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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 ^lHy)!&A  
    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! ]>X_E%`G<b  
     
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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  L8`v  
    function z = zernfun(n,m,r,theta,nflag) QEr<(wM-y  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. 4}H+hk8-  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N kvwnqaX  
    %   and angular frequency M, evaluated at positions (R,THETA) on the #gC [L=01  
    %   unit circle.  N is a vector of positive integers (including 0), and J p?XV<3Z  
    %   M is a vector with the same number of elements as N.  Each element m4/qxm"Dx:  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) ,6>3aD1w~q  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, gC1LQ!:;Oi  
    %   and THETA is a vector of angles.  R and THETA must have the same -pC'C%Q  
    %   length.  The output Z is a matrix with one column for every (N,M) s 47R,K$  
    %   pair, and one row for every (R,THETA) pair. aC,adNub  
    % 'zYS:W  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike /QQRy_Z1)  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), d,^O[9UWo  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral WoTeIkM9  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, O(-p md,  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized a3yNd  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. B7f<XBU6>  
    %  vD#U+  
    %   The Zernike functions are an orthogonal basis on the unit circle. G0 )[(s  
    %   They are used in disciplines such as astronomy, optics, and a`' >VCg  
    %   optometry to describe functions on a circular domain. 1$0Kvvg[  
    % ce;7  
    %   The following table lists the first 15 Zernike functions. GQbr}xX. #  
    % F!X0Wo=  
    %       n    m    Zernike function           Normalization cr!8Tp;2A  
    %       -------------------------------------------------- NVMn7H}>  
    %       0    0    1                                 1 j.&dHtp  
    %       1    1    r * cos(theta)                    2 nqy*>X`  
    %       1   -1    r * sin(theta)                    2 Q4cCg7|0  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) 6&$.E! z  
    %       2    0    (2*r^2 - 1)                    sqrt(3) 7fR5V  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) @AZNF+ \W$  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) $)#orZtzr  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) $}&a*c>  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) uz!8=,DFw  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) lAN&d;NU6Z  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) @ [;'b$T$  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) LACrg  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) pbx*Y`v  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) +@rFbsyJ.  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) E*YmHJ:k  
    %       -------------------------------------------------- lq9|tt6Z  
    % _mqU:?Q5  
    %   Example 1: bY P8  
    % a}@b2Wc*  
    %       % Display the Zernike function Z(n=5,m=1) 4!/QB6  
    %       x = -1:0.01:1; p:xyy*I  
    %       [X,Y] = meshgrid(x,x); d_`MS@2  
    %       [theta,r] = cart2pol(X,Y); d ~ M;  
    %       idx = r<=1; )]fiyXA  
    %       z = nan(size(X)); *ak0(yLn)  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); ZD&F ,2v  
    %       figure RnH?95n?{  
    %       pcolor(x,x,z), shading interp L/u|90) L  
    %       axis square, colorbar d#T5=5 #  
    %       title('Zernike function Z_5^1(r,\theta)') No7-fX1B  
    %  R[m-jUL  
    %   Example 2: ? $/::uo  
    % 7rdmj[vu  
    %       % Display the first 10 Zernike functions %NkiYiA  
    %       x = -1:0.01:1; )xcjQkb  
    %       [X,Y] = meshgrid(x,x); ;T^s&/>E  
    %       [theta,r] = cart2pol(X,Y); h ;uzbu  
    %       idx = r<=1; 7]rIq\bM  
    %       z = nan(size(X)); hrKeOwKHU  
    %       n = [0  1  1  2  2  2  3  3  3  3]; Qf_N,Bq{a  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; lj]M 1zEz&  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; +t,b/K(?]  
