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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 O>j_xW]V  
    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! ZW0gd7Wh  
     
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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 k9ThWo/#u  
    function z = zernfun(n,m,r,theta,nflag) T7!"gJ  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. j JxV)AIY  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ^MUSq(  
    %   and angular frequency M, evaluated at positions (R,THETA) on the ,(6U3W*bu  
    %   unit circle.  N is a vector of positive integers (including 0), and IU8/B+hM~  
    %   M is a vector with the same number of elements as N.  Each element "AzA|zk')"  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) oP$l(k  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, oTPPYi[r  
    %   and THETA is a vector of angles.  R and THETA must have the same I}#_Jt3R  
    %   length.  The output Z is a matrix with one column for every (N,M) d&dp#)._8  
    %   pair, and one row for every (R,THETA) pair. %)Pn<! L  
    %  4|9c+^%^  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 8%dE$smH  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), T w!]N%E  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral \UdHN=A&  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, CO` %eL ~  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized 2&f] v`|M|  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. VZ`L-P$AF  
    % OKo39 A\fu  
    %   The Zernike functions are an orthogonal basis on the unit circle. L@"1d.k_  
    %   They are used in disciplines such as astronomy, optics, and Yy$GfjJtL]  
    %   optometry to describe functions on a circular domain. TfD]`v`]   
    % LG0z|x(  
    %   The following table lists the first 15 Zernike functions. /$ -^k[%  
    % #sn2Vmi  
    %       n    m    Zernike function           Normalization &: i|;^^2  
    %       -------------------------------------------------- *vL2n>HH  
    %       0    0    1                                 1 Fo=hL  
    %       1    1    r * cos(theta)                    2 vgc #IEx@  
    %       1   -1    r * sin(theta)                    2 1 h.=c  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) WW'8&:x  
    %       2    0    (2*r^2 - 1)                    sqrt(3) PhHBmM GL  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) y&HfF~  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) (~R[K,G  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) T+OQa+E@P  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) _%M5 T  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) =@ '>|-w|  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) {Lex((  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) JF%eC}[d  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) O>Vb7`z0<  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) U4J9b p|  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) 5Av bKT  
    %       -------------------------------------------------- eY)JuJ?  
    % 7IrbwAGZ3  
    %   Example 1: {9tKq--@E9  
    % HC4vet  
    %       % Display the Zernike function Z(n=5,m=1) y<Hka'(%  
    %       x = -1:0.01:1; @l7~Zn  
    %       [X,Y] = meshgrid(x,x); td:GZ %  
    %       [theta,r] = cart2pol(X,Y); E4a`cGb  
    %       idx = r<=1; )575JY `6K  
    %       z = nan(size(X)); MeXzWLH  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); 0w0\TWz*   
    %       figure CCCd=s.  
    %       pcolor(x,x,z), shading interp %SG**7  
    %       axis square, colorbar Ow0-}Im~  
    %       title('Zernike function Z_5^1(r,\theta)') "f/Su(6{0  
    % O "jX|5  
    %   Example 2: G:W4<w  
    % P8hA<{UFS\  
    %       % Display the first 10 Zernike functions wABaNB=9;  
    %       x = -1:0.01:1; 82S?@%}#J  
    %       [X,Y] = meshgrid(x,x); [Yo3=(7J  
    %       [theta,r] = cart2pol(X,Y); O]"3o,/]G  
    %       idx = r<=1; &n_aMZ;  
    %       z = nan(size(X)); ?-40bb  
    %       n = [0  1  1  2  2  2  3  3  3  3]; Pc+8CuN?  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; k 8C[fRev  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; Ck71N3~W  
    %       y = zernfun(n,m,r(idx),theta(idx)); f`zH#{u  
    %       figure('Units','normalized') ,G";ny[$  
    %       for k = 1:10 cs'ylGH  
    %           z(idx) = y(:,k); ' }G! D  
    %           subplot(4,7,Nplot(k)) 8VbHZ9Q  
    %           pcolor(x,x,z), shading interp :xn/9y+s  
    %           set(gca,'XTick',[],'YTick',[]) < r6e23  
    %           axis square zh5$$*\  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 85>WK+=  
    %       end (zW;&A  
    % 8<,b5  
    %   See also ZERNPOL, ZERNFUN2. /%El0X  
    F\' ^DtB  
    %   Paul Fricker 11/13/2006 $$U Mc-Pq  
    ~hubh!d=  
    7+I%0U}m  
    % Check and prepare the inputs: wz!a;]agg  
    % ----------------------------- 0* G5Vd  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) }LXS!Ff:  
        error('zernfun:NMvectors','N and M must be vectors.') aNZJs<3;'D  
    end yZ {H  
    ~i`@  
    if length(n)~=length(m) cY%[UK$l  
        error('zernfun:NMlength','N and M must be the same length.') -JL  
    end <ej Wl%4  
    S >E|A %  
    n = n(:); BUH~aV  
    m = m(:); $U,`M"  
    if any(mod(n-m,2)) G8c 8`~t  
        error('zernfun:NMmultiplesof2', ... s[ {L.9Y  
              'All N and M must differ by multiples of 2 (including 0).') DU_38tz  
    end ',?9\xEB  
    JsNqijVC  
    if any(m>n) bU`Ih# q  
        error('zernfun:MlessthanN', ... 1-_op !N  
              'Each M must be less than or equal to its corresponding N.') 3j{VpacZY  
    end ('!{kVLT-  
    qT`sPEs;V  
    if any( r>1 | r<0 ) B;SN}I  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') S@AHI!"h=V  
    end DP2 ^(d<  
    vmI2o'zi  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ]ne&`uO  
        error('zernfun:RTHvector','R and THETA must be vectors.') zzf;3S?  
