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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 ?_ p3^kl  
    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! \GBv@  
     
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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 9x23## s  
    function z = zernfun(n,m,r,theta,nflag) yIA- +# r[  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. /Rf:Z.L  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N Y'mtMLfMc  
    %   and angular frequency M, evaluated at positions (R,THETA) on the :tdN#m6&  
    %   unit circle.  N is a vector of positive integers (including 0), and xN'$ Yh  
    %   M is a vector with the same number of elements as N.  Each element U]ynnw4  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) jH({Qc,97  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, I w~R@,  
    %   and THETA is a vector of angles.  R and THETA must have the same Xq@Bzya  
    %   length.  The output Z is a matrix with one column for every (N,M) Kejp7 okb  
    %   pair, and one row for every (R,THETA) pair. "A6m-xE~  
    % +Hgil  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike of659~EIW  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), |6v $!wBi  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral F2QFQX(j  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, x+EkL3{  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized u%!/-&?wF  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. L7;8:^  v  
    % :m]H?vq] \  
    %   The Zernike functions are an orthogonal basis on the unit circle. aS=-9P;v  
    %   They are used in disciplines such as astronomy, optics, and [MhKR }a  
    %   optometry to describe functions on a circular domain. \| &KD  
    % Ra) wlI x  
    %   The following table lists the first 15 Zernike functions. ^m~&2l\N=  
    % t-B5,,`  
    %       n    m    Zernike function           Normalization %D1 |0v8}  
    %       -------------------------------------------------- 70Jx[3vr  
    %       0    0    1                                 1 :e /*5ix  
    %       1    1    r * cos(theta)                    2 fG9 ;7KG  
    %       1   -1    r * sin(theta)                    2 `Y O(C<r-  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) 0xVw{k}1U  
    %       2    0    (2*r^2 - 1)                    sqrt(3) =gNPS 0H  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) FJ,"a%m/Q  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) s|IY t^  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) *IX<&u#  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) _NefzZWUJ  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) !6!Gx:  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) )G#mC0?PV  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) =' uePM")  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) *:bexDH  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) bd]9 kRq1K  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) 0vX4v)-^u  
    %       -------------------------------------------------- JTIt!E}P  
    % ;/:Sx/#s  
    %   Example 1: 3P@D!lV&K  
    % &S,_Z/BS;  
    %       % Display the Zernike function Z(n=5,m=1) *4/FN TC  
    %       x = -1:0.01:1; >)F "lR:o  
    %       [X,Y] = meshgrid(x,x);  J3`0i@  
    %       [theta,r] = cart2pol(X,Y); vd?Bk_d9k,  
    %       idx = r<=1; ?4A/?Z]ub  
    %       z = nan(size(X)); (\0 <|pW  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); w4(L@1  
    %       figure *Nm$b+  
    %       pcolor(x,x,z), shading interp U0gZf5;*  
    %       axis square, colorbar a`L:E'|B9  
    %       title('Zernike function Z_5^1(r,\theta)') _%q~K (::  
    % Q$uv \h;  
    %   Example 2: j$K*R."  
    %  />Q}0H g  
    %       % Display the first 10 Zernike functions z/u^  
    %       x = -1:0.01:1; ],_+J *  
    %       [X,Y] = meshgrid(x,x);  0j_kK  
    %       [theta,r] = cart2pol(X,Y); P q$0ih  
    %       idx = r<=1; dgL>7X=7  
    %       z = nan(size(X)); 9w$m\nV  
    %       n = [0  1  1  2  2  2  3  3  3  3]; I)tiXcJw  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; @/F61Ut  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; |>yWkq   
    %       y = zernfun(n,m,r(idx),theta(idx)); 9.8%Iw  
    %       figure('Units','normalized') V"m S$MN  
    %       for k = 1:10 U.KQjBi  
    %           z(idx) = y(:,k); MjU|XQS:  
    %           subplot(4,7,Nplot(k)) B*N1)J\5  
    %           pcolor(x,x,z), shading interp jMgXIK\  
    %           set(gca,'XTick',[],'YTick',[]) Hs*["zFc  
    %           axis square ,Cb3R|L8  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) #8|LPfA  
    %       end ?u|@,tQ[  
    % ;xZjt4M1  
    %   See also ZERNPOL, ZERNFUN2. '`3#FCg  
    )rq |t9kix  
    %   Paul Fricker 11/13/2006 C,An\lsT  
    yEq7ueJ'  
    T9C_=0(hn  
    % Check and prepare the inputs: )V\@N*L`ik  
    % ----------------------------- 7 !$[XD  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) h:nybLw?  
