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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 'q{d? K  
    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! ;/(<yu48  
     
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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 Q1u/QA:z7  
    function z = zernfun(n,m,r,theta,nflag) HpR(DG) ?  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. *ta?7uSiT  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 7\ kixfEg  
    %   and angular frequency M, evaluated at positions (R,THETA) on the s92SN F}g  
    %   unit circle.  N is a vector of positive integers (including 0), and J4q_}^/2w  
    %   M is a vector with the same number of elements as N.  Each element O" ,*N  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) W3 2]#M=  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, Tj,1]_`=V$  
    %   and THETA is a vector of angles.  R and THETA must have the same T8-,t];i  
    %   length.  The output Z is a matrix with one column for every (N,M) I@o42%w2  
    %   pair, and one row for every (R,THETA) pair. n_MY69W  
    % 6@geakq  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 0m&W: c  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), NT<vs"<B  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral /nVGr]t_pj  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, b&E9xD/;r  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized xPorlX)zW  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. MXGz_Db4'  
    % Kz2s{y~?  
    %   The Zernike functions are an orthogonal basis on the unit circle. #Bi8>S  
    %   They are used in disciplines such as astronomy, optics, and pn-`QB:{h  
    %   optometry to describe functions on a circular domain. qfl#ki`,  
    % KBy*QA  
    %   The following table lists the first 15 Zernike functions. /zZ";4  
    % y8CH=U[  
    %       n    m    Zernike function           Normalization "vN~7%  
    %       -------------------------------------------------- pF}WMt  
    %       0    0    1                                 1 HMPb%'U~  
    %       1    1    r * cos(theta)                    2 @w5x;uB|%G  
    %       1   -1    r * sin(theta)                    2 VJ()sbl{k  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) VVDd39q  
    %       2    0    (2*r^2 - 1)                    sqrt(3) )lDmYt7me  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) xJ|_R,>.H  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) o-r00H|  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) qB8R4wCf  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) CH+mzy  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) ^%jk.*  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) e|S_B*1*0  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) \9`76*X6 c  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) s2t9+ZA+s  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) +/4wioGm  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) R.$1aqA}  
    %       -------------------------------------------------- uo[W|Q  
    % p^THoF'~T  
    %   Example 1: r`5svY  
    % 5!*@gn  
    %       % Display the Zernike function Z(n=5,m=1) :DoE_  
    %       x = -1:0.01:1; y;xY74Nq  
    %       [X,Y] = meshgrid(x,x); )H| cri~D  
    %       [theta,r] = cart2pol(X,Y); II) K0<  
    %       idx = r<=1;  y)GH=@b  
    %       z = nan(size(X)); l[u=_uaYl  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); F0]xc  
    %       figure {?hpW+1,#  
    %       pcolor(x,x,z), shading interp |cIv&\ x  
    %       axis square, colorbar W2T6JFv  
    %       title('Zernike function Z_5^1(r,\theta)') ?3Y~q;I]O  
    % G7uYkJO  
    %   Example 2: O"V;otlC  
    % o#9 Q   
    %       % Display the first 10 Zernike functions lNba[;_  
    %       x = -1:0.01:1; jSd[  
    %       [X,Y] = meshgrid(x,x); cbaa*qoU  
    %       [theta,r] = cart2pol(X,Y); 35/K9l5  
    %       idx = r<=1; jU0E=;1  
    %       z = nan(size(X)); SBh"^q  
    %       n = [0  1  1  2  2  2  3  3  3  3]; 28x:]5=jb  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; Z )'gj  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; P]%)c6Uh  
    %       y = zernfun(n,m,r(idx),theta(idx)); y ]D[JX[  
    %       figure('Units','normalized') Nn='9s9F?}  
    %       for k = 1:10 Wf:LYL  
    %           z(idx) = y(:,k); br%l>Y\"  
    %           subplot(4,7,Nplot(k)) #$ooV1E  
    %           pcolor(x,x,z), shading interp 5N(OW:M  
    %           set(gca,'XTick',[],'YTick',[]) %_%Bb Qf  
    %           axis square O 8XHaVLg3  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) iOJ5KXrAO  
    %       end $(+#$F<eo+  
    % b!oj3|9  
    %   See also ZERNPOL, ZERNFUN2. ?4cj"i  
    Yaj}_M-  
    %   Paul Fricker 11/13/2006 }*?,&9/_)  
    E{BX $R_8  
    \[&&4CN{  
    % Check and prepare the inputs: s`gfz}/  
    % ----------------------------- 8F9x2CM-[C  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) qT~a`ou:  
        error('zernfun:NMvectors','N and M must be vectors.') 6_g:2=6S  
    end #7['M;_  
    MFQyB+Z  
    if length(n)~=length(m) b} FhC"'i  
        error('zernfun:NMlength','N and M must be the same length.') 2{<o1x,Ym  
    end (\UpJlW  
    -car>hQq  
    n = n(:); ?azcWf z0  
    m = m(:); qP BOt;N  
    if any(mod(n-m,2)) i2+_~$f  
        error('zernfun:NMmultiplesof2', ... <b:xyHS  
              'All N and M must differ by multiples of 2 (including 0).') 7~Z(dTdSG  
    end >R}G  
    ;z T3Fv\  
    if any(m>n) A DVUx}  
        error('zernfun:MlessthanN', ... `j8pgnY>5~  
              'Each M must be less than or equal to its corresponding N.') Ey=ymf.}  
    end N}>[To3  
    Xo$SQ0K  
    if any( r>1 | r<0 ) +U)4V}S)  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') 0< 93i   
    end {krBAz&  
    +o?;7  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) z(Z7[#.  
