切换到宽版
  • 广告投放
  • 稿件投递
  • 繁體中文
    • 10728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 0tqR wKL  
    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! jp^WsHI3  
     
    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 iB3 +KR  
    function z = zernfun(n,m,r,theta,nflag) pd>a6 lI`  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. {qWG^Db  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N f76|  
    %   and angular frequency M, evaluated at positions (R,THETA) on the KHus/M&0  
    %   unit circle.  N is a vector of positive integers (including 0), and h!N&gZ[0  
    %   M is a vector with the same number of elements as N.  Each element D^s0EW-E  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) fd"~[ z[  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, } o"_#\6  
    %   and THETA is a vector of angles.  R and THETA must have the same , q@(L  
    %   length.  The output Z is a matrix with one column for every (N,M) /9+A97{  
    %   pair, and one row for every (R,THETA) pair. Omh&)|Iql  
    % !bV(VRbu  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike a H|OA\<  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), tbzvO<~  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral Pv-V7`{  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, K# i*9sM  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized l&sO?P[ /  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. A[ECa{ v  
    % 0"<;You  
    %   The Zernike functions are an orthogonal basis on the unit circle. %M)oHX1p  
    %   They are used in disciplines such as astronomy, optics, and W3V{Xk|  
    %   optometry to describe functions on a circular domain. 'oiD#\t4  
    % g. Caapy  
    %   The following table lists the first 15 Zernike functions. x$5nLS2.  
    % )47j8jL  
    %       n    m    Zernike function           Normalization LJNie*  
    %       -------------------------------------------------- gj egzKU  
    %       0    0    1                                 1 So\|Ye  
    %       1    1    r * cos(theta)                    2 -m'3L7:  
    %       1   -1    r * sin(theta)                    2 Nzi/3r7m  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) R#7+  
    %       2    0    (2*r^2 - 1)                    sqrt(3) (LT\ IJSM  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) tY$ty0y-e  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) n#^?X  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) zsMw5C  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) "'}v0*[  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) b?2X>QJ  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10)  lGnql1(  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Q 9gFTLQ  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) yrE,,N%I  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10)  ny  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) V:F+HMBk  
    %       -------------------------------------------------- tgvpf /cQ  
    % S1az3VJI\  
    %   Example 1: o3i,B),K  
    % L VU)W^  
    %       % Display the Zernike function Z(n=5,m=1) -l40)^ E}  
    %       x = -1:0.01:1; /_:T\`5uO  
    %       [X,Y] = meshgrid(x,x); FU (}=5n  
    %       [theta,r] = cart2pol(X,Y); 4l%?mvA^m  
    %       idx = r<=1; tJ h3$K\  
    %       z = nan(size(X)); ;vI*ThzdD  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); EBIa%,  
    %       figure  *_ {l  
    %       pcolor(x,x,z), shading interp !rsa4t@ t  
    %       axis square, colorbar w `. T/  
    %       title('Zernike function Z_5^1(r,\theta)') N[a ljC-R  
    % 47C(\\  
    %   Example 2: *< $c =  
    % s}[A4`EWH  
    %       % Display the first 10 Zernike functions 5!SoN}$  
    %       x = -1:0.01:1; GTp?)nh^  
    %       [X,Y] = meshgrid(x,x); q lz9&w  
    %       [theta,r] = cart2pol(X,Y); rF8W(E_=  
    %       idx = r<=1; }rKJeOo^x?  
    %       z = nan(size(X)); <uBhi4  
    %       n = [0  1  1  2  2  2  3  3  3  3]; -40'[a9E  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; wuBlFUSg  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; GCP{Z]u  
    %       y = zernfun(n,m,r(idx),theta(idx)); _uO!N(k.  
    %       figure('Units','normalized') z\Pe{J  
    %       for k = 1:10 xs2,t*  
    %           z(idx) = y(:,k); 55>" R{q  
    %           subplot(4,7,Nplot(k)) .Ca"$2  
    %           pcolor(x,x,z), shading interp &J lpA<^s;  
    %           set(gca,'XTick',[],'YTick',[]) ,c,Xd  
    %           axis square `N|U"s;  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) _C< 6349w  
    %       end RjR&D?dc  
    % IdV,%d{  
    %   See also ZERNPOL, ZERNFUN2. .])>A')r  
    cX|[WT0[I  
    %   Paul Fricker 11/13/2006 zp7V\W; &  
    iA55yT+  
    $zk^yumdE  
    % Check and prepare the inputs: ,2 zt.aqB  
    % ----------------------------- Sk6b`W7$  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) sorSyuGr  
        error('zernfun:NMvectors','N and M must be vectors.') Q vv\+Jp^  
    end !G)mjvEe  
    la G$v-r  
    if length(n)~=length(m) F\-B3i%0  
        error('zernfun:NMlength','N and M must be the same length.') 5u2{n rc  
    end Vl5SL{+D  
    j!zA+hF (  
    n = n(:); EX`"z(L  
    m = m(:); P &;y] ,)E  
    if any(mod(n-m,2)) T-L|Q,-{-  
        error('zernfun:NMmultiplesof2', ...  ${A5-  
              'All N and M must differ by multiples of 2 (including 0).') pP|,7c5  
    end kZV^F*7  
    CCbkxHMf|!  
