Rl[M+Q  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) C/!.VMl^  
    %       -------------------------------------------------- <X:JMj+  
    % nt#9j',6Rn  
    %   Example 1: ]>t~Bcn m  
    %  u]P|  
    %       % Display the Zernike function Z(n=5,m=1) 9{*{Ba  
    %       x = -1:0.01:1; #;]#NqFX  
    %       [X,Y] = meshgrid(x,x); U!aM63F3  
    %       [theta,r] = cart2pol(X,Y); GtVT^u_   
    %       idx = r<=1; > S>*JP  
    %       z = nan(size(X)); zj1~[$  (  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); z uV%`n  
    %       figure  :\\NK/"  
    %       pcolor(x,x,z), shading interp 0O9b 7F  
    %       axis square, colorbar Vx h39eW  
    %       title('Zernike function Z_5^1(r,\theta)') d:@+dS  
    % i6WH^IQM  
    %   Example 2: Y%XF64)6  
    % bj pruJ`=  
    %       % Display the first 10 Zernike functions tk&AZb,sP  
    %       x = -1:0.01:1;  zm"  
    %       [X,Y] = meshgrid(x,x); {]k#=a4  
    %       [theta,r] = cart2pol(X,Y); m/KaWrw/)  
    %       idx = r<=1; Ghgn<YG  
    %       z = nan(size(X)); IZ=Z=k{  
    %       n = [0  1  1  2  2  2  3  3  3  3]; BJj'91B[d  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; ~_\Ra%  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; U.e!:f4{  
    %       y = zernfun(n,m,r(idx),theta(idx)); ~"#0rPT  
    %       figure('Units','normalized') hdPGqJE  
    %       for k = 1:10 5/=$p:E>  
    %           z(idx) = y(:,k); q)?%END  
    %           subplot(4,7,Nplot(k)) uUI#^ A  
    %           pcolor(x,x,z), shading interp k=]e7~!  
    %           set(gca,'XTick',[],'YTick',[]) (Q*q# U  
    %           axis square :_8K8Sa  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) &C9IR,&  
    %       end B\J[O5},  
    % Kh]es,$D  
    %   See also ZERNPOL, ZERNFUN2. ,:?ibE=  
    5pCicwea#  
    %   Paul Fricker 11/13/2006 -9b=-K.y  
    _3`G ZeGV  
    4uXGp sL  
    % Check and prepare the inputs: $*C }iJsF  
    % ----------------------------- Kxsd@^E  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) gP% <<yl  
        error('zernfun:NMvectors','N and M must be vectors.') !j6 k]BgZ  
    end TO6F  
    Y&6jFT_  
    if length(n)~=length(m) QVT0.GzR  
        error('zernfun:NMlength','N and M must be the same length.') '12m4quO  
    end q8{Bx03m6  
    xV> .]  
    n = n(:); 1=5"j]0hY  
    m = m(:); K *@?BE  
    if any(mod(n-m,2)) 'V&g"Pb  
        error('zernfun:NMmultiplesof2', ... K)'[^V Xh  
              'All N and M must differ by multiples of 2 (including 0).') Y=XDN:  
    end 3r~8:F"g  
    8-;.Ejz!\A  
    if any(m>n) x6/u+Urn  
        error('zernfun:MlessthanN', ... $bE" 3/uf  
              'Each M must be less than or equal to its corresponding N.') .x=abA$!9  
    end f7&ni#^Ztj  
    4@{;z4*`  
    if any( r>1 | r<0 ) {]IY; cL  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') mS%4  
    end AROHe  
    4Wl`hF  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) B&MDn']fV/  
        error('zernfun:RTHvector','R and THETA must be vectors.') WI1Y P0V  
    end +Z"Wa0wA  
    =c6d $  
    r = r(:); @1j*\gYz  
    theta = theta(:); \n}%RD-Ce  
    length_r = length(r); t]B`>SL3W  
    if length_r~=length(theta) [vr"FLM|9  
        error('zernfun:RTHlength', ... fHaF9o+/b  
              'The number of R- and THETA-values must be equal.') 3cJ'tRsp<  
    end |;J`~H"K  
    )a^&7  
    % Check normalization: Aw7N'0K9UN  
    % --------------------  KcT(/!  
