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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 v&qL r+_7  
    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! c`@";+|r  
     
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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 ve+bR   
    function z = zernfun(n,m,r,theta,nflag) S:XsO9:{  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. SpImd IpD  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N >(-A"jf  
    %   and angular frequency M, evaluated at positions (R,THETA) on the ]}kw'&  
    %   unit circle.  N is a vector of positive integers (including 0), and =Oq *9=v|  
    %   M is a vector with the same number of elements as N.  Each element 16>D?;2o(  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) d@p#{ -  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, vz~Oi  
    %   and THETA is a vector of angles.  R and THETA must have the same yVp,)T9  
    %   length.  The output Z is a matrix with one column for every (N,M) 7{]dh+)  
    %   pair, and one row for every (R,THETA) pair. Ia< V\$#  
    % ;?k<L\zaw  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ! Sw=ns7  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), /wax5FS'I,  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral DJ DQH\&  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, tXqX[Td`0g  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized m8;w7S7,j~  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. $_iE^zZaU^  
    % ]BUirJ,2  
    %   The Zernike functions are an orthogonal basis on the unit circle. O ,9^R  
    %   They are used in disciplines such as astronomy, optics, and @({=~ W^  
    %   optometry to describe functions on a circular domain. m^0vux  
    % %ioVNbrR7  
    %   The following table lists the first 15 Zernike functions. lKB9n}P  
    % co~NXpqg  
    %       n    m    Zernike function           Normalization W7Y@]QMX  
    %       -------------------------------------------------- S2e3d  
    %       0    0    1                                 1 =kfa1kD&{  
    %       1    1    r * cos(theta)                    2 33EF/k3vW  
    %       1   -1    r * sin(theta)                    2 x+j@YWDpG"  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) x1?mE)n]  
    %       2    0    (2*r^2 - 1)                    sqrt(3) w|6/i/X  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) )A xD|A  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) p_g`f9q6D  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) BvsSrse  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) 1*yxSU@uY  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) :SS \2  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) #-\5O  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 5 ty2e`~K  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) F3EAjO)ch  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) *!ng)3#  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) [8a(4]4  
    %       -------------------------------------------------- v\5O\ I ^  
    % }%8ZN :  
    %   Example 1: vX\9#Hj  
    % QM#Vl19>j(  
    %       % Display the Zernike function Z(n=5,m=1) '9Z`y_~)G  
    %       x = -1:0.01:1; pa1<=w  
    %       [X,Y] = meshgrid(x,x); xa@$cxt  
    %       [theta,r] = cart2pol(X,Y); NJQ)Ttt  
    %       idx = r<=1; 8W{M}>;[9  
    %       z = nan(size(X)); O-X(8<~H=  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); |~e"i<G#  
    %       figure OemY'M? ZQ  
    %       pcolor(x,x,z), shading interp W`_JERo  
    %       axis square, colorbar -R]0cefC<f  
    %       title('Zernike function Z_5^1(r,\theta)') ewU*5|*[  
    % jkx>o?s)z  
    %   Example 2: Lo%vG{yTr  
    % R\5,H!V9n  
    %       % Display the first 10 Zernike functions fwv^dEe  
    %       x = -1:0.01:1; Vf&U`K  
    %       [X,Y] = meshgrid(x,x); tg@61V?>  
    %       [theta,r] = cart2pol(X,Y); ["<Xh0_  
    %       idx = r<=1; hqvhnqQk  
    %       z = nan(size(X)); 0#9H;j<Op  
    %       n = [0  1  1  2  2  2  3  3  3  3]; u"=]cBRWL6  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; ?E"192 ,z@  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; 6[3Xe_  
    %       y = zernfun(n,m,r(idx),theta(idx)); $G`CXhbl  
    %       figure('Units','normalized') qC> tni%  
    %       for k = 1:10 Ohk\P;}  
    %           z(idx) = y(:,k); Q?i_Nl/|  
    %           subplot(4,7,Nplot(k)) } +}nrJv  
    %           pcolor(x,x,z), shading interp % -!%n= P  
    %           set(gca,'XTick',[],'YTick',[]) ~ tA ^K  
    %           axis square 1~c\J0h)d  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ng3ZK  
    %       end "00j]e.  