    %       y = zernfun(n,m,r(idx),theta(idx)); `/WxEu3  
    %       figure('Units','normalized') yP]>eLTSd  
    %       for k = 1:10 :P-H8*n""  
    %           z(idx) = y(:,k); 1`?o#w  
    %           subplot(4,7,Nplot(k)) X4o#kW  
    %           pcolor(x,x,z), shading interp uf?;;wg  
    %           set(gca,'XTick',[],'YTick',[]) ^KbR@Ah  
    %           axis square ft/k-64  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ' wl})  
    %       end Q b^{`  
    % m@ <,bZkl  
    %   See also ZERNPOL, ZERNFUN2. f hK<P_}  
    2(d  
    %   Paul Fricker 11/13/2006 H {=]94  
    f_ MK4  
    L$Z!  
    % Check and prepare the inputs: JcRxNH )<"  
    % ----------------------------- ?J$k 5;  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) !Mw/j`*  
        error('zernfun:NMvectors','N and M must be vectors.') `n6cpX5  
    end /'8%=$2Kw  
    FqJd  
    if length(n)~=length(m) N]yT/8  
        error('zernfun:NMlength','N and M must be the same length.') %rB,Gl:)g  
    end -)%\$z  
    'R1C-U3w,  
    n = n(:); PoZ$3V$(Lz  
    m = m(:); $-DW+|p.?^  
    if any(mod(n-m,2)) Hva!6vwO%O  
        error('zernfun:NMmultiplesof2', ... ]+G\1SN~  
              'All N and M must differ by multiples of 2 (including 0).') .;31G0<w2  
    end Sy?^+JdM/  
    \ MuKS4  
    if any(m>n) 0qR#o/~I  
        error('zernfun:MlessthanN', ... +;!^aNJ,  
              'Each M must be less than or equal to its corresponding N.') +Q"s!\5  
    end 3B[tbU(  
    PU ea`rE?R  
    if any( r>1 | r<0 ) ;xq;c\N  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') atZNX1LD[/  
    end N]8/l:@  
    >mQD/U  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) D7R;IA-w  
        error('zernfun:RTHvector','R and THETA must be vectors.') rG-x 3>b  
    end =Q;dYx%I5  
    _0[s]  
    r = r(:); c&#Q`m  
    theta = theta(:); [ )X(Qtk  
    length_r = length(r); |@rf#,hTDp  
    if length_r~=length(theta) b7'A5]X  
        error('zernfun:RTHlength', ... [C ezz5  
              'The number of R- and THETA-values must be equal.') Cjt].XR@  
    end 3-y2i/4}$  
    *` -  
    % Check normalization: 5!i\S[:  
    % -------------------- v*y,PY1*  
    if nargin==5 && ischar(nflag) 1jK2*y  
        isnorm = strcmpi(nflag,'norm'); WYvcN8F  
        if ~isnorm yqb$,$  
            error('zernfun:normalization','Unrecognized normalization flag.') }!kvoV)]1  
        end GOCe&?  
    else J"eE9FLM  
        isnorm = false; m 2%  
    end 9-Qu5L~  
    06af{FXsGb  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 2j^8{Agz  
    % Compute the Zernike Polynomials skLr6Cs|  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -<\hcV`&  
    W1}d6Sbg  
    % Determine the required powers of r: of_Om$  
    % ----------------------------------- (K kqyrb  
    m_abs = abs(m); E zU=q E  
    rpowers = []; R4f_Kio  
    for j = 1:length(n) mj& 4FQ#O*  
        rpowers = [rpowers m_abs(j):2:n(j)]; Gd2t^tc  
    end x*_'uPo S  
    rpowers = unique(rpowers); (ap,3$ hS  
    I!p[:.t7  
    % Pre-compute the values of r raised to the required powers, y $>U[^G[  
    % and compile them in a matrix: r[&/* ~xL  
    % ----------------------------- sc# q03  
    if rpowers(1)==0 csv;u'  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ?Hf8<C}3  
        rpowern = cat(2,rpowern{:}); D14i]  
        rpowern = [ones(length_r,1) rpowern]; pTcN8E&Unz  
    else &Y8S! W@4  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 3B ;aoejHm  
        rpowern = cat(2,rpowern{:}); }ILg_>uq[  
    end Xa=oEG  
    p M_oIH'8:  
    % Compute the values of the polynomials: 8CGjI?j  
    % -------------------------------------- ":Ll. =!  
