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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 _>=QZ`!r  
    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! )r|Pm-:A{  
     
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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 a=&{B'^G  
    function z = zernfun(n,m,r,theta,nflag) lSK<LytB  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. (>M? iB  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N w6<zPrA  
    %   and angular frequency M, evaluated at positions (R,THETA) on the F|! ib5  
    %   unit circle.  N is a vector of positive integers (including 0), and ;!Q}g19C  
    %   M is a vector with the same number of elements as N.  Each element "Kc1@EX=  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) 3#Qek2  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, rGUu K0L&  
    %   and THETA is a vector of angles.  R and THETA must have the same -W'T3_  
    %   length.  The output Z is a matrix with one column for every (N,M) ,]H2F']4Z  
    %   pair, and one row for every (R,THETA) pair. MCO`\"`l  
    % ukwO%JAr  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike +LB2V3UZ  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), zn2Qp  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral 3u@=]0ZN  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, cTu"Tu\Qw  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized \?~cJMN  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. (Y:?qy  
    % U C..)9  
    %   The Zernike functions are an orthogonal basis on the unit circle. `FHKQS5  
    %   They are used in disciplines such as astronomy, optics, and /M5R<rl  
    %   optometry to describe functions on a circular domain. ck\TTNA  
    % BVe c  
    %   The following table lists the first 15 Zernike functions. . l-eJ  
    % A| s\5"??  
    %       n    m    Zernike function           Normalization |$G|M=*LN  
    %       -------------------------------------------------- 4"d'iY  
    %       0    0    1                                 1 "fOxS\er  
    %       1    1    r * cos(theta)                    2 [Nv)37|W  
    %       1   -1    r * sin(theta)                    2 ..;ep2jSs  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) $9rQ w1#e  
    %       2    0    (2*r^2 - 1)                    sqrt(3) ~jDf,a2  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) |?<^4U8  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) aU?HIIA  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) cllnYvr3  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) ~fY\;  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) ,HECHA_"  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) u`Abko<D  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) N-YCOSUu  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) -W.bOr  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) h)pYV>!d  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) e!oL!Zg  
    %       -------------------------------------------------- ~=k?ea/>  
    % M+GtUE~"  
    %   Example 1: nNpXkI:  
    % `L7Cf&W\l8  
    %       % Display the Zernike function Z(n=5,m=1) O*udVE>  
    %       x = -1:0.01:1; 5#B M  
    %       [X,Y] = meshgrid(x,x); 4gh` >  
    %       [theta,r] = cart2pol(X,Y); |H&&80I  
    %       idx = r<=1; @BoZZ  
    %       z = nan(size(X)); s7"5NU-  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); ?L+|b5RS  
    %       figure sj8lvIY5  
    %       pcolor(x,x,z), shading interp \%Lj !\  
    %       axis square, colorbar PaZd^0'!Z  
    %       title('Zernike function Z_5^1(r,\theta)') bBgyLyg  
    % `9mc+  
    %   Example 2: *^i"q\n5(  
    % Z7J4r TA  
    %       % Display the first 10 Zernike functions pIl[)%F  
    %       x = -1:0.01:1; ,)PpE&  
    %       [X,Y] = meshgrid(x,x); Zy=DY  
    %       [theta,r] = cart2pol(X,Y); X]c>clk,  
    %       idx = r<=1; ()(^B}VK  
    %       z = nan(size(X)); v(~EO(n.  
    %       n = [0  1  1  2  2  2  3  3  3  3]; sfzDE&>'  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; w-P;E!gTt  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; XVzsqi*Z  
    %       y = zernfun(n,m,r(idx),theta(idx)); LX{mr{  
    %       figure('Units','normalized') Nn-EtM0w  
    %       for k = 1:10 _3zJ.%  
    %           z(idx) = y(:,k); 9{CajtN  
    %           subplot(4,7,Nplot(k)) oq[r+E-]$@  
    %           pcolor(x,x,z), shading interp {Lugdf'  
    %           set(gca,'XTick',[],'YTick',[]) BE)&.}l  
    %           axis square *X8Pa ;x  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) cQrXrij;!  
