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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 #1Mk9sxo  
    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! 7HPLD&WPt  
     
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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 e::5|6x  
    function z = zernfun(n,m,r,theta,nflag) #!#V!^ o  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. ibzYY"D:  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N @PwEom`a  
    %   and angular frequency M, evaluated at positions (R,THETA) on the ZfT%EPoZ:  
    %   unit circle.  N is a vector of positive integers (including 0), and } Q1$v~  
    %   M is a vector with the same number of elements as N.  Each element `RGZ-Q{_  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) :^%s oEi  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, ?P`wLS^;  
    %   and THETA is a vector of angles.  R and THETA must have the same ^%_B'X9  
    %   length.  The output Z is a matrix with one column for every (N,M) q,nj|9z V  
    %   pair, and one row for every (R,THETA) pair. R5]R pW=G  
    % L*FmJ{Yf  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike sbK 0OA  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), s^C*uP;R  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral A!^K:S:@  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, {(a@3m~a%  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized a]X6)6  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. N)poe2[  
    % 1<\cMY6  
    %   The Zernike functions are an orthogonal basis on the unit circle. yWzvE:!)  
    %   They are used in disciplines such as astronomy, optics, and u"T5m  
    %   optometry to describe functions on a circular domain. LV8,nTYvE  
    % o\|dm. "f  
    %   The following table lists the first 15 Zernike functions. nt;A7pI`  
    % 0?p_|X'_  
    %       n    m    Zernike function           Normalization ,6t0w|@-k  
    %       -------------------------------------------------- Fg#*rzA  
    %       0    0    1                                 1 }$qy_Esl  
    %       1    1    r * cos(theta)                    2 W@wT ,yJ8@  
    %       1   -1    r * sin(theta)                    2 ; UrwK  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) ?\vJ8H[bD  
    %       2    0    (2*r^2 - 1)                    sqrt(3) =Rb,`%  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) 00;=6q]TA  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) ?-@h Nrx  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8)  g<,v2A  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) E/U1g4S  
    %       3    3    r^3 * sin(3*theta)             sqrt(8)  o{-PT'  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) AO']Kmm  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ?WAlW,H>  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) &7@6Y{!/  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) P45q}v  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) JC=Bxv  
    %       -------------------------------------------------- N# ,4BU  
    % uN$X3Ls_  
    %   Example 1: %J|EDf ,M  
    % &q":o 'q  
    %       % Display the Zernike function Z(n=5,m=1) #G*z{BRQ  
    %       x = -1:0.01:1; $u3N ',&  
    %       [X,Y] = meshgrid(x,x); i}wu+<Mk  
    %       [theta,r] = cart2pol(X,Y); <EBp X   
    %       idx = r<=1; H[>_LYZ8  
    %       z = nan(size(X)); }1 _gemlf  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); .mok.f<G_m  
    %       figure c&0IJ7fZG  
    %       pcolor(x,x,z), shading interp PKjA@+  
    %       axis square, colorbar R8],}6,;E}  
    %       title('Zernike function Z_5^1(r,\theta)') tY[y?DJ  
    % m2_&rjGz  
    %   Example 2: q>Q|:g&:  
    % pM#:OlqC  
    %       % Display the first 10 Zernike functions }*R" yp  
    %       x = -1:0.01:1; Hfc^<q4a.  
    %       [X,Y] = meshgrid(x,x); {g @ *jo&  
    %       [theta,r] = cart2pol(X,Y); w:umr#  
    %       idx = r<=1; " g_\W  
    %       z = nan(size(X)); "\>3mVOb  
    %       n = [0  1  1  2  2  2  3  3  3  3]; *K+*0_  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; dUe"qH29s  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; 5mFi)0={y  
    %       y = zernfun(n,m,r(idx),theta(idx)); ZnJnjW PQ  
    %       figure('Units','normalized') =r_ S MTu  
    %       for k = 1:10 l|&|+u#  
    %           z(idx) = y(:,k); @8CD@SDv  
    %           subplot(4,7,Nplot(k)) Vm6^'1CY  
    %           pcolor(x,x,z), shading interp B' :ZX-Q)  
    %           set(gca,'XTick',[],'YTick',[]) hG ]jm  
    %           axis square Cog:6Gnw  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) T.(SBP  
    %       end J@Orrz2q#  
    % [{zekF~)@  
    %   See also ZERNPOL, ZERNFUN2. qlgh$9  
    <v2R6cj5  
    %   Paul Fricker 11/13/2006 {;-$;\D  
    2XXEg> CU  
    >K &b,o,[  
    % Check and prepare the inputs: u5,IH2BU  
    % ----------------------------- {K|{a  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) $K,aLcu  
        error('zernfun:NMvectors','N and M must be vectors.') :JN3@NsK  
    end HFDg@@  
    nB:Bw8U"Q  
    if length(n)~=length(m) tjTF?>^6|  
        error('zernfun:NMlength','N and M must be the same length.') RV($G8U  
    end }>OE"#si  
    >)5vsqGZaK  
    n = n(:); ~z'0~3  
    m = m(:); H~$|y9>qI  
    if any(mod(n-m,2)) =k8A7P  
        error('zernfun:NMmultiplesof2', ... 9<YB &:<  
              'All N and M must differ by multiples of 2 (including 0).') R1 wd Q8q  
    end '{+hti,Lh  
    +Rh'VZJs  
    if any(m>n)  (&gCVf  
        error('zernfun:MlessthanN', ... %(e=Q^=  
              'Each M must be less than or equal to its corresponding N.') DMf9wB  
    end Bo0y"W[+  
    K{iay g!k  
    if any( r>1 | r<0 ) {Ise (>V  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') u( o@_6  
    end stDn{x .  