    end %bM^/7  
    3t  
    r = r(:); pdcP;.   
    theta = theta(:); Ka[@-XH  
    length_r = length(r); L nQm2uF  
    if length_r~=length(theta) @agW{%R:.  
        error('zernfun:RTHlength', ... //c<p  
              'The number of R- and THETA-values must be equal.') 13oR-Stj|  
    end 9zdp 8?T  
    8no_xFA  
    % Check normalization: ?klV;+  
    % -------------------- n@pm5f  
    if nargin==5 && ischar(nflag) HGuY-f  
        isnorm = strcmpi(nflag,'norm'); +r7uIwi$@  
        if ~isnorm C$X )I~M  
            error('zernfun:normalization','Unrecognized normalization flag.') N3P!<J/tc  
        end O34'c_ fZ  
    else \Mk;Y  
        isnorm = false; IUX~dO  
    end mZ;W$y SO  
    "=l<%em  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \;0J6LBc  
    % Compute the Zernike Polynomials y'(bp=Nq  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% .@0i,7S  
    Dq/ _#&S  
    % Determine the required powers of r: K`!q1 g`  
    % ----------------------------------- >|<8QomD  
    m_abs = abs(m); xrbDqA.b  
    rpowers = []; (mq 7{ ;7y  
    for j = 1:length(n) =}S*]Me5  
        rpowers = [rpowers m_abs(j):2:n(j)]; 65FdA-4  
    end :Jp$_T&E  
    rpowers = unique(rpowers); 5#~ARk*?a  
    j%%l$i~  
    % Pre-compute the values of r raised to the required powers, #J AU5d  
    % and compile them in a matrix: NB]T~_?]*  
    % ----------------------------- v:s.V>{"S  
    if rpowers(1)==0 m?;aTSa  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); lk $S"OH!  
        rpowern = cat(2,rpowern{:}); \0%)eJ  
        rpowern = [ones(length_r,1) rpowern]; vkE[Ur>  
    else ZN)a}\]  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); '</  
        rpowern = cat(2,rpowern{:}); OtuOT=%  
    end o'.6gZ gk  
    |RqCw7  
    % Compute the values of the polynomials: 'T54k  
    % -------------------------------------- ]A}'jP  
    y = zeros(length_r,length(n)); w7Nb+/,sg  
    for j = 1:length(n) @";z?xj  
        s = 0:(n(j)-m_abs(j))/2; }{*((@GY}  
        pows = n(j):-2:m_abs(j); /p~Wk4'  
        for k = length(s):-1:1 Qh%(yL!  
            p = (1-2*mod(s(k),2))* ... ]JQk,<l5E  
                       prod(2:(n(j)-s(k)))/              ... wyO@oi Vn  
                       prod(2:s(k))/                     ... 5m")GWQaP@  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ]Xcqf9k  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); -Z&6PT7  
            idx = (pows(k)==rpowers); \LB =_W$  
            y(:,j) = y(:,j) + p*rpowern(:,idx); H27J kZ&  
        end x1)G!i  
         oD,f5Ci-  
        if isnorm B 95}_q  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); Fy-+? ~  
        end *JXiOs  
    end DKL< "#.7  
    % END: Compute the Zernike Polynomials xw-x<7  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% )L#C1DP#  
    Wt+aW  
    % Compute the Zernike functions: kvh}{@|-  
    % ------------------------------ 1 O+4A[cr  
    idx_pos = m>0; >8;Co]::kx  
    idx_neg = m<0; gO-C[j/  
    TRG(W^<F  
    z = y; !pI)i*V|  
    if any(idx_pos) Xz5 aTJ&  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); CQfrAk4mu  
    end q#B^yk|Y  
    if any(idx_neg) &F" Mkyf  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 4cK6B)X  
    end qPdNI1 |  
    0 1[LPN  
    % 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) Nd/iMV6V;  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. cA ;'~[  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated -CW&!oW  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive Lys4l$J]  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, }gL9G  
    %   and THETA is a vector of angles.  R and THETA must have the same xd8UdQ, lt  
    %   length.  The output Z is a matrix with one column for every P-value, s)<#a(!  
