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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 SRL-Z&M  
    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! <>=mCZ2  
     
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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 0=![fjm  
    function z = zernfun(n,m,r,theta,nflag) (lWq[0^N  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. A3+6 #?:;  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N x-_vl 9P)  
    %   and angular frequency M, evaluated at positions (R,THETA) on the *%A}x   
    %   unit circle.  N is a vector of positive integers (including 0), and 91d }, Mq:  
    %   M is a vector with the same number of elements as N.  Each element BSzkW}3q9  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) =2 jhII  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, vVVPw?Ww-  
    %   and THETA is a vector of angles.  R and THETA must have the same $f-hUOuyo  
    %   length.  The output Z is a matrix with one column for every (N,M) MR;X&Up6!  
    %   pair, and one row for every (R,THETA) pair. NQLiWz-q  
    % P))^vUt~  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike +nU.p/cK+\  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), FpVV4D  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral !B^K[2`)N  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, |t6~%6^8  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized 8MF2K6  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ?<~WO?  
    % m#H_*L0  
    %   The Zernike functions are an orthogonal basis on the unit circle. x$B&L`QV  
    %   They are used in disciplines such as astronomy, optics, and 2`Gv5}LfyR  
    %   optometry to describe functions on a circular domain. NFyMY#\]  
    % !OE*z $\  
    %   The following table lists the first 15 Zernike functions. V4K'R2t  
    % $>w/Cy  
    %       n    m    Zernike function           Normalization Y &f\VNlT  
    %       -------------------------------------------------- HL8eD^  
    %       0    0    1                                 1 >.DC!QV  
    %       1    1    r * cos(theta)                    2 .v])S}K  
    %       1   -1    r * sin(theta)                    2 4hAJ!7[A.  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) S; /. %  
    %       2    0    (2*r^2 - 1)                    sqrt(3) oXgdLtsu  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) OJ3UE(,I=  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) Ly #_?\bn  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) yrr) y  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) g22gIj]  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) 9&  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) I%;Jpe  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ZYMw}]#((E  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) qL 5>o>J  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) bToq$%sCg  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) X0uJNHO  
    %       -------------------------------------------------- {j SmoA  
    % b?VV'{4  
    %   Example 1:  .i/m  
    % npH?4S-8G  
    %       % Display the Zernike function Z(n=5,m=1) 2<r\/-#pU  
    %       x = -1:0.01:1; f8n V=AQ  
    %       [X,Y] = meshgrid(x,x); k`VM2+9h'^  
    %       [theta,r] = cart2pol(X,Y); 9M-K]0S(  
    %       idx = r<=1; *e{PxaF!C  
    %       z = nan(size(X)); (! KG)!  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); q``wt  
    %       figure X6@wkrf-  
    %       pcolor(x,x,z), shading interp s&tE_  
    %       axis square, colorbar ?<%=: Yh  
    %       title('Zernike function Z_5^1(r,\theta)') K-Mc6  
    % ;O=h$8]  
    %   Example 2: 7P**:b  
    % !:0v{ZQ  
    %       % Display the first 10 Zernike functions !1Y&Y@ze  
    %       x = -1:0.01:1; g>j| ]6  
    %       [X,Y] = meshgrid(x,x); ;L"!I3dM)  
    %       [theta,r] = cart2pol(X,Y); cxP&^,~  
    %       idx = r<=1; #&Is GyU  
    %       z = nan(size(X)); UY>v"M  
    %       n = [0  1  1  2  2  2  3  3  3  3]; s"~5']8  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; LN^f1/ b*  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; 0b-?q&*_  
    %       y = zernfun(n,m,r(idx),theta(idx)); Pq p *  
    %       figure('Units','normalized') 1mgLX_U9  
    %       for k = 1:10 {aOkV::  
