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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 v}w=I}<x  
    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! 4'L%Wz[6  
     
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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 ;dq AmBG{8  
    function z = zernfun(n,m,r,theta,nflag) K>H_q@-?f  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. Epm'u[wV  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N S&BJR!FQ  
    %   and angular frequency M, evaluated at positions (R,THETA) on the -u6`B -T  
    %   unit circle.  N is a vector of positive integers (including 0), and dm4dT59  
    %   M is a vector with the same number of elements as N.  Each element I<Vh Eo,  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) ]stAC3  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, cpV:y  
    %   and THETA is a vector of angles.  R and THETA must have the same HRF4 Ro  
    %   length.  The output Z is a matrix with one column for every (N,M) EFl[u+ 1tx  
    %   pair, and one row for every (R,THETA) pair. P<iS7Ys+  
    % ^FLuhLS\*  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike A]nDI:pO|  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), WZ"g:Khw  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral 5S:&^ A<  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, )9@I7QG?  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized = Mc]FCV  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. TI -#\v9  
    % n*\AB=|X  
    %   The Zernike functions are an orthogonal basis on the unit circle. yQQ[_1$pq  
    %   They are used in disciplines such as astronomy, optics, and |q$br-0+  
    %   optometry to describe functions on a circular domain. /wIev1Z!Y  
    % % ~%>3  
    %   The following table lists the first 15 Zernike functions. K%h83tm+  
    % %v++AcE  
    %       n    m    Zernike function           Normalization 7{oG4X!  
    %       -------------------------------------------------- Z@j$i\,`  
    %       0    0    1                                 1 KZV$rJ%G  
    %       1    1    r * cos(theta)                    2 l'N>9~f  
    %       1   -1    r * sin(theta)                    2 BaIh,iu  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) luZqW`?Bt  
    %       2    0    (2*r^2 - 1)                    sqrt(3) ;F@dN,Y  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) k07JMS?  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) AR\1w'  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) o?P(Fuf  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) $UlA_l29  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) S<+_yB?  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) zk]6|i$!I  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ZMJ\C|S:  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) tZ1iaYbvV  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) F0Xv84:O  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) d87pQ3e:&  
    %       -------------------------------------------------- <wTkPErUG  
    % kl[bDb1p  
    %   Example 1: ?Gr<9e2Eo  
    % #m9V) 1"wB  
    %       % Display the Zernike function Z(n=5,m=1) z x{\SU  
    %       x = -1:0.01:1; 6m21Y8N  
    %       [X,Y] = meshgrid(x,x); =Fea vyx  
    %       [theta,r] = cart2pol(X,Y); 5}e-~-  
    %       idx = r<=1; GpF,=:  
    %       z = nan(size(X)); C78d29  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx));  U^ BB|  
    %       figure ~I/7{B|yX  
    %       pcolor(x,x,z), shading interp ;3'}(_n  
    %       axis square, colorbar Pw /wAUt  
    %       title('Zernike function Z_5^1(r,\theta)') dQA J`9B  
    % ^~MHxF5d  
    %   Example 2: $y=sT({VVe  
    % M:?eK [h  
    %       % Display the first 10 Zernike functions -tx)7KV-  
    %       x = -1:0.01:1; 7w )#[^  
    %       [X,Y] = meshgrid(x,x); zE.4e&m%Z?  
