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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 rB?cm]G=  
    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! 28yxX431S  
     
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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 eMzCAO  
    function z = zernfun(n,m,r,theta,nflag) v2sU$M  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. :5J6rj;_  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N eov-"SJB  
    %   and angular frequency M, evaluated at positions (R,THETA) on the NkI:  
    %   unit circle.  N is a vector of positive integers (including 0), and I9>*Yy5RNS  
    %   M is a vector with the same number of elements as N.  Each element T_T{c+,Zd$  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) p> S/6 [X  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, ,eCXT=6  
    %   and THETA is a vector of angles.  R and THETA must have the same t7FQ.E,T  
    %   length.  The output Z is a matrix with one column for every (N,M) x~eEaD5m%J  
    %   pair, and one row for every (R,THETA) pair. SI5QdX  
    % >,Z{wxz J  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike "cM5=;  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), I1O?)x~  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral qw1J{xoHW  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 2s%M,Nb  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized !*6z=:J  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. =:eE!  
    % f*Js= hvO  
    %   The Zernike functions are an orthogonal basis on the unit circle. Al}PJz\  
    %   They are used in disciplines such as astronomy, optics, and l.l~K%P'h  
    %   optometry to describe functions on a circular domain.  H>6;I  
    % iZk``5tPE  
    %   The following table lists the first 15 Zernike functions. or`stBx  
    % 12dW:#[  
    %       n    m    Zernike function           Normalization ku8c)  
    %       -------------------------------------------------- V"iLeC  
    %       0    0    1                                 1 :X*LlN  
    %       1    1    r * cos(theta)                    2 G[j79o  
    %       1   -1    r * sin(theta)                    2 BxYA[#fd}  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) D7/Bp4I#o  
    %       2    0    (2*r^2 - 1)                    sqrt(3) |>GIPfVT  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) ^iS:mt  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) FoCkTp+/  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) *DzPkaYD>  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) .+h pxZ  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) 3 ?~+5DU  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) _1Gut"!{\  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) "\?G  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) *wcoDQ b;  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ,>v9 Y#U  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) v*'\w#  
    %       -------------------------------------------------- ,5*xE\9G  
    % :exuTn  
    %   Example 1: E,yK` mPp^  
    % (OQ @!R&  
    %       % Display the Zernike function Z(n=5,m=1) q.{/{9  
    %       x = -1:0.01:1; \w[%n0  
    %       [X,Y] = meshgrid(x,x); 1:UC\WW  
    %       [theta,r] = cart2pol(X,Y); F:GKnbY  
    %       idx = r<=1; F6VIH(  
    %       z = nan(size(X)); f`=T@nA  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); 5V8C+k)  
    %       figure 5>Yd\(`K  
    %       pcolor(x,x,z), shading interp SJ^?D8  
    %       axis square, colorbar B?Sfcq-  
    %       title('Zernike function Z_5^1(r,\theta)') 6*33k'=;F  
    % { BL1j  
    %   Example 2: n3j h\  
    % } /3pC a  
    %       % Display the first 10 Zernike functions 6'! {0 5=m  
    %       x = -1:0.01:1; OUO^/] J1S  
    %       [X,Y] = meshgrid(x,x); ){6)?[G  
    %       [theta,r] = cart2pol(X,Y); WVK-dBU  
    %       idx = r<=1; &novkkqY  
    %       z = nan(size(X)); X$Vz  
    %       n = [0  1  1  2  2  2  3  3  3  3]; pF+wH MhUe  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; <dPxy`_  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; m@yVG|eP#  
    %       y = zernfun(n,m,r(idx),theta(idx)); 4 xzJql  
    %       figure('Units','normalized') jZ,[{Z(N   
    %       for k = 1:10 lNVAKwW2#  
    %           z(idx) = y(:,k); x`vs-Y:P  
    %           subplot(4,7,Nplot(k)) #(g+jb0E  
    %           pcolor(x,x,z), shading interp ~(OIo7#;  
    %           set(gca,'XTick',[],'YTick',[]) ]Ja8i%LjOG  
    %           axis square lA-!~SM v"  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) - :z5m+  
    %       end B&k T#  
    % zTT  
    %   See also ZERNPOL, ZERNFUN2. C_ZD<UPA\  
    m{dyVE  
    %   Paul Fricker 11/13/2006 sxwW9_C  
    L^{;jgd&T9  
    P`I G9  
    % Check and prepare the inputs: 1$D`Z/N"A  
    % ----------------------------- :_,]?n  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) aX'g9E  
        error('zernfun:NMvectors','N and M must be vectors.') |abst&yp  
    end %g@3S!lK  
    s_Oh >y?Aq  
    if length(n)~=length(m) 05VOUa*pb  
        error('zernfun:NMlength','N and M must be the same length.') KSB_%OI1  
    end 'S4EKV]  
    /uXRZ  
    n = n(:); {F+M&+``  
    m = m(:); qTh='~m4[  
    if any(mod(n-m,2)) \M"^Oe{Dy?  
