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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 #BH]`A J  
    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! ajkpU.6E:  
     
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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 :#N]s  
    function z = zernfun(n,m,r,theta,nflag) #.,LWL]  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. #B_H/9f(  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N mK^E@uxN  
    %   and angular frequency M, evaluated at positions (R,THETA) on the }%y5<n*v\  
    %   unit circle.  N is a vector of positive integers (including 0), and {t]8#[lo  
    %   M is a vector with the same number of elements as N.  Each element ?+{_x^  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) dtV7YPz4+  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, _vAc/_ N  
    %   and THETA is a vector of angles.  R and THETA must have the same (H]NL   
    %   length.  The output Z is a matrix with one column for every (N,M) .`&k`  
    %   pair, and one row for every (R,THETA) pair. T*(mi{[T  
    % 4P7r\ hs  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike cF"}}c1*M  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), I%z,s{9p  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral Z:,`hW*A6  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, (7??5gjh  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized 8>I4e5Ym  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ^i@0P}K<  
    % , $cpm=1  
    %   The Zernike functions are an orthogonal basis on the unit circle. D'UIxc8  
    %   They are used in disciplines such as astronomy, optics, and _]0<G8|Rv  
    %   optometry to describe functions on a circular domain. F$YT4414  
    % A":cS }Ui  
    %   The following table lists the first 15 Zernike functions. <(45(6fQ  
    % >``  
    %       n    m    Zernike function           Normalization #aE>-81SS&  
    %       -------------------------------------------------- fM(~>(q&  
    %       0    0    1                                 1 p$Floubh]  
    %       1    1    r * cos(theta)                    2 CX]L'  
    %       1   -1    r * sin(theta)                    2 ' 'p<C)Q  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) .kfx\,lgm  
    %       2    0    (2*r^2 - 1)                    sqrt(3) ; 2aPhA  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) wf^p?=Ke  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) !R[~Z7b6  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) Vf$$e)  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) qtz~Y~h|>  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) #w!ewCvt  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) K}Q:L(SSr\  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) TALiH'w6|e  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) }E&:  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) bUuQ"!>ppu  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) NjO_Y t  
    %       -------------------------------------------------- 8RcLs1n/  
    % @E"lN  
    %   Example 1: K[Vj+qdyl  
    % ZT<VDcP{  
    %       % Display the Zernike function Z(n=5,m=1)  1%";|  
    %       x = -1:0.01:1; nJwP|P_  
    %       [X,Y] = meshgrid(x,x); G4\|bwh  
    %       [theta,r] = cart2pol(X,Y); 5>VX]nE3!  
    %       idx = r<=1; {r#uD5NJ/  
    %       z = nan(size(X)); JOwu_%  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); D8WKy  
    %       figure qu;$I'Ul%  
    %       pcolor(x,x,z), shading interp [|\#cVWs  
    %       axis square, colorbar x+[ATZ([  
    %       title('Zernike function Z_5^1(r,\theta)') >Udq{<]#r  
    % {"|la;*I  
    %   Example 2: m;ju@5X  
    % $s"-r9@q  
    %       % Display the first 10 Zernike functions m\MI 6/  
    %       x = -1:0.01:1; #@E:|^$1y  
    %       [X,Y] = meshgrid(x,x); ^-"tK:{  
    %       [theta,r] = cart2pol(X,Y); SErh"~[  
    %       idx = r<=1; ~ ^fb`f+%  
    %       z = nan(size(X)); I]WvcDJ}C  
    %       n = [0  1  1  2  2  2  3  3  3  3]; UQbk%K2  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; .S]*A b  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; hd`jf97*  
    %       y = zernfun(n,m,r(idx),theta(idx)); j rX`_Y  
    %       figure('Units','normalized') #JN4K>_4  
    %       for k = 1:10 /bLL!nD=^  
    %           z(idx) = y(:,k); 0#~k)>(7lR  
    %           subplot(4,7,Nplot(k)) Z tc\4  
    %           pcolor(x,x,z), shading interp f6{.Uq%SGp  
    %           set(gca,'XTick',[],'YTick',[]) #];ulDq  
    %           axis square {Ia$!q)  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) Zu94dFP  
    %       end $"?$r  
    % _`,ZI{.J^  
    %   See also ZERNPOL, ZERNFUN2. .eyJ<b9  
    [I7=]X  
    %   Paul Fricker 11/13/2006 . "7-f]!  
    Qkc 9X0J!  