    %       [theta,r] = cart2pol(X,Y); %{/0K<M  
    %       idx = r<=1; /eR@&!D '  
    %       z = nan(size(X)); 5n.4>yOY  
    %       n = [0  1  1  2  2  2  3  3  3  3]; )+w0NhJw  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; /H^bDUC :r  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; =KT7ZSTV  
    %       y = zernfun(n,m,r(idx),theta(idx)); I?-9%4 8iM  
    %       figure('Units','normalized') wlKpHd*  
    %       for k = 1:10 w _eu@R:u@  
    %           z(idx) = y(:,k); 4)9X) Qx  
    %           subplot(4,7,Nplot(k)) nC`#Hm.V%  
    %           pcolor(x,x,z), shading interp *goi^ Xp  
    %           set(gca,'XTick',[],'YTick',[]) R|NmkqTK~(  
    %           axis square 7"4|`y^#  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) +cy(}Vp  
    %       end /[nt=#+   
    % 9L:v$4{LU  
    %   See also ZERNPOL, ZERNFUN2. L6$,<}l  
    oB9Fas!N  
    %   Paul Fricker 11/13/2006 2T?t[;-  
    Q;r 0#"  
    */\dH<  
    % Check and prepare the inputs: v-G(bw3  
    % ----------------------------- 9FV#@uA}D  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) w/G5I )G  
        error('zernfun:NMvectors','N and M must be vectors.') pS%,wjb&P  
    end 5bmtUIj  
    [4xN:i  
    if length(n)~=length(m) Y<#7E;aL  
        error('zernfun:NMlength','N and M must be the same length.') IRo[|&c  
    end @292;qi  
    6t]oSxN  
    n = n(:); " I`YJEv  
    m = m(:); z=)5M*h  
    if any(mod(n-m,2)) 3)0*hq&83  
        error('zernfun:NMmultiplesof2', ... 6xz&Qi7w  
              'All N and M must differ by multiples of 2 (including 0).') l`$f@'k  
    end Pn7oQA\  
    qLYv=h$,  
    if any(m>n) 2b|vb}|t{  
        error('zernfun:MlessthanN', ...  |k 4+I  
              'Each M must be less than or equal to its corresponding N.') 8n~@Rj5  
    end zi*D8!_C  
    z eIBB  
    if any( r>1 | r<0 ) =Z-.4\3  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') >+oQxml6nI  
    end , st4K;-  
    zP=J5qOZ8  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) vgE5(fJh  
        error('zernfun:RTHvector','R and THETA must be vectors.') PVEEKKJP]J  
    end >b*Pd *f  
    $a5K  
    r = r(:); )sNtw Sl^  
    theta = theta(:); 3Q`'C7Pi  
    length_r = length(r); A ;kAAM  
    if length_r~=length(theta) Za}91z"  
        error('zernfun:RTHlength', ... QX(:!b  
              'The number of R- and THETA-values must be equal.') NmtBn^ t  
    end ?6j@EJ<2q  
    b:%>T PT  
    % Check normalization: nh9K(  
    % -------------------- C5sV-UMR  
    if nargin==5 && ischar(nflag) Ld`~^<B  
        isnorm = strcmpi(nflag,'norm'); ;#xhlR* ~  
        if ~isnorm 8%nTDSp&t  
            error('zernfun:normalization','Unrecognized normalization flag.') /Zv}u  
        end j<L!ONvJ1  
    else ',1rW  
        isnorm = false; f$WO{ J  
    end * 5P/&*c|  
    qVM]$V#e  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% yobi$mnsy!  
    % Compute the Zernike Polynomials XTeU 2I  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +U6! bu>C  
    l_f"}l  
    % Determine the required powers of r: tU)+q?Mw  
    % ----------------------------------- 80+" x3r  
    m_abs = abs(m); PiH#9X B  
    rpowers = []; 3rR(>}:[V  
    for j = 1:length(n) *4(.=k  
        rpowers = [rpowers m_abs(j):2:n(j)]; =~HX/]zF  
    end $6oLiYFX;  
    rpowers = unique(rpowers); 5Vvy:<.la  
    J-,T^Wv  
    % Pre-compute the values of r raised to the required powers, : wn![<`3q  
    % and compile them in a matrix: ^Y'>3o21f  
    % ----------------------------- O>k.sO <  
    if rpowers(1)==0 Jn:GqO  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); Vx#xq#wK  
        rpowern = cat(2,rpowern{:}); ,%ajIs"Gi  
        rpowern = [ones(length_r,1) rpowern]; %HSoQ?qA  
    else 14^t{  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); V25u'.'v  
        rpowern = cat(2,rpowern{:}); IPT}JX'  
    end OK2wxf  
    x3M`l|  
    % Compute the values of the polynomials: 74Kl!A  
    % -------------------------------------- beIEy(rA  
    y = zeros(length_r,length(n)); ?SQT;C3j(  
    for j = 1:length(n) 7qh_URt@  
        s = 0:(n(j)-m_abs(j))/2; ~^3B(feQ]  
        pows = n(j):-2:m_abs(j); 2oq>tnYyV[  
        for k = length(s):-1:1 !J6k\$r  
            p = (1-2*mod(s(k),2))* ... -i;#4@^t  
                       prod(2:(n(j)-s(k)))/              ... Wxg|jP$~   
                       prod(2:s(k))/                     ... a{)"KAP  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ~i(*.Z) \  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); _|s{G  
            idx = (pows(k)==rpowers); 3[Z?`X  
            y(:,j) = y(:,j) + p*rpowern(:,idx); I=lA7}  
        end OY@/18D<>  
         Z~P5SEg  
        if isnorm (2a~gQGD  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 4l z9z>J.V  
        end l[h??C`  
    end gWJLWL2  
    % END: Compute the Zernike Polynomials u/,m2N9cL  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% (F#Qunze  
    k-w._E <  
    % Compute the Zernike functions: \9 ^w M>U  
    % ------------------------------ pG|DT ?  