    %   and one row for every (R,THETA) pair. $DW3H1iW  
    % &NV[)6!  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike sChMIbq!Av  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) /h%<e  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) L1*P<Cb  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 9BB<. p  
    %   for all p. xbrxh-gV  
    % 3ydOBeY  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 %[4/UD=7  
    %   Zernike functions (order N<=7).  In some disciplines it is 9Qp39(l:  
    %   traditional to label the first 36 functions using a single mode 9"2.2li5$  
    %   number P instead of separate numbers for the order N and azimuthal %*P59%  
    %   frequency M. !c:Q+:,H  
    % a8aEZ724  
    %   Example: vTjgW?9  
    % 5bFE;Y;  
    %       % Display the first 16 Zernike functions a&)0_i:r  
    %       x = -1:0.01:1; wo7.y["$  
    %       [X,Y] = meshgrid(x,x); AY:3o3M  
    %       [theta,r] = cart2pol(X,Y); Mw7!w-1+  
    %       idx = r<=1; c\UVMyE  
    %       p = 0:15; zfL$z,zgf  
    %       z = nan(size(X)); =If% m9  
    %       y = zernfun2(p,r(idx),theta(idx)); MK[l*=\s  
    %       figure('Units','normalized') 4NbX! "0  
    %       for k = 1:length(p) )eGGA6G  
    %           z(idx) = y(:,k); bv0B  
    %           subplot(4,4,k) n1o/-UY  
    %           pcolor(x,x,z), shading interp CmRn  
    %           set(gca,'XTick',[],'YTick',[]) @/yRE^c  
    %           axis square Jl&bWp^3  
    %           title(['Z_{' num2str(p(k)) '}']) G ;V@oT  
    %       end @B ~! [l  
    % _~_04p  
    %   See also ZERNPOL, ZERNFUN. lO8.Q"mxo  
    ?AP2Opsl  
    %   Paul Fricker 11/13/2006 b$[_(QUw  
    I*@\pc}  
    Y\BB;"x1  
    % Check and prepare the inputs: VgZ<T,SuW  
    % ----------------------------- >5wA B  
    if min(size(p))~=1 KM 4w{  
        error('zernfun2:Pvector','Input P must be vector.') #NNj#  
    end .=rv,PWjZ  
    hSaS2RLF  
    if any(p)>35 2ko7t9y&  
        error('zernfun2:P36', ... Rp A76ug  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... - t4"BD  
               '(P = 0 to 35).']) rfYu8-  
    end 7GfgW02  
    ,oIZ5u{#,  
    % Get the order and frequency corresonding to the function number: F^_d8=67h  
    % ---------------------------------------------------------------- \l{*1lQ`  
    p = p(:); H#k"[eZ  
    n = ceil((-3+sqrt(9+8*p))/2); [ar:zl V8  
    m = 2*p - n.*(n+2); >NB}Bc  
    *]z.BZI:  
    % Pass the inputs to the function ZERNFUN: riZFcVsB  
    % ---------------------------------------- @iUzRsl  
    switch nargin cZ|D!1%  
        case 3 >?'q P ]  
            z = zernfun(n,m,r,theta); `NXyzT`:K  
        case 4  -6~*:zg,  
            z = zernfun(n,m,r,theta,nflag); 0-0 )E&2  
        otherwise _4o2AS:j  
            error('zernfun2:nargin','Incorrect number of inputs.') \bQ|O7s  
    end yZK1bnYG|I  
    pW:h\}%`n  
    % 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) kSU5  }  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. n#z^uq|v  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of bw%1*;n)  
    %   order N and frequency M, evaluated at R.  N is a vector of edGV[=]F  
    %   positive integers (including 0), and M is a vector with the _QvyFKAM  
    %   same number of elements as N.  Each element k of M must be a zIF &ZYP  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) oCy52Bm.!  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is r{\cm Ds  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix P7egT,Z  
    %   with one column for every (N,M) pair, and one row for every ez(4TtT  
    %   element in R. Da-F(^E  
    % hp-< 8Mf  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- G]P4[#5  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is ?jqZeO#W7  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to G8u8&|  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 e"r}I!.  