        error('zernfun:NMvectors','N and M must be vectors.') X/yq<_ g  
    end _p^ "l2%D/  
    N ~{N Nf Y  
    if length(n)~=length(m) @eJCr)#}  
        error('zernfun:NMlength','N and M must be the same length.') P.}d@qD{)  
    end hbJ>GSoZ,  
    ` y\)X C7  
    n = n(:); O\6U2b~  
    m = m(:); >#w;67he2  
    if any(mod(n-m,2)) !R=@Nr>  
        error('zernfun:NMmultiplesof2', ... {drc}BL_  
              'All N and M must differ by multiples of 2 (including 0).') Ho>Np&  
    end (k?H T'3)  
    );$99t  
    if any(m>n) t:2v`uk  
        error('zernfun:MlessthanN', ... 2yZr!Rb~*  
              'Each M must be less than or equal to its corresponding N.') E5w;75,  
    end iQ;p59wSzL  
    ,~1"50 Hp@  
    if any( r>1 | r<0 ) CIjc5^Y2  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') f8Iddm#  
    end w G%W{T$  
    xG9Sk  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) i"WYcF |  
        error('zernfun:RTHvector','R and THETA must be vectors.') k, HC"?K  
    end {FNkPX  
    ']r8q %  
    r = r(:); p;O%W@n"  
    theta = theta(:); |A%9c.DG.  
    length_r = length(r); 9;E=w+  
    if length_r~=length(theta) " 8xAe0-4  
        error('zernfun:RTHlength', ... i[o 2(d,  
              'The number of R- and THETA-values must be equal.') .T| }rB<c  
    end (N7 uaZ?Z  
    |eqBCZn  
    % Check normalization: *m~-8_ >;  
    % --------------------  c0oHE8@  
    if nargin==5 && ischar(nflag) *doNPp)m  
        isnorm = strcmpi(nflag,'norm'); ={qcDgn~C  
        if ~isnorm YmziHns`b  
            error('zernfun:normalization','Unrecognized normalization flag.') CKYg!\g(:  
        end rtV`Q[E  
    else P {TJ$  
        isnorm = false;  =<HDek  
    end .ZpOYhk  
    K^Awf6%  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% M:S-%aQ_<y  
    % Compute the Zernike Polynomials CU'JvVe3  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -V2\s  
    #BC"bY  
    % Determine the required powers of r: [#PE'i4  
    % ----------------------------------- `o[l%I\Q  
    m_abs = abs(m); 0 j.K?]f)h  
    rpowers = []; ~}Xus?e  
    for j = 1:length(n)  {>]\<  
        rpowers = [rpowers m_abs(j):2:n(j)]; ]A*}Dem*5  
    end '7Gv_G_  
    rpowers = unique(rpowers); qJhsMo2IH  
    t" .Ytz>  
    % Pre-compute the values of r raised to the required powers, YW7W6mWspS  
    % and compile them in a matrix: #z\ub5um  
    % ----------------------------- dzf2`@8#  
    if rpowers(1)==0 B,%Vy!o  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); "-J 5!y*,Y  
        rpowern = cat(2,rpowern{:}); RB5SK#z  
        rpowern = [ones(length_r,1) rpowern]; KZm&sk=QM-  
    else d#k(>+%=Q  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); *{g3ia  
        rpowern = cat(2,rpowern{:}); YR%iZ"`*+O  
    end +iVEA(0&$  
    p3Sh%=HE'  
    % Compute the values of the polynomials: :E:e ^$p  
    % -------------------------------------- I6>J.6luF9  
    y = zeros(length_r,length(n)); 8y;Rw#Dz  
    for j = 1:length(n) JK k0f9)  
        s = 0:(n(j)-m_abs(j))/2; 7]ieBUf S  
        pows = n(j):-2:m_abs(j); o[|[xuTm  
        for k = length(s):-1:1 nbi7r cT  
            p = (1-2*mod(s(k),2))* ... /%wS5IZ^  
                       prod(2:(n(j)-s(k)))/              ... Cf {F"o  
                       prod(2:s(k))/                     ... +v Bi7#&  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 5/meH[R\M  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); ]%Q!%uTh  
            idx = (pows(k)==rpowers); vQAFgG  
            y(:,j) = y(:,j) + p*rpowern(:,idx); ^h(wi`i  
        end R.~[$G!  
