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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 |ZC'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! >;lrH&  
     
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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 of GoaH*h  
    function z = zernfun(n,m,r,theta,nflag) J qmL|S)  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. ;JMmr-@  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 2Q7X"ek~[  
    %   and angular frequency M, evaluated at positions (R,THETA) on the 8F'm#0  
    %   unit circle.  N is a vector of positive integers (including 0), and yY*(!^S  
    %   M is a vector with the same number of elements as N.  Each element ?G<?: /CU  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) m. \JO  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, FUZuS!sJ  
    %   and THETA is a vector of angles.  R and THETA must have the same u#`51Hr$  
    %   length.  The output Z is a matrix with one column for every (N,M) ~3&hvm[IQ  
    %   pair, and one row for every (R,THETA) pair. 6'x3g2C/  
    % ^N7 C/" p  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike CJDNS21m  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ; xQhq*  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral yhI;FNSf  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, :6&#u.\u  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized :t;i2Ck  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. /{/mwS"W  
    % @,}tY ?>a  
    %   The Zernike functions are an orthogonal basis on the unit circle. +JM@kdE5b  
    %   They are used in disciplines such as astronomy, optics, and Rlm28  
    %   optometry to describe functions on a circular domain. U_.}V  
    % ^QG<_Dm]  
    %   The following table lists the first 15 Zernike functions. .JJ50p  
    % [0]J 2  
    %       n    m    Zernike function           Normalization Vg :''!4t2  
    %       -------------------------------------------------- kY6_n4  
    %       0    0    1                                 1 Eau V  
    %       1    1    r * cos(theta)                    2 'H4?V  
    %       1   -1    r * sin(theta)                    2 M;NIcM  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) yq<W+b/  
    %       2    0    (2*r^2 - 1)                    sqrt(3) "q!*RO'a  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) ZR"qrCSw`  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) e\f\CMb  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) vA[7i*D{w  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) !, rF(pz  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) !4<A|$mQ  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) cM4{ e^  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) E1`_[=8a9  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) 2$VSH&  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) e**'[3Y  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) #?eMEws  
    %       -------------------------------------------------- >6@,L+-6r  
    % `2^(Ss# )  
    %   Example 1: Kb-m  
    % _34%St!lg  
    %       % Display the Zernike function Z(n=5,m=1) GU9p'E  
    %       x = -1:0.01:1; Pj_DI)^  
    %       [X,Y] = meshgrid(x,x); oIMS >&  
    %       [theta,r] = cart2pol(X,Y); -w8?Ur1x:  
    %       idx = r<=1; tA'5ufj*:  
    %       z = nan(size(X)); -^;,m=4{3  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); ]scr@e  
    %       figure a<>cbP  
    %       pcolor(x,x,z), shading interp wlslG^^(!  
    %       axis square, colorbar I3izLi  
    %       title('Zernike function Z_5^1(r,\theta)') %K7;ePu  
    % aGws?<1$  
    %   Example 2: ='C;^ Bk  
    % D0MW~Y6{  
    %       % Display the first 10 Zernike functions =<zlg~i  
    %       x = -1:0.01:1; %da-/[  
    %       [X,Y] = meshgrid(x,x); Y?zo")  
    %       [theta,r] = cart2pol(X,Y); yS[HYq  
    %       idx = r<=1; qSD3]Dv"  
    %       z = nan(size(X)); Ir*{IVvej  
    %       n = [0  1  1  2  2  2  3  3  3  3]; gw%L M7yQR  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; a1[J>  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; yJlRW!@&:  
    %       y = zernfun(n,m,r(idx),theta(idx)); )KkV<$  
    %       figure('Units','normalized') "A5z!6T{  
    %       for k = 1:10 jqTK7b  
    %           z(idx) = y(:,k); d>c`hQ(V  
    %           subplot(4,7,Nplot(k)) i }Zz[b  
    %           pcolor(x,x,z), shading interp D$rn?@&g  
    %           set(gca,'XTick',[],'YTick',[]) |lu@rN  
    %           axis square aD6!x3c/  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) PGVp1TQ  
    %       end p6)6Gcx  
    % i=pfjC  
    %   See also ZERNPOL, ZERNFUN2. MBU4Awj  
    EU'rdG*t/R  
    %   Paul Fricker 11/13/2006 qzLD  
    s$0dLEa9  
    nr( C*E  
    % Check and prepare the inputs: }g|9P SbJ  
    % ----------------------------- Mii&doU  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 9i{(GO  
        error('zernfun:NMvectors','N and M must be vectors.') *KU:D Y{  
    end J9y}rGO  
    MU:v& sk  
    if length(n)~=length(m) !|9k&o  
        error('zernfun:NMlength','N and M must be the same length.') f'`y-]"V5)  
    end -rHqU|  
    qw)Ou]L=  
    n = n(:); 6{g&9~V  
    m = m(:); wsc=6/#u  
    if any(mod(n-m,2)) U^DR'X=  
        error('zernfun:NMmultiplesof2', ... A8AeM `  
              'All N and M must differ by multiples of 2 (including 0).') KF!d?  
