切换到宽版
  • 广告投放
  • 稿件投递
  • 繁體中文
    • 10950阅读
    • 9回复

    [求助]ansys分析后面型数据如何进行zernike多项式拟合? [复制链接]

    上一主题 下一主题
    离线niuhelen
     
    发帖
    19
    光币
    28
    光券
    0
    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 "87ghj_}  
    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! /ew Ukc8,  
     
    分享到
    离线phility
    发帖
    69
    光币
    11
    光券
    0
    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
    发帖
    69
    光币
    11
    光券
    0
    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
    发帖
    19
    光币
    28
    光券
    0
    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 * C6a?]  
    function z = zernfun(n,m,r,theta,nflag) (>I`{9x>6  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. d R]Q$CJ  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N mBG=jI "xh  
    %   and angular frequency M, evaluated at positions (R,THETA) on the !ZI7&r`u;  
    %   unit circle.  N is a vector of positive integers (including 0), and ulER1\W  
    %   M is a vector with the same number of elements as N.  Each element _Jt 2YZdA  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) K.z64/H:  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, I~9hx*!%%  
    %   and THETA is a vector of angles.  R and THETA must have the same y:v xE8$Q  
    %   length.  The output Z is a matrix with one column for every (N,M) )h8\u_U  
    %   pair, and one row for every (R,THETA) pair. e4z1`YLsG  
    % Zt&6Ua[Y}  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike D.1J_Y=9  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 8-Hsgf.*  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral wj1{M.EF\  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 3,Q^& 1  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized XFh>U7z.  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. en:4H   
    % f~_th @K  
    %   The Zernike functions are an orthogonal basis on the unit circle. n]u<!.X  
    %   They are used in disciplines such as astronomy, optics, and !E-Pa5s  
    %   optometry to describe functions on a circular domain. ]+m/;&0  
    % WzI8_uM  
    %   The following table lists the first 15 Zernike functions. ocyb5j  
    % `)Z!V?&!  
    %       n    m    Zernike function           Normalization 4L73]3&  
    %       -------------------------------------------------- VT.;:Q  
    %       0    0    1                                 1 QZ?=M@|f  
    %       1    1    r * cos(theta)                    2 5zIAhg@o:q  
    %       1   -1    r * sin(theta)                    2 \J6hI\/4^  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) a!mf;m  
    %       2    0    (2*r^2 - 1)                    sqrt(3) vc]cNz:mQ  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) ZDC9oX @  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) brZ sA Q+k  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) [M%9_CfZOy  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) JMS(9>+TA  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) "sKa`WN}  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) #%FN>v3e  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 9P<[7u  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) 2Gs$?}"a  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) PDLpNTBf  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) BnM4T~reOF  
    %       -------------------------------------------------- n 8pt\i0  
    % Hku!bJ  
    %   Example 1: {q3H5csFq  
    % SgEBh  
    %       % Display the Zernike function Z(n=5,m=1) R+~cl;#G6  
    %       x = -1:0.01:1; ~Gqno  
    %       [X,Y] = meshgrid(x,x); !P$'#5mr  
    %       [theta,r] = cart2pol(X,Y); ZK'-U,Y.H7  
    %       idx = r<=1; '/I:^9  
    %       z = nan(size(X)); 3~qR  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); olW|$?  