    Th8Q ~*v  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) -5qO}^i$a  
        error('zernfun:RTHvector','R and THETA must be vectors.') J\{)qJ*jp  
    end .DX#:?@4@Y  
    >Y,7>ahyt  
    r = r(:); l9jcoVo .  
    theta = theta(:); Hv=coS>g:  
    length_r = length(r); h!Q >h7  
    if length_r~=length(theta) F-R`'{ ka  
        error('zernfun:RTHlength', ... ~q4y'dBy*  
              'The number of R- and THETA-values must be equal.') /# eBDo  
    end rvG qUmSUs  
    Xmnq ZWB  
    % Check normalization: "s*{0'jo  
    % -------------------- q{@Wn]!k  
    if nargin==5 && ischar(nflag) Oh^X^*I$@  
        isnorm = strcmpi(nflag,'norm'); af_zZf!0  
        if ~isnorm F+6ZD5/  
            error('zernfun:normalization','Unrecognized normalization flag.') E`s_Dr}K  
        end 6RF01z|~_  
    else L"Gi~:z  
        isnorm = false; V|D;7  
    end y jpjJ  
    f"tO*/|`  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Q,4F=b  
    % Compute the Zernike Polynomials 4a 5n*6G!  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% .dfTv/n  
    #[si.rv->  
    % Determine the required powers of r: a} /Vu"  
    % ----------------------------------- *p-Fn$7\n  
    m_abs = abs(m); [X I5Bu ~  
    rpowers = []; :.~a[\C@V<  
    for j = 1:length(n) ! Q#b4f  
        rpowers = [rpowers m_abs(j):2:n(j)]; 3xe8DD  
    end b^xf ,`D  
    rpowers = unique(rpowers); wiVQMgi`  
    V.4j?\#%  
    % Pre-compute the values of r raised to the required powers, I*ej_cFQ^  
    % and compile them in a matrix: A/QVotcU  
    % ----------------------------- <|8 l;  
    if rpowers(1)==0 oaKf{$vg  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 4/jY;YN,2  
        rpowern = cat(2,rpowern{:}); dbLX}>  
        rpowern = [ones(length_r,1) rpowern]; k`t'P6 bU  
    else j@ "`!uPz  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); . 9 NS  
        rpowern = cat(2,rpowern{:}); 9,Mp/.T"\  
    end ELPJ}moWZ  
    cU>&E* wD  
    % Compute the values of the polynomials: 7^; OjO@8  
    % -------------------------------------- K c<z;  
    y = zeros(length_r,length(n)); ZChY:I$<  
    for j = 1:length(n) `8-aHPF-  
        s = 0:(n(j)-m_abs(j))/2; 5B2,=?+o  
        pows = n(j):-2:m_abs(j); (H F,p,h_  
        for k = length(s):-1:1 4"2/"D0  
            p = (1-2*mod(s(k),2))* ... 4Rm3'Ch  
                       prod(2:(n(j)-s(k)))/              ... C0W~Tk\C2  
                       prod(2:s(k))/                     ... SQ!lgm1bA  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... `SW " RLS3  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); GKSy|z  
            idx = (pows(k)==rpowers); +wSm6*j7=  
            y(:,j) = y(:,j) + p*rpowern(:,idx); VB#31T#q?  
        end vP4Ij  
         cg.e(@(  
        if isnorm oL@ou{iQ  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); >(CoXSV5  
        end :2My|3H\  
    end e^GW[lT  
    % END: Compute the Zernike Polynomials C{Ug ?hVP  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% B#MW`7c  
    d{hYT\7~1(  
    % Compute the Zernike functions: ]aRD6F:L  
    % ------------------------------ C]H <L#)ZU  
    idx_pos = m>0; $iPN5@F  
    idx_neg = m<0; TxvPfU?  