    end Q7UQwAN'  
    AP4s_X+=  
    if any(m>n) W3^^aD-  
        error('zernfun:MlessthanN', ... <KStl fX  
              'Each M must be less than or equal to its corresponding N.') h7m$P^=U  
    end %N\8!aXnf  
    :3J`+V}9;  
    if any( r>1 | r<0 ) ~(`MP<  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') E>2AG3)  
    end 8|+@A1)&4  
    1 .o0"  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) {W%XS E  
        error('zernfun:RTHvector','R and THETA must be vectors.') ^?A>)?Sq  
    end t~qAA\p}o  
    ' 8Q }pp`  
    r = r(:); 5a2;@ }%V  
    theta = theta(:); ygK,t*T20  
    length_r = length(r); xf|C{XV@H  
    if length_r~=length(theta) %/!f^PIwX  
        error('zernfun:RTHlength', ... A,7* 52U  
              'The number of R- and THETA-values must be equal.') !2/o]_K@+  
    end lACS^(  
    BgB0   
    % Check normalization: gzlRK^5  
    % -------------------- whGtVx|zR  
    if nargin==5 && ischar(nflag) 9PaV*S(\TR  
        isnorm = strcmpi(nflag,'norm'); 3J3wKw!`  
        if ~isnorm /L2.7`5  
            error('zernfun:normalization','Unrecognized normalization flag.') 5CH8;sMK  
        end }b{7+ + Ah  
    else p`!<yq2_  
        isnorm = false; 'mF&`BN}b  
    end 6J cXhlB`  
    @Yw42`> !s  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% _5OxESE  
    % Compute the Zernike Polynomials Vp1Nk#H  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% KsqS{VVCh  
    1].m4vC  
    % Determine the required powers of r: rf!i?vAe  
    % ----------------------------------- `?d` #) Ck  
    m_abs = abs(m); .5A .[ZY)  
    rpowers = []; Z8f?uF  
    for j = 1:length(n) )L_@l5l  
        rpowers = [rpowers m_abs(j):2:n(j)]; bY~V?yNgKM  
    end 6;M{suG|  
    rpowers = unique(rpowers); lj+&3<E  
     KcpQ[6\  
    % Pre-compute the values of r raised to the required powers, WP^wNi ~>  
    % and compile them in a matrix: 1DH P5q  
    % ----------------------------- 3,Iu!KB  
    if rpowers(1)==0 ]7q|) S\  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 3aJYl3:0B  
        rpowern = cat(2,rpowern{:}); z 1.vnGP  
        rpowern = [ones(length_r,1) rpowern]; z.tN<P7  
    else C[><m2T  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); Nkn2\ w  
        rpowern = cat(2,rpowern{:}); FyChH7  
    end dChMjaix  
    jFI`CA6P  
    % Compute the values of the polynomials: D23 c/8K  
    % -------------------------------------- SXNde@% {  
    y = zeros(length_r,length(n)); '<6DLtZl  
    for j = 1:length(n) on1B~?*D  
        s = 0:(n(j)-m_abs(j))/2; I`x[1%y2 F  
        pows = n(j):-2:m_abs(j); IUD@Kf]S  
        for k = length(s):-1:1 `1lGAKv  
            p = (1-2*mod(s(k),2))* ... sdN1BV2  
                       prod(2:(n(j)-s(k)))/              ... n-OQCz9Xl  
                       prod(2:s(k))/                     ... Qn;,OB k  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... eEYz A  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); VWk{?*Dp  
            idx = (pows(k)==rpowers); %kP=VUXj  
            y(:,j) = y(:,j) + p*rpowern(:,idx); [7,q@>:CS  
        end NFqGbA|  
         L08lkq,  
        if isnorm 7s Gf_`Z  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); lnK#q .]  
        end F4IU2_CnPD  
    end C>QWV[F  
    % END: Compute the Zernike Polynomials B'bOK`p  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [* |+ it+!  