    %       figure K A276#  
    %       pcolor(x,x,z), shading interp ,JEbd1Uf  
    %       axis square, colorbar 4TwQO$C  
    %       title('Zernike function Z_5^1(r,\theta)') JNFIT;L  
    % +]@Az.E  
    %   Example 2: T'fcc6D5p  
    % gs W0  
    %       % Display the first 10 Zernike functions )){xlFA}  
    %       x = -1:0.01:1; '?Jxt:<  
    %       [X,Y] = meshgrid(x,x); CZEW-PIhj  
    %       [theta,r] = cart2pol(X,Y); lZQ /W:OE  
    %       idx = r<=1; `PL[lP-<  
    %       z = nan(size(X)); 3?E&}J<n  
    %       n = [0  1  1  2  2  2  3  3  3  3]; [Lp,Hqi5  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; . /p|?pu  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; rz(0:vxwA  
    %       y = zernfun(n,m,r(idx),theta(idx)); ZE `lr+_Y  
    %       figure('Units','normalized') 0q9>6?=i  
    %       for k = 1:10 'lS `s(  
    %           z(idx) = y(:,k); <g9"Cr`  
    %           subplot(4,7,Nplot(k)) b%t+,0s|  
    %           pcolor(x,x,z), shading interp [ "xn5l E  
    %           set(gca,'XTick',[],'YTick',[]) d3]hyTqbtm  
    %           axis square IOK}+C0e  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) G]Jz"xH#  
    %       end kHJ96G  
    % 0"g@!gSrQ  
    %   See also ZERNPOL, ZERNFUN2. 1>r ,vD&  
    `Vq`z]}  
    %   Paul Fricker 11/13/2006 :h:@o h_=  
    t?^9HP1b_  
    gNx+>h`AF  
    % Check and prepare the inputs: +/?iCmW  
    % ----------------------------- :ZxLJK9x1  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) @gSkROCdC)  
        error('zernfun:NMvectors','N and M must be vectors.') IwpbfZ  
    end QT#6'>&7-b  
    iQ^: ])m>  
    if length(n)~=length(m) 5A&y]5-Q`  
        error('zernfun:NMlength','N and M must be the same length.') k*-NsNPw$  
    end d \>2  
    :Y)to/h  
    n = n(:); +ySY>`1k~  
    m = m(:); Napf"Av  
    if any(mod(n-m,2)) Ak~4|w-  
        error('zernfun:NMmultiplesof2', ... 2:$ k  
              'All N and M must differ by multiples of 2 (including 0).') &14W vAU  
    end @<_`2eW'/R  
    ,M3z!=oIGn  
    if any(m>n) :k46S<RE  
        error('zernfun:MlessthanN', ... AH.9A_dG  
              'Each M must be less than or equal to its corresponding N.') _eLVBG35z  
    end sa1mC  
    2r ];V'r  
    if any( r>1 | r<0 ) %B EC] h  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') 0zqj0   
    end )%du@a8  
    ke/_k/  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) @XOi62(  
        error('zernfun:RTHvector','R and THETA must be vectors.') hbuZaxo<  
    end OR+A_:c.D  
    Z{{ t^+XG  
    r = r(:); GH'O! }  
    theta = theta(:); vW' 5 ` %  
    length_r = length(r); "E*8h/4u  
    if length_r~=length(theta) |0{ i9 .=  
        error('zernfun:RTHlength', ... '=} Y2?(  
              'The number of R- and THETA-values must be equal.') Q:S\0cI0  
    end w1B<0'#  
    jeDlH6X'  
    % Check normalization: F>:%Cyo0!  