    idx_pos = m>0; ]C'r4Ch^  
    idx_neg = m<0; b9"Q.*c<Z^  
    2P]rJ  
    z = y; y|1-,u.$  
    if any(idx_pos) Ejn19{  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); Lo !kv*  
    end -lLq)  
    if any(idx_neg) h],_1!0  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); aA\v  
    end O*c +TiTb  
    >pn?~  
    % 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) ?n[+0a:8E  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. QCMt4`% 'u  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated &Tl3\T0D  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive _jP]ifu`  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, _d,_&7  
    %   and THETA is a vector of angles.  R and THETA must have the same (?oK+,v?L  
    %   length.  The output Z is a matrix with one column for every P-value, OCF= )#}qd  
    %   and one row for every (R,THETA) pair. hfVJg7-  
    % Pq !\6s@  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike 'Kc;~a  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) ^|OxlfS  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) (i&:=Bfn)  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 gh3_})8c  
    %   for all p. @7.Ews5Mke  
    % td{$ c6  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 v-g2k_ o|  
    %   Zernike functions (order N<=7).  In some disciplines it is _18) XR  
    %   traditional to label the first 36 functions using a single mode [N.4 i" Cd  
    %   number P instead of separate numbers for the order N and azimuthal rr9N(AoxW  
    %   frequency M. k${25*M!3  
    % $xNZ.|al  
    %   Example:  w8$8P  
    % Pe~[qETv  
    %       % Display the first 16 Zernike functions T[q2quXgk  
    %       x = -1:0.01:1; <D!"<&N  
    %       [X,Y] = meshgrid(x,x); _-^a8F>/19  
    %       [theta,r] = cart2pol(X,Y); FAo\`x  
    %       idx = r<=1; 8+^q9rLii  
    %       p = 0:15; O_*%_S}F&  
    %       z = nan(size(X)); AwUcU;"9>  
    %       y = zernfun2(p,r(idx),theta(idx)); 1H{J T op  
    %       figure('Units','normalized') xrf z-"n4  
    %       for k = 1:length(p) F7x]BeTM  
    %           z(idx) = y(:,k); B[epI3 R  
    %           subplot(4,4,k) 'de&9\  
    %           pcolor(x,x,z), shading interp 5$d>:" >  
    %           set(gca,'XTick',[],'YTick',[]) EWrIDZi  
    %           axis square yxik`vmH  
    %           title(['Z_{' num2str(p(k)) '}']) b<n*wH  
    %       end 3fM8W> *7  
    % Uyj6Ij_Pj)  
    %   See also ZERNPOL, ZERNFUN. *%E4 ,(T  
    _h6SW2:z!E  
    %   Paul Fricker 11/13/2006 e ^2n58  
    `-/-(v+ i  
    ]{s0/(EA  
    % Check and prepare the inputs: kNR -eG  
    % ----------------------------- e];lDa#4-Y  
    if min(size(p))~=1 gNUYHNzDM(  
        error('zernfun2:Pvector','Input P must be vector.') _(l?gj  
    end tp*.'p-SI  
    L`NY^  
    if any(p)>35 N:x--,2  
        error('zernfun2:P36', ... -Aaim`06bv  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... 3:C)1q  
               '(P = 0 to 35).']) k<Qhw)M8  
    end 1o`zAJ8|2  
    rP|~d}+I  
    % Get the order and frequency corresonding to the function number: ti'B}bH>'  
    % ---------------------------------------------------------------- + fS<YT  
    p = p(:); z?dd5.k  
    n = ceil((-3+sqrt(9+8*p))/2); GZH{"_$  
    m = 2*p - n.*(n+2); hz:h>Hwy  
    )Fon;/p  
    % Pass the inputs to the function ZERNFUN: sPX&XqWx  
    % ---------------------------------------- %|j`z?i|  
    switch nargin e`n+U-)z  
        case 3 GXC,p(vbE  
            z = zernfun(n,m,r,theta); 4Hy/K^Ci  
        case 4 <yl%q*gls  
            z = zernfun(n,m,r,theta,nflag); hh8Grl;  
        otherwise M8nfbc^  
            error('zernfun2:nargin','Incorrect number of inputs.') ;NU-\<Q{  
    end |;:g7eb  
    s@Dln Du .  