         ~+q1g[6  
        if isnorm  bGRt  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); i?00!t  
        end dP5x]'"x  
    end F3tps jQ  
    % END: Compute the Zernike Polynomials *@U{[J  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +H)'(<  
    5>k:PKHL  
    % Compute the Zernike functions:  Z>[7#;;  
    % ------------------------------ vOQ% f?%G\  
    idx_pos = m>0; 80xr zv  
    idx_neg = m<0; \2SbW7"/;P  
    Hbm 4oYN  
    z = y; %fS9F^AK  
    if any(idx_pos) \}jMC  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); lj4Fg*/Yn  
    end h$cm:uks  
    if any(idx_neg) ua\t5M5  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); d,<ni"  
    end %,>z`D,Hg  
    P4zo[R%4  
    % EOF zernfun
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) ;>6< u.N  
    %ZERNFUN2 Single-index Zernike functions on the unit circle.  q+P@2FL  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated f/Gx}x=  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive  Rr) 5 [  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, MKK ^-T  
    %   and THETA is a vector of angles.  R and THETA must have the same 1? >P3C  
    %   length.  The output Z is a matrix with one column for every P-value, @gUp9ZwtH  
    %   and one row for every (R,THETA) pair. 'Zx5+rM${}  
    % `Sod]bO +U  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike t],a1I.gk  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) FD=% 4#|  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) !MbzFs~  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 Qg>NJ\*Q  
    %   for all p. d;i|s[6ds`  
    % UG| /Px ]  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 h my%X`%j  
    %   Zernike functions (order N<=7).  In some disciplines it is $8EEtr,!  
    %   traditional to label the first 36 functions using a single mode 3m1g"  
    %   number P instead of separate numbers for the order N and azimuthal r(,U{bU<  
    %   frequency M. kVn RSg}R  
    % wj[yo S  
    %   Example: :X2_#qW#C  
    % 2& Q\W  
    %       % Display the first 16 Zernike functions rPxRGoR  
    %       x = -1:0.01:1; `/| *u  
    %       [X,Y] = meshgrid(x,x); >XN[KPTa  
    %       [theta,r] = cart2pol(X,Y); "N4^ ^~s  
    %       idx = r<=1; yOM/UdWq  
    %       p = 0:15; yD[d%w  
    %       z = nan(size(X)); y\Wn:RR1[  
    %       y = zernfun2(p,r(idx),theta(idx)); b,!C8rJ  
    %       figure('Units','normalized') VQ=  
    %       for k = 1:length(p) 5Cf!NNV  
    %           z(idx) = y(:,k); sz7*x{E  
    %           subplot(4,4,k) CEfqFn3^  
    %           pcolor(x,x,z), shading interp UmKE]1Yw4r  
    %           set(gca,'XTick',[],'YTick',[]) L!f~Am:#  
    %           axis square MT6p@b5  
    %           title(['Z_{' num2str(p(k)) '}']) g)Z8WH$;H3  
    %       end 2=cx`"a$  
    % W'G|sk  
    %   See also ZERNPOL, ZERNFUN. 8}%F`=Y0  
    !z?   
    %   Paul Fricker 11/13/2006 RB>=#03  
    )Q2Ap&  
    Bwg(f_[1  
    % Check and prepare the inputs: U32$ 9"  
    % ----------------------------- q~`hn(S  
    if min(size(p))~=1 VFE@qX|  
        error('zernfun2:Pvector','Input P must be vector.') .ARYCTyG  
    end bW yimr&B  
    "O$bq::(]e  
    if any(p)>35 [8ZDMe  
        error('zernfun2:P36', ... q` S ~w  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... hY}Q|-|  
               '(P = 0 to 35).']) EKc<|e,F  
    end '1u?-2  
    aIgexi,  
    % Get the order and frequency corresonding to the function number: }i9:k kfq2  
    % ---------------------------------------------------------------- N2:Hdu :  
    p = p(:); y_PA9#v7  
    n = ceil((-3+sqrt(9+8*p))/2); cXXZ'y>FP  
    m = 2*p - n.*(n+2); G1|1Z5r  
    ?XKX&ws  
    % Pass the inputs to the function ZERNFUN: +!).'  