    % -------------------- L(WOet('  
    if nargin==5 && ischar(nflag) \iMyo  
        isnorm = strcmpi(nflag,'norm'); Ugi5OKdj7)  
        if ~isnorm = cfm=+  
            error('zernfun:normalization','Unrecognized normalization flag.') ]Sta]}VQ  
        end $(>f8)Uku(  
    else PI7IBI  
        isnorm = false; oA3d^%(c  
    end X9'xn 0n;  
    ,0T)Oc|HL/  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% g'G8 3F  
    % Compute the Zernike Polynomials 'TEyP56  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% A9BxwQU#  
    I=YCQ VvA  
    % Determine the required powers of r: .]Ybp2`"U  
    % ----------------------------------- ,L-C(j  
    m_abs = abs(m); >.Q0 Tx!P  
    rpowers = []; y'rN5J:l  
    for j = 1:length(n) Qp&?L"U)2  
        rpowers = [rpowers m_abs(j):2:n(j)]; ida*]+ ~  
    end 'P/taEi=R  
    rpowers = unique(rpowers); P,1exgq9  
    /8p&Qf>lJ1  
    % Pre-compute the values of r raised to the required powers, yv${M u  
    % and compile them in a matrix: \r]('x3S  
    % ----------------------------- `2x34  
    if rpowers(1)==0 TczXHT}G  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); '?R=P  
        rpowern = cat(2,rpowern{:}); uAb 03Q  
        rpowern = [ones(length_r,1) rpowern]; A*Q[k 9B  
    else z1vni'%J  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ,iOZ |  
        rpowern = cat(2,rpowern{:}); t/Z!O z6ZE  
    end <t6 d)mJ%  
    [ i9[Mj  
    % Compute the values of the polynomials: xL&PJ /'  
    % -------------------------------------- ~}%&p& p  
    y = zeros(length_r,length(n)); ,%='>A  
    for j = 1:length(n) x=3I)}J(kn  
        s = 0:(n(j)-m_abs(j))/2; 0HPO" x3-O  
        pows = n(j):-2:m_abs(j); #f 9qlM32  
        for k = length(s):-1:1 /a%KS3>V*  
            p = (1-2*mod(s(k),2))* ... I:/4t^%  
                       prod(2:(n(j)-s(k)))/              ... *08+\ed"#  
                       prod(2:s(k))/                     ... 5xv,!/@  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... Z`"n:'&  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); 3dU#Ueu  
            idx = (pows(k)==rpowers); MVuP |&:n  
            y(:,j) = y(:,j) + p*rpowern(:,idx); (6[Wr}SW5  
        end (lWKy9eTy`  
         jhcuK:`L  
        if isnorm |bvGYsn_#=  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); A xR\ ned  
        end P59uALi  
    end M[vCpa  
    % END: Compute the Zernike Polynomials 573~-Jvx  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 8"pA9Mr  
    ]Qy,#p'~&H  
    % Compute the Zernike functions: "D!Dr1  
    % ------------------------------ 'w `d$c/p  
    idx_pos = m>0; `~KAk  
    idx_neg = m<0; tpz=} q  
    ~:s!].H  
    z = y; " #J}A0  
    if any(idx_pos) gTyW#verh$  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); }(rzH}X@  
    end {!tOI  
    if any(idx_neg) 'U*udkn 2]  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 95% :AQLV  
    end ILIRI[7 (  
    B4 <_"0  
    % EOF zernfun
    离线niuhelen
    发帖
    19
    光币
    28
    光券
    0
    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) UfO'.8*v  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. 79y'Ja+`j  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated Z|f^nH#-C  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive !/[AQ{**T!  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, 78\\8*  
    %   and THETA is a vector of angles.  R and THETA must have the same ;9 R40qi  
    %   length.  The output Z is a matrix with one column for every P-value, '$lw[1  
    %   and one row for every (R,THETA) pair. >l6XZQ >  
    % FUH *]U  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike WodF -bE  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) ((n5';|N  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) QZ_nQ3K  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 '[0 3L9  
    %   for all p. F&-5&'6G+  
    % NN?Bi=&9  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 `7+tPbjs  
