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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 5:n&G[Md  
    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! 0b*a2_|8k  
     
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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 Y|B/(  
    function z = zernfun(n,m,r,theta,nflag) #3/l4`/j  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. ^/g&Q  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N |ZRl.C/e  
    %   and angular frequency M, evaluated at positions (R,THETA) on the `L9o !OsQ  
    %   unit circle.  N is a vector of positive integers (including 0), and K h% x  
    %   M is a vector with the same number of elements as N.  Each element P<2yCovn`  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) k5}i^^.  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, qRB%G<H  
    %   and THETA is a vector of angles.  R and THETA must have the same NPS=?5p>  
    %   length.  The output Z is a matrix with one column for every (N,M) (<%i8xu 2  
    %   pair, and one row for every (R,THETA) pair. 4 &t6  
    % R^8Opf_UN  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike Bpk%,*$*)  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 2d1'!B zDA  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral KJ pM?:  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ASu9c2s  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized UdLC]  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. -@J;FjrXmP  
    % \LM'KD pP_  
    %   The Zernike functions are an orthogonal basis on the unit circle. #c!(97l6o  
    %   They are used in disciplines such as astronomy, optics, and BY \p?79  
    %   optometry to describe functions on a circular domain. 03rZz1  
    % 0U$6TDtmE  
    %   The following table lists the first 15 Zernike functions. C2Y&qX,  
    % =20Q! wcu  
    %       n    m    Zernike function           Normalization 8Q6il-  
    %       -------------------------------------------------- 5#2vSq!H  
    %       0    0    1                                 1 ;#Mq=Fr-SG  
    %       1    1    r * cos(theta)                    2 MGmtA(  
    %       1   -1    r * sin(theta)                    2 yY&(?6\{<<  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) PfuYT_p4s  
    %       2    0    (2*r^2 - 1)                    sqrt(3) 8{d`N|k  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) 1 1p\ z  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) 9)4N2=  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) Js=|r;'  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) ,#"AWQ  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) BB|{VwN  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) FAQ:0 L$G  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) |]m&LC  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) <!w-op2@ir  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) %@BQv 4oJ  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) ec]ksw6T+  
    %       --------------------------------------------------  |u$AzI  
    % -rH3rKtf~  
    %   Example 1: {{<o1{_H  
    % j?&FK  
    %       % Display the Zernike function Z(n=5,m=1) s V77WF  
    %       x = -1:0.01:1; pP".?|n  
    %       [X,Y] = meshgrid(x,x); Pq_Il9  
    %       [theta,r] = cart2pol(X,Y); |Ec$%  
    %       idx = r<=1; j+c)%  
    %       z = nan(size(X)); cF/FretoO  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); }wv$ #H[  
    %       figure -D(Ubk Pw  
    %       pcolor(x,x,z), shading interp `__CL )N|  
    %       axis square, colorbar Ok*:;G@  
    %       title('Zernike function Z_5^1(r,\theta)') c/x(v=LW  
    % M_XZOlW5  
    %   Example 2: }_gq vgI>p  
    % b(XhwkGVq  
    %       % Display the first 10 Zernike functions gK%&VzG4  
    %       x = -1:0.01:1; ,,G0}N@7s  
    %       [X,Y] = meshgrid(x,x); <`N\FM^vo  
    %       [theta,r] = cart2pol(X,Y); s*!2oj  
    %       idx = r<=1; # =322bnO  
    %       z = nan(size(X)); -6H)GK14b  
    %       n = [0  1  1  2  2  2  3  3  3  3]; c}{e,t  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; c9'#G>&h~^  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; >2v_fw  
    %       y = zernfun(n,m,r(idx),theta(idx)); +"p" ,Z  
    %       figure('Units','normalized') 'Lm.`U  
    %       for k = 1:10 4XKg3l1  
    %           z(idx) = y(:,k); `9wz:s QtP  
    %           subplot(4,7,Nplot(k)) G A7  
    %           pcolor(x,x,z), shading interp ^ #Wf  
    %           set(gca,'XTick',[],'YTick',[]) d[o =  
    %           axis square aG" UV\  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) i3Ffk+ |b  
    %       end {Qhv HV  
    % Z,d/FC#y(  
    %   See also ZERNPOL, ZERNFUN2. .:lzT"QXI  
    O&O1O> [p1  
    %   Paul Fricker 11/13/2006 !IGVN:E  
    x/4lD}Pw]  
    v =u|D$  
    % Check and prepare the inputs: Y&j6;2-Z  
    % ----------------------------- iYnw?4Y  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) I{RktO;1  
        error('zernfun:NMvectors','N and M must be vectors.') 2'x_zMV  
    end y k#:.5H  
    @RnGK 5  
    if length(n)~=length(m) 3Ys|M%N  
        error('zernfun:NMlength','N and M must be the same length.') d?S<h`{x   
    end ~pF'Qw" z|  
    w6E?TI  
    n = n(:); tq*Q|9j7VG  
    m = m(:); ,)QmQ ^/  
    if any(mod(n-m,2)) ]-AT(L >  
        error('zernfun:NMmultiplesof2', ... g`Rs;  
              'All N and M must differ by multiples of 2 (including 0).') fNK~z*  
    end ,Tr12#D:  
    )Z^( +  
    if any(m>n) /g8yc'{p  
        error('zernfun:MlessthanN', ... k(7! W  
              'Each M must be less than or equal to its corresponding N.') ^L'K?o  
    end lLg23k{'  
    ZPMEN,Dw  
    if any( r>1 | r<0 ) Bf-&[ 5N}  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') nY*ODL  
    end *3k~%RM%?  