    if nargin==5 && ischar(nflag) ;1~n|IY  
        isnorm = strcmpi(nflag,'norm'); *L<EGFP  
        if ~isnorm E?]$Y[KJKs  
            error('zernfun:normalization','Unrecognized normalization flag.') Ea4zC|;  
        end +P))*0(c_  
    else pauO_'j_1p  
        isnorm = false; >FeCa h Fn  
    end HDhkg-QC  
    B}7j20:Z  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% );HhV,$n  
    % Compute the Zernike Polynomials 3=wcA/"!  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% EwBrOq`C  
    ,L%]}8EL"  
    % Determine the required powers of r: whN<{AG  
    % ----------------------------------- bM'F8 Fi  
    m_abs = abs(m); J[}j8x?r  
    rpowers = []; &t UX(  
    for j = 1:length(n) LTf)`SN %'  
        rpowers = [rpowers m_abs(j):2:n(j)]; ce$ [H}rDB  
    end q>+!Ete1p  
    rpowers = unique(rpowers); y:E$n!  
    gR/?MJ(v  
    % Pre-compute the values of r raised to the required powers, kP5I+ B  
    % and compile them in a matrix: [m! P(o  
    % ----------------------------- wKJ|;o4;L  
    if rpowers(1)==0 eSZ':p  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); XnYX@p  
        rpowern = cat(2,rpowern{:}); (e;/Smol  
        rpowern = [ones(length_r,1) rpowern]; oHfr glGX  
    else `j3 OFC{7E  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); QUkP&sz  
        rpowern = cat(2,rpowern{:}); g\B ? |%  
    end n"?*"Ya  
    |A68+(3u  
    % Compute the values of the polynomials: |J@ &lBlq  
    % -------------------------------------- )M8,Tv*~  
    y = zeros(length_r,length(n)); ;P' 5RCqj  
    for j = 1:length(n) X6}W]  
        s = 0:(n(j)-m_abs(j))/2; jc3Q3Th/zn  
        pows = n(j):-2:m_abs(j); CY=lN5!J  
        for k = length(s):-1:1 M:.+^.h  
            p = (1-2*mod(s(k),2))* ...  rPr]f;  
                       prod(2:(n(j)-s(k)))/              ... Pc?"H!Hkn  
                       prod(2:s(k))/                     ... #t2N=3dOj  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ~[F7M{LS  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); y<HNAG j  
            idx = (pows(k)==rpowers); b*tb$F  
            y(:,j) = y(:,j) + p*rpowern(:,idx); R:l&2  
        end q`8 5-  
         ` ,SNqi  
        if isnorm yFd.tQs  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); @8w[Zo~  
        end tYgHJ~1L*  
    end =i}lh}(  
    % END: Compute the Zernike Polynomials gKQs:25  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 'cu14m_  
    \HrtPm`e  
    % Compute the Zernike functions: \)6AzCq  
    % ------------------------------ h=6D=6c  
    idx_pos = m>0; /]0SF_dZ  
    idx_neg = m<0; |aU8WRq  
    mcidA%  
    z = y; OVxg9  
    if any(idx_pos) @0 x   
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); V^!^wLLi  
    end d"E3ypPK  
    if any(idx_neg) 7}Mnv WP  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); XgXXBKf$  
    end X. Ur`X  
    #l`\'0`.  
    % 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) SHB'g){P  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. HCkfw+gaV  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated !%t2Z QJq  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive \Wg_ gA  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, 4Z=`;  
    %   and THETA is a vector of angles.  R and THETA must have the same oC} u  
    %   length.  The output Z is a matrix with one column for every P-value, ,uNJz-B8  
    %   and one row for every (R,THETA) pair. bYpeI(zK  
    % =vQ J2Rg  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike <}'=@a  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) (C uM*-  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) 0y/31hp  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 mN.[bz  
    %   for all p. Dm}M8`|X  
    % @^ti*`  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 ? * ,  
    %   Zernike functions (order N<=7).  In some disciplines it is _Yp~Oj  
    %   traditional to label the first 36 functions using a single mode |& jrU-(  
    %   number P instead of separate numbers for the order N and azimuthal XjP;O,x  
    %   frequency M.  f}*:wj  
    % !J!&JQ|  
    %   Example: QY<5o;m`  
    % r_,;[+!  