    %           z(idx) = y(:,k); MDO$m g  
    %           subplot(4,7,Nplot(k)) E4oz|2!m  
    %           pcolor(x,x,z), shading interp 4na8  
    %           set(gca,'XTick',[],'YTick',[]) L^0v\  
    %           axis square p{tK_ZBy]c  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) B$a-og(  
    %       end v#oi0-9o[  
    % #1/}3+=5B  
    %   See also ZERNPOL, ZERNFUN2. SoQR#(73HK  
    i*[n{=*l@  
    %   Paul Fricker 11/13/2006 WZewPn>#q  
    uO(w1Q"^  
    dl|gG9u4Q  
    % Check and prepare the inputs: W`)<vGn=Y  
    % ----------------------------- Le#spvV3J|  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) vF={9G  
        error('zernfun:NMvectors','N and M must be vectors.') 93Yn`Av;  
    end 1=)r@X/6d  
    /b[2lTC-e  
    if length(n)~=length(m) + ,4" u  
        error('zernfun:NMlength','N and M must be the same length.') ~(X(&  
    end mOBACTY^  
    TkRP3_b  
    n = n(:); 5J.0&Dda  
    m = m(:); F jrINxL7^  
    if any(mod(n-m,2)) N|Cs=-+  
        error('zernfun:NMmultiplesof2', ... W<,F28jI3v  
              'All N and M must differ by multiples of 2 (including 0).') f@ `*>"  
    end CboLH0Fa  
    ?u$u?j|N  
    if any(m>n) ! fl4"  
        error('zernfun:MlessthanN', ... p9[6^rjx8  
              'Each M must be less than or equal to its corresponding N.') YZwaD b  
    end X(AN)&L[  
    ;`j/D@H  
    if any( r>1 | r<0 ) #bnb ': f  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') +s[\g>i  
    end @4GA^h  
    ZCui Fm  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) &X>7n~@0  
        error('zernfun:RTHvector','R and THETA must be vectors.') (/{aJV  
    end  Lc2QXeo8  
    1 Y/$,Oa5  
    r = r(:); p .K*UP  
    theta = theta(:); nvq3*  
    length_r = length(r); 4B[D/kIg  
    if length_r~=length(theta) eEw.'B  
        error('zernfun:RTHlength', ... |(R5e  
              'The number of R- and THETA-values must be equal.') '-PC7"o  
    end 7=}F{U  
    -_A$DM!^=w  
    % Check normalization: }F=^O[  
    % -------------------- 6z,Dyy]tl  
    if nargin==5 && ischar(nflag) y-aRXF=W  
        isnorm = strcmpi(nflag,'norm'); ?A*Kg;IU  
        if ~isnorm oOU1{[  
            error('zernfun:normalization','Unrecognized normalization flag.') D{7w!z  
        end '0aG N<c  
    else 7'p8 a<x  
        isnorm = false; .TB"eUy  
    end @R6 ttx  
    <, @%*G1-  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% z%d#@w0X1  
    % Compute the Zernike Polynomials p3951-D  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% vDj;>VE2b  
    ^_5|BT@  
    % Determine the required powers of r: J >0b1  
    % ----------------------------------- 9.OA, 6  
    m_abs = abs(m); HTjkR*E  
    rpowers = []; ?8V UO x  
    for j = 1:length(n) z}4L=KR\v  
        rpowers = [rpowers m_abs(j):2:n(j)]; 8 ;gXg  
    end +b$S~0n   
    rpowers = unique(rpowers); D)b}f`  
    R[[ ,q:4  
    % Pre-compute the values of r raised to the required powers, n%%7KTqu  
    % and compile them in a matrix:  ht97s  
    % ----------------------------- \.{AAj^qD  
    if rpowers(1)==0 &m^@9E)S/  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); /8ynvhF#  
        rpowern = cat(2,rpowern{:}); @'FE2^~Jj  
        rpowern = [ones(length_r,1) rpowern]; ^z;JVrW  
    else "E*e2W  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ~W0(1# i  
        rpowern = cat(2,rpowern{:}); aE VsU|  
    end ,T{<vRj7_  
    %CnxjtTo  
    % Compute the values of the polynomials: i?@M  
    % -------------------------------------- >7Jr^o#|_x  
    y = zeros(length_r,length(n)); q ?j|K|%   
    for j = 1:length(n) "?}uQ5f  
        s = 0:(n(j)-m_abs(j))/2; . )XP\ m\  
        pows = n(j):-2:m_abs(j); #E7AmmqD%  
        for k = length(s):-1:1 G7LIdn=  
            p = (1-2*mod(s(k),2))* ... C|-pD  
                       prod(2:(n(j)-s(k)))/              ... Gc tsp2ndW  
                       prod(2:s(k))/                     ... TYns~X_PR  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 8AFczeg[[  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); -1|iz2^N  
            idx = (pows(k)==rpowers); +JyUe    
            y(:,j) = y(:,j) + p*rpowern(:,idx); n| !@1sd  
        end /1w2ehE<  
         QfjN"25_  
        if isnorm N!&:rK  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); ,Ds.x@p  
        end "UVFU-Z  
    end m6mwyom.  