    y = zeros(length_r,length(n)); 05[k@f$n  
    for j = 1:length(n) {b]V e/\  
        s = 0:(n(j)-m_abs(j))/2; :J;*]o:  
        pows = n(j):-2:m_abs(j); A}(Q^|6  
        for k = length(s):-1:1 %b3s|o3An  
            p = (1-2*mod(s(k),2))* ... -@G,Ry-\t  
                       prod(2:(n(j)-s(k)))/              ... J4^aD;j  
                       prod(2:s(k))/                     ... U=\!`_f':  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ~BD 80s:f  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); DH{^9HK  
            idx = (pows(k)==rpowers); Yuqt=\? #  
            y(:,j) = y(:,j) + p*rpowern(:,idx); $]S*(K3U ~  
        end ` @Tl7I\  
         tU >?j1  
        if isnorm ~jn~M_}K  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); -jQM h  
        end pv;ZR  
    end *Bm _  
    % END: Compute the Zernike Polynomials 4n, >EA85  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% tc<ly{ 1c  
    0GP\*Y8  
    % Compute the Zernike functions: gj-MkeI)  
    % ------------------------------ uQ'Izdm  
    idx_pos = m>0; b1ma(8{{{  
    idx_neg = m<0; 4vbtB2  
    k86j& .m_  
    z = y; Z@{e\sZ)  
    if any(idx_pos) T?V!%AqY:  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); k9vzxZ%s:  
    end 78-D/WY/X  
    if any(idx_neg) ?kKr/f4N  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); @<,YUp,%S  
    end r\DA&b  
    yV/A%y-P  
    % 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) _,{R3k  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. ~![J~CkPS  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated 7;x}W-`iF  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive CPt62j8  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, 9~ JeI/  
    %   and THETA is a vector of angles.  R and THETA must have the same Kdr7JQYzuz  
    %   length.  The output Z is a matrix with one column for every P-value, FQW{c3%qZ  
    %   and one row for every (R,THETA) pair. M'-Z"  
    % W&`{3L  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike ?4A$9H  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) S?K x:]  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) 2SV}mK U  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 sm9/sX!  
    %   for all p. 7x5wT ?2W  
    % -gK*&n~  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 Ka+N5 T.f  
    %   Zernike functions (order N<=7).  In some disciplines it is Gw1Rp  
    %   traditional to label the first 36 functions using a single mode o7=#ye&P  
    %   number P instead of separate numbers for the order N and azimuthal }Md5a%s<  
    %   frequency M. v_L2>Pa.  
    % iPeW;=-2Wk  
    %   Example: 'Tbdo >y  
    % Kd;Iu\4hv  
    %       % Display the first 16 Zernike functions Y7_2pGvZ  
    %       x = -1:0.01:1; @/f'i9?oM`  
    %       [X,Y] = meshgrid(x,x); U6x$R O!  
    %       [theta,r] = cart2pol(X,Y); unD.t  
    %       idx = r<=1; 3]wV 1<K  
    %       p = 0:15; I3,0vnE@  
    %       z = nan(size(X)); -50Qy[0."  