    % 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) ,qx^D  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. &&nbdu  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of &rGB58  
    %   order N and frequency M, evaluated at R.  N is a vector of F+"_]  
    %   positive integers (including 0), and M is a vector with the 85YUqVi9  
    %   same number of elements as N.  Each element k of M must be a >H^#!eaqw  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) (+c1.h  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is [\AOr`7  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix fuzB;Ea  
    %   with one column for every (N,M) pair, and one row for every [Ur\^wS  
    %   element in R. u\V^g   
    % lD[37U!  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- P#O2MiG  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is om`T/@_,  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to jUEgu  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 XEQTTD<  
    %   for all [n,m]. Jy5sZ }t[  
    % t%;w<1E  
    %   The radial Zernike polynomials are the radial portion of the P 0e-v0  
    %   Zernike functions, which are an orthogonal basis on the unit O&1qL)  
    %   circle.  The series representation of the radial Zernike RN[I%^$"  
    %   polynomials is xNzGp5H  
    % 3w</B- |nQ  
    %          (n-m)/2 s'h;a5Q1'Q  
    %            __ qT48Y  
    %    m      \       s                                          n-2s =}vT>b  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r odCt6Du  
    %    n      s=0 %8,$ILN  
    % 5BJ E  
    %   The following table shows the first 12 polynomials. PVsKI<  
    % z$e6T&u5B  
    %       n    m    Zernike polynomial    Normalization 0V4B Q:v  
    %       --------------------------------------------- ikW[lefTq  
    %       0    0    1                        sqrt(2) .E<nQWz 8  
    %       1    1    r                           2 z Fo11;*D  
    %       2    0    2*r^2 - 1                sqrt(6) vd{QFJ  
    %       2    2    r^2                      sqrt(6) Ut;`6t  
    %       3    1    3*r^3 - 2*r              sqrt(8) Zz0e4C  
    %       3    3    r^3                      sqrt(8) BH">#&j[  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) g w" \pD  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) GC{M"q|_  
    %       4    4    r^4                      sqrt(10) !R=@Nr>  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) $@>0;i ::  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) #;$]M4  
    %       5    5    r^5                      sqrt(12) j{@6y  
    %       --------------------------------------------- TxX=(7V  
    % ){*+s RBW  
    %   Example: u= NLR\  
    % &EfQ%r}C  
    %       % Display three example Zernike radial polynomials bC/":+s& p  
    %       r = 0:0.01:1; @1MnJP  
    %       n = [3 2 5]; +!/ATR%Uci  
    %       m = [1 2 1]; `gX@b^  
    %       z = zernpol(n,m,r); !y= R)k  
    %       figure 8R,<S-+v  
    %       plot(r,z) BmG(+;;&  
    %       grid on zxbf h/=  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') %2?+:R5.  
    % U ? +_\  
    %   See also ZERNFUN, ZERNFUN2. rm iOeS`:  
    u^1#9bAW8  
    % A note on the algorithm. }yz>(Pq  
    % ------------------------ j]Jgz<  
    % The radial Zernike polynomials are computed using the series JE=t e(a  
    % representation shown in the Help section above. For many special s6!6Oqh  
    % functions, direct evaluation using the series representation can qN $t_  
    % produce poor numerical results (floating point errors), because V!W.P  
    % the summation often involves computing small differences between \D7bTn  
    % large successive terms in the series. (In such cases, the functions Vw;Z0_C  
    % are often evaluated using alternative methods such as recurrence TSlB.pw%v  
    % relations: see the Legendre functions, for example). For the Zernike [9 W@<p  
    % polynomials, however, this problem does not arise, because the eU[g@Pq:Y  
    % polynomials are evaluated over the finite domain r = (0,1), and fpD$%.y'J  
    % because the coefficients for a given polynomial are generally all "& ,ov#  
    % of similar magnitude. P {TJ$  
    % E TT46%Y  
    % ZERNPOL has been written using a vectorized implementation: multiple O>~,RI!  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] K^Awf6%  
    % values can be passed as inputs) for a vector of points R.  To achieve =V^-@ji)b  
    % this vectorization most efficiently, the algorithm in ZERNPOL Gv:~P_vBH[  
    % involves pre-determining all the powers p of R that are required to Zxa.x?:?n  
    % compute the outputs, and then compiling the {R^p} into a single @(3F4Z.i%.  