    %       % Display the first 16 Zernike functions AOKC1iD%Y  
    %       x = -1:0.01:1; RjgJIVm(  
    %       [X,Y] = meshgrid(x,x); m__pQu:  
    %       [theta,r] = cart2pol(X,Y); .,#H]?Wil  
    %       idx = r<=1; X's<+hK&  
    %       p = 0:15; <-N2<s l  
    %       z = nan(size(X)); KUm?gFh  
    %       y = zernfun2(p,r(idx),theta(idx)); goF87^M  
    %       figure('Units','normalized') 34N~<-9AY  
    %       for k = 1:length(p) E]m?R 4  
    %           z(idx) = y(:,k);  QX<x2U  
    %           subplot(4,4,k) ~LOE^6C+~o  
    %           pcolor(x,x,z), shading interp )u=W?5%=}  
    %           set(gca,'XTick',[],'YTick',[]) mW{>  
    %           axis square z5~W >r  
    %           title(['Z_{' num2str(p(k)) '}']) WK7?~R%rq  
    %       end q TN)2G  
    % t&[<Dl/L  
    %   See also ZERNPOL, ZERNFUN. O1t$]k:  
    "8cI]~ V  
    %   Paul Fricker 11/13/2006 rcMf1\  
    Fzt?M  
    <m1v+cnqo  
    % Check and prepare the inputs: W-&V:S{<  
    % ----------------------------- XG C\6?L~  
    if min(size(p))~=1 Vq{3:QBR  
        error('zernfun2:Pvector','Input P must be vector.') 3b]M\ F9  
    end nu-&vX  
    6'@{ * u  
    if any(p)>35 E!mv}  
        error('zernfun2:P36', ... fG$LqzyqlK  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... pPo xx"y  
               '(P = 0 to 35).']) DU]KD%kl  
    end sO 6=w%l^  
    iT,7jd?6#  
    % Get the order and frequency corresonding to the function number: blIMrP%  
    % ---------------------------------------------------------------- /}L2LMIm  
    p = p(:); 3z2 OW@zL$  
    n = ceil((-3+sqrt(9+8*p))/2); 8 p[n>qV9  
    m = 2*p - n.*(n+2); S 593wfc  
    v}V[sIs}  
    % Pass the inputs to the function ZERNFUN: V(DY!f_%  
    % ---------------------------------------- 2xX:Q'\2  
    switch nargin kV5)3%?  
        case 3 "2sk1  
            z = zernfun(n,m,r,theta); Q1?*+]  
        case 4 9jEH"`qqk  
            z = zernfun(n,m,r,theta,nflag); rZaO^}u]  
        otherwise YE{t?Y\5  
            error('zernfun2:nargin','Incorrect number of inputs.') ]SRpMZ  
    end @v#P u_  
    H;=Fq+  
    % 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) M %!;5  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. B<m0YD?>~>  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of jj2\;b:a0  
    %   order N and frequency M, evaluated at R.  N is a vector of RK|*yt"f"  
    %   positive integers (including 0), and M is a vector with the 5j1d=h  
    %   same number of elements as N.  Each element k of M must be a wLXJ?iy3  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) 6xJffl  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is &EQhk9j  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix Rxd4{L )n  
    %   with one column for every (N,M) pair, and one row for every PKSfu++Z  
    %   element in R. 4#03x:/<\  
    % c!4F0(n4  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- 4r1\&sI$~  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is GN(<$,~g  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to Q]xkDr?   
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 .=#j dc/  
    %   for all [n,m]. K -rR)-rI  
    % Ytlzn%  
    %   The radial Zernike polynomials are the radial portion of the YoKyiO!   