    %       y = zernfun2(p,r(idx),theta(idx)); k;V (rf`  
    %       figure('Units','normalized') Hzc5BC  
    %       for k = 1:length(p) *F;W 1TF  
    %           z(idx) = y(:,k); ,+GS.]8<  
    %           subplot(4,4,k) /hp [ +K  
    %           pcolor(x,x,z), shading interp =Vat2'>+  
    %           set(gca,'XTick',[],'YTick',[]) dC(5I{I|  
    %           axis square ;FnU[Q`M#L  
    %           title(['Z_{' num2str(p(k)) '}']) K3xt,g  
    %       end SbXV'&M2AT  
    % ~i UG24v  
    %   See also ZERNPOL, ZERNFUN. 4Is Wp!`W  
    K,5_{pj  
    %   Paul Fricker 11/13/2006 r)|6H"n#]S  
    }Uue}VOA  
    !S#K6:  
    % Check and prepare the inputs: aJLc&o 8Yg  
    % ----------------------------- PS$g *x  
    if min(size(p))~=1 5Zuk`%O  
        error('zernfun2:Pvector','Input P must be vector.') wnjAiIE5  
    end Ia j`u  
    }6"l`$=Ev  
    if any(p)>35 6%Pvh- ~_  
        error('zernfun2:P36', ... xV"~?vD  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... sB0m^Y'  
               '(P = 0 to 35).']) cJ#n<Rsz  
    end KM9H<;A  
    Yy0U2N [i  
    % Get the order and frequency corresonding to the function number: s\e b  
    % ---------------------------------------------------------------- _'Rg7zHTp-  
    p = p(:); d >O/Zal  
    n = ceil((-3+sqrt(9+8*p))/2); ;sAe#b  
    m = 2*p - n.*(n+2); C9 j{:&  
    ',!>9Dj  
    % Pass the inputs to the function ZERNFUN: qNWSDZQ  
    % ---------------------------------------- %-<6Z9otc  
    switch nargin nh? JiH {  
        case 3 C8%MKNPd  
            z = zernfun(n,m,r,theta); 5[]Yxl  
        case 4 T0N6k acl  
            z = zernfun(n,m,r,theta,nflag); $!^C|,CS  
        otherwise piFZu/~Gq\  
            error('zernfun2:nargin','Incorrect number of inputs.') M7{_"9X{  
    end 5| Oj\L{  
    Y8c,+D,Ww  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) a:;7'w'  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. !x! 1H5"  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of DVf}='en8  
    %   order N and frequency M, evaluated at R.  N is a vector of 7a=ul:  
    %   positive integers (including 0), and M is a vector with the O$N;a9g  
    %   same number of elements as N.  Each element k of M must be a (}s& 84!  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) Dl,sl>{  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is LW#U+bv]Dq  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix y}?PyPz  
    %   with one column for every (N,M) pair, and one row for every ,L4zhhl!_  
    %   element in R. p+0gE5  
    % ?f6SKC  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- 9_8\xLk  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is @|s$ :;(=  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to ;j qF:Wl@  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 }^VikT]>1  
    %   for all [n,m]. y:_>R=sw  
    % ym_as8A*Q  
    %   The radial Zernike polynomials are the radial portion of the PF#<CF$=  
    %   Zernike functions, which are an orthogonal basis on the unit F_I!qcEQ  
    %   circle.  The series representation of the radial Zernike *.KVrS<B1  
    %   polynomials is My Ky*wD  
    % H@9QEj!Y  
    %          (n-m)/2 @/:7G.  
    %            __ 5<Uh2c  
    %    m      \       s                                          n-2s 5Ar gM%  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r *e<}hm Dr  
    %    n      s=0 4O.R=c2}7>  
    % &5XEjY>@  
    %   The following table shows the first 12 polynomials. t0r0{:  
    % g-B{K "z  
    %       n    m    Zernike polynomial    Normalization m)  rVzL  
    %       --------------------------------------------- [h !i{QD  
    %       0    0    1                        sqrt(2) fD3'Ye<R  
    %       1    1    r                           2 )WW*X6[k  
    %       2    0    2*r^2 - 1                sqrt(6) nK)1.KVN  
    %       2    2    r^2                      sqrt(6) sINf/mv+  
    %       3    1    3*r^3 - 2*r              sqrt(8) }0Uh<v@  
    %       3    3    r^3                      sqrt(8) {QaO\{J=  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) <m\<yZ2aa  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) JtsXMZz  
    %       4    4    r^4                      sqrt(10) |yz o|%]3  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) #5kclu%L$  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) w!v^6[!  