    % ---------------------------------------- A}fm).Wp@  
    switch nargin SQMl5d1d:  
        case 3 py6<QoGV  
            z = zernfun(n,m,r,theta); Z% +$<J  
        case 4 eP~bl   
            z = zernfun(n,m,r,theta,nflag); Xj, %t}  
        otherwise zC50 @S3|  
            error('zernfun2:nargin','Incorrect number of inputs.') @@R Mm$  
    end ?K$&|w%{3  
    iXWzIb}CJ-  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) hpf0fU  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. y&(#C:N  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of #7;?Ls  
    %   order N and frequency M, evaluated at R.  N is a vector of AojL4H|  
    %   positive integers (including 0), and M is a vector with the )at:Xm<s  
    %   same number of elements as N.  Each element k of M must be a \.2i?<BC  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) i]n2\v AG  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is re*Zs}(N\  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix <zCWLj3  
    %   with one column for every (N,M) pair, and one row for every GR|\OJ<2  
    %   element in R. B/X$ZQ0  
    % $SQ$2\iC  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- R;HE{q[ f  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is  Z 9:  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to 3cHYe  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 <M9NyD`  
    %   for all [n,m]. 4eWv).  
    % J0V m&TY  
    %   The radial Zernike polynomials are the radial portion of the 3JC uM_y  
    %   Zernike functions, which are an orthogonal basis on the unit F'MX9P  
    %   circle.  The series representation of the radial Zernike zgY VB}  
    %   polynomials is rC@VMe|0  
    % =%8 yEb*5#  
    %          (n-m)/2 0SvPr [ >  
    %            __ }etdXO_^  
    %    m      \       s                                          n-2s ?Uq"zq  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r OUWK  
    %    n      s=0 89>}`:xS^  
    % Tdh(J",d  
    %   The following table shows the first 12 polynomials. RP$u/x"b  
    % yF\yxdUX#  
    %       n    m    Zernike polynomial    Normalization \me5"ZU  
    %       --------------------------------------------- 7:B/ ?E  
    %       0    0    1                        sqrt(2) )W=O~g  
    %       1    1    r                           2 OPN\{<`*d  
    %       2    0    2*r^2 - 1                sqrt(6) r10VFaly  
    %       2    2    r^2                      sqrt(6) o2dO\$'  
    %       3    1    3*r^3 - 2*r              sqrt(8) "BsK' yo.  
    %       3    3    r^3                      sqrt(8) =?$~=1SL+  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) dQT[pNp:  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) ?98!2:'{9  
    %       4    4    r^4                      sqrt(10) [.4{s  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) (zFqb,P  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) s,r|p@^  
    %       5    5    r^5                      sqrt(12) c\n_[r  
    %       --------------------------------------------- hDi~{rbmc  
    % /a*){JQ5j  
    %   Example: ,c"J[$i$  
    % T9R# .y,  
    %       % Display three example Zernike radial polynomials H.ZF~Yu w  
    %       r = 0:0.01:1;  @_f^AQ  
    %       n = [3 2 5]; EMP|I^  
    %       m = [1 2 1]; |&"aZ!Kn  
    %       z = zernpol(n,m,r); 7d R?70Sz  
    %       figure P@PF" {S  
    %       plot(r,z) O:#YLmbCN  
    %       grid on |K_%]1*riC  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') i{m!v6j:  
    % |kK5:\H  
    %   See also ZERNFUN, ZERNFUN2. sJKr%2nVV  
    "a].v 8l!  
    % A note on the algorithm. tx7 zG.,  
    % ------------------------ M?YNK]   
    % The radial Zernike polynomials are computed using the series @\nQ{\^;  
    % representation shown in the Help section above. For many special ?PWg  
    % functions, direct evaluation using the series representation can ;@=3 @v  
    % produce poor numerical results (floating point errors), because h,FU5iK|  
    % the summation often involves computing small differences between zc8^#D2y&  
    % large successive terms in the series. (In such cases, the functions el`?:dY H  
    % are often evaluated using alternative methods such as recurrence 0 aH&M4  
    % relations: see the Legendre functions, for example). For the Zernike 2!0tD+B  
    % polynomials, however, this problem does not arise, because the Yw#fQFm  
    % polynomials are evaluated over the finite domain r = (0,1), and rX)&U4#[m  
    % because the coefficients for a given polynomial are generally all 0?$|F0U"J  