    %   Zernike functions (order N<=7).  In some disciplines it is 6S` ,j  
    %   traditional to label the first 36 functions using a single mode g=)U_DPRi  
    %   number P instead of separate numbers for the order N and azimuthal 0R.Gjz*Q  
    %   frequency M. hnlU,p&y3  
    % mOgx&ns;j  
    %   Example: !NQf< ch  
    % _AA`R`p;  
    %       % Display the first 16 Zernike functions '&&~IB4ud  
    %       x = -1:0.01:1; ZhxfI?i)l  
    %       [X,Y] = meshgrid(x,x); Va&KIHw  
    %       [theta,r] = cart2pol(X,Y); uBV^nUjS"m  
    %       idx = r<=1; Bx_8@+  
    %       p = 0:15; j>0SE  
    %       z = nan(size(X)); 'bd=,QW  
    %       y = zernfun2(p,r(idx),theta(idx)); ZfF`kD\  
    %       figure('Units','normalized') V1AEjh  
    %       for k = 1:length(p) xX[{E x   
    %           z(idx) = y(:,k); u&Ie%@:h9R  
    %           subplot(4,4,k) (*c`<|)  
    %           pcolor(x,x,z), shading interp %6vMpB`g  
    %           set(gca,'XTick',[],'YTick',[]) E$oA+n~  
    %           axis square [ 7CH(o1a&  
    %           title(['Z_{' num2str(p(k)) '}']) AF07KA#  
    %       end M]pel\{M  
    % oc,U4+T  
    %   See also ZERNPOL, ZERNFUN. Ra*k  
    gDjd{+LUo  
    %   Paul Fricker 11/13/2006 gPn%`_d5  
    U{.+*e18  
    =!Baz&#}  
    % Check and prepare the inputs: !~VR|n-  
    % ----------------------------- ?`BED6$`G9  
    if min(size(p))~=1 fNmG`Ke  
        error('zernfun2:Pvector','Input P must be vector.') ~gcst;  
    end S(YHwH":  
    2t~7eI%d  
    if any(p)>35 "J0Oa?  
        error('zernfun2:P36', ... *U5> j#,  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... M2;(+8 b  
               '(P = 0 to 35).']) x Jj8njuq4  
    end 2Q;Y@%G  
    EUYa =-  
    % Get the order and frequency corresonding to the function number: KX=:)%+  
    % ---------------------------------------------------------------- 5O4&BxQ~}  
    p = p(:); FK/ro91L  
    n = ceil((-3+sqrt(9+8*p))/2); OM#OPB rB  
    m = 2*p - n.*(n+2); tkUW)ScJ  
    n=Z[w5  
    % Pass the inputs to the function ZERNFUN: Cvu8X&y  
    % ---------------------------------------- a#~Z5>{  
    switch nargin :)3$&QdHT  
        case 3 [b\lcQ8O  
            z = zernfun(n,m,r,theta); vY TPZ@RL  
        case 4 .\hib. n3  
            z = zernfun(n,m,r,theta,nflag); .w*{=x0k  
        otherwise ;zxlwdfcr'  
            error('zernfun2:nargin','Incorrect number of inputs.') J 6d n~nPK  
    end iTtAj~dfZ  
    XiZ Zo  
    % EOF zernfun2
    离线niuhelen
    发帖
    19
    光币
    28
    光券
    0
    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) F|3FvxA  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. >p+gx,N  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of 2|_Jup  
    %   order N and frequency M, evaluated at R.  N is a vector of RAkFgC~  
    %   positive integers (including 0), and M is a vector with the do?n /<@o  
    %   same number of elements as N.  Each element k of M must be a <raqp Oo&  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) <t|9`l_XW  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is =[-- Hf  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix Iy 8E$B;  
    %   with one column for every (N,M) pair, and one row for every Zp(P)Obs#  
    %   element in R. pQ2)M8 gf  
    % T4, Zc  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- qt&"cw  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is ^OcfM_4pN  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to }4ghT(C}$  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 D;8V{Hs  
    %   for all [n,m]. n|`):sP  
    % ;<GTtt# D  
    %   The radial Zernike polynomials are the radial portion of the ;s/b_RN  
    %   Zernike functions, which are an orthogonal basis on the unit :phD?\!w8t  
    %   circle.  The series representation of the radial Zernike m ?tnk?oX  
    %   polynomials is gm8Tm$fY  
    % q,>F#A '  
    %          (n-m)/2 Z*Hxrw\!0  
    %            __ s^X/ Om  