    %       5    5    r^5                      sqrt(12) fR~0Fy Gp  
    %       --------------------------------------------- +P Dk>PdEt  
    % L N.:>,  
    %   Example: ARt{ 2|  
    % x}Lj|U$r<X  
    %       % Display three example Zernike radial polynomials GDe,n  
    %       r = 0:0.01:1; mqHH1}  
    %       n = [3 2 5]; a5saN5)H  
    %       m = [1 2 1]; <66%(J>  
    %       z = zernpol(n,m,r); F?"Gln~;  
    %       figure 3e47UquZ  
    %       plot(r,z) Ha/Qz'^S;  
    %       grid on SZLugyZ2Y  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') zPybP E8  
    % n7G`b'  
    %   See also ZERNFUN, ZERNFUN2. CGlEc  
    t 4zUj%F  
    % A note on the algorithm. nxBP@Td  
    % ------------------------ OFmHj]I7=  
    % The radial Zernike polynomials are computed using the series 4 qY  
    % representation shown in the Help section above. For many special )8rF'pxI  
    % functions, direct evaluation using the series representation can gq 3|vzNZ  
    % produce poor numerical results (floating point errors), because ]s^+/8d=  
    % the summation often involves computing small differences between I= <eCv  
    % large successive terms in the series. (In such cases, the functions e1E_$oJP  
    % are often evaluated using alternative methods such as recurrence lw}7kp4 2F  
    % relations: see the Legendre functions, for example). For the Zernike /SS~IhUX  
    % polynomials, however, this problem does not arise, because the febn?|@  
    % polynomials are evaluated over the finite domain r = (0,1), and !G[%; d  
    % because the coefficients for a given polynomial are generally all 60$;Q,]o  
    % of similar magnitude. Xx[,n-rA  
    % .I_<\h7  
    % ZERNPOL has been written using a vectorized implementation: multiple );JWrkpz  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] mIDVN  
    % values can be passed as inputs) for a vector of points R.  To achieve IQtQf_"e1  
    % this vectorization most efficiently, the algorithm in ZERNPOL z80FMulO  
    % involves pre-determining all the powers p of R that are required to vk X+{n  
    % compute the outputs, and then compiling the {R^p} into a single [ 't.x=  
    % matrix.  This avoids any redundant computation of the R^p, and Dt(D5A  
    % minimizes the sizes of certain intermediate variables. {5{VGAD&]>  
    % /FC HF#yK  
    %   Paul Fricker 11/13/2006 PWx%~U.8~j  
    w:deQ:k  
    >K4Nn(~ys  
    % Check and prepare the inputs: PMbZv%.,-  
    % ----------------------------- u=W[ S)w  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 6s~B2t:Y  
        error('zernpol:NMvectors','N and M must be vectors.') uQx/o ^  
    end ! M CV@5$  
    ilJ`_QN  
    if length(n)~=length(m) (.@peHu)#  
        error('zernpol:NMlength','N and M must be the same length.') uHZ4 @ w:  
    end Y7p@NG&1q  
    1 /{~t[*.  
    n = n(:); p o)lN[v  
    m = m(:); FI?J8a  
    length_n = length(n); Lhux~,EH  
    "h[)5V{  
    if any(mod(n-m,2)) h~9P3 4m  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') i?(cp["7  
    end inO;Uwlv  
    vw3[(_MV3_  
    if any(m<0) _^(1Qb[  
        error('zernpol:Mpositive','All M must be positive.')  \SQ4yc  
    end @SxZ>|r-|v  
    (T pnJq  
    if any(m>n) TS+jDs  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') #\Y`?  