    G_o/ lIz"  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) G's/Q-'[\  
        error('zernfun:RTHvector','R and THETA must be vectors.') MDB}G '  
    end LEhi/>T  
    huQ1A0(no  
    r = r(:); oOD|FrlY  
    theta = theta(:); 1/{:}9Z@  
    length_r = length(r); cKxJeM07  
    if length_r~=length(theta) TQEZ<B$  
        error('zernfun:RTHlength', ... V3m!dp]  
              'The number of R- and THETA-values must be equal.') ]ny(l#Hu:  
    end d3![b1  
    |_ @iaLE  
    % Check normalization: u_[Zu8  
    % -------------------- f{)*"  
    if nargin==5 && ischar(nflag) nBD7  
        isnorm = strcmpi(nflag,'norm'); {-E{.7  
        if ~isnorm T[7DJNdG6  
            error('zernfun:normalization','Unrecognized normalization flag.') e@q[Dv'mu  
        end Fj5^_2MU:  
    else %\^x3wP&o\  
        isnorm = false; *i\7dJ Dj  
    end >?DrC/  
    lS,Hr3Lz  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% "90}H0(+  
    % Compute the Zernike Polynomials  r>G$u  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  /!9949XV  
    7'o?'He-.2  
    % Determine the required powers of r: a{8GT2h`4  
    % ----------------------------------- mDq0 1fU4  
    m_abs = abs(m); '}OrFN  
    rpowers = []; Uvuvr_IP  
    for j = 1:length(n) ~k J#IA  
        rpowers = [rpowers m_abs(j):2:n(j)]; : i(h[0  
    end x##Iv|$  
    rpowers = unique(rpowers); p1&d@PF&&  
    F>}).qx  
    % Pre-compute the values of r raised to the required powers, oZ=e/\[K  
    % and compile them in a matrix: p"X\]g^jA>  
    % ----------------------------- ?ph"|LyL  
    if rpowers(1)==0 '6aH*B:}*;  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false);  dxU[>m;  
        rpowern = cat(2,rpowern{:}); _I -0[w  
        rpowern = [ones(length_r,1) rpowern]; WL7:22nSHa  
    else &zm5s*yNt  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); Y6CadC  
        rpowern = cat(2,rpowern{:}); p]>bN  
    end 6^wiEnA  
    ;j(xrPNb  
    % Compute the values of the polynomials: 57oY]NT?  
    % -------------------------------------- lE`ScYG  
    y = zeros(length_r,length(n)); t,H,*2  
    for j = 1:length(n) 1'g?B`  
        s = 0:(n(j)-m_abs(j))/2; \q,w)BE  
        pows = n(j):-2:m_abs(j); P EbB0GL  
        for k = length(s):-1:1 'LX=yL]I  
            p = (1-2*mod(s(k),2))* ... &B3kzs  
                       prod(2:(n(j)-s(k)))/              ... kTnvD|3_!P  
                       prod(2:s(k))/                     ... `t8e2?GH  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 0)84Z.k  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); m t*v@'l.  