    %    m      \       s                                          n-2s q^+NhAMz  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r u~T$F/]k>  
    %    n      s=0 OcS`Fxs  
    % fvAV[9/-  
    %   The following table shows the first 12 polynomials. \ A UtGP  
    % w`!foPE  
    %       n    m    Zernike polynomial    Normalization S^"e5n2  
    %       --------------------------------------------- =gv/9ce)3  
    %       0    0    1                        sqrt(2) >-o:> 5  
    %       1    1    r                           2 !?M_%fNE  
    %       2    0    2*r^2 - 1                sqrt(6) ok _{8z\#  
    %       2    2    r^2                      sqrt(6) umrI4.1c  
    %       3    1    3*r^3 - 2*r              sqrt(8) A,[m=9V  
    %       3    3    r^3                      sqrt(8) P FFw$\j  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) \ ozy_s[  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) v@s`l#  
    %       4    4    r^4                      sqrt(10) j/ IZm)\  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) ~\IDg/9 Cj  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) aJ_Eh(cF  
    %       5    5    r^5                      sqrt(12) F+9`G[  
    %       --------------------------------------------- &rWJg6/  
    % eQIi}\`  
    %   Example: T.dO0$,Q@$  
    % Kd1\D!#!6  
    %       % Display three example Zernike radial polynomials )Bvu[r Uy  
    %       r = 0:0.01:1; ur[bh  
    %       n = [3 2 5]; ul&7hHp_u%  
    %       m = [1 2 1]; Q&a<9e&  
    %       z = zernpol(n,m,r); A2bV[+Q  
    %       figure 6oBt<r?CJ  
    %       plot(r,z) SO<K#HfE$?  
    %       grid on Ri0+nJ6  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') gbZX'D  
    % G7JZP T  
    %   See also ZERNFUN, ZERNFUN2. LKY Q?  
    m:7bynT{  
    % A note on the algorithm. sgsMlZ3/  
    % ------------------------ ^Wz{su2  
    % The radial Zernike polynomials are computed using the series 'Em($A (  
    % representation shown in the Help section above. For many special O]!DNN  
    % functions, direct evaluation using the series representation can 5X-{|r3q  
    % produce poor numerical results (floating point errors), because ? D'-{/<4  
    % the summation often involves computing small differences between lv&wp@  
    % large successive terms in the series. (In such cases, the functions t!N >0]:mo  
    % are often evaluated using alternative methods such as recurrence m&8_i`%<  
    % relations: see the Legendre functions, for example). For the Zernike Q{kuB+s  
    % polynomials, however, this problem does not arise, because the *~X\c Z  
    % polynomials are evaluated over the finite domain r = (0,1), and YI+|6s[  
    % because the coefficients for a given polynomial are generally all #2=30  
    % of similar magnitude. PrxXL/6  
    % $OuA<-  
    % ZERNPOL has been written using a vectorized implementation: multiple @#">~P|Hp  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] "0mR*{nF  
    % values can be passed as inputs) for a vector of points R.  To achieve D=]P9XDvb.  
    % this vectorization most efficiently, the algorithm in ZERNPOL LYv2ll`XP  
    % involves pre-determining all the powers p of R that are required to 5=e@yIr'#  
    % compute the outputs, and then compiling the {R^p} into a single 0\A[a4crj  
    % matrix.  This avoids any redundant computation of the R^p, and #2iA-5  
    % minimizes the sizes of certain intermediate variables. Ok\UIi~  
    % 07&S^ X^/  
    %   Paul Fricker 11/13/2006 S8t9Ms: k  
    J{I?t~u  
    #,C{?0!  
    % Check and prepare the inputs: F"I@=R-n  
    % ----------------------------- I115Rp0  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) \!Pm^FD .  
        error('zernpol:NMvectors','N and M must be vectors.') S5M t?v|K  
    end XZJx3!~fm  
    NU"X*g-x^  
    if length(n)~=length(m) dXQWT@$y!E  
        error('zernpol:NMlength','N and M must be the same length.') H6QQ<~_&  
    end Ft7l/  
    `. Z".  