    end q|]CA  
    !x R9I0V5  
    if any( r>1 | r<0 ) 3]JJCaf  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') srN7  
    end b?iPQ$NyQ  
    I|&DXF  
    if ~any(size(r)==1) ?32~%?m  
        error('zernpol:Rvector','R must be a vector.') Q) aZ0 Pt  
    end }ixCbuD  
    e4Ibj/  
    r = r(:); BI,]pf;GWv  
    length_r = length(r); C)yw b6  
    4V,p\$;  
    if nargin==4 ._]Pz 6  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); (#>X*~6  
        if ~isnorm s=jH1^  
            error('zernpol:normalization','Unrecognized normalization flag.') <h#W*a  
        end %$o[,13=  
    else tx<^PV2  
        isnorm = false; sDg1nKw(  
    end 1@'I eywg  
    *74/I>i  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% )XSHKPTQ1  
    % Compute the Zernike Polynomials "eTALRL'o  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% fNx!'{o"  
    (r?hD*2r  
    % Determine the required powers of r:  _fn7-&6  
    % ----------------------------------- ^.@%n1I"5y  
    rpowers = []; Io IhQ  
    for j = 1:length(n) v\G 7V  
        rpowers = [rpowers m(j):2:n(j)]; d#d&CJAfr  
    end |/xA5_-N  
    rpowers = unique(rpowers); n;=FD;}j+  
    eF7I 5k4  
    % Pre-compute the values of r raised to the required powers, ,:K{  
    % and compile them in a matrix: w1zMY:9  
    % ----------------------------- l+y;>21sTu  
    if rpowers(1)==0 WB'1_a  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); eG2'W  
        rpowern = cat(2,rpowern{:}); 6/g 82kqpk  
        rpowern = [ones(length_r,1) rpowern]; ]i,o+xBKH  
    else FaWc:GsfB  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); m-~3c]pA  
        rpowern = cat(2,rpowern{:}); ppLLX1S  
    end wmR~e  
    lqrI*@>Tz  
    % Compute the values of the polynomials: >tMI%r  
    % -------------------------------------- jvR(e"  
    z = zeros(length_r,length_n); 'ks  .TS&  
    for j = 1:length_n ui`EODhA(  
        s = 0:(n(j)-m(j))/2; T]/>c  
        pows = n(j):-2:m(j); y}={S,z%22  
        for k = length(s):-1:1 :YkAp9civ  
            p = (1-2*mod(s(k),2))* ... L|s\IM1g  
                       prod(2:(n(j)-s(k)))/          ... S{MB$JA  
                       prod(2:s(k))/                 ... /D|q-`*K  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... sHPeAa22  
                       prod(2:((n(j)+m(j))/2-s(k))); #J5_z#-Q;  
            idx = (pows(k)==rpowers); M5:*aCN6P  
            z(:,j) = z(:,j) + p*rpowern(:,idx); >P}XCAU  
        end  Q3bU"f  
         km^ZF<.@  
        if isnorm b9T6JS j  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); {w"Cr0F,  
        end 1#AxFdm1  
    end {(Mmv[y  
    \3K7)o^  
    % EOF zernpol
    离线niuhelen
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    只看该作者 6楼 发表于: 2011-03-12
    这三个文件,我不知道该怎样把我的面型节点的坐标及轴向位移用起来,还烦请指点一下啊,谢谢啦!
    离线li_xin_feng
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    只看该作者 7楼 发表于: 2012-09-28
    我也正在找啊
    离线guapiqlh
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    只看该作者 8楼 发表于: 2014-03-04
    我也一直想了解这个多项式的应用,还没用过呢
    离线phoenixzqy
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    只看该作者 9楼 发表于: 2014-04-22
    回 guapiqlh 的帖子
    guapiqlh:我也一直想了解这个多项式的应用,还没用过呢 (2014-03-04 11:35)  XBkaum4j  
    ?~t5>PEonv  
    数值分析方法看一下就行了。其实就是正交多项式的应用。zernike也只不过是正交多项式的一种。 8AY;WL:;  
    5p S$rf  
    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)