    %       end &?Z<"+B8S  
    % Yd]  
    %   See also ZERNPOL, ZERNFUN2. m*vz   
    dZuPR  
    %   Paul Fricker 11/13/2006 `Ln1g@  
    (je`sV  
    OXS.CFZM  
    % Check and prepare the inputs: kJpr:4;@_  
    % ----------------------------- lY[\eQ 1:  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Wn&9R j  
        error('zernfun:NMvectors','N and M must be vectors.') hCob^o  
    end FZtT2Z4&i  
    D*t[5,~j  
    if length(n)~=length(m) iHeu<3O  
        error('zernfun:NMlength','N and M must be the same length.') )WsR 8tk  
    end =55V<VI  
    @T] G5|\ok  
    n = n(:); uTNy{RBD+  
    m = m(:); dpcU`$kt  
    if any(mod(n-m,2)) RmJ|g<  
        error('zernfun:NMmultiplesof2', ... Uj^Y\w-@Z  
              'All N and M must differ by multiples of 2 (including 0).') 7ea%mg\  
    end #6mr'e1  
    i4lB ]k  
    if any(m>n) Au"BDP  
        error('zernfun:MlessthanN', ... !im%t9  
              'Each M must be less than or equal to its corresponding N.') W4"1H0s`l  
    end $ZlzS`XF7  
    s:ojlmPb  
    if any( r>1 | r<0 ) jJAr #|  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') y=zs6HaS  
    end FTu<$`!1L  
     `l  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) o.wXaS8  
        error('zernfun:RTHvector','R and THETA must be vectors.') ?dmw z4k0  
    end )3^#CD  
    &/?OP)N,}  
    r = r(:); )kIjZ  
    theta = theta(:); MbeK{8~E%l  
    length_r = length(r); oxLO[js  
    if length_r~=length(theta) _ygdv\^Tet  
        error('zernfun:RTHlength', ... 4iY <7l8  
              'The number of R- and THETA-values must be equal.') ]L?WC  
    end Awe'MGp%  
    -qG7,t  
    % Check normalization: 2]}e4@{  
    % -------------------- )h1 `?q:5  
    if nargin==5 && ischar(nflag) H[N~)3x  
        isnorm = strcmpi(nflag,'norm'); vj"['6Xa  
        if ~isnorm S2?)Sb`  
            error('zernfun:normalization','Unrecognized normalization flag.') B-V   
        end W?0u_F  
    else +/r h8?  
        isnorm = false; 2[Xe:)d  
    end o<rbC < U  
    =z'533C  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 4F+G;'JV  
    % Compute the Zernike Polynomials pIY3ft\  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - XB[2h  
    Ni#y=cb  
    % Determine the required powers of r: :@S=0|:j  
    % ----------------------------------- ~>$z1o&}.  
    m_abs = abs(m); R^rA.7T  
    rpowers = []; n +dRAIqB  
    for j = 1:length(n) *}Rd%'  
        rpowers = [rpowers m_abs(j):2:n(j)]; :AyZe7:(D  
    end rLcXo %w  
    rpowers = unique(rpowers); \b?O+;5Cj  
    a KIS%M#Y  
    % Pre-compute the values of r raised to the required powers, >Sm#-4B-  
    % and compile them in a matrix: $it>*%  
    % ----------------------------- ,&jjp eZP  
    if rpowers(1)==0 Y^gIvX  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ;V^I>-fnm  
        rpowern = cat(2,rpowern{:}); ^ ?T,>ZI  
        rpowern = [ones(length_r,1) rpowern]; \>+BvF  
    else `!.c_%m2  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); \$ :)Ka  
        rpowern = cat(2,rpowern{:}); t}gK)"g  
    end 4}Hf"L[ l  
    EI@ep~  
    % Compute the values of the polynomials: RMa#z [{0  
    % -------------------------------------- hcQv!!Q"k$  
    y = zeros(length_r,length(n)); SpZmwa #\  
    for j = 1:length(n) &sGLm~m#  
        s = 0:(n(j)-m_abs(j))/2; "~T06!F45  
        pows = n(j):-2:m_abs(j); fw0Z- 9*  
        for k = length(s):-1:1 EiWd =jDm  
            p = (1-2*mod(s(k),2))* ... s_76)7  
                       prod(2:(n(j)-s(k)))/              ... uQkQ#'e|  
                       prod(2:s(k))/                     ... E /V`NqC  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... Y 4*?QBYA  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); >u=nGeO  
            idx = (pows(k)==rpowers); -3C$br  
            y(:,j) = y(:,j) + p*rpowern(:,idx); (Jk:Qz5  
        end yJw4!A 1!  
         cQ/T:E7$`  
        if isnorm ^7C,GaDsn  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 2^&5D,}0  
        end yj9 Ad*.  
    end 1JN/oq;  
    % END: Compute the Zernike Polynomials =4 W jb  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \>4x7mF!  