    % PGJh>[ s  
    %   See also ZERNPOL, ZERNFUN2. SYY x>1;8`  
    +PjTT6  
    %   Paul Fricker 11/13/2006 e'.BTt58Y  
    94+^K=lAX  
    ;[}OZt  
    % Check and prepare the inputs: &T,|?0>~=J  
    % ----------------------------- 4{YA['  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ?Ts]zO%%Z  
        error('zernfun:NMvectors','N and M must be vectors.') b!]O]dk#  
    end (<eLj Q  
    yMz#e0k  
    if length(n)~=length(m) |@X^_L.!  
        error('zernfun:NMlength','N and M must be the same length.') (N9-YP?qm  
    end wuH*a3(  
    + +}!Gfc?s  
    n = n(:); R.rc h2  
    m = m(:); jg710.v:  
    if any(mod(n-m,2)) 'Gn>~m  
        error('zernfun:NMmultiplesof2', ... <{kPa_`'  
              'All N and M must differ by multiples of 2 (including 0).') >J7slDRo  
    end }ssV"5M  
    m[}k]PB>  
    if any(m>n) -i`jS_-Cv-  
        error('zernfun:MlessthanN', ... _ p\L,No  
              'Each M must be less than or equal to its corresponding N.') ]eKuR"ob0  
    end uCDe>Q4@/  
    ;d6Dm)/(  
    if any( r>1 | r<0 ) BYq80Vk%@  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') UH!(`Z\C  
    end r @4A% ql<  
    y(J~:"}7)  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ?}KRAtJ8  
        error('zernfun:RTHvector','R and THETA must be vectors.') a<<4gXx  
    end xJCx zJ  
    kkq1:\pZ]a  
    r = r(:); `j>5W<5q\  
    theta = theta(:); SY +0~5E  
    length_r = length(r); #%"G[B  
    if length_r~=length(theta) EB[T 5{  
        error('zernfun:RTHlength', ... u}iuf_  
              'The number of R- and THETA-values must be equal.') ..} P$  
    end 9r?Z'~,Za  
    spV7\Gs.@  
    % Check normalization: j L|6i-?!  
    % -------------------- .g8*K "  
    if nargin==5 && ischar(nflag) 4-yK!LR  
        isnorm = strcmpi(nflag,'norm'); L!cOg8Z  
        if ~isnorm KR>)Ek  
            error('zernfun:normalization','Unrecognized normalization flag.') PQ 4mNjXN  
        end S~Gse+*  
    else ?@ oF@AEx=  
        isnorm = false; X<%D@$  
    end /pj[c;aO  
    v&d1ACctJ  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% '#+&?6p  
    % Compute the Zernike Polynomials Z mJ<h&  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% p/ ITg  
    [Z$H <m{c-  
    % Determine the required powers of r: iJzBd7  
    % ----------------------------------- TPN+jK  
    m_abs = abs(m); cyCh^- <l@  
    rpowers = []; } k2 Q  
    for j = 1:length(n) Pu3oQDldV  
        rpowers = [rpowers m_abs(j):2:n(j)]; %hVR|K|J  
    end &*v\t\]  
    rpowers = unique(rpowers); sM-,95H  
    Wlc&QOfF  
    % Pre-compute the values of r raised to the required powers, /.SG? 5t4  
    % and compile them in a matrix: ["3dr@T9Z  
    % ----------------------------- w)EY j+L  
    if rpowers(1)==0 AQ'%}(#0  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); fp [gKRSF  
        rpowern = cat(2,rpowern{:}); ]}v]j`9m%  
        rpowern = [ones(length_r,1) rpowern]; <A,V/']  
    else pkn^K+<n,  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); {:1j>4m 2  
        rpowern = cat(2,rpowern{:}); ` g]  
    end 9+@h2"|N4*  
    T-] {gc  
    % Compute the values of the polynomials: WE) *~5  
    % -------------------------------------- +hN>Q $E  
    y = zeros(length_r,length(n)); "`% ,l|D  
    for j = 1:length(n) %B$ftsYXmu  
        s = 0:(n(j)-m_abs(j))/2; 0i*V?  
        pows = n(j):-2:m_abs(j); +bznKy!  
        for k = length(s):-1:1 & P-8_I  
            p = (1-2*mod(s(k),2))* ... 0-Mzb{n5  
                       prod(2:(n(j)-s(k)))/              ... Q4u.v,sE  
                       prod(2:s(k))/                     ... {+67<&g  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... &aRL}#U  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); Xz^nm\  
            idx = (pows(k)==rpowers); MfL7|b)  
            y(:,j) = y(:,j) + p*rpowern(:,idx); J0!V(  
        end KsKE#])&l  
         $*0-+h  
        if isnorm -#ZLu.  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); =.3#l@E!C  
        end ~F,~^r!Jtu  
    end c""&He4zp  
    % END: Compute the Zernike Polynomials >$D!mraih  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% h&'|^;FM  
    FOk&z!xYKd  
    % Compute the Zernike functions: m'"r<]pB*4  
    % ------------------------------ Y9^l|,bm5  
    idx_pos = m>0; 0+CcNY9  
    idx_neg = m<0; t{>K).'  
