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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 l q\'  
    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! 3dX=xuQ%/  
     
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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 .Ca"$2  
    function z = zernfun(n,m,r,theta,nflag) :Wyn+  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. gqP -E  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N H9Y2n 0  
    %   and angular frequency M, evaluated at positions (R,THETA) on the VjA wn}eO  
    %   unit circle.  N is a vector of positive integers (including 0), and {[M0y*^64$  
    %   M is a vector with the same number of elements as N.  Each element  .6O52E  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) KMxNH,5  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, `2B*CMW{  
    %   and THETA is a vector of angles.  R and THETA must have the same 9*}iBs  
    %   length.  The output Z is a matrix with one column for every (N,M) ^eT DD  
    %   pair, and one row for every (R,THETA) pair. wMH[QYb<*  
    % P3>..fhoW  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike [K/O5_  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), tr6jh=  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral N_u&3CG  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, QHHW(InG<  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized ZQ,fm`y\  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. e-[>( n/[  
    % LJRg>8  
    %   The Zernike functions are an orthogonal basis on the unit circle. YMc8Q\*B  
    %   They are used in disciplines such as astronomy, optics, and ]&Y#) ebs  
    %   optometry to describe functions on a circular domain. D~G5]M,}$  
    % Xt</ -`  
    %   The following table lists the first 15 Zernike functions. $$haVY&  
    % u-AWJc+F.  
    %       n    m    Zernike function           Normalization G0_&gx`  
    %       -------------------------------------------------- {l&Ltruhz  
    %       0    0    1                                 1 d&}pgb-Md  
    %       1    1    r * cos(theta)                    2 ,vY)n6  
    %       1   -1    r * sin(theta)                    2 !Gln Q`T  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) OOEV-=  
    %       2    0    (2*r^2 - 1)                    sqrt(3) 2Pbe~[  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) E:uReT  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) dO>k5!ge|:  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) G{@C"H[$<  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) qSFc=Wwc  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) 1vB-M6(  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) ayV6m  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) jP1$qhp  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) Sg-g^ dIN1  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) |ZS 57c:  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) NJn&>/vM  
    %       -------------------------------------------------- 6BDt.bG  
    % u~" siH  
    %   Example 1: k4S} #!  
    % p]wP36<S!  
    %       % Display the Zernike function Z(n=5,m=1) k/df(cs  
    %       x = -1:0.01:1; 4rI:1 yGt@  
    %       [X,Y] = meshgrid(x,x); ?k [%\jq{a  
    %       [theta,r] = cart2pol(X,Y); (7IqY1W  
    %       idx = r<=1; C@*%AY  
    %       z = nan(size(X)); *f79=x  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); 'I&|1I^  
    %       figure _Ny8j~  
    %       pcolor(x,x,z), shading interp ;(K  
    %       axis square, colorbar 1s Br.+p  
    %       title('Zernike function Z_5^1(r,\theta)') Hl4\M]]/&  
    % 7N>oY$&)  
    %   Example 2: 3>i>@n_  
    % u FMIY(vB  
    %       % Display the first 10 Zernike functions *Wzwbwg  
    %       x = -1:0.01:1; JxjP@nr  
    %       [X,Y] = meshgrid(x,x); Iph3%RaE  
    %       [theta,r] = cart2pol(X,Y); :bwM]k*$  
    %       idx = r<=1; 1<`9HCm  
    %       z = nan(size(X)); 6^Ph '  
    %       n = [0  1  1  2  2  2  3  3  3  3];  VJ3hC[  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; WElrk:b  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; mKV'jm0  
    %       y = zernfun(n,m,r(idx),theta(idx)); XdcG0D^  
    %       figure('Units','normalized') K>kLUcC7Z  
    %       for k = 1:10 \ZS\i4  
    %           z(idx) = y(:,k); JL.5QzA  
    %           subplot(4,7,Nplot(k)) Yrpxy.1=F5  
    %           pcolor(x,x,z), shading interp 7U,k 2LS  
    %           set(gca,'XTick',[],'YTick',[]) u,f A!  
    %           axis square 3@G;'|z  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) &} ,*\Oj  
    %       end 5^>n5u/  
    % 9SY(EL  
    %   See also ZERNPOL, ZERNFUN2. :y\09)CJK  
    Gfv(w=rr?  