            idx = (pows(k)==rpowers); /bw-*  
            y(:,j) = y(:,j) + p*rpowern(:,idx); hQk mB|];5  
        end P(Lwpa,S  
         * T~sR'K+|  
        if isnorm L72GF5+!!  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); D QZS%)  
        end !Q?4sAB  
    end nbYaYL?&  
    % END: Compute the Zernike Polynomials 0~-+5V  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% mq "p"iI  
    '-*r&:  
    % Compute the Zernike functions: :bh[6 F  
    % ------------------------------ ;J`X0Vl$  
    idx_pos = m>0; ?r@ZTuq#  
    idx_neg = m<0; 6Qo6 T][  
    .a^/r'?  
    z = y; 'DIE#l`  
    if any(idx_pos) N[mOJa:  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); qItI):9U  
    end p;'vOb  
    if any(idx_neg) %Cr- cR0  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 8G@FX $$Q  
    end O_:Q#  
    J^?O] |  
    % 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) Cz(PjS  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. nd?m+C&W  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated oL~Yrb%R  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive I4)vJ0  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, w(@`g/b  
    %   and THETA is a vector of angles.  R and THETA must have the same x0 #+yP  
    %   length.  The output Z is a matrix with one column for every P-value, LD5'4,%-  
    %   and one row for every (R,THETA) pair. 7X.1QSuE  
    % EYQ!ELuF  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike ?^7~|?v  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) QoW3*1o  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) :DZiDJ@  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 w8on3f;6n#  
    %   for all p. l%)XPb2$J  
    % M\]E;C'"U  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 Nn^el' S'  
    %   Zernike functions (order N<=7).  In some disciplines it is i0 R=P[  
    %   traditional to label the first 36 functions using a single mode l==T3u r  
    %   number P instead of separate numbers for the order N and azimuthal Hnaq+ _]  
    %   frequency M. <ImeZ'L7  
    % 6$z UFIk  
    %   Example: d`xqs,0f  
    % Z`f _e?  
    %       % Display the first 16 Zernike functions k82'gJ;MC=  
    %       x = -1:0.01:1; E^qKkl  
    %       [X,Y] = meshgrid(x,x); +I')>6  
    %       [theta,r] = cart2pol(X,Y); C/cyqxVl}  
    %       idx = r<=1; (O&b:D/Y  
    %       p = 0:15; QR#,n@fE  
    %       z = nan(size(X)); ;xRyONt  
    %       y = zernfun2(p,r(idx),theta(idx)); Z]6D0b  
    %       figure('Units','normalized') W}e5 4-lu  
    %       for k = 1:length(p) < /}[x2w?]  
    %           z(idx) = y(:,k); &Y,Rm78  
    %           subplot(4,4,k) M\GS&K$lq  
    %           pcolor(x,x,z), shading interp B^OhL!*tI  
    %           set(gca,'XTick',[],'YTick',[]) {4Isz-P  
    %           axis square @tP,l$O&  
    %           title(['Z_{' num2str(p(k)) '}']) Qejzp/2  
    %       end 5yQgGd)  
    % vz _U  
    %   See also ZERNPOL, ZERNFUN. ZE1#{u~[y  
    ru U|  
    %   Paul Fricker 11/13/2006 )PwDP  
    aH~il!K  
    Ufk7%`  
    % Check and prepare the inputs: OU[Sm7B  
    % ----------------------------- fu|I(^NV  
    if min(size(p))~=1 > vahj,CZZ  
        error('zernfun2:Pvector','Input P must be vector.') $`riB$v  
    end aR3W9  
    }b{N[  
    if any(p)>35 t4Z.b 5g  
        error('zernfun2:P36', ... ;TR.UUT  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... .z9JoQ  
               '(P = 0 to 35).'])  g6~uf4;  
    end c-3? D;  
    4u;W1=+Vn  
    % Get the order and frequency corresonding to the function number: Vw,dHIe(3  
    % ---------------------------------------------------------------- AKHi$Bk  
    p = p(:); )QKZI))G0  
    n = ceil((-3+sqrt(9+8*p))/2); >yaz  
    m = 2*p - n.*(n+2); yNqrL?i  
    LX3 5Lt  
    % Pass the inputs to the function ZERNFUN: P3:hGmk8|j  
    % ---------------------------------------- p3-sEIw}Ru  
    switch nargin UrtN3icph  
        case 3 !W6]+  
            z = zernfun(n,m,r,theta); >Rr]e`3wG  
        case 4 NTn-4iJy  
            z = zernfun(n,m,r,theta,nflag); a~{mRh  
        otherwise e06r5%|.%  
            error('zernfun2:nargin','Incorrect number of inputs.') 8 /\rmf\  
    end *f,EDSN1@d  
    X2EC+<  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) f):|Ad|  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. Q.!D2RZc  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of 7H=/FT?e]  
    %   order N and frequency M, evaluated at R.  N is a vector of @Gl=1  
    %   positive integers (including 0), and M is a vector with the n}YRE`>D  