        error('zernfun:RTHvector','R and THETA must be vectors.') Zc'^iDAY  
    end /@B2-.w  
    Qk >9o  
    r = r(:); C8x9 Jrc  
    theta = theta(:); lffw "  
    length_r = length(r); vi28u xc  
    if length_r~=length(theta) nyetK  
        error('zernfun:RTHlength', ... BA[ uO3\4  
              'The number of R- and THETA-values must be equal.') ,^RZ1tLz  
    end IhRdn1&  
    6-z(34&N  
    % Check normalization: g(9kc<`3'D  
    % -------------------- Gt)ij?~  
    if nargin==5 && ischar(nflag) /24}>oAH  
        isnorm = strcmpi(nflag,'norm'); hpgOsF9Lh  
        if ~isnorm yf7|/M  
            error('zernfun:normalization','Unrecognized normalization flag.') l(W?]{C[%  
        end C^;>HAK|F  
    else $01csj  
        isnorm = false; TF9A4  
    end W,"Re,`H  
    S+"Bq:u"  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% E]v?:!!ds  
    % Compute the Zernike Polynomials a?yU;IKJ  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {Kf5a m  
    TB-dV'w  
    % Determine the required powers of r: e{9~m  
    % ----------------------------------- /EG'I{oC  
    m_abs = abs(m); Y'5(exW  
    rpowers = []; cUr!U\X[  
    for j = 1:length(n) w51l;2$des  
        rpowers = [rpowers m_abs(j):2:n(j)]; N6v?Qzvi  
    end 7377g'jL  
    rpowers = unique(rpowers); ?J,,RK.  
    e"_kH_7sv  
    % Pre-compute the values of r raised to the required powers, *{P/3yH  
    % and compile them in a matrix: Oxa8ue?  
    % ----------------------------- &=MVX>[  
    if rpowers(1)==0 <nb%$2r1  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 8d2\H*a9~  
        rpowern = cat(2,rpowern{:}); H>W8F2VT  
        rpowern = [ones(length_r,1) rpowern]; C fM[<w   
    else YYT#{>&  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ;b:'i& r  
        rpowern = cat(2,rpowern{:}); D6H?*4f]  
    end R7U%v"F>`  
    O@4J=P=w  
    % Compute the values of the polynomials: gO)":!_n W  
    % -------------------------------------- e#,(a  
    y = zeros(length_r,length(n)); DIw_"$'At  
    for j = 1:length(n) lx=tOfj8  
        s = 0:(n(j)-m_abs(j))/2; g8l6bh$}  
        pows = n(j):-2:m_abs(j); P%H  Dz  
        for k = length(s):-1:1 ~\8(+qIv%f  
            p = (1-2*mod(s(k),2))* ... kiyc^s  
                       prod(2:(n(j)-s(k)))/              ... -- FzRO{D  
                       prod(2:s(k))/                     ... gnjhy1o  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... s5rD+g]E`  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); wMj #.Jh  
            idx = (pows(k)==rpowers); o<%0|n_O&  
            y(:,j) = y(:,j) + p*rpowern(:,idx); /aMOZ=,q}  
        end ~ !!\#IX  
         TYb$+uY  
        if isnorm B~7!v${  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 3F@P$4!#l  
        end o{! :N>(  
    end ]gg(Z!|iQ  
    % END: Compute the Zernike Polynomials vXRY/Zzj1  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% )Z:D}r8[  
    =u?aP}zc  
    % Compute the Zernike functions: [!yA#{xl,  
    % ------------------------------ ~mARgv  
    idx_pos = m>0; B ~N3k  
    idx_neg = m<0; \0d'y#Gp*  
    q :TNf\/o  
    z = y; e1LIk1`p  
    if any(idx_pos) |5tZ*$nGa  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); xE w\'tH  
    end 4|E^ #C  
    if any(idx_neg) uBa<5YDF  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); R-j*fO}  
    end Jp_#pV*}:  
    uT4|43< G  
    % 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) Vv}R S@4U  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. e))L&s  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated 9+^)?JUYll  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive .{h"0<x  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, 3WJk04r  
    %   and THETA is a vector of angles.  R and THETA must have the same ERV]N:(  
    %   length.  The output Z is a matrix with one column for every P-value, kFWwz^x  
    %   and one row for every (R,THETA) pair. ovTL'j!  