    Fdw[CYHz  
    z = y; O}-7 V5  
    if any(idx_pos) I3Lsj}69  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); h%s  
    end T/;hIX:R  
    if any(idx_neg) \.a .'l  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); nc~d*K\!  
    end [J`G`s!  
    Zsogx}i-  
    % EOF zernfun
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    function z = zernfun2(p,r,theta,nflag) T:">,* |  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. PJ@,01  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated 8VmN? "5v  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive a.IF%hP0xo  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, AV4HX\`{P0  
    %   and THETA is a vector of angles.  R and THETA must have the same U_;J.{n  
    %   length.  The output Z is a matrix with one column for every P-value, =k= 2~ j  
    %   and one row for every (R,THETA) pair. /VO@>Hoh  
    % Qf>Pb$c$U  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike )xx/di  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) &] F|U3  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) ].P(/~FS9  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 h&M RQno  
    %   for all p. Qz(T[H5%W  
    % (OcNC/9  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 !TL}~D:J  
    %   Zernike functions (order N<=7).  In some disciplines it is xO-U]%oq  
    %   traditional to label the first 36 functions using a single mode <1+6O[>{  
    %   number P instead of separate numbers for the order N and azimuthal >MWpYp  
    %   frequency M. {dx /p-Tv  
    % v6-~fcX0G  
    %   Example: s|j<b#<xQ  
    % fEG3b#t N  
    %       % Display the first 16 Zernike functions Z)i1?#  
    %       x = -1:0.01:1; u?3NBc$~A  
    %       [X,Y] = meshgrid(x,x); T5jG IIa  
    %       [theta,r] = cart2pol(X,Y); ]|t.wr3AU  
    %       idx = r<=1; -0o6*?[Z  
    %       p = 0:15; zO5u{  
    %       z = nan(size(X)); fk7Cf"[w  
    %       y = zernfun2(p,r(idx),theta(idx)); LL[#b2CKa  
    %       figure('Units','normalized') .hlQ?\  
    %       for k = 1:length(p) n~ >h4=h  
    %           z(idx) = y(:,k); #G  +  
    %           subplot(4,4,k) Ipz 1+ #s'  
    %           pcolor(x,x,z), shading interp \_Kt6=  
    %           set(gca,'XTick',[],'YTick',[]) BZ;}ROmqk  
    %           axis square EcU'*  
    %           title(['Z_{' num2str(p(k)) '}']) /1W7<']>xV  
    %       end 9^QYuf3O  
    % X ' #$e{  
    %   See also ZERNPOL, ZERNFUN. -j`!(IJ  
    q= yZx)  
    %   Paul Fricker 11/13/2006 ZE8/ m")  
    Qyv'nx0=  
    fAM D2C  
    % Check and prepare the inputs: %-ZR~*  
    % ----------------------------- nKh%E-c  
    if min(size(p))~=1 { 0%TMiVf  
        error('zernfun2:Pvector','Input P must be vector.') ,|. *,  
    end mfngbFa1  
    {Bq"$M!Y  
    if any(p)>35 F!)M<8jL&9  
        error('zernfun2:P36', ... g| ._n  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... xZP>g  
               '(P = 0 to 35).']) <p^*Ydx  
    end YQ @dl  
    uZo`IKJ  
    % Get the order and frequency corresonding to the function number: mS:j$$]u  
    % ---------------------------------------------------------------- c8-69hb?  
    p = p(:); Im?= e  
    n = ceil((-3+sqrt(9+8*p))/2); "y~muE:.  