    zxvowM  
    % Compute the Zernike functions: iPrAB*  
    % ------------------------------ PSa"u5O  
    idx_pos = m>0; |R(rb-v  
    idx_neg = m<0; *1_A$14 l  
    `Dv &.  
    z = y; y#5;wb<1  
    if any(idx_pos) RQ[6svfP  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 8wA'a'V.  
    end 1iE*-K%Q  
    if any(idx_neg) ,y/N^^\  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); +6 x:+9S  
    end CB?,[#r5f  
    tNCKL. yU  
    % 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) ZNG{:5u,  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. j0[9Cj^%c  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated t~FOaSt  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive e5fzV.'5  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, n7K\\|X  
    %   and THETA is a vector of angles.  R and THETA must have the same d!]_n|B@9  
    %   length.  The output Z is a matrix with one column for every P-value, < 5#}EiT5  
    %   and one row for every (R,THETA) pair. &(HIBF'O  
    % `?:X-dh_  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike b n<}  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) z[ ml;?  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) UI.>BZ6}  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 Zw"K69A)  
    %   for all p. *>p#/'_E  
    % [\e2 ID;  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 `=cOTn52  
    %   Zernike functions (order N<=7).  In some disciplines it is ;]Bkw6 o  
    %   traditional to label the first 36 functions using a single mode :u>9H{a  
    %   number P instead of separate numbers for the order N and azimuthal En/EQ\T@F  
    %   frequency M. Og~3eL[1%C  
    %  6,;7iA]  
    %   Example: >0qe*4n|M  
    % DG"Z:^`*  
    %       % Display the first 16 Zernike functions Vy[ m%sEP  
    %       x = -1:0.01:1; I<ta2<h  
    %       [X,Y] = meshgrid(x,x); iSxuor ^;  
    %       [theta,r] = cart2pol(X,Y); Rc k k  
    %       idx = r<=1; "kT?9&  
    %       p = 0:15; EAM2t|M G.  
    %       z = nan(size(X)); fbK`A?5K  
    %       y = zernfun2(p,r(idx),theta(idx)); *(sFr E  
    %       figure('Units','normalized') s4{WPU9  
    %       for k = 1:length(p) #y&O5    
    %           z(idx) = y(:,k); <rpXhcR  
    %           subplot(4,4,k) Da"GYEC  
    %           pcolor(x,x,z), shading interp G;3N"az  
    %           set(gca,'XTick',[],'YTick',[]) ,a /<t"  
    %           axis square )mw&e}jRV  
    %           title(['Z_{' num2str(p(k)) '}']) .IJgkP)!]  
    %       end Y^@Nvt$<K  
    % Iz[T.$9  
    %   See also ZERNPOL, ZERNFUN. Xm! ;  
    J#^oUq  
    %   Paul Fricker 11/13/2006 V,CVMbn/%N  
    52JtEt7E  
    J< U,~ra\  
    % Check and prepare the inputs: hO@VYO   
    % ----------------------------- vCh/%7+  
    if min(size(p))~=1 1Xi.OGl  
        error('zernfun2:Pvector','Input P must be vector.') Iq[Z5k(K  
    end ;,yjkD[mWE  
    9(;I+.;8k  
    if any(p)>35 u:0M,Ye  
        error('zernfun2:P36', ...  1ZF>e`t8  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... T[YGQT|B  
               '(P = 0 to 35).']) }}'0r2S  
    end mt(2HBNoz  
    qJZ5w }  
    % Get the order and frequency corresonding to the function number:  )6 _+  
    % ---------------------------------------------------------------- T1Q c?5K^  
    p = p(:); xi =\]  
    n = ceil((-3+sqrt(9+8*p))/2); dNL<O   
    m = 2*p - n.*(n+2); oJEUNgY&  
    aA-gl9  
    % Pass the inputs to the function ZERNFUN: `:I<Jp  
    % ---------------------------------------- \Y{^Q7!>:8  
    switch nargin =7U_ jDME  
        case 3 j%qBNoT~  
            z = zernfun(n,m,r,theta); #K3`$^0 s  
        case 4 ny]R,D0  
            z = zernfun(n,m,r,theta,nflag); 1/H9(2{L  
        otherwise xC,;IS k,  
            error('zernfun2:nargin','Incorrect number of inputs.')  :nHa-N3  
    end &~xzp^&  
    EhOy<f[4W  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) ; SS/bS|  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M.  Nx8~Rn  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of /*+P}__k  
    %   order N and frequency M, evaluated at R.  N is a vector of o"Dk`L2  
    %   positive integers (including 0), and M is a vector with the >2[\WF*"X  