    if any(m>n) B#HV20\?v  
        error('zernfun:MlessthanN', ... k1ipvKxp:8  
              'Each M must be less than or equal to its corresponding N.') ^=eq .(>  
    end 9 9Ba{qj  
    cZNi~  
    if any( r>1 | r<0 ) 0lX)Cl  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') pyUNRqp  
    end I#"t'=9H  
    j2RRSz&9  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) vS7/~:C  
        error('zernfun:RTHvector','R and THETA must be vectors.') |HrM_h<X  
    end K^"w]ii=  
    cU7rq j_  
    r = r(:); Hze-Ob8  
    theta = theta(:); Z} c'Bm(  
    length_r = length(r); El~-M`Gf  
    if length_r~=length(theta) xj0cgK|!  
        error('zernfun:RTHlength', ... cqeR<len  
              'The number of R- and THETA-values must be equal.') k/df(cs  
    end 4rI:1 yGt@  
    g d z  
    % Check normalization: ;*y|8od B  
    % -------------------- X Y~;)<s_  
    if nargin==5 && ischar(nflag) %4j&H!y-w;  
        isnorm = strcmpi(nflag,'norm'); LYp'vZ!  
        if ~isnorm D`~JbKV5@^  
            error('zernfun:normalization','Unrecognized normalization flag.') HbNYP/MN3  
        end #2h+dk$1  
    else _ e6a8  
        isnorm = false; ?3`q+[:  
    end sa_R$ /H  
    Ej' 7h~=v  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% #MUiL=  
    % Compute the Zernike Polynomials %m oJF1  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% vKU`C?,L  
    3AK(dC[ri  
    % Determine the required powers of r: c\M#5+1j  
    % ----------------------------------- , Hn7(^t  
    m_abs = abs(m); D,k(~  
    rpowers = []; I[ai:   
    for j = 1:length(n) HeCcF+  
        rpowers = [rpowers m_abs(j):2:n(j)]; :v`o6x8  
    end =K :(&6f<t  
    rpowers = unique(rpowers); IeVLn^?+:  
    J2r1=5HS  
    % Pre-compute the values of r raised to the required powers, *9J1$Wa  
    % and compile them in a matrix: WM/#.  
    % ----------------------------- $'^&\U~?  
    if rpowers(1)==0 kGm:VYf%  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); #Rc5c+/(  
        rpowern = cat(2,rpowern{:}); )%s +?  
        rpowern = [ones(length_r,1) rpowern]; )!cI|tovs  
    else |=\91fP68`  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); .8]Y-  
        rpowern = cat(2,rpowern{:}); X:Q$gO?[4  
    end Y&s2C%jT  
    6)ycmu;!$  
    % Compute the values of the polynomials: Uh.Sc:trA  
    % -------------------------------------- ;+ G9-  
    y = zeros(length_r,length(n)); s;J\Kc?"|  
    for j = 1:length(n) va5FxF*%  
        s = 0:(n(j)-m_abs(j))/2; 4b4QbJ$  
        pows = n(j):-2:m_abs(j); CN/IH   
        for k = length(s):-1:1  Vu [:A  
            p = (1-2*mod(s(k),2))* ... _S"f_W  
                       prod(2:(n(j)-s(k)))/              ... R uLvG+  
                       prod(2:s(k))/                     ... |q_ !. a  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... a}>Dz 1R  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); ]ZTcOf  
            idx = (pows(k)==rpowers); =E y`M#t;  
            y(:,j) = y(:,j) + p*rpowern(:,idx); g]|_ `  
        end 7UKYmJk.  
         kM!V .e[g  
        if isnorm B;!f<"a8  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); ) r9b:c\  
        end y? "@v.  