    m = 2*p - n.*(n+2); :otY;n-  
    ,qe]fo >  
    % Pass the inputs to the function ZERNFUN: Tr+h$M1_Ja  
    % ---------------------------------------- I mPu}  
    switch nargin 8|5Gv  
        case 3 {1Ju} =69  
            z = zernfun(n,m,r,theta); c?.r"5#  
        case 4  :Hzz{'  
            z = zernfun(n,m,r,theta,nflag); @.Z[M  
        otherwise *K+jsVDY  
            error('zernfun2:nargin','Incorrect number of inputs.') '&-5CpDUs  
    end Mhv1K|4s  
    ]&C:>  
    % 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) Gur8.A;Y  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. {cR_?Y@  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of MAE7A"l a  
    %   order N and frequency M, evaluated at R.  N is a vector of $ \Q<K@{  
    %   positive integers (including 0), and M is a vector with the a>o"^%x  
    %   same number of elements as N.  Each element k of M must be a Sf  024  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) E-UB -"6  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is !,cQ'*<W8-  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix FOOQ'o[}  
    %   with one column for every (N,M) pair, and one row for every @-'/__cgt  
    %   element in R. /S:w&5e  
    % R'Kt=.s<  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- J)9 AnGWe  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is 5h`m]#YEG  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to +1otn~(E  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 V";mWws+?#  
    %   for all [n,m]. 5f;n<EP y  
    % &Ki> h  
    %   The radial Zernike polynomials are the radial portion of the K0tV'Ml#"  
    %   Zernike functions, which are an orthogonal basis on the unit Jj2g5={  
    %   circle.  The series representation of the radial Zernike ; cGv] A+  
    %   polynomials is i{o#3  
    % $Y8>_6%+T  
    %          (n-m)/2 F>(qOH.I  
    %            __ cC^W2\  
    %    m      \       s                                          n-2s vuYO\u+ud  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r 8)L'rW{q#  
    %    n      s=0 y,MPGW_  
    % jCU=+b=  
    %   The following table shows the first 12 polynomials. _Zh2eXWdjM  
    % GwcI0~5  
    %       n    m    Zernike polynomial    Normalization Q;4}gUmI$  
    %       --------------------------------------------- R<"2%oY  
    %       0    0    1                        sqrt(2) !Tv?%? 2l  
    %       1    1    r                           2 K@B" ]6  
    %       2    0    2*r^2 - 1                sqrt(6) 1TKEm9j]u  
    %       2    2    r^2                      sqrt(6) ^'m\D;  
    %       3    1    3*r^3 - 2*r              sqrt(8) u3U4UK  
    %       3    3    r^3                      sqrt(8) "gFxfWIA  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) _b>F#nD,'%  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) >BBl 7  
    %       4    4    r^4                      sqrt(10) %1Yz'AiW[  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) j:7* 3@f  
    %       5    3    5*r^5 - 4*r^3            sqrt(12)  }VF#\q  
    %       5    5    r^5                      sqrt(12) OkLz^R?d  
    %       --------------------------------------------- r]v&t  
    % N? M   
    %   Example: w#oGX  
    % %B@ !  
    %       % Display three example Zernike radial polynomials $30oc Tt{  
    %       r = 0:0.01:1; k!T|)\nc+  
    %       n = [3 2 5]; M)L/d_4ka  
    %       m = [1 2 1]; *RBV'b  
    %       z = zernpol(n,m,r); <3b'm*  
    %       figure grr'd+_e  
    %       plot(r,z) d^PD#&"g  
    %       grid on _IWxYp  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') "u_i[[y  
    % 1!vPc93 $$  
    %   See also ZERNFUN, ZERNFUN2. <j;]!qFR  
    ^EF'TO$  
    % A note on the algorithm. W Zq,()h  
    % ------------------------ qpI]R  
    % The radial Zernike polynomials are computed using the series xq2V0Jp1u  
    % representation shown in the Help section above. For many special W;4Lkk$  
    % functions, direct evaluation using the series representation can 3QW_k5o  
    % produce poor numerical results (floating point errors), because ylu2R0] (  
    % the summation often involves computing small differences between a5|@R<iF  
    % large successive terms in the series. (In such cases, the functions KF_?'X0=  
    % are often evaluated using alternative methods such as recurrence WSRy%#  
    % relations: see the Legendre functions, for example). For the Zernike N>0LQ MI  
    % polynomials, however, this problem does not arise, because the b(l0js  
    % polynomials are evaluated over the finite domain r = (0,1), and ygN>"eP  
