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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 }y|% wym  
    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! ,~K_rNNZ  
     
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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 <NJ7mR}  
    function z = zernfun(n,m,r,theta,nflag) xVl90ak  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. jC\R8_  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N x<ENN>mW1  
    %   and angular frequency M, evaluated at positions (R,THETA) on the /$9/,5|EA  
    %   unit circle.  N is a vector of positive integers (including 0), and DdSUB  
    %   M is a vector with the same number of elements as N.  Each element p{-1%jQ}]  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) ;m`I}h<  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, ]iz5VI@  
    %   and THETA is a vector of angles.  R and THETA must have the same (|6q N  
    %   length.  The output Z is a matrix with one column for every (N,M) jzPC9  
    %   pair, and one row for every (R,THETA) pair. DV%tby  
    % Tu6he8Q-  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 10[~ki-1;  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), h M8G"b  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral ^k)f oD  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, [ B (lJz  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized U{O\  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. Q ?Nzt;)!.  
    % Q z/pz_}  
    %   The Zernike functions are an orthogonal basis on the unit circle. oO UVU}H  
    %   They are used in disciplines such as astronomy, optics, and 2j"%}&  
    %   optometry to describe functions on a circular domain. n]o+KT\  
    % *|=&MU*+  
    %   The following table lists the first 15 Zernike functions. k~vmHb  
    % L>L4%?  
    %       n    m    Zernike function           Normalization r+lY9 l  
    %       -------------------------------------------------- ol YSr .Q`  
    %       0    0    1                                 1 A?7%q^;E  
    %       1    1    r * cos(theta)                    2 NA3yd^sr  
    %       1   -1    r * sin(theta)                    2 ?%LD1 <ya  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) T\WNT#My  
    %       2    0    (2*r^2 - 1)                    sqrt(3) 3oKqj>  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) *508PY  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) q7)$WXe2LM  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) Maxnk3n  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) >`NM?KP s  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) .K7A!;  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) h:GOcLYM@X  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 1L9^N  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) vj_oMmjKw  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) c:$:j,i}  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) 7"[lWC!As5  
    %       -------------------------------------------------- q2 f/#"k  
    % z)]EB6uRg  
    %   Example 1: _Ng*K]0/E  
    % nRo`O  
    %       % Display the Zernike function Z(n=5,m=1) ~/#?OLj(T  
    %       x = -1:0.01:1; z`Q5J9_<cV  
    %       [X,Y] = meshgrid(x,x);  JA)gM  
    %       [theta,r] = cart2pol(X,Y); K9P"ncMt  
    %       idx = r<=1; P"]+6sm&es  
    %       z = nan(size(X)); %-*vlNC)  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); \W\6m0-x  
    %       figure H\b5]q %  
    %       pcolor(x,x,z), shading interp Q O?ha'Sl  
    %       axis square, colorbar 05zHLj  
    %       title('Zernike function Z_5^1(r,\theta)') bF Vd v&  
    % Mb9q<4  
    %   Example 2: Z8#I  
    % $x)'_o}e  
    %       % Display the first 10 Zernike functions m3XH3FgKz  
    %       x = -1:0.01:1; )N6R#   
    %       [X,Y] = meshgrid(x,x); Mu( Y6  
    %       [theta,r] = cart2pol(X,Y); QbNv+Eu5  
    %       idx = r<=1; |l? ALP_g  
    %       z = nan(size(X)); PRLV1o1#  
    %       n = [0  1  1  2  2  2  3  3  3  3]; XVLuhw i  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; _F*w ,b$8  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; ,G:4H%?  