    %   Paul Fricker 11/13/2006 X:_<Y_JT  
    N=#4L$@-  
    7$ d}!S  
    % Check and prepare the inputs: Q!K`e)R  
    % ----------------------------- M`~!u/D7  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) $_)=8"Sn  
        error('zernfun:NMvectors','N and M must be vectors.') +jtA&1cf  
    end Ivsb<qzG  
    "IG+V:{ou  
    if length(n)~=length(m) nX._EC  
        error('zernfun:NMlength','N and M must be the same length.') W}h|K:-S  
    end _S"f_W  
    R uLvG+  
    n = n(:); |q_ !. a  
    m = m(:); {]^2R>0Q  
    if any(mod(n-m,2)) S8%n.<OB  
        error('zernfun:NMmultiplesof2', ... -l "U"U"F  
              'All N and M must differ by multiples of 2 (including 0).') t^.'>RwW|  
    end |z~LzSJv  
    < A?<N?%o  
    if any(m>n) t}Ss=0dJO  
        error('zernfun:MlessthanN', ... Zm(dY*z5:J  
              'Each M must be less than or equal to its corresponding N.') 7 jjU  
    end 6Nt$ZYS  
    Wr>(#*r7q  
    if any( r>1 | r<0 ) =Y9\DeIZ  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') YUscz!rM  
    end H] k'?;  
    [T`}yb@  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) S5_t1wqBJ  
        error('zernfun:RTHvector','R and THETA must be vectors.') u g\w\b  
    end 5 lTD]d  
    #dc1pfL!y{  
    r = r(:); ;LRW 8Wd  
    theta = theta(:); m_* R.a  
    length_r = length(r); ioV_oR9I  
    if length_r~=length(theta) dn,gZ"<  
        error('zernfun:RTHlength', ... /APcL5:=  
              'The number of R- and THETA-values must be equal.') `tE^jqrke5  
    end O"*`'D|hK  
    Q> 8pP\ho  
    % Check normalization: aq Mc6N`z  
    % -------------------- $ [7 Vgs  
    if nargin==5 && ischar(nflag) R#(G%66   
        isnorm = strcmpi(nflag,'norm'); @T&t.|`  
        if ~isnorm \ZD[ !w7  
            error('zernfun:normalization','Unrecognized normalization flag.') ^7aN2o3{  
        end by86zX  
    else ?t rV72D  
        isnorm = false; uLN[*D  
    end hVP IHQt  
    \t3qS eWc/  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% J!h^egP  
    % Compute the Zernike Polynomials KrKu7]If6#  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% }B q^3?,#{  
    f v LC_'M  
    % Determine the required powers of r: B9X8  
    % ----------------------------------- e*]r  
    m_abs = abs(m); 9<s4yZF@x  
    rpowers = []; ~p*1:ij  
    for j = 1:length(n) ;=jr0\|e  
        rpowers = [rpowers m_abs(j):2:n(j)]; N[Sb#w`[/  
    end LdTdQ,s<  
    rpowers = unique(rpowers); C+%K6/J(  
    [s` G^  
    % Pre-compute the values of r raised to the required powers, 0{) $SY  
    % and compile them in a matrix: v-`h>J!Nx  
    % ----------------------------- 7@~tVxB;  
    if rpowers(1)==0 7Kf}O6nE  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); cDV ^8 R  
        rpowern = cat(2,rpowern{:}); ]Kde t"+  
        rpowern = [ones(length_r,1) rpowern]; Vq ^]s $'  
    else :reTJQwr  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); vR>o}%`  
        rpowern = cat(2,rpowern{:}); v6uxxsI>Hm  
    end )1F<6R  
    h`5)2n+P  
    % Compute the values of the polynomials: I*\^,ow  
    % -------------------------------------- Bct"X#W|&  
    y = zeros(length_r,length(n)); uQeu4$k!  
    for j = 1:length(n) QH@>icAb  
        s = 0:(n(j)-m_abs(j))/2; eThy+  
        pows = n(j):-2:m_abs(j); Yrn"saVc,  
        for k = length(s):-1:1 F}X0',   
            p = (1-2*mod(s(k),2))* ... mBk5+KyT  
                       prod(2:(n(j)-s(k)))/              ... !/I0i8T  
                       prod(2:s(k))/                     ... 4TRG.$2[  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... qpqokK  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); {CUk1+  
            idx = (pows(k)==rpowers); 2t1I3yA'{z  
            y(:,j) = y(:,j) + p*rpowern(:,idx); {G*QY%j^  
        end H:S,\D?%2x  
         ZR3nK0  
        if isnorm MZv\ C  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); S~F`  
        end p!W[X%`)  
    end y,m2(V  
    % END: Compute the Zernike Polynomials }zMf7<C  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {'bip`U.  