    % of similar magnitude. zoi0Z  
    % Hk;;+'-  
    % ZERNPOL has been written using a vectorized implementation: multiple 4/~x+tdc  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] ?|kbIZP(  
    % values can be passed as inputs) for a vector of points R.  To achieve 1iY4|j;ahV  
    % this vectorization most efficiently, the algorithm in ZERNPOL Soq#cl'll-  
    % involves pre-determining all the powers p of R that are required to t3<8n;'y:  
    % compute the outputs, and then compiling the {R^p} into a single ~(v5p"]dj  
    % matrix.  This avoids any redundant computation of the R^p, and %JrZMs>  
    % minimizes the sizes of certain intermediate variables. hy~[7:/<I&  
    % ~2\Sn-`  
    %   Paul Fricker 11/13/2006 EA(4xj&:U  
    ["f6Ern  
    MoN0w.V  
    % Check and prepare the inputs: Wz.iDRFl  
    % ----------------------------- }O7sP^  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) {,JO}Dmu5  
        error('zernpol:NMvectors','N and M must be vectors.') QP.Lq }  
    end rlR!Tc>  
    (9RfsV4^  
    if length(n)~=length(m) ]?+i6 [6U  
        error('zernpol:NMlength','N and M must be the same length.') MrB#=3pT  
    end HhQ0>  
    ;+XrCy!.)L  
    n = n(:); `2]0 X#R  
    m = m(:); >I\B_q  
    length_n = length(n); Ez~5ax7x  
    2, )>F"R  
    if any(mod(n-m,2)) m|W17LhW{  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') V3ozaVk;  
    end '>t&fzD0  
    &PE%tm  
    if any(m<0) K7`6G[RMb  
        error('zernpol:Mpositive','All M must be positive.') F8Ety^9>9  
    end d~qQ_2M[G  
    F:q4cfL6  
    if any(m>n) sR1_L/.  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') ]uox ^HC  
    end >fWGiFmlk  
    '27$x&6>S  
    if any( r>1 | r<0 ) _Z]l=5d  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') AMjr[!44 @  
    end ^'E^*R  
    ,5v'hG  
    if ~any(size(r)==1) 86)2\uan  
        error('zernpol:Rvector','R must be a vector.') c+3`hVV  
    end P6.PjK!Ar  
    J-tqEK*  
    r = r(:); 'Wnh1|z  
    length_r = length(r); nSyLt6zn\  
    ~Pw9[ycn3  
    if nargin==4 =F$?`q`  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); eZOR{|z  
        if ~isnorm 4& cQW)  
            error('zernpol:normalization','Unrecognized normalization flag.') [tk x84M8  
        end x3cjyu<K  
    else ~'lT8 n_  
        isnorm = false; syB pF:`-W  
    end C33Jzn's  
    F2}Fuupb.  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ]]K?Q )9x  
    % Compute the Zernike Polynomials fX`u"`o5  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% J[:#(c&c!1  
    XiTi3vCe  
    % Determine the required powers of r: zN!W_2W*  
    % ----------------------------------- Hi={(Z5tC4  
    rpowers = []; LHA^uuBN}  
    for j = 1:length(n) d.+  
        rpowers = [rpowers m(j):2:n(j)]; 8c.>6 Hy  
    end yS~Y"#F!.  
    rpowers = unique(rpowers); `f}s<At  
    bK%F_v3'  
    % Pre-compute the values of r raised to the required powers, dh`s^D6Q>  
    % and compile them in a matrix: w>j5oz}  
    % ----------------------------- ~|Vq v{  
    if rpowers(1)==0 <&b,%O  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); [T r7SU#x  
        rpowern = cat(2,rpowern{:}); b\ED<'  
        rpowern = [ones(length_r,1) rpowern]; _MC',p&  
    else y[$UeE"0  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); }&=l)\e  
        rpowern = cat(2,rpowern{:}); *d C|X  
    end ^$P_B-C N  
    Ld*Ds!*'/  
    % Compute the values of the polynomials: =hTJp/L  
    % -------------------------------------- a?+C]u?_D  
    z = zeros(length_r,length_n); I[&x-}w  
    for j = 1:length_n M _< |n  
        s = 0:(n(j)-m(j))/2; P 2_!(FZ<l  
        pows = n(j):-2:m(j); [8za=B/  
        for k = length(s):-1:1 |_p7vl"  
            p = (1-2*mod(s(k),2))* ... !O"2)RU1  
                       prod(2:(n(j)-s(k)))/          ... L?nhm=D  
                       prod(2:s(k))/                 ... YaS!YrpI  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... WDghlC6g!l  
                       prod(2:((n(j)+m(j))/2-s(k))); {2q"9Ox"  
            idx = (pows(k)==rpowers); ?VotIruR  
            z(:,j) = z(:,j) + p*rpowern(:,idx); $O\m~r4  
        end Zuzwc[Z1  
         u_WUJ_  
        if isnorm J'WzEgCnU  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); EwzcB\m  
        end i}8OaX3x  
    end R-zS7Jyox  
    h!dij^bD  
    % 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)  4%yeEc ;z  
    /55 3v;l<  
    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。 YuO!Y9iEm  
    Z~w?Qm:/  
    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)