    % because the coefficients for a given polynomial are generally all qe?Qeh(!X  
    % of similar magnitude. /Y,r@D  
    % Oa! m  
    % ZERNPOL has been written using a vectorized implementation: multiple  A^ViDP  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] ,9d9_c.T  
    % values can be passed as inputs) for a vector of points R.  To achieve OiF{3ae(  
    % this vectorization most efficiently, the algorithm in ZERNPOL _-O cc=Z  
    % involves pre-determining all the powers p of R that are required to gw^'{b  
    % compute the outputs, and then compiling the {R^p} into a single 2:Q(Gl`<l  
    % matrix.  This avoids any redundant computation of the R^p, and }k7_'p&yk  
    % minimizes the sizes of certain intermediate variables.  Hy]  
    % VevNG *  
    %   Paul Fricker 11/13/2006 'f+NW &   
    zPR8f-Uvw  
    FbAW_Am(  
    % Check and prepare the inputs: _1aGtX|W  
    % ----------------------------- dQD$K|aUp  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) _\\ -md:  
        error('zernpol:NMvectors','N and M must be vectors.') 6V1:qp/6  
    end )u*^@Wo  
    }^Gd4[(,g  
    if length(n)~=length(m) ^z~~VBv  
        error('zernpol:NMlength','N and M must be the same length.') oZN'H T  
    end px=]bALU  
    .po>qb6  
    n = n(:); e"k/d<  
    m = m(:); _okWQvdH  
    length_n = length(n); "$|Zr  
    $'{=R 45Z  
    if any(mod(n-m,2)) $ J1f.YE  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') sZg6@s=  
    end X:EEPGE  
    };b1ahaG  
    if any(m<0) Qs9OC9X1  
        error('zernpol:Mpositive','All M must be positive.') }Cj8  
    end 6o=G8y  
    wvN`R  
    if any(m>n) BI/&dKM  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') q/PNJ#<  
    end DMn4ll|  
     &;c>O  
    if any( r>1 | r<0 ) ;a r><w  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') 8Lz]Z h=ZU  
    end ^zr^ N?a  
    XRtD< jlA"  
    if ~any(size(r)==1) qf#)lyr<D6  
        error('zernpol:Rvector','R must be a vector.') ]*N1t>fb  
    end ^YlI>_3s  
    lG:kAtx4  
    r = r(:); .c+9P<VmC}  
    length_r = length(r); -SCM:j%h  
    S,{tV=&m]  
    if nargin==4 Am"(+>W21  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); S,jZ3^  
        if ~isnorm n V&cC  
            error('zernpol:normalization','Unrecognized normalization flag.') t;NV $!!  
        end ny*i+4Mb  
    else vScjq5 "p  
        isnorm = false; -c*\o3)  
    end I G ~`i I  
    M 4yI`dr6  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% C! 9}  
    % Compute the Zernike Polynomials i=S~(gp  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% W 7sn+g \  
    KP]"P*? ?  
    % Determine the required powers of r: uLR<FpM  
    % ----------------------------------- B?bW1  
    rpowers = []; aZS7sV28  
    for j = 1:length(n) g>JLDQdc  
        rpowers = [rpowers m(j):2:n(j)]; Ib=x~za@n  
    end }G V X>p  
    rpowers = unique(rpowers); I/6)3 su%  
    1q7tiMvV-  
    % Pre-compute the values of r raised to the required powers, 0#_'o ,  
    % and compile them in a matrix: fmX!6Kv  
    % ----------------------------- `G9 l  
    if rpowers(1)==0 H`9Uf)  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); I\O\,yPhhP  
        rpowern = cat(2,rpowern{:}); (Z] HX@"{J  
        rpowern = [ones(length_r,1) rpowern]; 6%G-Vs]*2  
    else Mkxi~p%<r  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); p'_%aVm7  
        rpowern = cat(2,rpowern{:}); $g0+,ll[6  
    end o5U(i  
    zP\7S}p7%  
    % Compute the values of the polynomials: w;6bD'.>;  
    % -------------------------------------- zAzP,1$?  
    z = zeros(length_r,length_n); Z @ dC+0[=  
    for j = 1:length_n 6w8" >~)Z  
        s = 0:(n(j)-m(j))/2; 2Os1C}m  
        pows = n(j):-2:m(j); j$7|XM6  
        for k = length(s):-1:1 B;>{0 s  
            p = (1-2*mod(s(k),2))* ... : 18KR*;p  
                       prod(2:(n(j)-s(k)))/          ... &#`l;n:]+  
                       prod(2:s(k))/                 ... /AY4M;}p  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... \_V-A f{6  
                       prod(2:((n(j)+m(j))/2-s(k))); Rhc-q|Lz8  
            idx = (pows(k)==rpowers); w'MGA  
            z(:,j) = z(:,j) + p*rpowern(:,idx); RD7^&  
        end aT!'}GjL  
         OJ|r6  
        if isnorm x+8_4>,>Y7  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); yZr M.%V  
        end "5R~(+~<@  
    end ?'86d_8  
    K_)eWf0a  
    % 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)  TYy.jFT-  
    Hla0 5N' 4  
    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。 q5?# 3T=  
    gvL f|+m  
    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)