    %   same number of elements as N.  Each element k of M must be a uq>\pO&P  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) <=D\Ckmb  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is FKOTv2  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix )r)3.|wJm  
    %   with one column for every (N,M) pair, and one row for every avb'J^}f  
    %   element in R. fU )@Lj1Wo  
    % 8 pQx6QE  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- /7nircXj@  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is 2k}" 52  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to |Mj2lZS  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 NT^m.o~4  
    %   for all [n,m]. 7ktSj}7W]  
    % 6,707h  
    %   The radial Zernike polynomials are the radial portion of the %V+hm5Q  
    %   Zernike functions, which are an orthogonal basis on the unit b=5"*=T{+  
    %   circle.  The series representation of the radial Zernike SpM|b5c5  
    %   polynomials is {!lC$SlJ  
    % Y~P1r]piB  
    %          (n-m)/2 eCDwY:t`  
    %            __ A{HP*x~t  
    %    m      \       s                                          n-2s <Be:fnPX7  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r o1m+4.-  
    %    n      s=0 |# _F  
    % V!Px975P  
    %   The following table shows the first 12 polynomials. Y;X_E7U  
    % E^)>9f7  
    %       n    m    Zernike polynomial    Normalization aDV~T24  
    %       --------------------------------------------- +:a#+]g  
    %       0    0    1                        sqrt(2) \; 9log<Z  
    %       1    1    r                           2 ~]MACG:'  
    %       2    0    2*r^2 - 1                sqrt(6) KlMSkdmW  
    %       2    2    r^2                      sqrt(6) ^dR="N  
    %       3    1    3*r^3 - 2*r              sqrt(8) qHZ!~Kq,"'  
    %       3    3    r^3                      sqrt(8) m#\I&(l+  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) 9vQI ~rz?  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) ZU=om Rh5  
    %       4    4    r^4                      sqrt(10) <sALA~p|0  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) gQ < >S  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) |6cz r  
    %       5    5    r^5                      sqrt(12) ;qA(!`h+  
    %       --------------------------------------------- -x~4@~  
    % eucacXiZ  
    %   Example: [tKH'}/s=  
    % P}2i[m.*,  
    %       % Display three example Zernike radial polynomials zS9HR1  
    %       r = 0:0.01:1; v%ldg833l  
    %       n = [3 2 5]; ?06+"Z  
    %       m = [1 2 1]; ftr8~*]O  
    %       z = zernpol(n,m,r); CAbeb+O  
    %       figure 4Bn <L&@/  
    %       plot(r,z) Ft<6`C  
    %       grid on CFLWo1  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') ~t>i+{J KE  
    % !-cO 0c!  
    %   See also ZERNFUN, ZERNFUN2. F}f/cG<X  
    ii3{HJ*C  
    % A note on the algorithm. tdCD!rV`{  
    % ------------------------ 1}7Q2Ad w  
    % The radial Zernike polynomials are computed using the series hT=6XO od4  
    % representation shown in the Help section above. For many special W Ai91K@  
    % functions, direct evaluation using the series representation can L[D<e?j  
    % produce poor numerical results (floating point errors), because ;R_H8vp  
    % the summation often involves computing small differences between fEWXC|"  
    % large successive terms in the series. (In such cases, the functions !kTI@103Wd  
    % are often evaluated using alternative methods such as recurrence R_vF$X'Ow  
    % relations: see the Legendre functions, for example). For the Zernike j>}<FW-N  
    % polynomials, however, this problem does not arise, because the e5s=@-[  
    % polynomials are evaluated over the finite domain r = (0,1), and z0jF.ub  
    % because the coefficients for a given polynomial are generally all QY^v*+lr\  
    % of similar magnitude. @@ 1Sxv_  
    % 1ti9FQ  
    % ZERNPOL has been written using a vectorized implementation: multiple Knjg`f  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] !,(6uO%  
    % values can be passed as inputs) for a vector of points R.  To achieve -p-<mC@<&S  
    % this vectorization most efficiently, the algorithm in ZERNPOL "PK`Ca@`v  