    %   Zernike functions, which are an orthogonal basis on the unit H,X|-B  
    %   circle.  The series representation of the radial Zernike Jd>~gA}l  
    %   polynomials is qM(}|fMbN  
    % x^ f)I|t  
    %          (n-m)/2 @9gZH_ur>E  
    %            __ f`uRC-B/  
    %    m      \       s                                          n-2s .="X vVdkp  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r :BF? r  
    %    n      s=0 }bjZeh.  
    % ilL0=[2  
    %   The following table shows the first 12 polynomials. 1jl !VU6  
    % `R[ZY!=+  
    %       n    m    Zernike polynomial    Normalization S13cQ?4  
    %       --------------------------------------------- oY18a*_>M1  
    %       0    0    1                        sqrt(2) A]/o-S_  
    %       1    1    r                           2 -[V-f> :  
    %       2    0    2*r^2 - 1                sqrt(6) ssi7)0  
    %       2    2    r^2                      sqrt(6) "n!yK  
    %       3    1    3*r^3 - 2*r              sqrt(8) cqNK`3:.j  
    %       3    3    r^3                      sqrt(8) (8JU!lin  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) ~.m<`~u  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) m.e]tTe  
    %       4    4    r^4                      sqrt(10) vjGQ!xF  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) )#}>,,S  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) -1g :3'% P  
    %       5    5    r^5                      sqrt(12) 3yZmW$E.  
    %       --------------------------------------------- dw bR,K  
    % @LKQ-<dZG  
    %   Example: FM@iIlY"  
    % Ic#xz;elM  
    %       % Display three example Zernike radial polynomials )|F|\6:ne  
    %       r = 0:0.01:1; bV_nYpo  
    %       n = [3 2 5]; #.bW9j/  
    %       m = [1 2 1]; #&&T1;z"#  
    %       z = zernpol(n,m,r); Ma[EgG  
    %       figure p~qe/  
    %       plot(r,z) i6`"e[aT[o  
    %       grid on 1sQIfX#2f  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') 9t"Rw ns  
    % V8?}I)#(7  
    %   See also ZERNFUN, ZERNFUN2. g7*)|FOb  
    y/lF1{}5  
    % A note on the algorithm. 7;"0:eX  
    % ------------------------ {*ak>Wud  
    % The radial Zernike polynomials are computed using the series /U~|B.z@6  
    % representation shown in the Help section above. For many special |cPHl+$nh.  
    % functions, direct evaluation using the series representation can ~&?bU]F  
    % produce poor numerical results (floating point errors), because %qP[+N&  
    % the summation often involves computing small differences between ^OR0Vp>L  
    % large successive terms in the series. (In such cases, the functions '$K E= Jy  
    % are often evaluated using alternative methods such as recurrence =!O->C:  
    % relations: see the Legendre functions, for example). For the Zernike J!6FlcsZm  
    % polynomials, however, this problem does not arise, because the !>8~R2  
    % polynomials are evaluated over the finite domain r = (0,1), and FK|O^- >B  
    % because the coefficients for a given polynomial are generally all 1j<(?MT-  
    % of similar magnitude. h+f>#O+:  
    % kN1MPd4Yh  
    % ZERNPOL has been written using a vectorized implementation: multiple \N>-+r  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] AZtS4]4G)  
    % values can be passed as inputs) for a vector of points R.  To achieve E$e7(D  
    % this vectorization most efficiently, the algorithm in ZERNPOL `@]s[1?f  
    % involves pre-determining all the powers p of R that are required to [I $+wWW_  
    % compute the outputs, and then compiling the {R^p} into a single k*.]*]   
    % matrix.  This avoids any redundant computation of the R^p, and 3^ Yc%  
    % minimizes the sizes of certain intermediate variables. \oQ]=dDCd%  
    % yBIlwN`kB  
    %   Paul Fricker 11/13/2006 xvr5$x|h  
    I5$P9UE+^9  
    Nk`UQ~g$  
    % Check and prepare the inputs: uy'ghF  
    % -----------------------------  `zwz  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) KhCP9(A=Qo  
        error('zernpol:NMvectors','N and M must be vectors.') }{aGh I~<  
    end Y Y:Bw W:  
    S;Bk/\2  
    if length(n)~=length(m) [uq>b|`R G  
        error('zernpol:NMlength','N and M must be the same length.')  R$a<=  
    end XZ rI w  
    QFyL2Xes/  
    n = n(:); T5BZD +Ta  
    m = m(:); S)rZE*~2  
    length_n = length(n); h`fVQN.3  
    ~xH&"1  
    if any(mod(n-m,2)) f0Q6sVZHa  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') g;IlS*Ld  
    end Gn]36~)*H  
    e _vsiT  
    if any(m<0) 0P^h6Vat  
        error('zernpol:Mpositive','All M must be positive.') WA{igj@\  
    end GN.O a$  
    A]1Nm3@  
    if any(m>n) $ |4C]Me (  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') zd?@xno  
    end ZS Med(//b  
    D)_ C@*q  
    if any( r>1 | r<0 ) +)Te)^&v%  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') q!K :N?  