    >'jkL5l  
    % Check and prepare the inputs: e{^^u$C1.e  
    % ----------------------------- -vc ,O77z"  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) e7plL^^`  
        error('zernfun:NMvectors','N and M must be vectors.') FRXaPod  
    end w}e_ 17A  
    86a,J3C[  
    if length(n)~=length(m) { _Y'%Ggh  
        error('zernfun:NMlength','N and M must be the same length.') q(Ow:3&  
    end qq@]xdl  
    &>G8DvfJ9  
    n = n(:); 9_ ~9?5PU  
    m = m(:); N0N%~3  
    if any(mod(n-m,2)) qx*N-,M%k(  
        error('zernfun:NMmultiplesof2', ... 9Q\RCl_1  
              'All N and M must differ by multiples of 2 (including 0).') 8~g~XUl  
    end U~dqxR"Q  
    FtlJ3fB@  
    if any(m>n) A+FQmLS  
        error('zernfun:MlessthanN', ... B9H.8+~(  
              'Each M must be less than or equal to its corresponding N.') mP?}h  
    end 9#kk5)J  
    SL +\{V2  
    if any( r>1 | r<0 ) }g:'K  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') 7p>T6jK)  
    end MM( ,D& Z  
    D[4%CQ1m  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) yV31OBC:  
        error('zernfun:RTHvector','R and THETA must be vectors.') E )2/Vn2  
    end Q5_,`r`  
    0wAB;|~*62  
    r = r(:); u`Kc\B Sn  
    theta = theta(:); S"`{ JCW$  
    length_r = length(r); ~RZN+N  
    if length_r~=length(theta) bL{D*\HF  
        error('zernfun:RTHlength', ... Ds{bYK_y  
              'The number of R- and THETA-values must be equal.') <vu~EY0.  
    end p4kK" \ln  
    3Q2NiYg3  
    % Check normalization: n8D'fvY  
    % -------------------- i+lq:St  
    if nargin==5 && ischar(nflag) uLNOhgSUf  
        isnorm = strcmpi(nflag,'norm'); k0TQFx.A  
        if ~isnorm )Lk2tvr  
            error('zernfun:normalization','Unrecognized normalization flag.') ,mz7!c9H^a  
        end #Yy5@A}`o  
    else eKU4"XTk  
        isnorm = false; Ec}9R3 m  
    end ?9?o8!  
    Ok}e|b[D  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% n7zM;@{7  
    % Compute the Zernike Polynomials "chf \ -!$  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% gV*4{ d`  
    i.7$~}  
    % Determine the required powers of r: L:Faq1MG  
    % ----------------------------------- $}EARW9  
    m_abs = abs(m); "cbJ{ G1pk  
    rpowers = []; !"aGo1 $$  
    for j = 1:length(n) )]Sf|@K]  
        rpowers = [rpowers m_abs(j):2:n(j)]; T~4HeEG>uH  
    end K)h<#F  
    rpowers = unique(rpowers); nFro#qx  
    {7v|\6@e3  
    % Pre-compute the values of r raised to the required powers, Z+4Mo*#  
    % and compile them in a matrix: ZvK3Su)f1  
    % ----------------------------- ?*<1B  
    if rpowers(1)==0 u/N_62sk5  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false);  U8% IpI;  
        rpowern = cat(2,rpowern{:}); VRHS 4  
        rpowern = [ones(length_r,1) rpowern]; &?']EcU5h9  
    else {yi!vw  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); >z,Y%A  
        rpowern = cat(2,rpowern{:}); Upm#:i|"  
    end `[ZA#8Ma  
    z_8Bl2tl  
    % Compute the values of the polynomials: 'uwq^b_  
    % -------------------------------------- b'xBPTN  
    y = zeros(length_r,length(n)); v~p?YYOm<  
    for j = 1:length(n) R80|q#h,]  
        s = 0:(n(j)-m_abs(j))/2; TBHd)BhI.  
        pows = n(j):-2:m_abs(j); @#9xSs#  
        for k = length(s):-1:1 ~u?rjkSFoh  
            p = (1-2*mod(s(k),2))* ... AAF;M}le,  
                       prod(2:(n(j)-s(k)))/              ... z,VXH ?.Zo  
                       prod(2:s(k))/                     ... YG>Eop  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... IEfm>N-]  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); Ysi@wK-LnF  
            idx = (pows(k)==rpowers); dO-Zj#%7z8  
            y(:,j) = y(:,j) + p*rpowern(:,idx); c3\p@}  
        end 6O@Lx ]t  
         8"u.GL.  
        if isnorm 4dh> B>Q  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); {4%ddJn[.)  