    end [Uli>/%JB  
    % END: Compute the Zernike Polynomials H?uukmZl  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ANMYX18M  
    Gy!P,a)z  
    % Compute the Zernike functions: .Pw%DZ'  
    % ------------------------------ 3sFeP &  
    idx_pos = m>0; wVqd$nsY"  
    idx_neg = m<0; Kd3QqVJBz1  
    Q.k :\m*h  
    z = y; )p8I @E  
    if any(idx_pos) "}b'E#  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 1zjaR4Tf  
    end . uR M{Bs  
    if any(idx_neg) =XT)J6z^"  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); xMI+5b8  
    end aV>aiR=  
    m&IsDAn  
    % EOF zernfun
    离线niuhelen
    发帖
    19
    光币
    28
    光券
    0
    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) <y)E>Fl  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. TfRGA (+#  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated = .oHnMX2M  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive }rbZ&IN\?E  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, ?_q+&)4-o  
    %   and THETA is a vector of angles.  R and THETA must have the same 4/*H.Fl  
    %   length.  The output Z is a matrix with one column for every P-value, E'c%d[:H,  
    %   and one row for every (R,THETA) pair. {8@\Ij  
    % G> \T bx  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike 5PZN^\^  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) kWL.ewTiex  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) Y)b@0'  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 ]hE="z=n  
    %   for all p. 1H{jy^sP7  
    % ~rv})4h  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 B@t'U=@7  
    %   Zernike functions (order N<=7).  In some disciplines it is I^3:YVR&  
    %   traditional to label the first 36 functions using a single mode &ZJgQ-Pc(m  
    %   number P instead of separate numbers for the order N and azimuthal "0&N}  
    %   frequency M. C3VLV&wF  
    % ?Zz'|.l@  
    %   Example: 8Fq_i-u  
    % <]G${y*;  
    %       % Display the first 16 Zernike functions u&]vd /  
    %       x = -1:0.01:1; x &=9P e(  
    %       [X,Y] = meshgrid(x,x); D59T?B|BdD  
    %       [theta,r] = cart2pol(X,Y); $'"8QOnJ?k  
    %       idx = r<=1; ~}%~oT  
    %       p = 0:15; 1u}nm;3  
    %       z = nan(size(X)); vtxvS3   
    %       y = zernfun2(p,r(idx),theta(idx)); ohQAA h  
    %       figure('Units','normalized') \u{8Bak0  
    %       for k = 1:length(p) Ya Y8 `M{  
    %           z(idx) = y(:,k); YQ(Po!NI\'  
    %           subplot(4,4,k) +S~.c;EK  
    %           pcolor(x,x,z), shading interp \ijMw  
    %           set(gca,'XTick',[],'YTick',[]) ?o[L7JI  
    %           axis square %_gho  
    %           title(['Z_{' num2str(p(k)) '}']) S~F`  
    %       end p!W[X%`)  
    % )\ 0F7Z  
    %   See also ZERNPOL, ZERNFUN. 5/I_w0  
    ,&]MOe4@>  
    %   Paul Fricker 11/13/2006 SR7j\1a/2A  
    Xm_$ dZ  
    v[S-Pi1  
    % Check and prepare the inputs: 61K"(r~  
    % ----------------------------- Hs?zq  
    if min(size(p))~=1 ?m"|QS!!K  
        error('zernfun2:Pvector','Input P must be vector.') 'BqZOZw  
    end wu~hqd  
    wH6u5*$p  
    if any(p)>35 k%Vv?{g  
        error('zernfun2:P36', ... HKmcQM  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... =mt?C n}  
               '(P = 0 to 35).']) Yx)o:#2  
    end /c-nE3+rn  
    KCR N}`^  
    % Get the order and frequency corresonding to the function number: M##';x0  
    % ---------------------------------------------------------------- JMyTwj[7  
    p = p(:); bEH de*q(  
    n = ceil((-3+sqrt(9+8*p))/2); %XRN]tsu  
    m = 2*p - n.*(n+2); H;KDZO9W  
    e~\QE0Oe:  
    % Pass the inputs to the function ZERNFUN: aXR%;]<Dw  
    % ---------------------------------------- VOgi7\  
    switch nargin h)x_zZ%>o  
        case 3 4%ZM:/  
            z = zernfun(n,m,r,theta); Q/^A #l[  
        case 4 G=d(*+& B  
            z = zernfun(n,m,r,theta,nflag); oXYMoi  
        otherwise | {P|.  
            error('zernfun2:nargin','Incorrect number of inputs.') iI?{"}BZ  
    end .p@N:)W6  
    3<(q }  
    % 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) vA&Vu"}S  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. H7&xLYQ2  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of zICI_*~  
    %   order N and frequency M, evaluated at R.  N is a vector of k`YYZt]@  
    %   positive integers (including 0), and M is a vector with the W)=%mdxW0  
    %   same number of elements as N.  Each element k of M must be a d/U."V}  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) ST',4 Oph5  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is v 1.*IV5Y  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix $RO$}!  