    % involves pre-determining all the powers p of R that are required to [X\<C '<  
    % compute the outputs, and then compiling the {R^p} into a single mj7Em&  
    % matrix.  This avoids any redundant computation of the R^p, and C$<['D?8  
    % minimizes the sizes of certain intermediate variables. z6Xn9  
    % ,ix>e  
    %   Paul Fricker 11/13/2006 "!AbH<M;@  
    Fv )H;1V  
    k-;A9!^h  
    % Check and prepare the inputs: ] 'B4O1  
    % ----------------------------- 0dKv%X#\  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) -Wt (t2  
        error('zernpol:NMvectors','N and M must be vectors.') A7hWAq  
    end hmG^l4B.T  
    *#| lhf'  
    if length(n)~=length(m) pR,eus;8  
        error('zernpol:NMlength','N and M must be the same length.')  {ch+G~oS  
    end H6vO}pq) r  
    H!81Pq~  
    n = n(:); n a3st*3V_  
    m = m(:); a9sbB0q-K@  
    length_n = length(n); ?j:g.a+U  
    q=J8SvSRl  
    if any(mod(n-m,2))  (%\tE  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') % k}+t3aF  
    end 'Cp]Q@]\  
    v6#i>n~x,  
    if any(m<0) q qFN4AO  
        error('zernpol:Mpositive','All M must be positive.') H7}@56  
    end b;~?a#Z}  
    sq;nUA=  
    if any(m>n) d,:3;:CR  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') r/e} DYL&  
    end 5_yu4{@;y  
    rF:l+I]  
    if any( r>1 | r<0 ) _enS_R  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') -nL!#R{e  
    end [a2Q ^ab  
    FDQP|,  
    if ~any(size(r)==1) Frd`u .I  
        error('zernpol:Rvector','R must be a vector.') 8IQqDEY^  
    end q Xj]O3 mm  
    'X(Sn3  
    r = r(:); Tsc2;I  
    length_r = length(r); Ae3=o8p  
    DFvj  
    if nargin==4 L_(Y[!  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); $Ao iH{f  
        if ~isnorm 11Y4oS  
            error('zernpol:normalization','Unrecognized normalization flag.') 1!"iN~  
        end tg#d.(  
    else 9'I$8Su  
        isnorm = false; in>.Tax*  
    end U+7!Vpq  
    FrL ;1zt  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +h?Rb3=S  
    % Compute the Zernike Polynomials AY)R2> fW%  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% N&YQZ^o  
    dxk~  
    % Determine the required powers of r: i^_?C5  
    % ----------------------------------- dkI(&/  
    rpowers = []; ^sb+|b  
    for j = 1:length(n) -D^.I  
        rpowers = [rpowers m(j):2:n(j)]; UkzLUok]U  
    end Bm:N@wg  
    rpowers = unique(rpowers); =Dc9|WuHN  
    {+  @M!  
    % Pre-compute the values of r raised to the required powers, ,Z aPY  
    % and compile them in a matrix: hutdw>  
    % ----------------------------- 3+tr_psH  
    if rpowers(1)==0 3uJ>:,~r  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); F0 ^kUyF|  
        rpowern = cat(2,rpowern{:}); v#ERXIrf  
        rpowern = [ones(length_r,1) rpowern]; c3X8Wi7m  
    else VU ,tCTXz  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); i8 fUzg)  
        rpowern = cat(2,rpowern{:}); ,mm9X\ '  
    end -h1FrDBt  
    &8'QD~  
    % Compute the values of the polynomials: Y V#|qb  
    % -------------------------------------- P d)<Iw^<  
    z = zeros(length_r,length_n); y1*z," dx  
    for j = 1:length_n !^_G~`r$2J  
        s = 0:(n(j)-m(j))/2; q%\rj?U_  
        pows = n(j):-2:m(j); T*v@hbJ  
        for k = length(s):-1:1 %o4HCzId<  
            p = (1-2*mod(s(k),2))* ... n}!D)Gx  
                       prod(2:(n(j)-s(k)))/          ... y - Ge"mY  
                       prod(2:s(k))/                 ... DfX}^'#m+  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... \h UE, ^  
                       prod(2:((n(j)+m(j))/2-s(k))); $,DX^I%!  
            idx = (pows(k)==rpowers); 6,:`esl  
            z(:,j) = z(:,j) + p*rpowern(:,idx); 3X]\p}]z  
        end ^ e4y:#Nu  
         C Y K W4  
        if isnorm 8^)K|+_'m  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); ;&?l1Vu  
        end Yt#e[CYnu  
    end n=tg{_9f%  
    HS[N]'dc  
    % 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)  S;[*5g6a&x  
    &k /uR;yw  
    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。 /HsJyp+t  
    /RuGh8qzP  
    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)