    %   for all [n,m]. H7Y}qP5X  
    % 4bAgbx-^  
    %   The radial Zernike polynomials are the radial portion of the $nn~K  
    %   Zernike functions, which are an orthogonal basis on the unit R%#c~NOO  
    %   circle.  The series representation of the radial Zernike n!eg"pL  
    %   polynomials is }S#.Pw%  
    % X'k w5P!sq  
    %          (n-m)/2 LWVO%@)w  
    %            __ w-B\AK?}  
    %    m      \       s                                          n-2s T"QY@#E  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r 30DpIkf  
    %    n      s=0 * uEU9fX  
    % v/m`rc]e  
    %   The following table shows the first 12 polynomials. P*aD2("Z  
    % N e^#5T  
    %       n    m    Zernike polynomial    Normalization @b>]q$)(}  
    %       --------------------------------------------- ]m4LY.SQ  
    %       0    0    1                        sqrt(2) //J:p,AF  
    %       1    1    r                           2 T&R`s+7  
    %       2    0    2*r^2 - 1                sqrt(6) e\yj>tQJg  
    %       2    2    r^2                      sqrt(6) V?_%Y<|L  
    %       3    1    3*r^3 - 2*r              sqrt(8) xje{ kx#  
    %       3    3    r^3                      sqrt(8) b%oma{I=.c  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) c'G\AbUVjE  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) `/HygC6  
    %       4    4    r^4                      sqrt(10) 20fCWVw}?}  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) 8ZIv:nO$  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) s &.Z;X  
    %       5    5    r^5                      sqrt(12) e<p$Op  
    %       --------------------------------------------- CC)9Ks\  
    % A%GJ|h,i  
    %   Example: 8p5'}Lq  
    % ]$L[3qA.  
    %       % Display three example Zernike radial polynomials .s!:p pwl  
    %       r = 0:0.01:1; JQtH },T r  
    %       n = [3 2 5]; plf<O5'  
    %       m = [1 2 1]; ;8T<L[ ^U  
    %       z = zernpol(n,m,r); xS(sRx+A  
    %       figure "|Pl(HX  
    %       plot(r,z) t =*K?'ly  
    %       grid on FdSaOod8  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') cYp}$  
    % 'sA&Pm  
    %   See also ZERNFUN, ZERNFUN2. w+MdQ@'5  
    @"~\[z5  
    % A note on the algorithm. ]Yj>~k:K  
    % ------------------------ {c J6Lq&  
    % The radial Zernike polynomials are computed using the series %b*%'#iK  
    % representation shown in the Help section above. For many special E$1^}RGT)  
    % functions, direct evaluation using the series representation can gRFC n6Q  
    % produce poor numerical results (floating point errors), because Ym6ec|9;  
    % the summation often involves computing small differences between $bo^UYZ6  
    % large successive terms in the series. (In such cases, the functions gO/(/e>P  
    % are often evaluated using alternative methods such as recurrence x$Dv&4  
    % relations: see the Legendre functions, for example). For the Zernike 2tbqmWw/s  
    % polynomials, however, this problem does not arise, because the H,I}R  
    % polynomials are evaluated over the finite domain r = (0,1), and cpy"1=K~M  
    % because the coefficients for a given polynomial are generally all kDz.{Ih  
    % of similar magnitude. W+eN%w5  
    % ~"wD4Ue  
    % ZERNPOL has been written using a vectorized implementation: multiple 4ku/3/ 6  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] e"2QV vB  
    % values can be passed as inputs) for a vector of points R.  To achieve OP&[5X+Y  
    % this vectorization most efficiently, the algorithm in ZERNPOL 68!]q(!6F  
    % involves pre-determining all the powers p of R that are required to 7 dzE"m  
    % compute the outputs, and then compiling the {R^p} into a single VsmL#@E  
    % matrix.  This avoids any redundant computation of the R^p, and a(7ryl~c=  
    % minimizes the sizes of certain intermediate variables. L,ra=SVF  
    % BmM,vllO  
    %   Paul Fricker 11/13/2006 HUghl2L.<  
    Lg?'1dg  
    `u}x:f !  