        end "{jVsih0  
    end Af^9WJ  
    % END: Compute the Zernike Polynomials D9n+eZ  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% B\`${O(  
    u R!'v  
    % Compute the Zernike functions: ZV07;`I  
    % ------------------------------ Zh?n;n}  
    idx_pos = m>0; YT@H^=  
    idx_neg = m<0; C{6m?6  
    t V7{j'If  
    z = y; Pfm B{  
    if any(idx_pos) \ow(4O#  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 4XeO^#  
    end E/E|*6R  
    if any(idx_neg) Wx8;+!2Q/  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Z,F1n/7  
    end J!'IkC$>  
    X0KUnxw  
    % EOF zernfun
    离线niuhelen
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) y!\q ', F  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. +ZRm1q   
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated g;y*F;0@  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive M!\6Fl{ b  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, JOki4N  
    %   and THETA is a vector of angles.  R and THETA must have the same QmsS,Zljo  
    %   length.  The output Z is a matrix with one column for every P-value, 'gk^NAG2^E  
    %   and one row for every (R,THETA) pair. az Oib=3fz  
    % 4{" v  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike o^BX:\}  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) PC)V".W 1  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) 3d_g@x#9  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 ab<7jfFIa  
    %   for all p. lq27^K  
    % @Lm(bW  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 ?/KkN3Y_j[  
    %   Zernike functions (order N<=7).  In some disciplines it is JZD&u6tB   
    %   traditional to label the first 36 functions using a single mode .r-kH&)"GU  
    %   number P instead of separate numbers for the order N and azimuthal A2B]E,JMp  
    %   frequency M. w)gMJX/0yw  
    % E^m2:J]G  
    %   Example: cLMFC1=b  
    % ;B"S*wYMN  
    %       % Display the first 16 Zernike functions N3Z6o.k  
    %       x = -1:0.01:1; AT I2  
    %       [X,Y] = meshgrid(x,x); $`/F5R!  
    %       [theta,r] = cart2pol(X,Y); _[J>GfQd  
    %       idx = r<=1; SvD:UG  
    %       p = 0:15; "=9)|{=m  
    %       z = nan(size(X)); }4xz,oN  
    %       y = zernfun2(p,r(idx),theta(idx)); Dn;$4Dak(  
    %       figure('Units','normalized') Oxh . &  
    %       for k = 1:length(p) 2iWxx:e  
    %           z(idx) = y(:,k); K.6xNQl{}  
    %           subplot(4,4,k) "ODs.m oq  
    %           pcolor(x,x,z), shading interp W; ?'  
    %           set(gca,'XTick',[],'YTick',[]) })ic@ Mmd$  
    %           axis square g U v`G  
    %           title(['Z_{' num2str(p(k)) '}']) )<%IY&\  
    %       end 7p,!<X}%  
    % `,FvYA"  
    %   See also ZERNPOL, ZERNFUN. rh(77x1|(G  
    {hKf 'd9E  
    %   Paul Fricker 11/13/2006 \H.1I=<  
    i>@"&  
    ^g n7DiIPH  
    % Check and prepare the inputs: 'FGf#l<  
    % ----------------------------- $:0?"?o);  
    if min(size(p))~=1 }m-+EUEo9  
        error('zernfun2:Pvector','Input P must be vector.') VXu1Y xY  
    end v iM6q<Ht  
    iYzm<3n?  
    if any(p)>35 3 e<sNU?  
        error('zernfun2:P36', ... tje   
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... >*[Bq;  
               '(P = 0 to 35).']) Kj*:G!r0.:  
    end Twr<MXa  
    HVcd< :g0  
    % Get the order and frequency corresonding to the function number: z T#j.v  
    % ---------------------------------------------------------------- LXcH<)  
    p = p(:); Fu#mMn0c  
    n = ceil((-3+sqrt(9+8*p))/2); /g]m,Y{OI  
    m = 2*p - n.*(n+2); ZG)%vB2c  
    a`uHkRX )U  
    % Pass the inputs to the function ZERNFUN: `$N AK  
    % ---------------------------------------- (8OaXif  
    switch nargin i.*Utm`1"e  
        case 3 <YBA 7i  
            z = zernfun(n,m,r,theta); JGKiVBN  
        case 4 ?W4IAbT\G  
            z = zernfun(n,m,r,theta,nflag); uE{nnNZy  
        otherwise Z,m;eCLG]  
            error('zernfun2:nargin','Incorrect number of inputs.') K~Hp%.  