    ~(R=3  
    z = y; Py;5z  
    if any(idx_pos) p~w] ~\  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 'gf[Wjb,%  
    end cACIy yQ  
    if any(idx_neg) [^"*I.Z_  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); A)U"F&tvm  
    end n#">k%bD  
    YmC}q20;  
    % 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) I/(U0`%  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. cp Ot?XYR~  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated #Pg#\v|7#>  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive % G= cKM  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, Xe_ <]|  
    %   and THETA is a vector of angles.  R and THETA must have the same .) Ej#mk  
    %   length.  The output Z is a matrix with one column for every P-value, hQ|mow@Zmz  
    %   and one row for every (R,THETA) pair. Ic0Sb7c  
    % %tVU Rj  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike AY52j  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) |?88EG@05  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) 76w[X=Fv  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 Tksv7*5$  
    %   for all p. _x.2&S89  
    % <W0(!<U  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 xQ"uC!Gu4  
    %   Zernike functions (order N<=7).  In some disciplines it is Z,)H f  
    %   traditional to label the first 36 functions using a single mode M+&~sX*a  
    %   number P instead of separate numbers for the order N and azimuthal a[K&;)  
    %   frequency M. ql@2<V{  
    % LLgw1 @-D  
    %   Example: >>{):r Z  
    % ^&<M""Z  
    %       % Display the first 16 Zernike functions li%@HdA!  
    %       x = -1:0.01:1; *O$|,EsY  
    %       [X,Y] = meshgrid(x,x); je mb/ :E  
    %       [theta,r] = cart2pol(X,Y); OL\-SQ&  
    %       idx = r<=1; A$wC !P|;  
    %       p = 0:15; AW r2Bv  
    %       z = nan(size(X)); #2^0z`-\_z  
    %       y = zernfun2(p,r(idx),theta(idx)); :gsRJy1  
    %       figure('Units','normalized') lj]M 1zEz&  
    %       for k = 1:length(p) +<\LY(o  
    %           z(idx) = y(:,k); j55_wx@cA  
    %           subplot(4,4,k) yP]>eLTSd  
    %           pcolor(x,x,z), shading interp ->I.D?p  
    %           set(gca,'XTick',[],'YTick',[]) Cj>HMB}  
    %           axis square &0T.o,&y  
    %           title(['Z_{' num2str(p(k)) '}']) {qw'gJmX  
    %       end G `|7NL   
    % --]blP7  
    %   See also ZERNPOL, ZERNFUN. gxO~44"  
    {gzQ/|}#z-  
    %   Paul Fricker 11/13/2006 XuP%/\  
    4S42h_9  
    =Z.0-C>W  
    % Check and prepare the inputs: jsAx;Z:QT  
    % ----------------------------- e;vI XJE  
    if min(size(p))~=1 diXWm-ZKL  
        error('zernfun2:Pvector','Input P must be vector.') B*G]Dr)e  
    end 4(htdn6\  
    QI[WXx p  
    if any(p)>35 B9"d7E#wHF  
        error('zernfun2:P36', ... l=xG<)Okb  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... q#j[0,^ $  
               '(P = 0 to 35).']) L$Z!  
    end \8#[AD*@s2  
    ILCh1=?{9r  
    % Get the order and frequency corresonding to the function number: <J d!`$  
    % ---------------------------------------------------------------- l'l&Zqd  
    p = p(:); 4 6v C/  
    n = ceil((-3+sqrt(9+8*p))/2); fOMaTnm'  
    m = 2*p - n.*(n+2); p{oz}}  
    vs8[352  
    % Pass the inputs to the function ZERNFUN: -Jd7  
    % ---------------------------------------- |gV~U~A]  
    switch nargin F+Kju2  
        case 3 -$!r+4|q  
            z = zernfun(n,m,r,theta); yPmo1|'X>d  
        case 4 \2u7>fU!  