    % Check and prepare the inputs: k|_2aQ02  
    % ----------------------------- Om^/tp\  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) n>4S P_[E7  
        error('zernpol:NMvectors','N and M must be vectors.') |]q=D1/A  
    end O<3,n;56Z  
    4 * OU  
    if length(n)~=length(m) kmoJ`W} N  
        error('zernpol:NMlength','N and M must be the same length.') n,F00Y R  
    end Zb 2  
    /Ow?nWSt  
    n = n(:); m=9 N^_  
    m = m(:); Q*8-d9C  
    length_n = length(n); 4yA`);r62  
    f+920/>!Z  
    if any(mod(n-m,2)) -b$OHFL  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') AAW7@\q.  
    end |FFC8R%@]u  
    d.AjH9 jg  
    if any(m<0) (*ng$z Z$  
        error('zernpol:Mpositive','All M must be positive.') k~R[5W|'  
    end T]W -g  
    mig3.is  
    if any(m>n) =#tQIhX`  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') gp HwiFc  
    end #H`y1zm  
    a%!XLyq  
    if any( r>1 | r<0 ) [Mz;:/  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') s]c$]&IGG  
    end KV_Ga8hs  
    a,~P_B|@  
    if ~any(size(r)==1) F4Uk+|]Bu  
        error('zernpol:Rvector','R must be a vector.') {wP|b@(1t  
    end As|/ O7%  
    udFju&!W  
    r = r(:); 1%"` =$q%  
    length_r = length(r); JNaW> X$K  
    \2pFFVT  
    if nargin==4 |esjhf}H>v  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); G 7]wg>*  
        if ~isnorm )^H9C"7T  
            error('zernpol:normalization','Unrecognized normalization flag.') W~3tQ!  
        end l;A_Aii(  
    else <&:&qn gg  
        isnorm = false; Mj B[5:s  
    end kW&Z%k  
    3+3m`%G  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Q ~JKKq  
    % Compute the Zernike Polynomials 1`lFF_stkP  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% fR4l4 GU?)  
    &.hRVW(  
    % Determine the required powers of r: L_"(A #H:  
    % ----------------------------------- n-3j$x1Ne  
    rpowers = []; ,,@`l\Pgd  
    for j = 1:length(n) NCnId}BT  
        rpowers = [rpowers m(j):2:n(j)]; =jc8=h[F<  
    end Lc<xgN+cJ  
    rpowers = unique(rpowers); K9Xd? ]a  
    HFuaoS+b*  
    % Pre-compute the values of r raised to the required powers, "GI&S%F  
    % and compile them in a matrix: WgJAr73 l  
    % ----------------------------- Us,[x Q  
    if rpowers(1)==0 (V.,~t@  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 7/_ VE  
        rpowern = cat(2,rpowern{:});  \t# 9zn>  
        rpowern = [ones(length_r,1) rpowern]; w"agn}CK  
    else Ln2C#Uf  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); i i@1!o  
        rpowern = cat(2,rpowern{:}); v\(m"|4(i  
    end k(z<Bm  
    4c[)}8\  
    % Compute the values of the polynomials: MW$H/:3  
    % -------------------------------------- |vN@2h(|"  
    z = zeros(length_r,length_n); ](>7h _2B  
    for j = 1:length_n ` .(S#!gw  
        s = 0:(n(j)-m(j))/2; C6UMc} 9h  
        pows = n(j):-2:m(j); '0')6zW5s  
        for k = length(s):-1:1 }u_EXP8M  
            p = (1-2*mod(s(k),2))* ... w2N3+Tkg  
                       prod(2:(n(j)-s(k)))/          ... ESxC{ "  
                       prod(2:s(k))/                 ... B9IXa;  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... A?D"j7JD=L  
                       prod(2:((n(j)+m(j))/2-s(k))); =j+oKGkoCa  
            idx = (pows(k)==rpowers); %}MA5 t]o  
            z(:,j) = z(:,j) + p*rpowern(:,idx); } ndvV~*1  
        end O {6gNR,*  
         (_qBsng:  
        if isnorm NQ!N"C3u  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); s i "`  
        end 'CX.qxF1;p  
    end \As oeeF  
    Dg>'5`&  
    % EOF zernpol
    离线niuhelen
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    只看该作者 6楼 发表于: 2011-03-12
    这三个文件,我不知道该怎样把我的面型节点的坐标及轴向位移用起来,还烦请指点一下啊,谢谢啦!
    离线li_xin_feng
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    只看该作者 7楼 发表于: 2012-09-28
    我也正在找啊
    离线guapiqlh
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    只看该作者 8楼 发表于: 2014-03-04
    我也一直想了解这个多项式的应用,还没用过呢
    离线phoenixzqy
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    只看该作者 9楼 发表于: 2014-04-22
    回 guapiqlh 的帖子
    guapiqlh:我也一直想了解这个多项式的应用,还没用过呢 (2014-03-04 11:35)  #V<`U:.  
    eQ$N:]  
    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。 oJP< 'l1  
    90Xt_$_}s  
    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)