    >HTbegi  
    % Compute the Zernike functions: ?IYY'fS"  
    % ------------------------------ B 0)]s<<  
    idx_pos = m>0; p25Fn`}H  
    idx_neg = m<0; TbhH&kG)1  
    t})$lM  
    z = y; 30F!kP*E  
    if any(idx_pos) \7Cg,Xn  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); O`W%Tr  
    end 'ks{D(`  
    if any(idx_neg) s& yk  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); /"#4T^7&  
    end `  2%6V)s  
    $3P`DJo  
    % 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) sH!O0WL  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. w2"]Pl  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated TZB+lj1  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive 1'KishHK=  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, <8o(CA\  
    %   and THETA is a vector of angles.  R and THETA must have the same 1]OSWCEm*[  
    %   length.  The output Z is a matrix with one column for every P-value, }NmNanW^  
    %   and one row for every (R,THETA) pair. 45+{nN[  
    % x8N|($1  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike TT}]wZ  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) }GZbo kWg.  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) 1;r69e  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 ;#anZC;  
    %   for all p. Plo,XU  
    % !,&yyx.  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 JdNF-64ky  
    %   Zernike functions (order N<=7).  In some disciplines it is FLr ;`3  
    %   traditional to label the first 36 functions using a single mode %5B%KCCN  
    %   number P instead of separate numbers for the order N and azimuthal wD9a#AgEd  
    %   frequency M. lpC @I^:  
    % ecHP &Z$  
    %   Example: =!.m GW-Q}  
    % tS?lB05TOR  
    %       % Display the first 16 Zernike functions h%}( h2 W  
    %       x = -1:0.01:1; p+w8$8)  
    %       [X,Y] = meshgrid(x,x); Fwfo2   
    %       [theta,r] = cart2pol(X,Y);  v[,Src  
    %       idx = r<=1; X;GfPw.m  
    %       p = 0:15; i@$*Csj\9*  
    %       z = nan(size(X)); F:T GsV#  
    %       y = zernfun2(p,r(idx),theta(idx)); #@//7Bf%  
    %       figure('Units','normalized') t&RruwN_;  
    %       for k = 1:length(p) 9]yW_]P  
    %           z(idx) = y(:,k); Fr:5$,At7-  
    %           subplot(4,4,k) =nRuY '  
    %           pcolor(x,x,z), shading interp u<Xog$esu  
    %           set(gca,'XTick',[],'YTick',[]) .ER98  
    %           axis square ygViPz<J  
    %           title(['Z_{' num2str(p(k)) '}']) VXKT\9g3A  
    %       end 8A2 z 5Aa  
    % Ot9V< D6h  
    %   See also ZERNPOL, ZERNFUN. HD-Erop  
    (FVX57  
    %   Paul Fricker 11/13/2006 cyF4iG'M,y  
    N6/T#UVns  
    ltA/  
    % Check and prepare the inputs: tYe:z:7l?<  
    % ----------------------------- U}AX0*S  
    if min(size(p))~=1 j6>tH"i  
        error('zernfun2:Pvector','Input P must be vector.') A WJWtUa  
    end @.$MzPQQI  
    x>3@R0A 1:  
    if any(p)>35 5K.+CO<  
        error('zernfun2:P36', ... ;VzMU ;j  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... r0\f;q  
               '(P = 0 to 35).']) C1B'#F9EO  
    end n9oR)&:o  
    Y1\K;;X  
    % Get the order and frequency corresonding to the function number: a6nlt? 1?D  
    % ---------------------------------------------------------------- ycpE=fso'  
    p = p(:); Spj9H?m  
    n = ceil((-3+sqrt(9+8*p))/2); y-+G wa3  
    m = 2*p - n.*(n+2); |B[eJq  
    xFb3O|TC  
    % Pass the inputs to the function ZERNFUN: [.cq{6-  
    % ---------------------------------------- &Ocu#Cb  
    switch nargin >)c9|e=8  
        case 3 !#WqA9<  
            z = zernfun(n,m,r,theta); <r\I"z$  
        case 4 \< 65??P  
            z = zernfun(n,m,r,theta,nflag); 'mV:@].le  
        otherwise 6 =>G#  
            error('zernfun2:nargin','Incorrect number of inputs.') ]VjLKFb~U  
    end c> ~:dcy  
    q=0 pQ1>  
    % 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) RVF F6N^  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. n;OHH{E{  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of L@A9{,9Pl  