    4v9d& m!<  
    % Compute the Zernike functions: Y<_;8%S  
    % ------------------------------ :4r*Jju<V  
    idx_pos = m>0; )G*xI`(@  
    idx_neg = m<0; q w @g7  
    fT YlIT9  
    z = y; bKEiS8x  
    if any(idx_pos) gSe3S-Lt  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); WYIv&h<h"  
    end !1Ht{cA0  
    if any(idx_neg) \p^'[B(O77  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ZzxWKIE'c  
    end FbXur-et^  
    s(r4m/  
    % 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) lNTbd"}$:  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. %AmyT  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated lbC,*U^  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive !'B='].  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, R@U4Ae{+  
    %   and THETA is a vector of angles.  R and THETA must have the same | /n  
    %   length.  The output Z is a matrix with one column for every P-value, g{f7 } gTG  
    %   and one row for every (R,THETA) pair. uQ7lC~  
    % pF(6M3>IN  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike B>@l(e)b  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2)  GInw7  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) 1MmEP  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 *]nk{jo2  
    %   for all p. 9!.S9[[N  
    % ,H1K sN  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 k= &n>P  
    %   Zernike functions (order N<=7).  In some disciplines it is amQTPNI  
    %   traditional to label the first 36 functions using a single mode ^x_$%8  
    %   number P instead of separate numbers for the order N and azimuthal WQbjq}RfI  
    %   frequency M. /Z[HU{4  
    % X#zp,7j?  
    %   Example: T6."j_  
    % cIcu=U  
    %       % Display the first 16 Zernike functions ^;tB,7:*V  
    %       x = -1:0.01:1; WdB\n/BWB  
    %       [X,Y] = meshgrid(x,x); i1'G_bo4F7  
    %       [theta,r] = cart2pol(X,Y); %jHe_8=o  
    %       idx = r<=1; GRaU]Z]ck  
    %       p = 0:15; ?Iq{6O>D.  
    %       z = nan(size(X));  ) TRUx  
    %       y = zernfun2(p,r(idx),theta(idx)); 5"X@<;H%  
    %       figure('Units','normalized')  +cKOIMu9  
    %       for k = 1:length(p) 7 p1B"%  
    %           z(idx) = y(:,k); ^Ai QNL}  
    %           subplot(4,4,k) *z.rOY= 8  
    %           pcolor(x,x,z), shading interp \jmZ t*c  
    %           set(gca,'XTick',[],'YTick',[]) ` U-vXP  
    %           axis square @;N(3| n7  
    %           title(['Z_{' num2str(p(k)) '}']) ;cZp$ xb3  
    %       end w'E?L`c  
    % $cU7)vmK`  
    %   See also ZERNPOL, ZERNFUN. rm-;Z<  
    tGzp= PyA  
    %   Paul Fricker 11/13/2006 e m0 hTxb  
    i0J`{PbI  
    v`]y:Ku|wR  
    % Check and prepare the inputs: *aFY+.;U`  
    % ----------------------------- dZI["FeO&d  
    if min(size(p))~=1 gXM+N(M-  
        error('zernfun2:Pvector','Input P must be vector.') Z`U+ a  
    end 9X +dp  
    yqVoedN  
    if any(p)>35 X8-x$07)  
        error('zernfun2:P36', ... \CV HtV  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... Y=g]\%-PB  
               '(P = 0 to 35).']) 6 jm@`pYbE  
    end !l Egta[Ql  
    |I29m`  
    % Get the order and frequency corresonding to the function number: +r9neS.l  
    % ---------------------------------------------------------------- y7%SHYC p[  
    p = p(:); h+A+>kC5  
    n = ceil((-3+sqrt(9+8*p))/2); ~(7ct*U~  
    m = 2*p - n.*(n+2); ST;o^\B  
    B2'TRXIm1U  
    % Pass the inputs to the function ZERNFUN: 0L1sF'ZN  
    % ---------------------------------------- Yl:[b{Py  
    switch nargin GN:|b2 "  
        case 3 j>uu3ADd2  
            z = zernfun(n,m,r,theta); WiZTE(NM`  
        case 4 u6Wan*I?  