    %   with one column for every (N,M) pair, and one row for every T1 MY X  
    %   element in R. M<`|CVl  
    % ?9KGnOVu  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- 5M){!8"S)#  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is XW~bu2%{7"  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to B;t=B_oK  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 4<v;1   
    %   for all [n,m]. \V%_hl  
    % 8tc*.H{^+  
    %   The radial Zernike polynomials are the radial portion of the /L~m#HxWU  
    %   Zernike functions, which are an orthogonal basis on the unit 4ke^*g K<  
    %   circle.  The series representation of the radial Zernike :)c80`-E  
    %   polynomials is  Y7Gs7  
    % cf;Ht^M\  
    %          (n-m)/2 Y E1Hpeb  
    %            __ 284zmZZ  
    %    m      \       s                                          n-2s j}WByaZ&  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r (JZ".En#X  
    %    n      s=0 JLm @Ag  
    % ~"dhu]^  
    %   The following table shows the first 12 polynomials. #g v4  
    % %_f;G+fK\p  
    %       n    m    Zernike polynomial    Normalization {d!Y3+I%G  
    %       --------------------------------------------- );JJ2Jlkd  
    %       0    0    1                        sqrt(2) Yp5L+~J[  
    %       1    1    r                           2 Wmz`&nsn[  
    %       2    0    2*r^2 - 1                sqrt(6) AK/:I>M  
    %       2    2    r^2                      sqrt(6) SkP[|g'56  
    %       3    1    3*r^3 - 2*r              sqrt(8) &RY)o^g[4  
    %       3    3    r^3                      sqrt(8) R@`rT*lJ  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) a6nlt? 1?D  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) ycpE=fso'  
    %       4    4    r^4                      sqrt(10) Spj9H?m  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) y-+G wa3  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) |B[eJq  
    %       5    5    r^5                      sqrt(12) xFb3O|TC  
    %       --------------------------------------------- %e=!nRc  
    % |*\C{b  
    %   Example: ElR)Gd_8  
    % bkv/I{C>?  
    %       % Display three example Zernike radial polynomials u{C)qb5Pu  
    %       r = 0:0.01:1; ~@9zil41  
    %       n = [3 2 5]; ->oz#  
    %       m = [1 2 1]; dgc&[  
    %       z = zernpol(n,m,r); `XMM1y>V9>  
    %       figure v\0^mp  
    %       plot(r,z) @ss):FwA  
    %       grid on pXW`+<g0  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') !Q)3-u  
    % HeS'~Z$  
    %   See also ZERNFUN, ZERNFUN2. V SAafux  
    )I9aC~eAD  
    % A note on the algorithm. z7=fDe -  
    % ------------------------ 80&D""  
    % The radial Zernike polynomials are computed using the series ,wK 1=7  
    % representation shown in the Help section above. For many special J/kH%_ >Ir  
    % functions, direct evaluation using the series representation can o# {#r@,i  
    % produce poor numerical results (floating point errors), because I'InZ0J2  
    % the summation often involves computing small differences between 14l; *  
    % large successive terms in the series. (In such cases, the functions K H}t:m+h  
    % are often evaluated using alternative methods such as recurrence s[V `e2O  
    % relations: see the Legendre functions, for example). For the Zernike gCV rC  
    % polynomials, however, this problem does not arise, because the e,Zv]Cym  
    % polynomials are evaluated over the finite domain r = (0,1), and MSYN1  
    % because the coefficients for a given polynomial are generally all `:5,e/5,  
    % of similar magnitude. [.3sE  
    % yq6LH   
    % ZERNPOL has been written using a vectorized implementation: multiple g :i*O^c @  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] 4ftj>O  
    % values can be passed as inputs) for a vector of points R.  To achieve 2"M_sL  
    % this vectorization most efficiently, the algorithm in ZERNPOL sU%" azc  
    % involves pre-determining all the powers p of R that are required to AM/lbMr  
    % compute the outputs, and then compiling the {R^p} into a single \+]O*Bm&`8  