    % END: Compute the Zernike Polynomials yzsab ^]  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% e( X|3h|  
    X"MU3]  
    % Compute the Zernike functions: Vy<HA*  
    % ------------------------------ V7Yaks  
    idx_pos = m>0; &} 6KPA;  
    idx_neg = m<0; R,2P3lv1v@  
    *>8ce-PV  
    z = y; U977#M Xf  
    if any(idx_pos) LtgXShp_!  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 2;3f=$3  
    end o(kM9G|  
    if any(idx_neg) *LC+ PZV@  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); sJx+8 -  
    end m} ?rJ  
    o|pT;1a"  
    % EOF zernfun
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) RkEN ,xWE  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. 2S3lsp5!  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated R8ONcG  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive C#V ~Y  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, <bck~E  
    %   and THETA is a vector of angles.  R and THETA must have the same 3-n1 9[zk  
    %   length.  The output Z is a matrix with one column for every P-value, 4674SzL  
    %   and one row for every (R,THETA) pair. &)F*@C-  
    % YV4#%I!<  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike |C%Pjl^YkV  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) 2I1uX&g  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) p{dwZ_gl  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 M6r^L6$N  
    %   for all p. Pl=]Srw  
    % b#)U UGmI  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 s MN*RKer  
    %   Zernike functions (order N<=7).  In some disciplines it is 98jN)Nl,oD  
    %   traditional to label the first 36 functions using a single mode 9Mp$8-=>7  
    %   number P instead of separate numbers for the order N and azimuthal <Peebv&v  
    %   frequency M. /.Nov  
    % f;SC{2f  
    %   Example: ;^Sr"v6r>u  
    % 1'v5/   
    %       % Display the first 16 Zernike functions s^OO^%b  
    %       x = -1:0.01:1; hJz):d>Im  
    %       [X,Y] = meshgrid(x,x); ixm&aW6<  
    %       [theta,r] = cart2pol(X,Y); ]j~"mFAP  
    %       idx = r<=1; ^\:8w0Y^  
    %       p = 0:15; 2 !" XzdD  
    %       z = nan(size(X)); KfCoe[Vv  
    %       y = zernfun2(p,r(idx),theta(idx)); &5{xXWJK  
    %       figure('Units','normalized') . v@>JZC  
    %       for k = 1:length(p) ~9\WFF/  
    %           z(idx) = y(:,k); iJxQB\x  
    %           subplot(4,4,k) i|)Su4Dw  
    %           pcolor(x,x,z), shading interp z\ss4  
    %           set(gca,'XTick',[],'YTick',[]) Q^K"8 ;  
    %           axis square +z9@:L  
    %           title(['Z_{' num2str(p(k)) '}']) ; |/leu8  
    %       end >N\0"F7.  
    % j;_c+w!P  
    %   See also ZERNPOL, ZERNFUN. OU4pjiLx  
    Awv`)"RAR  
    %   Paul Fricker 11/13/2006 RC|!+ TD  
    YKbCdLQ  
    \AUI|M;'  
    % Check and prepare the inputs: p R dk>Ph  
    % ----------------------------- ./j,Z$|  
    if min(size(p))~=1 p,pR!qC>  
        error('zernfun2:Pvector','Input P must be vector.') *=ZsqOHwG  
    end Hd7,ZHj3 ^  
    S_T^G` [  
    if any(p)>35 lJP1XzN_  
        error('zernfun2:P36', ... R`";Z$~{  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... kc'pN&]r:  
               '(P = 0 to 35).']) LWsP ya  
    end $P7iRM]  
    UX<Qcjm$e  
    % Get the order and frequency corresonding to the function number: YJS{i  
    % ---------------------------------------------------------------- e7fiGl  
    p = p(:); v1k)hFjPK  
    n = ceil((-3+sqrt(9+8*p))/2); 49Df?sx  
    m = 2*p - n.*(n+2); ~1m2#>  
    7J28JK  
    % Pass the inputs to the function ZERNFUN: !{n<K:x1  
    % ---------------------------------------- _ ~RpGX  
    switch nargin w:Jrmx  
        case 3 LIU} a5  
            z = zernfun(n,m,r,theta); KD1=Y80P  
        case 4 v]% WH~>  
            z = zernfun(n,m,r,theta,nflag); S|rgCh!h  
        otherwise 9%ii '{  
            error('zernfun2:nargin','Incorrect number of inputs.') B()/.w?A  
    end =z?%;4'|  
    nhSb~QqEh  
    % 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)  `G1&Z]z  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. v{i7h|e  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of T,fI BD:  
    %   order N and frequency M, evaluated at R.  N is a vector of #U=X NU}k  
    %   positive integers (including 0), and M is a vector with the 9p 4"r^  
    %   same number of elements as N.  Each element k of M must be a H4OhIxK  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) I9o6k?$K  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is w|mb4AyL{?  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix a</D_66  
    %   with one column for every (N,M) pair, and one row for every 'tN25$=V&W  
    %   element in R. M,j(=hRJ/E  
    % =5D nR  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- =S[yE]v^  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is sfr(/mp(  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to iFSJL,QZ3  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 KucV3-I  
    %   for all [n,m]. d1!i(MaV!  