            z = zernfun(n,m,r,theta,nflag); >,h{`  
        otherwise >d *`K  
            error('zernfun2:nargin','Incorrect number of inputs.') %5DM ew  
    end 3,X8 5`v^  
    ezCJq`b  
    % 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) )fxn bBz{  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. B]: |;d  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of /BD'{tZ]Sl  
    %   order N and frequency M, evaluated at R.  N is a vector of ]!@=2kG4  
    %   positive integers (including 0), and M is a vector with the -mn/Yv  
    %   same number of elements as N.  Each element k of M must be a *|<~IQg  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) u[Si=)`VPk  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is D~URY_[A  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix C"B'Dj  
    %   with one column for every (N,M) pair, and one row for every }!R*Q`m  
    %   element in R. R! On  
    % Y:L[Iz95o  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly-  _cj=}!I  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is _DT,iF*6  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to bt_c$TN  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 eEP{?F^I[  
    %   for all [n,m]. .{*l,  
    % (GC5r#AnS  
    %   The radial Zernike polynomials are the radial portion of the ,(zV~-:9  
    %   Zernike functions, which are an orthogonal basis on the unit 2f@Cy+W'[  
    %   circle.  The series representation of the radial Zernike 8ih_S2Cd  
    %   polynomials is Ui"{0%  
    % N6\rjYx+7  
    %          (n-m)/2 h6^|f%\w*i  
    %            __ 9H/R@i[E  
    %    m      \       s                                          n-2s |iX>hJSl  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r dcD#!v\0  
    %    n      s=0 Q"nw.FjUG  
    % dE_"|,:  
    %   The following table shows the first 12 polynomials. b1jDbiH&  
    % H!u8+  
    %       n    m    Zernike polynomial    Normalization 6 U[VoUU   
    %       --------------------------------------------- law$LL  
    %       0    0    1                        sqrt(2) -m=A1~|7  
    %       1    1    r                           2 W Zm8!Y  
    %       2    0    2*r^2 - 1                sqrt(6) KYJP`va6k  
    %       2    2    r^2                      sqrt(6) =_zo  
    %       3    1    3*r^3 - 2*r              sqrt(8) 2IRARZ,3  
    %       3    3    r^3                      sqrt(8) E;x-O)(&  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) G9LWnyQt  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) {FKr^)g  
    %       4    4    r^4                      sqrt(10) #$-?[c$>  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) : [328X2  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) v @0G^z|  
    %       5    5    r^5                      sqrt(12) U5H%wA['m  
    %       --------------------------------------------- 5QuRwu_  
    % e98QT9  
    %   Example: UH}lKc=t  
    % W&R67ff|  
    %       % Display three example Zernike radial polynomials +~aIT=i3  
    %       r = 0:0.01:1; AG9DJ{T  
    %       n = [3 2 5]; 5[jS(1a`c  
    %       m = [1 2 1]; buN@O7\  
    %       z = zernpol(n,m,r); ![\P/1p  
    %       figure B ;E"VS0  
    %       plot(r,z) saP%T~  
    %       grid on 8>#ZU]cG  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') Ao}<a1f  
    % #)xlBq4cZ  
    %   See also ZERNFUN, ZERNFUN2. Ys}^ hy  
    Ui (nMEon  
    % A note on the algorithm. WQ[n K5#  
    % ------------------------ r(uo-/7z  
    % The radial Zernike polynomials are computed using the series BO7HJF)a  
    % representation shown in the Help section above. For many special iz^uj  
    % functions, direct evaluation using the series representation can YW&K,)L@  
    % produce poor numerical results (floating point errors), because /7Pqy2sgE  
    % the summation often involves computing small differences between YX-j|m|  
    % large successive terms in the series. (In such cases, the functions }EkL[H!  