        error('zernfun:NMmultiplesof2', ... K[>@'P}y  
              'All N and M must differ by multiples of 2 (including 0).') I<(.i!-x  
    end P[GX}~_k  
    Q}?N4kg  
    if any(m>n) %*6oUb  
        error('zernfun:MlessthanN', ... LLn{2,jfQ  
              'Each M must be less than or equal to its corresponding N.') H@2"ove-uC  
    end Ma=6kX]  
    mn; 7o~4  
    if any( r>1 | r<0 ) !Xx<~l IC  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') {q tc \O  
    end >6l;/J  
    P`3s\8[Q  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) jFNs=D&(  
        error('zernfun:RTHvector','R and THETA must be vectors.') <\d|=>;  
    end <.=#EV^i  
    j #I:6yA3  
    r = r(:); ?%xhe  
    theta = theta(:); NBqV0>vR  
    length_r = length(r); H MjeGO.i  
    if length_r~=length(theta) ,8=`*  
        error('zernfun:RTHlength', ... lO2T/1iMTW  
              'The number of R- and THETA-values must be equal.') Y( $Ji12  
    end UrhM)h?%  
    !V]MLA`  
    % Check normalization: Z ]aK'  
    % -------------------- U!\2K~  
    if nargin==5 && ischar(nflag) i2FD1*=/?  
        isnorm = strcmpi(nflag,'norm'); ;]&~D +XH  
        if ~isnorm u3*NO )O  
            error('zernfun:normalization','Unrecognized normalization flag.') "0'*q<8  
        end eN]>l  
    else (,Ja  
        isnorm = false; };"+ O  
    end <K,% y(]  
    5qd_>UHp  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {7=WU4$  
    % Compute the Zernike Polynomials G !1~i*P$u  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% AvrL9D  
    wTlK4R#  
    % Determine the required powers of r: vcw>v={x  
    % ----------------------------------- bCA2ik  
    m_abs = abs(m); J+71FP`ZH  
    rpowers = []; ]|,q|c,  
    for j = 1:length(n) Z&dr0w8  
        rpowers = [rpowers m_abs(j):2:n(j)]; a/QtJwIV  
    end so!w!O@@  
    rpowers = unique(rpowers); 5@+4  
    {K45~ha9!m  
    % Pre-compute the values of r raised to the required powers, JQ"`9RNb  
    % and compile them in a matrix: ?E+:]j_  
    % ----------------------------- .# 6n  
    if rpowers(1)==0 g5tjj.  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); @e! Zc3  
        rpowern = cat(2,rpowern{:});  (# 6<k  
        rpowern = [ones(length_r,1) rpowern]; |*tWF! D6`  
    else m"gni #  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); s&dO/}3uR]  
        rpowern = cat(2,rpowern{:}); ^)f{q)to  
    end ~!]&>n;=G  
    _{LN{iqDv  
    % Compute the values of the polynomials: %@}o'=[  
    % -------------------------------------- t."g\;  
    y = zeros(length_r,length(n)); W\@?e32  
    for j = 1:length(n) ?Oy'awf_  
        s = 0:(n(j)-m_abs(j))/2; bBUbw*DF)  
        pows = n(j):-2:m_abs(j); w4e%-Ln  
        for k = length(s):-1:1 t&GA6ML#s  
            p = (1-2*mod(s(k),2))* ... E`Jp(gK9F  
                       prod(2:(n(j)-s(k)))/              ... NP K#].F  
                       prod(2:s(k))/                     ... OUEI~b1  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... J [ YtA  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); Gb(C#,xbK  
            idx = (pows(k)==rpowers); r0\cc6  
            y(:,j) = y(:,j) + p*rpowern(:,idx); _0'm4?"  