    % *uhQP47B  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike 0X5cn 0L^  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) M% \ T5  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) &,k!,<IF  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 3- Kgz  
    %   for all p. );7 d_#  
    % j#n ]q{s4  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 *x|%Nua"  
    %   Zernike functions (order N<=7).  In some disciplines it is N3|:MMl  
    %   traditional to label the first 36 functions using a single mode A _zCSRF,  
    %   number P instead of separate numbers for the order N and azimuthal d#rr7O  
    %   frequency M. Lj<TzPzg*  
    % F,wB6Cw  
    %   Example: `Npa/Q  
    % j8` B  
    %       % Display the first 16 Zernike functions {r&mNbz  
    %       x = -1:0.01:1; #ODP+>-IjB  
    %       [X,Y] = meshgrid(x,x); {fR\yWkt?  
    %       [theta,r] = cart2pol(X,Y); t<p#u=jOa  
    %       idx = r<=1; vPNbV  
    %       p = 0:15; h9H z6 >  
    %       z = nan(size(X)); K$,Zg  
    %       y = zernfun2(p,r(idx),theta(idx)); T(D6'm:X  
    %       figure('Units','normalized') rUb{iU;~m  
    %       for k = 1:length(p) ZL6HD n!  
    %           z(idx) = y(:,k); gu(:'5cX  
    %           subplot(4,4,k) c`!e#w  
    %           pcolor(x,x,z), shading interp d&FXndC4F  
    %           set(gca,'XTick',[],'YTick',[]) .KA-=$~J1  
    %           axis square 3U@jw,K!{A  
    %           title(['Z_{' num2str(p(k)) '}']) )[Tm[o?Y.  
    %       end xSmG,}3mF  
    % SH?McBxS  
    %   See also ZERNPOL, ZERNFUN. F] c\Qt  
    -qIi.]/f"9  
    %   Paul Fricker 11/13/2006 JY,$B-l  
    ;'n%\*+fHH  
    .dlsiBh  
    % Check and prepare the inputs: !cyrt<  
    % ----------------------------- 9Y:I)^ek  
    if min(size(p))~=1 !/XNpQP  
        error('zernfun2:Pvector','Input P must be vector.') w.?4}'DK  
    end } {1IB  
    = j1Jl^[  
    if any(p)>35 og}Ri!^  
        error('zernfun2:P36', ... gXdMGO>  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... Tz @=N]D  
               '(P = 0 to 35).']) " ]S  
    end @|b-X? `  
    W@T \i2r$z  
    % Get the order and frequency corresonding to the function number: Jl~ *@0(  
    % ---------------------------------------------------------------- z{rV|vQ  
    p = p(:); QoZV 6  
    n = ceil((-3+sqrt(9+8*p))/2); X0;u7g2Yz  
    m = 2*p - n.*(n+2); h3?>jE=H  
    ( s3k2Z  
    % Pass the inputs to the function ZERNFUN: zZ=SAjT QP  
    % ---------------------------------------- a2Ak?W1  
    switch nargin f$C{Z9_SX  
        case 3 ~Gu$E qQ  
            z = zernfun(n,m,r,theta); 8fXiadP#  
        case 4  k[r^@|  
            z = zernfun(n,m,r,theta,nflag); YRu@; `  
        otherwise ~1uQyt  
            error('zernfun2:nargin','Incorrect number of inputs.') e|]e\Or>  
    end k>($[;k|b  
    !Km[Qw k-  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) F/oqYk9`  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. ?:"ABkL|+Y  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of mLhM_=  
    %   order N and frequency M, evaluated at R.  N is a vector of f^F;`;z  
    %   positive integers (including 0), and M is a vector with the rwP#Yj[BK+  
    %   same number of elements as N.  Each element k of M must be a -<#) ]um  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) 7gC?<;\0  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is h4;kjr}h}  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix 1/}H 0\9'  
    %   with one column for every (N,M) pair, and one row for every S/]\GG{  
    %   element in R. b80#75Bj>  
    %  &K/?#  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- FLi'}C  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is J2z/XHS  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to <*( Z}p  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 i~';1 .g  
    %   for all [n,m]. n5}]C{s'  
    % I*u3 e  
    %   The radial Zernike polynomials are the radial portion of the '(vZfzc{J  
    %   Zernike functions, which are an orthogonal basis on the unit 0AB a&'h  
    %   circle.  The series representation of the radial Zernike K\K& K~Z  