    end .J#'k+>  
    R+sT &d  
    if ~any(size(r)==1) ajbe7#}  
        error('zernpol:Rvector','R must be a vector.') HDyf]2N*N  
    end od;-D~  
    K,f:X g!:  
    r = r(:); mgxIxusR  
    length_r = length(r); gjF5~ `  
    8 WP>u8&  
    if nargin==4 ,N/@=As9$  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); X/]@EF  
        if ~isnorm oL4W>b )  
            error('zernpol:normalization','Unrecognized normalization flag.') ] !UYl  
        end ~/:vr  
    else :-&|QVH  
        isnorm = false; wZrFu(_  
    end XLpP*VH3  
    f *ZU a  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Pms@!yce  
    % Compute the Zernike Polynomials SpH|<L3  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% wAMg"ImJ  
    B`tq*T%  
    % Determine the required powers of r: MsB >3  
    % ----------------------------------- SNEhP5!  
    rpowers = []; e~h>b.~  
    for j = 1:length(n) ! VwU=5  
        rpowers = [rpowers m(j):2:n(j)]; Z['.RF'`  
    end #mH@ /6,#[  
    rpowers = unique(rpowers); h\RX/C!+  
    5 s7BUT  
    % Pre-compute the values of r raised to the required powers, E}^V@ :j>  
    % and compile them in a matrix: ?7{U=1gb$  
    % ----------------------------- '1<Z"InU  
    if rpowers(1)==0 0SpB 2>_  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); }A9#3Y|F  
        rpowern = cat(2,rpowern{:}); QxT'\7f  
        rpowern = [ones(length_r,1) rpowern]; wcHk]mLM  
    else %lKw+D  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); GR,2^]<{  
        rpowern = cat(2,rpowern{:}); 6fwNlC/9  
    end ~f QrH%@  
    x"r0<RK  
    % Compute the values of the polynomials:  yCX5 5:  
    % -------------------------------------- p l)":}/)  
    z = zeros(length_r,length_n); g/?Vl2W  
    for j = 1:length_n *4O=4F)x  
        s = 0:(n(j)-m(j))/2; ^[ae )}  
        pows = n(j):-2:m(j); ktu?-?#0,  
        for k = length(s):-1:1 u#05`i:Z  
            p = (1-2*mod(s(k),2))* ... z%&FLdXgW+  
                       prod(2:(n(j)-s(k)))/          ... ;W|kc</R*  
                       prod(2:s(k))/                 ... [V}vd@*k  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... T/$ gnn  
                       prod(2:((n(j)+m(j))/2-s(k))); xBHf~:!  
            idx = (pows(k)==rpowers); l;F"m+B!$  
            z(:,j) = z(:,j) + p*rpowern(:,idx); iUKjCq02  
        end OjU{r N*  
         $KcAB0 B8  
        if isnorm t]c<HDCK  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); $e^"Inhtqp  
        end NP>v @jO  
    end ,@"yr>Q9#6  
    5!0iK9O  
    % 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)  4FneP i~i  
    {? yRO]  
    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。 :~s"]*y  
    4 JBfA,  
    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)