            z = zernfun(n,m,r,theta,nflag); &</)k|.A6\  
        otherwise 1a9' *[  
            error('zernfun2:nargin','Incorrect number of inputs.') -)%\$z  
    end (w-"1(  
    "6]oi*_8  
    % 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) 7Vf XE/  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. At>e4t2@  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of &5jc &CS  
    %   order N and frequency M, evaluated at R.  N is a vector of u9:sj  
    %   positive integers (including 0), and M is a vector with the 2KXF XR  
    %   same number of elements as N.  Each element k of M must be a 4grV2xtX  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) yq, qS0Fo  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is 6&+dpr&c~=  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix bZSt<cH3  
    %   with one column for every (N,M) pair, and one row for every >l}v _k*~B  
    %   element in R. HziQ%QR  
    % :hO B  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- '3^_:E5y  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is Y".?j5f?  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to ?/}IDwuh  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 0<A*I{,4L  
    %   for all [n,m]. Q~^v=ye  
    % 860y9wzU  
    %   The radial Zernike polynomials are the radial portion of the %>nAPO+e  
    %   Zernike functions, which are an orthogonal basis on the unit =WF@S1  
    %   circle.  The series representation of the radial Zernike xz:  
    %   polynomials is ;t N@  
    % Lniz>gSc  
    %          (n-m)/2 WjVm{7?{  
    %            __ Ku6ndc  
    %    m      \       s                                          n-2s O 8l`1  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r kFQo[O]  
    %    n      s=0 PtPx(R3  
    % {2xc/   
    %   The following table shows the first 12 polynomials. C^JtJv  
    % g:^Hex?Yfd  
    %       n    m    Zernike polynomial    Normalization 7F]oK0l_  
    %       --------------------------------------------- Ai%Wt-  
    %       0    0    1                        sqrt(2) Y}}1]}VIK  
    %       1    1    r                           2 |p/[sD+M  
    %       2    0    2*r^2 - 1                sqrt(6) q%s<y+  
    %       2    2    r^2                      sqrt(6) 5!i\S[:  
    %       3    1    3*r^3 - 2*r              sqrt(8) @B)5Ho  
    %       3    3    r^3                      sqrt(8) =J-&usX  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) abVEi[nP  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) 5[6{o$I  
    %       4    4    r^4                      sqrt(10) J{;\TNkJ  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) Ng&K5Z/  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) L.9@rwfI  
    %       5    5    r^5                      sqrt(12) {Y+e|B0  
    %       --------------------------------------------- z/o&r`no  
    % Y qKQm+G  
    %   Example: $*fEgU% c  
    % x 'i~o'  
    %       % Display three example Zernike radial polynomials J"eE9FLM  
    %       r = 0:0.01:1; UZJs!#P  
    %       n = [3 2 5]; mg3YKHNG  
    %       m = [1 2 1]; @ uL4'@Ej  
    %       z = zernpol(n,m,r); 0 4ceDe  
    %       figure ],<pZ1V;  
    %       plot(r,z) z9Y}[ pN  
    %       grid on Y>i Qp/k:  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') ){J,Z*&  
    % jTsQsHq   
    %   See also ZERNFUN, ZERNFUN2. K?S5C8  
    W1}d6Sbg  
    % A note on the algorithm. '8^>Z.~V  
    % ------------------------ sVS),9\}  
    % The radial Zernike polynomials are computed using the series 30cb+)h(  
    % representation shown in the Help section above. For many special U4NA'1yo  
    % functions, direct evaluation using the series representation can wx/*un%2  
    % produce poor numerical results (floating point errors), because x1+V  
    % the summation often involves computing small differences between Cj)*JZV G  
    % large successive terms in the series. (In such cases, the functions  ~Ctq  
    % are often evaluated using alternative methods such as recurrence (Ixmg=C6y  
    % relations: see the Legendre functions, for example). For the Zernike s=&x%0f%  
    % polynomials, however, this problem does not arise, because the k _Bz@^J  
    % polynomials are evaluated over the finite domain r = (0,1), and . P! pC  
    % because the coefficients for a given polynomial are generally all NW*$+u%/R  
    % of similar magnitude. J.,7d ,  
    % L# .vbf  
    % ZERNPOL has been written using a vectorized implementation: multiple d-UQc2r  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] $Ehe8,=fj  
    % values can be passed as inputs) for a vector of points R.  To achieve }(UU~V  
    % this vectorization most efficiently, the algorithm in ZERNPOL CVgVyy^  
    % involves pre-determining all the powers p of R that are required to dJ,,yA*  
    % compute the outputs, and then compiling the {R^p} into a single 4^w`] m  
    % matrix.  This avoids any redundant computation of the R^p, and )K 0rPnYV  
    % minimizes the sizes of certain intermediate variables. kSqMI'89  
    % ?Hf8<C}3  
    %   Paul Fricker 11/13/2006 ftqeiZ 2  
    hLSas#B>  
    ~hQTxLp  
    % Check and prepare the inputs: -H](2}  
    % ----------------------------- u`Zj~ t  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) $@ZrGT  
        error('zernpol:NMvectors','N and M must be vectors.') tc<HA7vpt~  
    end >f~y2YAr  
    VL@eR9}9K  
    if length(n)~=length(m) X_J(P?  