        end }>MP{67Dm  
         hLb;5u&!kW  
        if isnorm B{7Kzwh;  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); ]y3pE}R  
        end kOs(?=  
    end yicO!:bM  
    % END: Compute the Zernike Polynomials T-4/d5D[  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% L([E98fo  
    r<;l{7lY_  
    % Compute the Zernike functions: 4$yV%[j  
    % ------------------------------ ]g{hhP3>  
    idx_pos = m>0; W8w3~  
    idx_neg = m<0; m7$8k@r  
    Q)09]hP[Xj  
    z = y; G 9DJa_]X  
    if any(idx_pos) 3/X-Cr+d  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); *)limqe3"$  
    end G 1]"s@8(  
    if any(idx_neg) 2Y400  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); yiiyqL*E  
    end sK+ (v  
    81~Kpx  
    % 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) )h#]iGVN}  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. vu=me?m?(  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated ~A6"sb=  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive fX_#S|DlSG  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, [`d$X^<y;  
    %   and THETA is a vector of angles.  R and THETA must have the same WzjL-a(  
    %   length.  The output Z is a matrix with one column for every P-value, >*IN  
    %   and one row for every (R,THETA) pair. ~ |6dH  
    % WvujcmOf  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike }^9]jSq5  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) #?dUv#  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) P''X_1oMC  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 'l~6ErBSg  
    %   for all p. r!7Y'|  
    % cB#nsu>  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 \#CM <%  
    %   Zernike functions (order N<=7).  In some disciplines it is -T7%dLHY  
    %   traditional to label the first 36 functions using a single mode ;6ky5}z  
    %   number P instead of separate numbers for the order N and azimuthal J{`eLmTu  
    %   frequency M. 98fu>>*G{  
    % ` @8`qXg  
    %   Example: EM@ ;3.IO  
    % 3\AM=`  
    %       % Display the first 16 Zernike functions TI=h_%mO  
    %       x = -1:0.01:1; 1~J5uB4  
    %       [X,Y] = meshgrid(x,x); ZPHXzi3j  
    %       [theta,r] = cart2pol(X,Y); P-CB;\  
    %       idx = r<=1; 2edBQYWd  
    %       p = 0:15; ,w{m3;]_%  
    %       z = nan(size(X)); XF|WCZUnY%  
    %       y = zernfun2(p,r(idx),theta(idx)); NBjeH tT  
    %       figure('Units','normalized') AVG>_$<  
    %       for k = 1:length(p) t|V0x3X  
    %           z(idx) = y(:,k); *:_P8G;  
    %           subplot(4,4,k) B<7/,d'  
    %           pcolor(x,x,z), shading interp EATu KLP\  
    %           set(gca,'XTick',[],'YTick',[]) y:d{jG^  
    %           axis square @m~RtC-Q  
    %           title(['Z_{' num2str(p(k)) '}']) ;Wc4qJ.@  
    %       end /4$4h;_8  
    % fj>C@p  
    %   See also ZERNPOL, ZERNFUN. >`'O7.R  
    g%xGOA  
    %   Paul Fricker 11/13/2006 xY\ 0 zQ  
    ]"F5;p; y  
    R^*K6Ad  
    % Check and prepare the inputs: Wkzs<y"  
    % ----------------------------- w8iR|TV  
    if min(size(p))~=1 >O7~h[FN  
        error('zernfun2:Pvector','Input P must be vector.') 6_gnEve h  
    end Vw#{C>  
    w~Ff%p@9  
    if any(p)>35 g> S*<  
        error('zernfun2:P36', ... AW,OH SXh6  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... DNkWOY#{  
               '(P = 0 to 35).']) ?":'O#E  
    end !:CJPM6j3  
    PUdM[-zjh  
    % Get the order and frequency corresonding to the function number: 3=  -pG  
    % ---------------------------------------------------------------- s '?GH  
    p = p(:); s%pfkoOY%  
    n = ceil((-3+sqrt(9+8*p))/2); Gi FXX  
    m = 2*p - n.*(n+2); re &E{  
    ,xI%A, (,;  
    % Pass the inputs to the function ZERNFUN: is?2DcSl5  
    % ---------------------------------------- [xb]Wf  
    switch nargin X|DO~{-au  
        case 3 #~L h#  
            z = zernfun(n,m,r,theta); RthT \%R  
        case 4 bEV<iZDq%  
            z = zernfun(n,m,r,theta,nflag); ^Q+i=y{W  
        otherwise Avlz=k1*  
            error('zernfun2:nargin','Incorrect number of inputs.') <spZ! #o  
    end Yw; D:Y(  
    r\`+R"  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) JPn$FQD  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. NS)}6OI3~"  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of 7Q w|!  