    %   polynomials is 8b/$Qp4d  
    % J"r?F0  
    %          (n-m)/2 BSm"]!D8*  
    %            __ :33@y%>L  
    %    m      \       s                                          n-2s `ASDUgx Mq  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r '_c/CNs  
    %    n      s=0 .+:iAnf  
    % 9j 2t|D4uT  
    %   The following table shows the first 12 polynomials. v=llg ^  
    % t13V>9to  
    %       n    m    Zernike polynomial    Normalization \g}]u(zg%  
    %       --------------------------------------------- y7HFmGM  
    %       0    0    1                        sqrt(2) Os9SfL  
    %       1    1    r                           2 6 U.Jaai:  
    %       2    0    2*r^2 - 1                sqrt(6) h^3gYL7O6  
    %       2    2    r^2                      sqrt(6) 82LE9<4A  
    %       3    1    3*r^3 - 2*r              sqrt(8) F^%w%E\  
    %       3    3    r^3                      sqrt(8) j`_S%E%X  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) F-m%d@P&X  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) d/d)MoaJ*t  
    %       4    4    r^4                      sqrt(10) B;9,Qbb  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) Dz}i-tw+  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) &=@{`2&  
    %       5    5    r^5                      sqrt(12) &_L@hsm  
    %       --------------------------------------------- !Tnjha*  
    % wps/{h,  
    %   Example: "Z@P&jl  
    % !ku}vTe  
    %       % Display three example Zernike radial polynomials 63fYX"  
    %       r = 0:0.01:1; gVG^R02#<k  
    %       n = [3 2 5]; Xh"9Bcjf  
    %       m = [1 2 1];  ~~>m  
    %       z = zernpol(n,m,r); "| '~y}v_  
    %       figure "| nXR8t.r  
    %       plot(r,z) L)'G_)Sl  
    %       grid on 0%f}Q7*R  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') V(S7mA:T  
    % T@W:@,34  
    %   See also ZERNFUN, ZERNFUN2. ^6W}ZLp  
    I5"wa:Z  
    % A note on the algorithm. (5$Ge$  
    % ------------------------ "tyRnUP  
    % The radial Zernike polynomials are computed using the series 3BM z{ny=  
    % representation shown in the Help section above. For many special b**vUt\  
    % functions, direct evaluation using the series representation can MzvhE0ab  
    % produce poor numerical results (floating point errors), because ?mH=3 :~  
    % the summation often involves computing small differences between :C5w5 Vnj  
    % large successive terms in the series. (In such cases, the functions *V&M5  
    % are often evaluated using alternative methods such as recurrence H oQb.Z  
    % relations: see the Legendre functions, for example). For the Zernike ";/]rwHa)  
    % polynomials, however, this problem does not arise, because the B!'K20"gF  
    % polynomials are evaluated over the finite domain r = (0,1), and do" m=y  
    % because the coefficients for a given polynomial are generally all lelmX  
    % of similar magnitude. <'\Nv._2a  
    % Jn hdZa  
    % ZERNPOL has been written using a vectorized implementation: multiple <'=!f6Wh  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] A{_CU-,  
    % values can be passed as inputs) for a vector of points R.  To achieve S1=P-Ao  
    % this vectorization most efficiently, the algorithm in ZERNPOL W2{w<<\$3}  
    % involves pre-determining all the powers p of R that are required to S#ryEgc]  
    % compute the outputs, and then compiling the {R^p} into a single dgVGP_~  
    % matrix.  This avoids any redundant computation of the R^p, and ~ 5}t;  
    % minimizes the sizes of certain intermediate variables. <#0i*PM_  
    % J^8j|%h%e  
    %   Paul Fricker 11/13/2006 7C|AiSH  
    P& 1$SWNyW  
    - (s0f  
    % Check and prepare the inputs: YnpN -Y%g  
    % ----------------------------- S}C[  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) YJ~<pH  
        error('zernpol:NMvectors','N and M must be vectors.') c]pz&  
    end  +P(*S  
    Fo3*PcUv  
    if length(n)~=length(m) U5"u h} 3  
        error('zernpol:NMlength','N and M must be the same length.') )"TVR{I%B  
    end =z}PR1X!  