    %       y = zernfun(n,m,r(idx),theta(idx)); TZP{=v<  
    %       figure('Units','normalized') N1Z8I:  
    %       for k = 1:10 YH[_0!JY^  
    %           z(idx) = y(:,k); O}`01A!u;  
    %           subplot(4,7,Nplot(k)) 4l1=l#\S  
    %           pcolor(x,x,z), shading interp Gzfb|9 ,q  
    %           set(gca,'XTick',[],'YTick',[]) v\k,,sI  
    %           axis square F@*lR(4C  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) pd;-z  
    %       end WV@Tm$ r  
    % xh6x B|Z  
    %   See also ZERNPOL, ZERNFUN2. %~;Q_#CR/K  
    [s34N+vU  
    %   Paul Fricker 11/13/2006 __fR #D  
    6C0_. =7#  
    W@C56fCa  
    % Check and prepare the inputs: 0;H6b=  
    % ----------------------------- u20b+c4  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 6uXW`/lvX  
        error('zernfun:NMvectors','N and M must be vectors.') IX*S:7S[  
    end CU;nrd"  
    )[)]@e  
    if length(n)~=length(m) -5cH$]1\  
        error('zernfun:NMlength','N and M must be the same length.') R>U<8z"i  
    end D{4hNO  
    /C:'qhY,  
    n = n(:); 5Hm!5:ZB  
    m = m(:); `eWc p^|  
    if any(mod(n-m,2)) j~E +6f \  
        error('zernfun:NMmultiplesof2', ... ?iLd5 Z  
              'All N and M must differ by multiples of 2 (including 0).') ]18ygqt  
    end -h@0 1  
    H/3Zdj 9  
    if any(m>n) N39nJqo>"  
        error('zernfun:MlessthanN', ... D,n}Qf!GYk  
              'Each M must be less than or equal to its corresponding N.') BXo|CITso  
    end V0 F30rK  
    KYu(H[a  
    if any( r>1 | r<0 ) tv OAN|+F  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') w~U`+2a3  
    end Inc:t_  
    iW}l[g8sw!  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) J4}\V$ysN  
        error('zernfun:RTHvector','R and THETA must be vectors.') rH9}nL  
    end hcgc =$^  
    )h0E$*  
    r = r(:); IOkC[([  
    theta = theta(:); S@g/Tn  
    length_r = length(r); 0c61q Q6  
    if length_r~=length(theta) o$ce1LO?|N  
        error('zernfun:RTHlength', ... uvDoo6'  
              'The number of R- and THETA-values must be equal.') gc@#O#K~h^  
    end @sHw+to|p)  
    ~Ex.Yp8.  
    % Check normalization: & fSc{/  
    % -------------------- VMIX$#  
    if nargin==5 && ischar(nflag) $XQxWH|  
        isnorm = strcmpi(nflag,'norm'); = (gmd>N  
        if ~isnorm bjBeiKH  
            error('zernfun:normalization','Unrecognized normalization flag.') _`_IUuj$E  
        end 8q [c  
    else }=hoATs  
        isnorm = false; <i'u96  
    end I26gGp  
    d BMe`hM)  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% vaRwh E:  
    % Compute the Zernike Polynomials .W :  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 9J7J/]7f  
    NVJ&C]H6  
    % Determine the required powers of r: +qUkMx  
    % ----------------------------------- RF5q5<0  
    m_abs = abs(m); 48 CI8[T  
    rpowers = []; ZSRR lkU  
    for j = 1:length(n) %L j0  
        rpowers = [rpowers m_abs(j):2:n(j)]; 9*|3E"Vr  
    end !p,hy `  
    rpowers = unique(rpowers); 5 Y Q  
    #t@x6Vt  
    % Pre-compute the values of r raised to the required powers, M7DLs;sD  
    % and compile them in a matrix: %A62xnX  
    % ----------------------------- :@ E1Pun?  
    if rpowers(1)==0 4`6c28K0?  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 3_MS'&M  
        rpowern = cat(2,rpowern{:}); (.,'}+1  
        rpowern = [ones(length_r,1) rpowern]; Q+d.%qhc  
    else _@!QY   
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); z,bX.*.-  
        rpowern = cat(2,rpowern{:}); ,|.8nk"  
    end KR=d"t Qw  
    [vWkAJ'K  
    % Compute the values of the polynomials: 9$+^"ilk  
    % -------------------------------------- Y=a v8Y|`  
    y = zeros(length_r,length(n)); "%E-X:Il#  
    for j = 1:length(n) :4 j a@~  
        s = 0:(n(j)-m_abs(j))/2; @fqV0l!GR  
        pows = n(j):-2:m_abs(j); ?+n&hHRg  
        for k = length(s):-1:1 -XVEV  
            p = (1-2*mod(s(k),2))* ... wb6L? t  
                       prod(2:(n(j)-s(k)))/              ... u  m: 0y,  
                       prod(2:s(k))/                     ... i_=?eUq%q/  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... hza> jR  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); KQ4kZN  
            idx = (pows(k)==rpowers); xHJ8?bD p  
            y(:,j) = y(:,j) + p*rpowern(:,idx); .Iw ur;/\  
        end :}@C9pqr2  
         dG\U)WA(p  
        if isnorm +Y>"/i. N  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); h `\$sT!Z  
        end @S}/g/+2  
    end 1Xy8|OFc[  
    % END: Compute the Zernike Polynomials 0]T.Lh$3  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% uu}`warW  
    ietRr!$.  