    %   same number of elements as N.  Each element k of M must be a b2ZKhS8  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) p-;*K(#X  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is g<tr |n  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix ['{mW4i  
    %   with one column for every (N,M) pair, and one row for every ZX'/[wAN)  
    %   element in R. eM{+R^8  
    % 38rC; 6  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- %kyvt t  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is ':J[KWuV  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to ;Q\Duj  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 IY+P Yad  
    %   for all [n,m]. \QQw1c+  
    % {wK98>$a  
    %   The radial Zernike polynomials are the radial portion of the N U\B  
    %   Zernike functions, which are an orthogonal basis on the unit ](B+ilr   
    %   circle.  The series representation of the radial Zernike Kc0KCBd8];  
    %   polynomials is +1_NB;,e  
    % wOn.m  
    %          (n-m)/2 7RDfhKdb  
    %            __ j^>J*gLM}W  
    %    m      \       s                                          n-2s s )\%%CM  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r  >.0B%  
    %    n      s=0 F!'y47QD  
    % y&UcTE2;%(  
    %   The following table shows the first 12 polynomials. Q.@9"&)t  
    % <-FAF:6$@@  
    %       n    m    Zernike polynomial    Normalization 8L^5bJ  
    %       --------------------------------------------- MoavA 3`  
    %       0    0    1                        sqrt(2) ' 4ftclzL  
    %       1    1    r                           2 yd'>Mw  
    %       2    0    2*r^2 - 1                sqrt(6) QT&2&#Z  
    %       2    2    r^2                      sqrt(6) R8sj>.I9j  
    %       3    1    3*r^3 - 2*r              sqrt(8) g>cp;co9g  
    %       3    3    r^3                      sqrt(8) }[\l$sS  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) bU7n1pzW,o  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) P|l62!m<   
    %       4    4    r^4                      sqrt(10) 1=}+NK!  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) u%}zLwMH  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) !Qy%sY  
    %       5    5    r^5                      sqrt(12) wL\OAM6R  
    %       --------------------------------------------- zT 9"B  
    % JgEPzHgx  
    %   Example: !g'kWE[  
    % 'H0uvvhOp  
    %       % Display three example Zernike radial polynomials *?:V)!.2z  
    %       r = 0:0.01:1; -c{O!z6sX  
    %       n = [3 2 5]; \C#X Kk$OE  
    %       m = [1 2 1]; \oA>%+]5  
    %       z = zernpol(n,m,r); B<%cqz@  
    %       figure N2#Wyt8MC  
    %       plot(r,z) +`}QIp0  
    %       grid on 5_!s\5  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') AY<(`J{  
    % yS'W ss  
    %   See also ZERNFUN, ZERNFUN2. W US[hx,  
    fh,kbn==r?  
    % A note on the algorithm. r~nD%H:}P  
    % ------------------------ 0%&ZR=y(G  
    % The radial Zernike polynomials are computed using the series U[,."w]T  
    % representation shown in the Help section above. For many special > mk>VM  
    % functions, direct evaluation using the series representation can 7@oM?r7td  
    % produce poor numerical results (floating point errors), because ~.7/o0'+  
    % the summation often involves computing small differences between e ?sMOBPlv  
    % large successive terms in the series. (In such cases, the functions lJb1{\|.,  
    % are often evaluated using alternative methods such as recurrence @cRR  
    % relations: see the Legendre functions, for example). For the Zernike v#c'p^T  
    % polynomials, however, this problem does not arise, because the {%Cb0Zh  
    % polynomials are evaluated over the finite domain r = (0,1), and zZp0g^;.?  
    % because the coefficients for a given polynomial are generally all 79`OB##  
    % of similar magnitude. !LJEo>D  
    % /Z^"[Ke  
    % ZERNPOL has been written using a vectorized implementation: multiple ut j7"{'k|  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] Cw$0XyO  
    % values can be passed as inputs) for a vector of points R.  To achieve JJe8x4  
    % this vectorization most efficiently, the algorithm in ZERNPOL \no6]xN;  
    % involves pre-determining all the powers p of R that are required to 08czP-)OZ  
    % compute the outputs, and then compiling the {R^p} into a single E mG':K(  
    % matrix.  This avoids any redundant computation of the R^p, and lDsT?yHS`Z  
    % minimizes the sizes of certain intermediate variables. -+y lJo[D  
    % fJ<I|ZZ  
    %   Paul Fricker 11/13/2006 /~~A2.=.  