    H&s`Xr  
    n = n(:); YKe&Ph.  
    m = m(:); uzp\V 39  
    length_n = length(n); hWly8B[I  
    SS/vw%  
    if any(mod(n-m,2)) e=LrgRy+  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') (aYu[ML  
    end Jxl'!8t  
    D5"5`w=C  
    if any(m<0) $#V'm{Hh  
        error('zernpol:Mpositive','All M must be positive.') 8L[+$g`  
    end &S="]*Z  
    RxP~%oADw  
    if any(m>n) !$Uo$?gC  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') 4j3q69TZR  
    end +"84.PZ  
    X1w11Z7o  
    if any( r>1 | r<0 ) HD<$0M|  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') K qJE?caw  
    end WSpF/Wwc  
     ]j0+4w  
    if ~any(size(r)==1) ?q6#M&|j/I  
        error('zernpol:Rvector','R must be a vector.') | >}CoR7  
    end co,0@.i  
    cK|Uwzif d  
    r = r(:); @. sn  
    length_r = length(r); +kWWx#L#  
    &wi+)d  
    if nargin==4 F,vkk{Z>  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); 7fq Q  
        if ~isnorm ;[o:VuTs  
            error('zernpol:normalization','Unrecognized normalization flag.') w!UF^~  
        end ET^?>YsA  
    else ]D nAW'm  
        isnorm = false; JOuy_n  
    end TJYhgna  
    v,n 8$,  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *Y8 5ev q  
    % Compute the Zernike Polynomials l]wfL;u  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% bF9.k  
    5_y w  
    % Determine the required powers of r: Q>L(=j2t  
    % ----------------------------------- x((u  
    rpowers = []; <5L99<E  
    for j = 1:length(n) ]$#bNt/p  
        rpowers = [rpowers m(j):2:n(j)]; DD/B\  
    end $mK;{9Z  
    rpowers = unique(rpowers); Uic  
    \i?bt0bM  
    % Pre-compute the values of r raised to the required powers, hXFT(J=  
    % and compile them in a matrix: S\ak(<X  
    % ----------------------------- a6zWg7 PN  
    if rpowers(1)==0 In4VS:dD  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); kmW/{I9,ua  
        rpowern = cat(2,rpowern{:}); @@ @}FV&  
        rpowern = [ones(length_r,1) rpowern]; <IIz-6*V  
    else U _pPI$ =  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); Lp%J:ogV`  
        rpowern = cat(2,rpowern{:}); p+Q9?9  
    end Tf=1p1!3  
    e 6wevK\  
    % Compute the values of the polynomials: ")9^  
    % -------------------------------------- qbQdx Kk  
    z = zeros(length_r,length_n); ?Xpk"N7  
    for j = 1:length_n h>>~Bi  
        s = 0:(n(j)-m(j))/2; t[;-gi,,  
        pows = n(j):-2:m(j); 6  _V1s1F  
        for k = length(s):-1:1 pj7a l;  
            p = (1-2*mod(s(k),2))* ... F,as>X#  
                       prod(2:(n(j)-s(k)))/          ... a`:F07r  
                       prod(2:s(k))/                 ... !d 4DTo  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... m2~`EL>  
                       prod(2:((n(j)+m(j))/2-s(k))); <FR!x#!   
            idx = (pows(k)==rpowers); #"oLz"{  
            z(:,j) = z(:,j) + p*rpowern(:,idx); }@.@k6`n  
        end W)Mz1v #s  
         l ?b*T#uIk  
        if isnorm zk1]?  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); y%9Hu  
        end +P+h$gQ  
    end -p0*R<t  
    1Z?uT[kR  
    % 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)  1;B&R89}  
    =1VZcLNt  
    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。 f87XE";:A  
    jaavh6h)  
    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)