    %   order N and frequency M, evaluated at R.  N is a vector of G~7 i@Zs  
    %   positive integers (including 0), and M is a vector with the ._9 n~=!  
    %   same number of elements as N.  Each element k of M must be a !QI\Fz?  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) %M|,b!eF  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is wCf~O'XLw  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix 9[c%J*r   
    %   with one column for every (N,M) pair, and one row for every P + "Y  
    %   element in R. b1XRC`Gy  
    % S& #U!#@  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- vsWHk7 9  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is )Or  .;  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to *'Y@3vKE  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 %?^6).aEK  
    %   for all [n,m]. z@Q@^ &0Mr  
    % [%Bf< J<  
    %   The radial Zernike polynomials are the radial portion of the $;M:TpX  
    %   Zernike functions, which are an orthogonal basis on the unit h7*W *Bd  
    %   circle.  The series representation of the radial Zernike @yXfBML?]  
    %   polynomials is <<](XgR(  
    % r_e7a6  
    %          (n-m)/2 ^EG\iO2X  
    %            __  c gzwx  
    %    m      \       s                                          n-2s I+>%uShm  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r 6 5y+Z  
    %    n      s=0 mbnV[  
    % {!|}=45Z  
    %   The following table shows the first 12 polynomials. ^<e@uNGg  
    % %>-@K|:gS  
    %       n    m    Zernike polynomial    Normalization ~8"8w(CG*I  
    %       --------------------------------------------- [gy*`@w  
    %       0    0    1                        sqrt(2) X|0R= n]  
    %       1    1    r                           2 \0lnxLA  
    %       2    0    2*r^2 - 1                sqrt(6) pj4!:{.;  
    %       2    2    r^2                      sqrt(6) Hqnxq  
    %       3    1    3*r^3 - 2*r              sqrt(8) 2aJS{[  
    %       3    3    r^3                      sqrt(8) YEkh3FrbwH  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) ^Q*atU  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) L-B<nl  
    %       4    4    r^4                      sqrt(10) +w@M~?>  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) lrf v+  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) qd8n2f  
    %       5    5    r^5                      sqrt(12) &E xYXI  
    %       --------------------------------------------- \#o2\!@`  
    % 9j W2  
    %   Example: FnJ?C&xK  
    % V $z} K  
    %       % Display three example Zernike radial polynomials {hln?'  
    %       r = 0:0.01:1; p!k7C&]E  
    %       n = [3 2 5]; lds- T  
    %       m = [1 2 1]; 54 >-  
    %       z = zernpol(n,m,r); vad12WrG<  
    %       figure >.dWjb6t  
    %       plot(r,z) \J+*  
    %       grid on "4vy lHIo  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') TuW%zF/  
    % ^Y-]*8;]  
    %   See also ZERNFUN, ZERNFUN2. tmqY2.   
    p-1 3H0Kt  
    % A note on the algorithm. asY[8r?U  
    % ------------------------ (JM4R8fR&  
    % The radial Zernike polynomials are computed using the series JaB<EL-9r2  
    % representation shown in the Help section above. For many special /dv<qp  
    % functions, direct evaluation using the series representation can \:'%9 x  
    % produce poor numerical results (floating point errors), because 4 C}bJzZ  
    % the summation often involves computing small differences between pb#?l6x$+  
    % large successive terms in the series. (In such cases, the functions k)TSR5A  
    % are often evaluated using alternative methods such as recurrence $Of0n` e  
    % relations: see the Legendre functions, for example). For the Zernike nLV9<M Zm  
    % polynomials, however, this problem does not arise, because the ooUk O  
    % polynomials are evaluated over the finite domain r = (0,1), and ?nP*\8  
    % because the coefficients for a given polynomial are generally all "M|zv  
    % of similar magnitude. `7/Y@}n  
    % gi 0W;q  
    % ZERNPOL has been written using a vectorized implementation: multiple |&Ym@Jyj  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] 0ez(A  
    % values can be passed as inputs) for a vector of points R.  To achieve TDd{.8qf  