    % are often evaluated using alternative methods such as recurrence Wq<oP  
    % relations: see the Legendre functions, for example). For the Zernike 4s9@4  
    % polynomials, however, this problem does not arise, because the iJ^}{-  
    % polynomials are evaluated over the finite domain r = (0,1), and Gg ~0>XS  
    % because the coefficients for a given polynomial are generally all Gt*K:KT=L  
    % of similar magnitude. K;sC#9m  
    % ?2~fvMWu  
    % ZERNPOL has been written using a vectorized implementation: multiple 2XeyNX  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] OzrIiahz/  
    % values can be passed as inputs) for a vector of points R.  To achieve |TM n  
    % this vectorization most efficiently, the algorithm in ZERNPOL r|4D.O]  
    % involves pre-determining all the powers p of R that are required to 0{z8pNrc  
    % compute the outputs, and then compiling the {R^p} into a single 3w"JzC@  
    % matrix.  This avoids any redundant computation of the R^p, and 1_q!E~)  
    % minimizes the sizes of certain intermediate variables. oj Y.6w  
    % Q;y5E`G  
    %   Paul Fricker 11/13/2006 W0sLMHq  
    tB~#;:g  
    e( @< /W  
    % Check and prepare the inputs: ZM-P  
    % ----------------------------- cYg J}(>}  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) qna!j|90Lp  
        error('zernpol:NMvectors','N and M must be vectors.') ]goJ- &  
    end (:OMt2{r  
    R3_OCM_*  
    if length(n)~=length(m) 06%-tAq:  
        error('zernpol:NMlength','N and M must be the same length.') o [V8h @K)  
    end P8By~f32_  
    4sQm"XgE  
    n = n(:); 9M27;"gK  
    m = m(:); 1mJUl x  
    length_n = length(n); ^`id/  
    k6ry"W3  
    if any(mod(n-m,2)) !;*flr`/  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') 83Rs1}*  
    end H i8V=+  
    B//*hH >F  
    if any(m<0) _d3Z~cH  
        error('zernpol:Mpositive','All M must be positive.') ^;a .;wR  
    end ( `bb1gz  
    PbZ%[F  
    if any(m>n) %\48hSe  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') [T =>QS@g  
    end +%\Ci!%b  
    \h#aPG<yo  
    if any( r>1 | r<0 ) P8X9bW~GQ  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') (["kbPma  
    end &gr 8;O:0  
    m0ra  
    if ~any(size(r)==1) ,.ivdg( /  
        error('zernpol:Rvector','R must be a vector.') J?J4<l9  
    end KMy"DVqE  
    _";w*lg}  
    r = r(:); & tT6.@kH  
    length_r = length(r); _ncBq;j{  
    <lIm==U<-  
    if nargin==4 !q,'k2= b,  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); ])F+ C/Px1  
        if ~isnorm -~8PI2  
            error('zernpol:normalization','Unrecognized normalization flag.') eEVB   
        end "*<vE7  
    else =Mwuhk|*  
        isnorm = false; SJP3mq/^K  
    end %8u9:Cl):  
    XV%R Mr6  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% iy]L"7&Z2  
    % Compute the Zernike Polynomials P!f0&W  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% `mteU"{bx  
    fK=0?]s}I  
    % Determine the required powers of r: ,i}EGW,9q  
    % ----------------------------------- <bhGpLh-E  
    rpowers = []; %/e'6g<  
    for j = 1:length(n) SS%Bde&<{  
        rpowers = [rpowers m(j):2:n(j)]; Xn=yC Pi  
    end :]F66dh+  
    rpowers = unique(rpowers); lG0CCOdQ  
    }(u:K}8  
    % Pre-compute the values of r raised to the required powers, r-$xLe7a  
    % and compile them in a matrix: )C?H m^ #  
    % ----------------------------- MMKN^a"GA  
    if rpowers(1)==0  \8C<nh  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); AFL'Ox]0  
        rpowern = cat(2,rpowern{:}); +%9Re5R  
        rpowern = [ones(length_r,1) rpowern]; ]QJ5JtD-  
    else t^MTR6y+8  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); jSvq1$U  
        rpowern = cat(2,rpowern{:}); 0/ 33Z Oc  
    end _GxC|d  
    ($<&H>j0  
    % Compute the values of the polynomials: "!P h  
    % -------------------------------------- V*rLGY#  
    z = zeros(length_r,length_n); 3AdYZ7J  
    for j = 1:length_n 5SUO`4L  
        s = 0:(n(j)-m(j))/2; {jYVA~.|Z  
        pows = n(j):-2:m(j);  tM\BO0  
        for k = length(s):-1:1 d.w]\  
            p = (1-2*mod(s(k),2))* ... ~Sb)i f  
                       prod(2:(n(j)-s(k)))/          ... \Db`RvEmR  
                       prod(2:s(k))/                 ...  ]'% iR  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... >F7HKwg}Z  
                       prod(2:((n(j)+m(j))/2-s(k))); rQT%~oM:  
            idx = (pows(k)==rpowers); _Cz98VqRk  
            z(:,j) = z(:,j) + p*rpowern(:,idx);  ~ e?af  
        end +2:HgW  
         _XP}f x7$C  
        if isnorm ]}'bRq*]  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); 2 ^"j]g>mj  
        end X(E`cH |  
    end L;*7p9  
    w+ ')wyB  
    % 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},Y0QXw  
    )`|`PB  
    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。 ~S :8M<aB  
    A2PeI"y  
    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)