    %   order N and frequency M, evaluated at R.  N is a vector of z,+m[x=/N  
    %   positive integers (including 0), and M is a vector with the _rjBc ;a  
    %   same number of elements as N.  Each element k of M must be a 'Y)/~\FI  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) g i4  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is 5 ,ZRP'oI  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix uUS)#qM |  
    %   with one column for every (N,M) pair, and one row for every ^!uO(B&  
    %   element in R. 1t2cY;vJ  
    % i+ic23$4M  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- aBlbg3q  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is k?-S`o%Q  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to i./Y w  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 e "_"vbk  
    %   for all [n,m]. 2np-Fc{S  
    % 8IOj[&%0  
    %   The radial Zernike polynomials are the radial portion of the l?/gW D^  
    %   Zernike functions, which are an orthogonal basis on the unit .v l="<  
    %   circle.  The series representation of the radial Zernike k2uBaj]  
    %   polynomials is n/-N;'2J  
    % _IKQ36=  
    %          (n-m)/2 a71}y;W  
    %            __ )"~=7)~<^  
    %    m      \       s                                          n-2s v>k b^38  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r C$y fMK,,N  
    %    n      s=0 =n)#!i  
    % 9OZ>y0)K~  
    %   The following table shows the first 12 polynomials. Dauo(Uhuo  
    % g TD%4V  
    %       n    m    Zernike polynomial    Normalization YiNo#M91  
    %       --------------------------------------------- vGyppm[0  
    %       0    0    1                        sqrt(2) Tvrc%L(]  
    %       1    1    r                           2 nOr"K;C  
    %       2    0    2*r^2 - 1                sqrt(6) qAvvXs=5  
    %       2    2    r^2                      sqrt(6) ;]u1~  
    %       3    1    3*r^3 - 2*r              sqrt(8) skSNzF7'  
    %       3    3    r^3                      sqrt(8) qL~Pjr>cF  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) ?a8nz, zb  
    %       4    2    4*r^4 - 3*r^2            sqrt(10)  qKx59  
    %       4    4    r^4                      sqrt(10) !g/_ w  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) !$XO U'n  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) bY&YSlO  
    %       5    5    r^5                      sqrt(12) 6(sfpK'  
    %       --------------------------------------------- b@F_7P%  
    % $"(3MnR  
    %   Example: o"f%\N0_8  
    % LA>dkPB  
    %       % Display three example Zernike radial polynomials '[xut1{  
    %       r = 0:0.01:1; h!~|6nj  
    %       n = [3 2 5]; fl!1AKSn@N  
    %       m = [1 2 1]; "$4hv6 s  
    %       z = zernpol(n,m,r); ]X4RnV55Q  
    %       figure :e52hK1[T  
    %       plot(r,z) m(h/:JZ\  
    %       grid on ZS|Z98  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') N6f%>3%1|.  
    % >4#tkv>S.  
    %   See also ZERNFUN, ZERNFUN2. wfWS-pQ  
    l.yJA>\24I  
    % A note on the algorithm. F ^[M  
    % ------------------------ P'gT6*an,"  
    % The radial Zernike polynomials are computed using the series b^;N>zx  
    % representation shown in the Help section above. For many special jCam,$oE  
    % functions, direct evaluation using the series representation can }% FDm@+  
    % produce poor numerical results (floating point errors), because |)*m[_1  
    % the summation often involves computing small differences between $o{F  
    % large successive terms in the series. (In such cases, the functions VQ/ <09e  
    % are often evaluated using alternative methods such as recurrence ]Oig ..LJ  
    % relations: see the Legendre functions, for example). For the Zernike Qdh"X^^  
    % polynomials, however, this problem does not arise, because the i:AjWC@]  
    % polynomials are evaluated over the finite domain r = (0,1), and %y!   
    % because the coefficients for a given polynomial are generally all 'aLPTVM^  
    % of similar magnitude. oR4fK td  
    % mJ7 `.  