    % DlMe5=n -u  
    %   The radial Zernike polynomials are the radial portion of the .%'(9E  
    %   Zernike functions, which are an orthogonal basis on the unit e@@?AB$n(  
    %   circle.  The series representation of the radial Zernike J68j=`Y  
    %   polynomials is UV}73Sp  
    % Mcw4!{l`  
    %          (n-m)/2 l ?Y_~Wuw  
    %            __ oHM ]  
    %    m      \       s                                          n-2s >Sa*`q3J  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r W$JebW<z(  
    %    n      s=0 `<^VR[Mx  
    % . .QB~  
    %   The following table shows the first 12 polynomials. oRN-xng  
    % }MR1^  
    %       n    m    Zernike polynomial    Normalization C\_zdADUb%  
    %       --------------------------------------------- a m-b!l!q^  
    %       0    0    1                        sqrt(2) s57N) 0kP  
    %       1    1    r                           2 JJV0R}z?TV  
    %       2    0    2*r^2 - 1                sqrt(6) IUGz =%[  
    %       2    2    r^2                      sqrt(6) r8xyd"Axy  
    %       3    1    3*r^3 - 2*r              sqrt(8) 6{x,*[v  
    %       3    3    r^3                      sqrt(8) eZ a:o1y  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) 3qHQX?a  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) eRbGZYrJ  
    %       4    4    r^4                      sqrt(10) 0Q1FL MLV  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) _2fkb=2@  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) ?3z-_8#  
    %       5    5    r^5                      sqrt(12) |VOg\[f  
    %       --------------------------------------------- l=`L7| ^/d  
    % Kzy/9  
    %   Example: e{({|V '  
    % |( (zTf  
    %       % Display three example Zernike radial polynomials 8pM>Co!  
    %       r = 0:0.01:1; Gx?+9C V  
    %       n = [3 2 5]; QVZD/shq  
    %       m = [1 2 1]; d lH$yub  
    %       z = zernpol(n,m,r); d {lP  
    %       figure RVtQ20e";r  
    %       plot(r,z) a\kb^D=T  
    %       grid on Ap&)6g   
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') IWVlrGyM  
    % `Yc _5&"  
    %   See also ZERNFUN, ZERNFUN2. x+? 9C  
    '"pd  
    % A note on the algorithm. ]!1OH |Ad  
    % ------------------------ y<W8Q<9  
    % The radial Zernike polynomials are computed using the series hlvt$Jwq  
    % representation shown in the Help section above. For many special F}Mhs17!|  
    % functions, direct evaluation using the series representation can ,p{`pma  
    % produce poor numerical results (floating point errors), because p\wJD1s  
    % the summation often involves computing small differences between JnD {J`:  
    % large successive terms in the series. (In such cases, the functions N\t1T(C|  
    % are often evaluated using alternative methods such as recurrence KHKS$D  
    % relations: see the Legendre functions, for example). For the Zernike t^=U*~  
    % polynomials, however, this problem does not arise, because the 7>o .0  
    % polynomials are evaluated over the finite domain r = (0,1), and pl*~kG=  
    % because the coefficients for a given polynomial are generally all L^kp8o^$  
    % of similar magnitude. `T ^G^7&  
    % &zL#hBE  
    % ZERNPOL has been written using a vectorized implementation: multiple fbrp#G71y  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] ?{o/I\\  
    % values can be passed as inputs) for a vector of points R.  To achieve >QQ(m\a$  
    % this vectorization most efficiently, the algorithm in ZERNPOL m:tiY [c>W  
    % involves pre-determining all the powers p of R that are required to l2v_?j-)x  
    % compute the outputs, and then compiling the {R^p} into a single Q+|{Bs)6i1  
    % matrix.  This avoids any redundant computation of the R^p, and Q>}2cDl  
    % minimizes the sizes of certain intermediate variables. ;SwC&.I  
    % 5`^o1nGO'  