    n = n(:); 0'",4=c#V  
    m = m(:); lU3wIB  
    length_n = length(n); n&lLC&dL  
    i[swOY z]X  
    if any(mod(n-m,2)) 1l{n`gR  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') -i4gzak  
    end !+U.)u9 '  
    7UfyOOFa  
    if any(m<0) l!2.)F`x  
        error('zernpol:Mpositive','All M must be positive.') B Xp3u|t  
    end =W?c1EPLCx  
    a?dM8zAnc  
    if any(m>n) mj pH)6aD0  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') O`4X[r1LD  
    end qW9|&GuZ$  
    hsh W5j  
    if any( r>1 | r<0 ) n=~?BxB  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') :Li)]qN.I  
    end i?]!8Ji  
    1'iRx,  
    if ~any(size(r)==1) IdM ;N  
        error('zernpol:Rvector','R must be a vector.') Wl{Vz  
    end ?k-IS5G  
    gNJ\*]SY  
    r = r(:); `|4{|X*U.  
    length_r = length(r); -HOCxR  
    [(1O_X(M  
    if nargin==4 6 BMn7m?  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); a fjC~}  
        if ~isnorm mdwY48b  
            error('zernpol:normalization','Unrecognized normalization flag.') tSjK=1"}  
        end %rYt; 7B  
    else p[RD[&#b  
        isnorm = false; DWDe5$^{  
    end D6D*RTi4  
    E yuc~[  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @-wAR=k7  
    % Compute the Zernike Polynomials hd900LA}  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ({)_[dJ'  
    jhHb[je~{4  
    % Determine the required powers of r: 6*%lnd+_  
    % ----------------------------------- }I]9I _S  
    rpowers = []; 0kDT:3  
    for j = 1:length(n) dg-pwWqN  
        rpowers = [rpowers m(j):2:n(j)]; Ofn:<d  
    end B $HQFdTli  
    rpowers = unique(rpowers); ~V<je b  
    ;9rQN3J$gn  
    % Pre-compute the values of r raised to the required powers, 2- )Ml*  
    % and compile them in a matrix: |KA8qQI]%  
    % ----------------------------- dJkT Hmw  
    if rpowers(1)==0 gpPktp2  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 1oC/W?l^  
        rpowern = cat(2,rpowern{:}); <1ai0]  
        rpowern = [ones(length_r,1) rpowern]; 0X@5W$x  
    else q.*qZ\;K  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); :x_l"y"  
        rpowern = cat(2,rpowern{:}); 8`*(lKiL  
    end Vi]D](^!  
    d;f,vN(  
    % Compute the values of the polynomials: ar{Yq  
    % -------------------------------------- b{(:'.  
    z = zeros(length_r,length_n); >$}nKPC,Y  
    for j = 1:length_n |^FDsJUN  
        s = 0:(n(j)-m(j))/2; r+>9O  
        pows = n(j):-2:m(j); y};qo'dlt  
        for k = length(s):-1:1 Q_ $AGF  
            p = (1-2*mod(s(k),2))* ... H`fkds  
                       prod(2:(n(j)-s(k)))/          ... Cu >pql<O  
                       prod(2:s(k))/                 ... vWow^g  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... mv5!fp_*7  
                       prod(2:((n(j)+m(j))/2-s(k))); ,TL~];J'  
            idx = (pows(k)==rpowers); ]RxNSr0e  
            z(:,j) = z(:,j) + p*rpowern(:,idx); Um%E/0j  
        end DRw%~  
         O'(qeN<^w  
        if isnorm i2y?CI  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); _,vJ0{*  
        end ~]N% {;F}  
    end =\`9\Gd  
    c sYICLj  
    % EOF zernpol
    离线niuhelen
    发帖
    19
    光币
    28
    光券
    0
    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
    发帖
    59
    光币
    0
    光券
    0
    只看该作者 7楼 发表于: 2012-09-28
    我也正在找啊
    离线guapiqlh
    发帖
    856
    光币
    846
    光券
    0
    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
    发帖
    4352
    光币
    5478
    光券
    1
    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)  ]7fqVOiOu  
    X#Ajt/XQ  
    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。 VLkK6W.u  
    LKFL2|af  
    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)