    % ZERNPOL has been written using a vectorized implementation: multiple hVROzGZk  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] 5mDVFb 3a  
    % values can be passed as inputs) for a vector of points R.  To achieve z2"2tFK  
    % this vectorization most efficiently, the algorithm in ZERNPOL 8Q#t\$RY  
    % involves pre-determining all the powers p of R that are required to /5a$@%  
    % compute the outputs, and then compiling the {R^p} into a single Ma n^\gkCi  
    % matrix.  This avoids any redundant computation of the R^p, and F%Ro98?{  
    % minimizes the sizes of certain intermediate variables. {^2({A#&  
    % 1"*Nb5s  
    %   Paul Fricker 11/13/2006 N}eU.#L  
    E5v|SFD  
    #J'Z5)i|  
    % Check and prepare the inputs: |% la  
    % ----------------------------- 6C@0[Q\ER  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 7H/! rx  
        error('zernpol:NMvectors','N and M must be vectors.') 0Ax>gj-`  
    end )1X' W  
    ]QjXh >  
    if length(n)~=length(m) HCfS)`  
        error('zernpol:NMlength','N and M must be the same length.') #S/pYP`7  
    end tF{{cd  
    bdNY7|j`  
    n = n(:); \= )[  
    m = m(:); x`/m>~_  
    length_n = length(n); ,.1&Ff)S  
    38zR\@'j]4  
    if any(mod(n-m,2)) 6x`\ J2x  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') Q{(,/}kA-  
    end =6L :I x  
    ?eY chVq  
    if any(m<0) i2\\!s  
        error('zernpol:Mpositive','All M must be positive.') [:/7OM  
    end 2J>A;x_?  
    kV]%Q3t  
    if any(m>n) Vj9`[1}1Z  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') U?+30{hb  
    end >G w%r1)  
    s%0[DO3NV  
    if any( r>1 | r<0 ) $! fz~  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') hx2!YNx !  
    end 3P<Zzt%eT  
    7csl1|U  
    if ~any(size(r)==1) yE!7`c.[u  
        error('zernpol:Rvector','R must be a vector.') mt&JgA/  
    end ocDVCCkxg  
    0t/z "  
    r = r(:); Z7k1fv:S^  
    length_r = length(r); 7J)Hwl  
    b5iJ m-  
    if nargin==4 l[!C-Tq  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); !W6    
        if ~isnorm s$3eJ|  
            error('zernpol:normalization','Unrecognized normalization flag.') V< 9em7  
        end FB^dp}  
    else ?!Th-Cc&m  
        isnorm = false; h&^/, G  
    end N5I W@?4  
    3GuMiht5  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% S+bWD7  
    % Compute the Zernike Polynomials VN55!l'OV  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% VS@rM<K{  
    ;74 DT  
    % Determine the required powers of r: rym\5 `)  
    % ----------------------------------- eyl) uR  
    rpowers = []; tz \:r>3vI  
    for j = 1:length(n) |M{,}.*CU  
        rpowers = [rpowers m(j):2:n(j)]; !WVabdt  
    end hH@o|!y  
    rpowers = unique(rpowers); P.2.Ge|  
    bI6V &Dd  
    % Pre-compute the values of r raised to the required powers, p|O-I&Xd  
    % and compile them in a matrix: CI3_lWax%  
    % ----------------------------- JOoLHZQ1v  
    if rpowers(1)==0 .ubbNp_LU  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); /%@RO^P  
        rpowern = cat(2,rpowern{:}); ]Rys=.!  
        rpowern = [ones(length_r,1) rpowern]; :_b =Km<  
    else 9zGKQ|X)  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); HS7 G_  
        rpowern = cat(2,rpowern{:}); oXZ@*   
    end ~bm2_/RL  
    l Ib>t  
    % Compute the values of the polynomials: AFF>r#e  
    % -------------------------------------- }A&Xxh!Fwo  
    z = zeros(length_r,length_n); CSg5i&A=  
    for j = 1:length_n VL,?91qwe  
        s = 0:(n(j)-m(j))/2; W0,"V'C  
        pows = n(j):-2:m(j); :!*;0~#  
        for k = length(s):-1:1 $hY]EB  
            p = (1-2*mod(s(k),2))* ... -*{(#k$  
                       prod(2:(n(j)-s(k)))/          ... CIs1*:Q9  
                       prod(2:s(k))/                 ... SoON@h/  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... n<(5B|~y  
                       prod(2:((n(j)+m(j))/2-s(k))); LW8{a&  
            idx = (pows(k)==rpowers); Y_iF$ m/R  
            z(:,j) = z(:,j) + p*rpowern(:,idx); /)OO)B-r  
        end #(wz l  
         6"c!tJc7j  
        if isnorm 'rx,f  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); }n&JZ`8<s  
        end >j~70 ?  
    end 'H-YFB$l  
    ba:du |Ec  
    % 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)  1w^[Eno$$  
    N9<eU!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。 TI  
    r&DK> H  
    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)