    % this vectorization most efficiently, the algorithm in ZERNPOL rj6#1kt  
    % involves pre-determining all the powers p of R that are required to oh$Q6G  
    % compute the outputs, and then compiling the {R^p} into a single Ur*6Gi6  
    % matrix.  This avoids any redundant computation of the R^p, and wm+/e#'&  
    % minimizes the sizes of certain intermediate variables. ID#I`}h.k  
    % Ug&,Y/tFw2  
    %   Paul Fricker 11/13/2006 q$aaA`E%  
    R'S0 zp6  
    271&i  
    % Check and prepare the inputs: -!c"k}N=  
    % ----------------------------- qIld;v8w"g  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) T0&f8  
        error('zernpol:NMvectors','N and M must be vectors.') C -iK$/U  
    end ;]_o4e6\p  
    [,TkFbDq"J  
    if length(n)~=length(m) {J^lX/D  
        error('zernpol:NMlength','N and M must be the same length.') 4C FB"?n0  
    end 1UKg=A-q  
    ( H6c{'&  
    n = n(:); :>+s0~  
    m = m(:); cK 06]-Y  
    length_n = length(n); 1x[)/@.'f  
    _1U1(^)  
    if any(mod(n-m,2)) ?wO-cnl  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') 6P';DB  
    end =C~/7N,lW]  
    .|/~op4;  
    if any(m<0) W^s ;Bi+Nw  
        error('zernpol:Mpositive','All M must be positive.') gB<3-J1R  
    end QcgfBsv96  
    y K"kEA[;  
    if any(m>n) q `pP$i:  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') )KP5Wud X  
    end _)\c&.p]f  
    ;&ASkI  
    if any( r>1 | r<0 ) }Q";aU0^  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') 48Mpf=f`  
    end KCWc`Oz  
    /c,(8{(O  
    if ~any(size(r)==1) p ZZc:\fJ  
        error('zernpol:Rvector','R must be a vector.') X=> =5'  
    end e6!LSx}y  
    2 aL)  
    r = r(:); $]8h $  
    length_r = length(r); *W kIq>  
    i F+vl]  
    if nargin==4 $#]]K  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); 7PkJ-JBA  
        if ~isnorm Mb]rY>B4  
            error('zernpol:normalization','Unrecognized normalization flag.') mdw7}%5V  
        end EI^06q4x  
    else :hM/f  
        isnorm = false; 0C>%LJ8r  
    end &-mX ,   
    !tp1:'KG  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 8KRba4[  
    % Compute the Zernike Polynomials g>J<%z, }2  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% AhNq/?Q Q~  
    AW,53\ 0  
    % Determine the required powers of r: 6qaulwV4t  
    % ----------------------------------- Jm42b4  
    rpowers = []; >ss/D^YS  
    for j = 1:length(n) :duo#w"K  
        rpowers = [rpowers m(j):2:n(j)]; R%'^gFk 8  
    end HB7;0yt`:  
    rpowers = unique(rpowers); PnoPb k[<  
    |M+<m">E  
    % Pre-compute the values of r raised to the required powers, )LyojwY_g  
    % and compile them in a matrix: o";Z$tAJkC  
    % ----------------------------- {>F7CT'G6  
    if rpowers(1)==0 |gU(s  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); d.P\fPSD  
        rpowern = cat(2,rpowern{:}); Rb{U+/gq  
        rpowern = [ones(length_r,1) rpowern]; O/<K!;(@?  
    else nI*v820,  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); @U2qD  J6  
        rpowern = cat(2,rpowern{:}); }5 (Ho$S(  
    end Q^#;WASi  
    8:/e GM  
    % Compute the values of the polynomials: ph-ATJ"  
    % -------------------------------------- Et/&^&=\-  
    z = zeros(length_r,length_n); D &/L:  
    for j = 1:length_n di>cMS 4 c  
        s = 0:(n(j)-m(j))/2; Ck!VV2U#  
        pows = n(j):-2:m(j); OdB?_.+$  
        for k = length(s):-1:1 dx+hhg\L  
            p = (1-2*mod(s(k),2))* ... 4Z/Q=Mq2  
                       prod(2:(n(j)-s(k)))/          ... `YI f_a{  
                       prod(2:s(k))/                 ... ruazOmnn~  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... EMfdBY5  
                       prod(2:((n(j)+m(j))/2-s(k))); o!!yd8~*r  
            idx = (pows(k)==rpowers); pb=cBZ$  
            z(:,j) = z(:,j) + p*rpowern(:,idx); ,Y>Bex_v  
        end Y2?.}ZO  
         &Y^WP?HS  
        if isnorm f?'JAC*  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); fOMvj%T@2  
        end P [k$vD  
    end uIDuGrt  
    KFFSv{m[  
    % 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)  e9E\% p  
    @K:N,@yq  
    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。 nL?oTze*p  
    &,.Y9; b  
    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)