    end tV,Y38e  
    Q[N6#C:(4  
    % 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) Lg[_9 `\  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. J 4EG  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of vo_m$/O  
    %   order N and frequency M, evaluated at R.  N is a vector of b:uMO N,H  
    %   positive integers (including 0), and M is a vector with the &jHnM^nQ  
    %   same number of elements as N.  Each element k of M must be a .oFkx*Ln  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) jP9)utEm6  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is H,H=y},  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix Z@I.socA  
    %   with one column for every (N,M) pair, and one row for every A<zSh }eh6  
    %   element in R. FF"`F8-w>Z  
    % `kd P)lI `  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- 8UyYN$7V  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is b{o%`B*  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to "`AIU}[_I  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 vUXas*s4  
    %   for all [n,m]. tOK lCc  
    % <ZV !fn  
    %   The radial Zernike polynomials are the radial portion of the ?=;dNS@i@  
    %   Zernike functions, which are an orthogonal basis on the unit _ ecKX</Q  
    %   circle.  The series representation of the radial Zernike v<z%\`y  
    %   polynomials is {-( B  
    % x xh(VQdg  
    %          (n-m)/2 _f8<t=R  
    %            __ *hp3w  
    %    m      \       s                                          n-2s N| dwuBW  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r lxpi   
    %    n      s=0 +8 avA:o  
    % > T,^n {_v  
    %   The following table shows the first 12 polynomials. jc|"wN]  
    % (@H'7,  
    %       n    m    Zernike polynomial    Normalization G:e 9}  
    %       --------------------------------------------- dM{xPpnx  
    %       0    0    1                        sqrt(2) 8uR4ZE*  
    %       1    1    r                           2 .OHjn|  
    %       2    0    2*r^2 - 1                sqrt(6) g pN{1  
    %       2    2    r^2                      sqrt(6) `)[bu  
    %       3    1    3*r^3 - 2*r              sqrt(8) mRT`'fxK  
    %       3    3    r^3                      sqrt(8) (0Xgv3wd  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) ! `yg bI.  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) ]R8}cbtU  
    %       4    4    r^4                      sqrt(10) !'()QtvC<  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) fCL5Et  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) 0?]*-wvp  
    %       5    5    r^5                      sqrt(12) BK>uJv-qU  
    %       ---------------------------------------------  2L~[dn.s  
    % ei>8{v&g  
    %   Example: LB|FVNW/S  
    % 5sD\4g)HK  
    %       % Display three example Zernike radial polynomials c[4  H  
    %       r = 0:0.01:1; k{3:$, b  
    %       n = [3 2 5]; jj,Y:  
    %       m = [1 2 1]; U[1Ir92:  
    %       z = zernpol(n,m,r); R<+K&_  
    %       figure C ebl"3Q  
    %       plot(r,z) y5L%_ {n  
    %       grid on <6=kwV6  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') ^d!(8vh  
    % `b^eRnpR  
    %   See also ZERNFUN, ZERNFUN2. X0Q};,  
    G'/36M@  
    % A note on the algorithm. ^w eU\  
    % ------------------------ 0^VA,QkQ\  
    % The radial Zernike polynomials are computed using the series jFBLElE  
    % representation shown in the Help section above. For many special ssv4#8p3  
    % functions, direct evaluation using the series representation can xeqAFq=9?  
    % produce poor numerical results (floating point errors), because S.bB.<  
    % the summation often involves computing small differences between MXWCYi  
    % large successive terms in the series. (In such cases, the functions 9|Cu2  
    % are often evaluated using alternative methods such as recurrence b$kCyOg  
    % relations: see the Legendre functions, for example). For the Zernike Tti]H9g_  
    % polynomials, however, this problem does not arise, because the IG?044Y  
    % polynomials are evaluated over the finite domain r = (0,1), and Re3vW re  
    % because the coefficients for a given polynomial are generally all v Dgf}  
    % of similar magnitude. LEoL6ga  
    % __\Tv>Y  
    % ZERNPOL has been written using a vectorized implementation: multiple P%hi*0pwZ  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] +@wa?"  