    % Compute the Zernike functions: t7w-TJvP  
    % ------------------------------ z\fW )/  
    idx_pos = m>0; YDQ:eebg(  
    idx_neg = m<0; `^7:7Wr]=  
    R_1)mPQ^P  
    z = y; C.J`8@a]?  
    if any(idx_pos) zL:&Q<  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); PiMKu|,3  
    end o84UFhm   
    if any(idx_neg) %G;0T;0L  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); )0#j\ B  
    end (O\U /daB  
    h+,'B&=|_  
    % 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) 1/B]TT  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. 3&'2aW   
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated %.mEBI=hs  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive C ye T]y  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, GCDwWCxh  
    %   and THETA is a vector of angles.  R and THETA must have the same M!1U@6n!=)  
    %   length.  The output Z is a matrix with one column for every P-value, lT2 4JhJ#  
    %   and one row for every (R,THETA) pair. X1+ wX`f  
    % Xka<I3UD5  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike 2 {bhA5L  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) -fE.<)m=!  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) ]r4bRK[1  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 Cyq?5\a  
    %   for all p. BZK2$0  
    % y$X(S\W  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 q\%cFB}  
    %   Zernike functions (order N<=7).  In some disciplines it is tz26=8  
    %   traditional to label the first 36 functions using a single mode ,LD m8   
    %   number P instead of separate numbers for the order N and azimuthal UtnZNdl v  
    %   frequency M. !b8uLjd;  
    % :ygWNK[ 6D  
    %   Example: S,)d(g3>  
    % 62)d22  
    %       % Display the first 16 Zernike functions E@-ta):  
    %       x = -1:0.01:1; OS-sk!  
    %       [X,Y] = meshgrid(x,x); Z(j{F<\jS  
    %       [theta,r] = cart2pol(X,Y); ~ 3^='o  
    %       idx = r<=1; T*?s@$)m4  
    %       p = 0:15; )p<WDiX1!e  
    %       z = nan(size(X)); .N,&Uv-  
    %       y = zernfun2(p,r(idx),theta(idx)); tF*szf|$-  
    %       figure('Units','normalized') 7p.>\YtoR}  
    %       for k = 1:length(p) %Y]=1BRk}  
    %           z(idx) = y(:,k); +Y}V3(w9X  
    %           subplot(4,4,k) ;}qhc l+  
    %           pcolor(x,x,z), shading interp +k.%PO0np  
    %           set(gca,'XTick',[],'YTick',[]) :4:N f  
    %           axis square ?+~cA^-3T  
    %           title(['Z_{' num2str(p(k)) '}']) |?d#eQ9a  
    %       end Kz jC/1sd  
    % .Obn&S  
    %   See also ZERNPOL, ZERNFUN. > @_im6  
    (hB?  
    %   Paul Fricker 11/13/2006 1|{bDlmt  
    D-2.fjo9!  
    0fm*`4Q  
    % Check and prepare the inputs: UH? p]4Nz  
    % ----------------------------- eujK4s  
    if min(size(p))~=1 lhH`dG D  
        error('zernfun2:Pvector','Input P must be vector.') ST5V!jz  
    end iYJZvN  
    .1yT*+`  
    if any(p)>35 6KHN&P  
        error('zernfun2:P36', ... BHA923p?  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... ;{#^MD MB  
               '(P = 0 to 35).']) <q (z>*-e  
    end U!(@q!>G  
    ~j>D=!  