    b'r</ncZ  
    2i0 .x  
    % Check and prepare the inputs: Cfs2tN  
    % ----------------------------- = y @*vl   
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) NVnId p  
        error('zernpol:NMvectors','N and M must be vectors.') {#`O'F>  
    end *Ri\7CqU"6  
    c~``)N  
    if length(n)~=length(m) I-Q@v`  
        error('zernpol:NMlength','N and M must be the same length.') }_mVXjF  
    end ~F"<Nq  
    Ah 2*7@U  
    n = n(:); A>\5fO  
    m = m(:); S4 j5-  
    length_n = length(n); DplS\}='s  
    atiyQuT6Wh  
    if any(mod(n-m,2)) 6;:z?Q  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') 2x5^kN7  
    end z( \4{Y  
    OI^??joQ  
    if any(m<0) ^/~ZP?%]  
        error('zernpol:Mpositive','All M must be positive.') XQ3"+M_KG  
    end t]IHQ8  
    #7Fdmnu`  
    if any(m>n) whi#\>i  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') fV#,<JG  
    end ObPXVqG"?  
    ='vD4}"j  
    if any( r>1 | r<0 ) %1oB!+tv  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') {=%,NwPs  
    end Kpg?' !I  
    6o0}7T%6  
    if ~any(size(r)==1) !F:ANoaS  
        error('zernpol:Rvector','R must be a vector.') ,xw1B-dx  
    end **V8a-@  
    K'Y/0:"*  
    r = r(:); <Hf3AB;#4  
    length_r = length(r); a,|Hn  
    5rb<u>e{  
    if nargin==4 2U|"]tpM&  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); %*zV&H   
        if ~isnorm 2$OV`qy@?  
            error('zernpol:normalization','Unrecognized normalization flag.') J`]9 n>G  
        end )IVk4|  
    else 7{Lp/z%r  
        isnorm = false; 1Q_Q-Z  
    end Cag^$nj  
    a<0q%A x  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% z:a7)z  
    % Compute the Zernike Polynomials ?edf$-"z/  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  J8-K  
    O3V.4tp  
    % Determine the required powers of r: ?y-@c]  
    % ----------------------------------- BG6.,'~7o  
    rpowers = []; AGl#f\_^  
    for j = 1:length(n) ;hPVe _/  
        rpowers = [rpowers m(j):2:n(j)]; CNe(]HIOH  
    end Q45gC28x  
    rpowers = unique(rpowers); ]=o1to-  
    ;Fo7 -kK  
    % Pre-compute the values of r raised to the required powers, **$kW bS  
    % and compile them in a matrix: <0VC`+p<)  
    % ----------------------------- ,.kmUd  
    if rpowers(1)==0 /Xq|S O  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); `_f&T}]  
        rpowern = cat(2,rpowern{:}); k8\ KCKql  
        rpowern = [ones(length_r,1) rpowern]; L@'2}7N1%  
    else %+ nM4)h  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ;m7~!m)  
        rpowern = cat(2,rpowern{:}); deQ {  
    end a@Vk(3Rx_  
    k`#E#1niN  
    % Compute the values of the polynomials: _&(L{cFx6  
    % -------------------------------------- ^W(ue]j}o  
    z = zeros(length_r,length_n); LF `]=.Q  
    for j = 1:length_n <ne?;P1L  
        s = 0:(n(j)-m(j))/2; ,SPgop'  
        pows = n(j):-2:m(j); 2ql)]Skg6  
        for k = length(s):-1:1 4X",:B}  
            p = (1-2*mod(s(k),2))* ... ZbiC=uh  
                       prod(2:(n(j)-s(k)))/          ... <"K2t Tg.  
                       prod(2:s(k))/                 ... ]cv/dY#  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... pTaC$Ne  
                       prod(2:((n(j)+m(j))/2-s(k))); /Xj{]i3{  
            idx = (pows(k)==rpowers); Wy\^}  
            z(:,j) = z(:,j) + p*rpowern(:,idx); Rp;"]Q&b  
        end QW'*^^  
         w:9`R<L  
        if isnorm ePZ Ai"k  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); .Tm.M7  
        end KwV!smi2  
    end JB%_&gX)v  
    Ie K+  
    % 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)  ?B;7J7T  
    ;92xSe"Ww  
    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。 @v@F%JCZ  
    "P@ SR`v#  
    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)