    % values can be passed as inputs) for a vector of points R.  To achieve ln#Jb&u  
    % this vectorization most efficiently, the algorithm in ZERNPOL _@[M0t}g_  
    % involves pre-determining all the powers p of R that are required to Z }(,OZh  
    % compute the outputs, and then compiling the {R^p} into a single d Ybb>rlu  
    % matrix.  This avoids any redundant computation of the R^p, and X8 uVet]D~  
    % minimizes the sizes of certain intermediate variables. ^NB @wuf7  
    % c0v;r4Jo#j  
    %   Paul Fricker 11/13/2006 JHpaDy*  
    &CCp@" +  
    EGYYSoBLU  
    % Check and prepare the inputs: :jB8Q$s  
    % ----------------------------- |tC`rzo  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) `<>Emc8Z  
        error('zernpol:NMvectors','N and M must be vectors.') ` V}e$  
    end `a}!t=~#w  
    l$$N~FN  
    if length(n)~=length(m) b&BSigrvou  
        error('zernpol:NMlength','N and M must be the same length.') f!;4 -.p`  
    end RkVU^N"  
    +Z0@z^6\  
    n = n(:); "V!y"yQ  
    m = m(:); [m7^Euury  
    length_n = length(n); f)x^s$H  
    ;W6P$@'zs  
    if any(mod(n-m,2)) 'ojI_%9<  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') 1df }gG  
    end :*V1jp+  
    t0XM#9L  
    if any(m<0) 2 fp\s5%J}  
        error('zernpol:Mpositive','All M must be positive.') @N?A 0S/  
    end =}txcA+  
    :Hdn&a i  
    if any(m>n) y>T:fu  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') %\8E{M:  
    end pj.}VF!d  
    Sns`/4S?6Z  
    if any( r>1 | r<0 ) ,"!t[4p=f  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') |,c\R"8xS  
    end HE0UcP1U  
    NLHF3h=?1p  
    if ~any(size(r)==1) @l~zn%!X  
        error('zernpol:Rvector','R must be a vector.') xh[De}@  
    end `~'yy q  
    5\Sm^t|Tx  
    r = r(:); J%c4-'l  
    length_r = length(r); t(FI Bf3  
    |T:' G  
    if nargin==4 t><AaYij_  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); /7Z;/|oU  
        if ~isnorm .JIn(  
            error('zernpol:normalization','Unrecognized normalization flag.') W|_^Oe<  
        end a>jiq8d]4  
    else DDh$n?2fd  
        isnorm = false; x}I'W?g  
    end =H&@9=D*  
    &Pu}"M$[MH  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% _JZS;8WYR  
    % Compute the Zernike Polynomials _Wn5* Pi%Z  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -}nTwx:|5u  
    GmoY~}cg~  
    % Determine the required powers of r: p3Uus''V4  
    % ----------------------------------- uXPvl5(Y?  
    rpowers = []; ,zmGKn#n2  
    for j = 1:length(n) m{itMZ@  
        rpowers = [rpowers m(j):2:n(j)]; T\\Q!pY  
    end ni$7)YcF  
    rpowers = unique(rpowers); ,&$w*D%  
    S'"(zc3 =  
    % Pre-compute the values of r raised to the required powers, L7i^?40  
    % and compile them in a matrix: g:bw;6^ u  
    % ----------------------------- _:|/4.]`_  
    if rpowers(1)==0 ^DZ(T+q,  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); /(-X[[V  
        rpowern = cat(2,rpowern{:}); {J-kcD!bz`  
        rpowern = [ones(length_r,1) rpowern]; lTOO`g  
    else ts rcX  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); FL -yt  
        rpowern = cat(2,rpowern{:}); rdd%"u+  
    end oW]~\vp^0  
    \yeo-uN8  
    % Compute the values of the polynomials: .<8kDyi m  
    % -------------------------------------- V6%J9+DK  
    z = zeros(length_r,length_n); m}Z=m8  
    for j = 1:length_n  A i`  
        s = 0:(n(j)-m(j))/2; bbevy!m  
        pows = n(j):-2:m(j); !Oj]. WQ  
        for k = length(s):-1:1 {%!.aQ,  
            p = (1-2*mod(s(k),2))* ... :p^7XwX%w  
                       prod(2:(n(j)-s(k)))/          ... Z~O1$,Z  
                       prod(2:s(k))/                 ... 7I>@PV N  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... C FqteY"  
                       prod(2:((n(j)+m(j))/2-s(k))); 9L+dN%C  
            idx = (pows(k)==rpowers); ]AjDe]  
            z(:,j) = z(:,j) + p*rpowern(:,idx); <0!/7*;#ZT  
        end k!Y7 Rc{"  
         C^>txui8  
        if isnorm UR;F W`  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); 8RC7 Ei  
        end ~4YU  
    end ~Xa8\>  
    I8=p_Ie  
    % 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)  pq<302uBQ  
    xp+Z%0D  
    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。  Fu@2gd  
    5u(B]_r.  
    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)