        error('zernpol:NMlength','N and M must be the same length.') &_<!zJ;Hn  
    end 8(GJz ~y  
    ,a@jg&Mb]  
    n = n(:); $mDlS  
    m = m(:); vt2A/9_Z%  
    length_n = length(n); } |? W  
    G+W0X  
    if any(mod(n-m,2)) 0bI} s`sr  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') b~EA&dc  
    end 9d >AnTf&H  
    ,gbQqoLV  
    if any(m<0) `BKV/Xl  
        error('zernpol:Mpositive','All M must be positive.') J*r%b+  
    end v^Pjvv=  
    uY$BZEuAZ  
    if any(m>n) JQ"w{O  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') =bl6:  
    end 47^R  
    fpPHw)dTd  
    if any( r>1 | r<0 ) J4^aD;j  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') A{KF<Omu  
    end <lHVch"(^$  
    q5ja \  
    if ~any(size(r)==1) YhFd0A?]  
        error('zernpol:Rvector','R must be a vector.') l]__!X  
    end rh 7%<xb>  
    nv2p&-e+  
    r = r(:); 1usLCG>w{  
    length_r = length(r); $]S*(K3U ~  
    j21nh> d  
    if nargin==4 z!:%Hbh=  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); KMogwulG  
        if ~isnorm t73" d#+  
            error('zernpol:normalization','Unrecognized normalization flag.') G[z4 $0f  
        end <gbm 1iEe  
    else 3!0~/8!f@  
        isnorm = false; pv;ZR  
    end T9O3$1eqfo  
    o[E|xw  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% &Uu8wFbIJ  
    % Compute the Zernike Polynomials K&>+<bJ_  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {S5H H"  
    7C@%1kL  
    % Determine the required powers of r: O7D61~G]  
    % ----------------------------------- z,q1TU9  
    rpowers = []; Ij9ezNZT=  
    for j = 1:length(n) "YePd * W  
        rpowers = [rpowers m(j):2:n(j)]; Yl0_?.1 z  
    end b1ma(8{{{  
    rpowers = unique(rpowers); eg Zb)pP  
    5Cjh%rj(jl  
    % Pre-compute the values of r raised to the required powers, `hE@S |4  
    % and compile them in a matrix: 68-2EWq  
    % ----------------------------- y?@(%PTp  
    if rpowers(1)==0 Y5?OJO{h"  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); &'d3Yt  
        rpowern = cat(2,rpowern{:}); kKEs >a  
        rpowern = [ones(length_r,1) rpowern]; KBkS>0;X  
    else nN$aZSb`  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 2u?k;"]V  
        rpowern = cat(2,rpowern{:}); 97SOa.@  
    end ym.:I@b?6  
    (,!G$~Sy  
    % Compute the values of the polynomials: #Qnl,lf  
    % -------------------------------------- >* >}d%  
    z = zeros(length_r,length_n); "-a CF  
    for j = 1:length_n 65||]l  
        s = 0:(n(j)-m(j))/2; N#zh$0!8bJ  
        pows = n(j):-2:m(j); MclW!CmJ  
        for k = length(s):-1:1 U9eb&nd  
            p = (1-2*mod(s(k),2))* ... }6!*H!  
                       prod(2:(n(j)-s(k)))/          ... .P5OUK  
                       prod(2:s(k))/                 ... akB+4?+s)  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... S<bsrS*$  
                       prod(2:((n(j)+m(j))/2-s(k))); &D*22R4{CX  
            idx = (pows(k)==rpowers); +Ssu^ >D  
            z(:,j) = z(:,j) + p*rpowern(:,idx); JuTIP6 /G  
        end #<EMG|&(  
         N497"H</  
        if isnorm o1?bqVF;6  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); 9E-]S'Z  
        end ?KMGk]_<  
    end 2~(\d\k  
    rVF7!|&  
    % 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)  >J) 9&?  
    \5P 5N]]  
    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。 8e"MP\0V  
    ^y.|KA3[  
    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)