    % Get the order and frequency corresonding to the function number: c$:1:B9\  
    % ---------------------------------------------------------------- WOLuw%  
    p = p(:); ^03j8Pc-c  
    n = ceil((-3+sqrt(9+8*p))/2); {2&m`D bm  
    m = 2*p - n.*(n+2); 6"/WZmOp  
    fX~'Zk\u  
    % Pass the inputs to the function ZERNFUN: >j*;vG5T  
    % ---------------------------------------- `'I{U5;e  
    switch nargin h6_(?|:-(  
        case 3 5| B(\wqG  
            z = zernfun(n,m,r,theta); jN31hDg<z  
        case 4 bWWXc[O2&(  
            z = zernfun(n,m,r,theta,nflag); "9!d]2.-Vk  
        otherwise 1x)ZB~L  
            error('zernfun2:nargin','Incorrect number of inputs.') K+*Q@R D  
    end S4UM|`  
    q:\g^_!OGA  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) te|? )j  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. Q?b14]6im  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of Vd^g9  
    %   order N and frequency M, evaluated at R.  N is a vector of uvDzKMw~R  
    %   positive integers (including 0), and M is a vector with the fmqb` %  
    %   same number of elements as N.  Each element k of M must be a C+[%7vF1  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) )J]9 lW&y  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is [^CV>RuO  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix Y3.$G1{#0w  
    %   with one column for every (N,M) pair, and one row for every q6Rr.A  
    %   element in R. :Z`:nq.a  
    % &|>S|  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- m>USD? i  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is o#) {1<0vg  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to 'c2W}$q  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 **9x?s  
    %   for all [n,m]. :NJ_n6E  
    % ]]7 mlQ  
    %   The radial Zernike polynomials are the radial portion of the j',W 64  
    %   Zernike functions, which are an orthogonal basis on the unit 1b=lpw 1}  
    %   circle.  The series representation of the radial Zernike W} WI; cI  
    %   polynomials is {3;AwhN0H  
    % \(226^|j  
    %          (n-m)/2 dS^T$sz.co  
    %            __ `E+Jnu,jC  
    %    m      \       s                                          n-2s R_M?dEtE>  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r SJD@&m%?[  
    %    n      s=0 #/PAA  
    % ~ wg:!VWA)  
    %   The following table shows the first 12 polynomials. zvABU+{jD  
    % V5+SWXZ  
    %       n    m    Zernike polynomial    Normalization SGb;!T *  
    %       --------------------------------------------- B8E'ddUw  
    %       0    0    1                        sqrt(2) oFjIA!  
    %       1    1    r                           2 ;^t{Il'j  
    %       2    0    2*r^2 - 1                sqrt(6) ~l;[@jsw F  
    %       2    2    r^2                      sqrt(6) r0p w_j  
    %       3    1    3*r^3 - 2*r              sqrt(8) d%l{V6  
    %       3    3    r^3                      sqrt(8) /mo(_  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) h-iJlm  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) {vAE:W.s  
    %       4    4    r^4                      sqrt(10) :W b j\  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) Gy=B&boZ  
    %       5    3    5*r^5 - 4*r^3            sqrt(12)  hi g2  
    %       5    5    r^5                      sqrt(12) xsWur(>]  
    %       --------------------------------------------- Y*mbjyt[?X  
    % v<Bynd-  
    %   Example: nUkaz*4qU  
    % j*~T1i  
    %       % Display three example Zernike radial polynomials <uj 8lctmP  
    %       r = 0:0.01:1; J2uZmEt  
    %       n = [3 2 5]; AwQ?l(iZ"p  
    %       m = [1 2 1]; (O0Urm  
    %       z = zernpol(n,m,r); 2^?:&1:  
    %       figure >X*Mio8P#  
    %       plot(r,z) 4CGPO c  
    %       grid on NcY608C  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') bWOS `5  
    % N},n `Yl.  
    %   See also ZERNFUN, ZERNFUN2. Jx'i2&hGN  
    '\jd#Kn'h  
    % A note on the algorithm. l<M'=-Y  
    % ------------------------ mKYeD%Pm*  
    % The radial Zernike polynomials are computed using the series 6e7{Iy  
    % representation shown in the Help section above. For many special N!*_La=TuH  
    % functions, direct evaluation using the series representation can Z@hD(MS(C  
    % produce poor numerical results (floating point errors), because aZ\UrV4,  
    % the summation often involves computing small differences between fu~ +8CE.  
    % large successive terms in the series. (In such cases, the functions uc|45Zxt  
    % are often evaluated using alternative methods such as recurrence CbN!1E6).  
    % relations: see the Legendre functions, for example). For the Zernike WxF:~{  
    % polynomials, however, this problem does not arise, because the 7RZh<A>m  
    % polynomials are evaluated over the finite domain r = (0,1), and #r3l[ bKK  
    % because the coefficients for a given polynomial are generally all .^b;osAU  
    % of similar magnitude. Wb S4pdA  
    % 8i?l02  
    % ZERNPOL has been written using a vectorized implementation: multiple y\&>Z yOY  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] 52o x`t|  
    % values can be passed as inputs) for a vector of points R.  To achieve *OQG 4aWy  
    % this vectorization most efficiently, the algorithm in ZERNPOL LzYO$Ir:g  
    % involves pre-determining all the powers p of R that are required to ak:c rrkx  
    % compute the outputs, and then compiling the {R^p} into a single ,^S@EDq  
    % matrix.  This avoids any redundant computation of the R^p, and '= l[;Q^Q  
    % minimizes the sizes of certain intermediate variables. s: 3z'4oX  
    % +iI&c s  
    %   Paul Fricker 11/13/2006 Q,80Hor#J  
    j2 !3rI  
    1T:Y0  
    % Check and prepare the inputs: 3"rzb]=R  
    % ----------------------------- -j&Tc` j_  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Bn!$UUC  
        error('zernpol:NMvectors','N and M must be vectors.') bGorH=pb5R  
    end Znetzm=0  
    8/K!SpM*d  
    if length(n)~=length(m) Wa?; ^T  
        error('zernpol:NMlength','N and M must be the same length.') 0q&'(-{s1  
    end JsotOic%  
    itzyCw2|#  
    n = n(:); !~h}8'a?  