    %   Paul Fricker 11/13/2006 ~KjJ\b)R  
    3 K/Df#  
    $<@\-vYvr@  
    % Check and prepare the inputs: I"L;L?\S  
    % ----------------------------- B,$l4m4  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Rz%e>)  
        error('zernpol:NMvectors','N and M must be vectors.') Z{-Lc68  
    end P=AS>N^yaL  
    XY7Qa!>7j  
    if length(n)~=length(m) @`u?bnx]e  
        error('zernpol:NMlength','N and M must be the same length.') uE_c4Hp  
    end wWW~_zP0  
    9G?ldp8  
    n = n(:); AWr}"r?s  
    m = m(:); qcB){p+UQ  
    length_n = length(n); A Ayv  
    8``;0}'PC  
    if any(mod(n-m,2)) Lrz3   
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') -Q e~)7  
    end ;uI~BV*3  
    HP2wtN{Zs  
    if any(m<0) Pd=,$UQp  
        error('zernpol:Mpositive','All M must be positive.') l?N`{ ,1^  
    end ucYkxi`x  
    f*((;*n ;  
    if any(m>n) 6/ g%\ka  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') T(X:Yw  
    end n"{X!(RIcx  
    JV"NZvjN7d  
    if any( r>1 | r<0 ) 4z4v\IpB  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') M.%shrJ/  
    end PB'0?b}fab  
    O??vm?eo  
    if ~any(size(r)==1) ,krS-.  
        error('zernpol:Rvector','R must be a vector.') </oY4$l'  
    end ,4F,:w  
    uZjI?Z.A  
    r = r(:); Z_z#QX>=D  
    length_r = length(r); K!{5 [G  
    DQ!J!ltQ  
    if nargin==4 AY2:[ 5cm  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); *$,+`+  
        if ~isnorm D!> d0k,Y  
            error('zernpol:normalization','Unrecognized normalization flag.') v#w_eqg  
        end E:A!wS`"  
    else cf8-]G?tK  
        isnorm = false; s3t!<9[m  
    end ;_JH:}j  
    W|c.l{A5Q  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% =G>(~+EA  
    % Compute the Zernike Polynomials d+2daKi  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% `7Ug/R<  
    Agy <j   
    % Determine the required powers of r: L/r{xS  
    % ----------------------------------- >q( 5ir  
    rpowers = []; 8[5|_Eh+  
    for j = 1:length(n) hY8#b)l~lu  
        rpowers = [rpowers m(j):2:n(j)]; 1 p\Ak  
    end hw,^G5m  
    rpowers = unique(rpowers); n.$(}A  
    (O5)wej   
    % Pre-compute the values of r raised to the required powers, =I4.Gf"~f  
    % and compile them in a matrix: Z!\@%`0$  
    % ----------------------------- :EHQ .^  
    if rpowers(1)==0 l8wF0|  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); w=J4zkWk  
        rpowern = cat(2,rpowern{:}); 2w1tK  
        rpowern = [ones(length_r,1) rpowern]; gbGTG(:1S  
    else vjK, I9  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); Vewzo1G2  
        rpowern = cat(2,rpowern{:}); =MSu3<y,  
    end Z;<ep@gy~  
    moO _-@i  
    % Compute the values of the polynomials: jxY-u+B  
    % -------------------------------------- Fj=NiZ=  
    z = zeros(length_r,length_n); gue(C(~.k_  
    for j = 1:length_n +WF.wP?y  
        s = 0:(n(j)-m(j))/2; B=zMYi  
        pows = n(j):-2:m(j); Pz473d  
        for k = length(s):-1:1 -<oZ)OfU  
            p = (1-2*mod(s(k),2))* ... b=LF%P  
                       prod(2:(n(j)-s(k)))/          ... c^S&F9/U*  
                       prod(2:s(k))/                 ... ]h@{6N'oNS  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... Dd/}Ya(Gi  
                       prod(2:((n(j)+m(j))/2-s(k))); 4 X`^{~  
            idx = (pows(k)==rpowers); JSjYC0e  
            z(:,j) = z(:,j) + p*rpowern(:,idx); lgT?{,>RkW  
        end L>nO:`>h  
         X <xqT  
        if isnorm _i@x@:_l  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); `Cj,HI_/*  
        end N(R,8GF5G  
    end  c!D> {N  
    WEC-<fN|Y\  
    % 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)  k{!iDZr&f,  
    * N2#{eF&]  
    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。 MX%|hIOpr  
    zV9 =  
    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)