    % matrix.  This avoids any redundant computation of the R^p, and -\,VGudM}  
    % minimizes the sizes of certain intermediate variables. /Ynt<S9"  
    % 0w}{(P;  
    %   Paul Fricker 11/13/2006 &kx\W)  
    *vs~SzF$  
    "1%*'B^}bw  
    % Check and prepare the inputs: v=MzI#0L  
    % ----------------------------- 5KaSWw/  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) W-XN4:,qI  
        error('zernpol:NMvectors','N and M must be vectors.') *1v_6<;2i<  
    end 8Mb$+^zU  
    R `Q?J[e  
    if length(n)~=length(m) yu_gNro L  
        error('zernpol:NMlength','N and M must be the same length.') 7b,AQ9  
    end h@Dw'w  
    Dauo(Uhuo  
    n = n(:); ^Kum%<[i  
    m = m(:); _w%s(dzk  
    length_n = length(n); |wJ),h8/  
    x`3. Wu\  
    if any(mod(n-m,2)) !Iko0#4i  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') U]mO7HK  
    end ;]u1~  
    L]NYYP-  
    if any(m<0) 't ;/,+:V  
        error('zernpol:Mpositive','All M must be positive.') gyg|Tno  
    end WiwwCKjSa  
    jL2MW(d^Q  
    if any(m>n) =ZrjK=K  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') ]o!&2:'N`  
    end J ZNyC!u  
    2}@*Ki7  
    if any( r>1 | r<0 ) ^ CX,nj_(  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') /gF]s_  
    end LA>dkPB  
    '[xut1{  
    if ~any(size(r)==1) h!~|6nj  
        error('zernpol:Rvector','R must be a vector.') `F@f?*s:  
    end roL]v\tr  
    ]X4RnV55Q  
    r = r(:); \O,j}O'  
    length_r = length(r); su%Z{f)#  
    ~.!?5(AH8z  
    if nargin==4 5 u"nxT   
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); uvNnW}G4  
        if ~isnorm " gwm23Rpj  
            error('zernpol:normalization','Unrecognized normalization flag.') :az!H"4W/  
        end s}<)B RZi  
    else 0n7HkDo  
        isnorm = false; wsna5D6i  
    end =7H.F:BBG  
    ?|gGsm+  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% $)Jc-V 6E  
    % Compute the Zernike Polynomials YDdLDE  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   -kV|  
    2>`m<&y  
    % Determine the required powers of r: EPR(i#xU  
    % ----------------------------------- !~Ax  
    rpowers = []; Py}!C@e  
    for j = 1:length(n) U_.n=d~B  
        rpowers = [rpowers m(j):2:n(j)]; ?%_]rr9  
    end 38OIFT  
    rpowers = unique(rpowers); *yL|}  
    t Qp* '  
    % Pre-compute the values of r raised to the required powers, /0X0#+kn  
    % and compile them in a matrix: }u38:(^`ai  
    % ----------------------------- LZ3rr-  
    if rpowers(1)==0 aEV|>K=6Y'  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); vK[v eFH  
        rpowern = cat(2,rpowern{:}); WX+< 4j  
        rpowern = [ones(length_r,1) rpowern]; Z 5{*? 2  
    else fbi H   
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); zDKLo 3:  
        rpowern = cat(2,rpowern{:}); O1l4gduN|i  
    end ,dGFX]P  
    l;"ub^AH  
    % Compute the values of the polynomials: DtI%-I.  
    % -------------------------------------- k4]R]=Fh.  
    z = zeros(length_r,length_n); ksxO<Y  
    for j = 1:length_n ]Hv*^Bak  
        s = 0:(n(j)-m(j))/2; ZA 99vO  
        pows = n(j):-2:m(j); e2,<,~_K6  
        for k = length(s):-1:1 Q;{D8 #!  
            p = (1-2*mod(s(k),2))* ... ft*G*.0kO  
                       prod(2:(n(j)-s(k)))/          ... :*^(OnIe  
                       prod(2:s(k))/                 ... >Rx8 0  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... \" 5F;J  
                       prod(2:((n(j)+m(j))/2-s(k))); \xa36~hh40  
            idx = (pows(k)==rpowers); 1o"y%*"  
            z(:,j) = z(:,j) + p*rpowern(:,idx); GN}9$:  
        end q[Sp|C6x  
         PaU@T!v  
        if isnorm Q&:92f\y  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); ORFr7a'K  
        end Q_UCF'f;}  
    end uL qpbn  
    2- |j  
    % 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
    851
    光币
    831
    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)  O5n] 4)<  
    aNf3 R;*  
    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。 <:}AC{I  
    u2Z^iY  
    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)