    m = m(:); z)uuxNv[R  
    length_n = length(n); !Kv@\4  
    BATG FS&  
    if any(mod(n-m,2)) \%p34K\  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') +}-@@,  
    end oTT7M`P3h  
    "]\+?  
    if any(m<0) .$#rV?7  
        error('zernpol:Mpositive','All M must be positive.') mXXt'_"  
    end w}CmfR  
    1 `KN]Nt  
    if any(m>n) +d|mR9^([  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') M'!U<Y -  
    end CA +uKM^"6  
    Reu*Pe  
    if any( r>1 | r<0 ) gR@C0  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') %e@#ux m  
    end r/*=%~*  
    ){Ob,LEU&  
    if ~any(size(r)==1) ox ;  
        error('zernpol:Rvector','R must be a vector.') j+HHQd7Y  
    end gsn)Wv$h  
    1LJUr"6]  
    r = r(:); 1 1cWy+8D  
    length_r = length(r); ;.bm6(;  
    ,()0' h}n  
    if nargin==4 _.-;5M-  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); n!qV>k9Y  
        if ~isnorm r$wxk 4%Rz  
            error('zernpol:normalization','Unrecognized normalization flag.') qL94SW;  
        end $k$4% 7  
    else _FwK-?4E-  
        isnorm = false; }=!,o  
    end D0N9Ksq  
    {f{ZHi|  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% K x~|jq  
    % Compute the Zernike Polynomials J sEa23  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% kD}Y|*]5-5  
    &  =/  
    % Determine the required powers of r: wYlf^~#"  
    % ----------------------------------- 7L^%x3-|&  
    rpowers = []; ^S6u<,  
    for j = 1:length(n) U w4>v:  
        rpowers = [rpowers m(j):2:n(j)]; w eQYQrN  
    end C4NTh}6t T  
    rpowers = unique(rpowers); />j';6vi  
    Q%-di=  
    % Pre-compute the values of r raised to the required powers, {%#)5l)  
    % and compile them in a matrix: CZbYAxNl  
    % ----------------------------- Vn~UB#]'3  
    if rpowers(1)==0 $\k0Nup}  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 0dh=fcb  
        rpowern = cat(2,rpowern{:}); FV,4pi  
        rpowern = [ones(length_r,1) rpowern]; $fgf Y8  
    else !s@Rok  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); @2*]"/)*0  
        rpowern = cat(2,rpowern{:}); 4hw@yTUo  
    end [NFNzwUB  
    6K-5g/hL  
    % Compute the values of the polynomials: ( 9]_ HW[  
    % -------------------------------------- f>ZyI{  
    z = zeros(length_r,length_n); P+)DsZ0ig  
    for j = 1:length_n %_Yx<wR%  
        s = 0:(n(j)-m(j))/2; 4Bl{WyMJ|  
        pows = n(j):-2:m(j); *:O.97q@h  
        for k = length(s):-1:1 J6<rX[ yZe  
            p = (1-2*mod(s(k),2))* ... Z;h<6[(  
                       prod(2:(n(j)-s(k)))/          ... s{w[b\rA  
                       prod(2:s(k))/                 ... +t2SzQ j>  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... i>[_r,-\[  
                       prod(2:((n(j)+m(j))/2-s(k))); mE^o-9/  
            idx = (pows(k)==rpowers); F_bF  
            z(:,j) = z(:,j) + p*rpowern(:,idx); HV/cc"  
        end =tU{7i*+  
         7n90f2"m  
        if isnorm 4 QZ?}iz  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); ^'jEnN(  
        end s[gKc'  
    end oBUxKisW  
    'Z`$n8  
    % 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)  YnV/M,U  
    +?`b=6e(`  
    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。 GIHpSy`z  
    6ew "fCrH!  
    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)