    % matrix.  This avoids any redundant computation of the R^p, and `o[l%I\Q  
    % minimizes the sizes of certain intermediate variables. W>K^55'  
    % (_T{Z>C/J  
    %   Paul Fricker 11/13/2006 apvcWF%  
    J|`0GDSn  
    +y GQt3U  
    % Check and prepare the inputs: rE3dHJN;  
    % ----------------------------- *g/klK  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) at: li  
        error('zernpol:NMvectors','N and M must be vectors.') d1b] +AG4  
    end c{z$^)A/  
     is'V%q  
    if length(n)~=length(m) oQ$yr^M  
        error('zernpol:NMlength','N and M must be the same length.') "mlQ z4D)5  
    end ;G[V:.o-  
    Dw-d`8*  
    n = n(:); $Ome]+0  
    m = m(:); #Y'eS'lv4  
    length_n = length(n); d2rs+-  
    $v^hzC  
    if any(mod(n-m,2)) !?2)a pM  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') ]<;,HGO  
    end XzUGlrp:Y#  
    $l7^-SK`E  
    if any(m<0) C?PQ>Q!f-  
        error('zernpol:Mpositive','All M must be positive.') y.rN(  
    end IGlR,tw_/  
    )!T~l(g  
    if any(m>n) iI3:<j l  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') +v Bi7#&  
    end 5/meH[R\M  
    ]%Q!%uTh  
    if any( r>1 | r<0 ) vQAFgG  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') ^h(wi`i  
    end !l:GrT8J  
    0RUk^  
    if ~any(size(r)==1) 2MkrVQQ9g  
        error('zernpol:Rvector','R must be a vector.') qQ@| Cj  
    end / f%mYL  
     @/2Kfr  
    r = r(:); gQ1 obT"|  
    length_r = length(r); hHs/Qtq  
    Q8p6n  
    if nargin==4 @u~S!(7.Wi  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); 2*#|t: (c  
        if ~isnorm @Nu2 :~JO  
            error('zernpol:normalization','Unrecognized normalization flag.') _z\/{  
        end 5!-TLwl`j\  
    else $( hT{C,K  
        isnorm = false; n3^(y"q  
    end Z8$}Rpo  
    5]xuU.w'  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 7|rH9Bc{U  
    % Compute the Zernike Polynomials 3h@]cWp  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% RNg?o [S  
    Lvk}%,S8t  
    % Determine the required powers of r: nJD GNm,  
    % ----------------------------------- 12$0-@U  
    rpowers = []; 8@3K, [Mo  
    for j = 1:length(n) Xn7G2Yp  
        rpowers = [rpowers m(j):2:n(j)]; IwYeKN6s  
    end \Mf>X\}  
    rpowers = unique(rpowers); #&3,T1i`  
    @[GV0*yz$  
    % Pre-compute the values of r raised to the required powers, h`[$ Bp  
    % and compile them in a matrix: Ni$'# W?t  
    % ----------------------------- Q eeV<  
    if rpowers(1)==0 bIQ,=EA1  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); b#j:)PA0C  
        rpowern = cat(2,rpowern{:}); v.0qE}' |  
        rpowern = [ones(length_r,1) rpowern]; bO~y=Pa \  
    else -,bFGTvYQ  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); !_W:%t)g  
        rpowern = cat(2,rpowern{:}); ou,[0B3n0  
    end #-{<d% qk  
    xtV+Le%  
    % Compute the values of the polynomials: FX:`7c]:9  
    % -------------------------------------- w.qtSW6M+  
    z = zeros(length_r,length_n); Y&|Z*s+ +}  
    for j = 1:length_n j,IRUx13f  
        s = 0:(n(j)-m(j))/2; xR7ZqTcw  
        pows = n(j):-2:m(j); [W[{ 4 Xu  
        for k = length(s):-1:1 <-lM9}vd  
            p = (1-2*mod(s(k),2))* ... )^(*B6;z5  
                       prod(2:(n(j)-s(k)))/          ... Sp `l>BL  
                       prod(2:s(k))/                 ... {X{R]  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... st'T._  
                       prod(2:((n(j)+m(j))/2-s(k))); h my%X`%j  
            idx = (pows(k)==rpowers); ;vx5 =^7P  
            z(:,j) = z(:,j) + p*rpowern(:,idx); TnW`#.f  
        end `oRyw6Sko  
         C@M-_Ud>Q  
        if isnorm V&Y`?Edc  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); n@p]v*  
        end ('J/Ww<  
    end R2bqhSlF  
    fN vQ.;  
    % 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)  `BmnXWMgx  
     hh4R  
    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。 #ss/mvc3  
    n1%2 sV)>  
    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)