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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 P" c@V,.  
    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! QQ?t^ptv  
     
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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 zS:2?VXxq  
    function z = zernfun(n,m,r,theta,nflag) ]9_gbQ   
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. 6uD<E  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N !<TkX/O  
    %   and angular frequency M, evaluated at positions (R,THETA) on the ]x)!Kd2>  
    %   unit circle.  N is a vector of positive integers (including 0), and !h1:AW_iz  
    %   M is a vector with the same number of elements as N.  Each element "U^m~N9k{  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) rp\`uj*D  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, ]R Ah['u|  
    %   and THETA is a vector of angles.  R and THETA must have the same `M~R4lr  
    %   length.  The output Z is a matrix with one column for every (N,M) 7"eK<qJ  
    %   pair, and one row for every (R,THETA) pair. s(py7{ ^K  
    % )bM,>x  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike LZ wCe$1  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), g}!{_z  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral JDf>Qg{  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 6y!U68L;B  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized U4 *u|A  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. G,>YzjMY`  
    % 0{vT`e'  
    %   The Zernike functions are an orthogonal basis on the unit circle. 7c"Csq/]I  
    %   They are used in disciplines such as astronomy, optics, and \^6[^\@[  
    %   optometry to describe functions on a circular domain. k.C&6*l!5;  
    % nA0%M1a  
    %   The following table lists the first 15 Zernike functions. %%ouf06.|  
    % %Bw:6Y4LZ  
    %       n    m    Zernike function           Normalization t+ w{uwEY  
    %       -------------------------------------------------- X<5fn+{]S:  
    %       0    0    1                                 1 /4O))}TX  
    %       1    1    r * cos(theta)                    2 wU|@fm"  
    %       1   -1    r * sin(theta)                    2 ~~Bks{"BS  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) N!c FUZ5]  
    %       2    0    (2*r^2 - 1)                    sqrt(3) R*vQvO%)h  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) S'5)K  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) j4,y+ 9U  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) 0g30nr)  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) : %& E58  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) Iuz_u2"C  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) (o*YGYC  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) PP{ 9Y Vr  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) =Rx4ZqTI|  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ~;9n6U  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) ,=\.L_'  
    %       -------------------------------------------------- Btxtu"]nJo  
    % +YZo-tE  
    %   Example 1: 8\68NG6o  
    % <oJ?J^  
    %       % Display the Zernike function Z(n=5,m=1) {ol7*%u  
    %       x = -1:0.01:1; $ (;:4  
    %       [X,Y] = meshgrid(x,x); KANR=G   
    %       [theta,r] = cart2pol(X,Y); A:ts_*  
    %       idx = r<=1; pMT7/y-  
    %       z = nan(size(X)); ~-Kx^3(#  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); 27 XM&ZrZ  
    %       figure 9HO9>^  
    %       pcolor(x,x,z), shading interp K@*+;6y@  
    %       axis square, colorbar B!pz0K*uG  
    %       title('Zernike function Z_5^1(r,\theta)') \t)va:y  
    % 7)QZ<fme  
    %   Example 2: 3N$@K"qM#  
    % 3" m]A/6C}  
    %       % Display the first 10 Zernike functions -XXsob}/8  
    %       x = -1:0.01:1; i=\)[;U  
    %       [X,Y] = meshgrid(x,x); C]2-V1,ZX  
    %       [theta,r] = cart2pol(X,Y); RAl/p9\A+  
    %       idx = r<=1; ic`BDkNO  
    %       z = nan(size(X)); rwJ U;wy  
    %       n = [0  1  1  2  2  2  3  3  3  3]; ~(v5p"]dj  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; UstUPO  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; (Ff}Y.4  
    %       y = zernfun(n,m,r(idx),theta(idx)); <L8|Wz  
    %       figure('Units','normalized') EA(4xj&:U  
    %       for k = 1:10 {Vj&i.2,  
    %           z(idx) = y(:,k); k*?T^<c3  
    %           subplot(4,7,Nplot(k)) Wz.iDRFl  
    %           pcolor(x,x,z), shading interp V K6D  
    %           set(gca,'XTick',[],'YTick',[]) {,JO}Dmu5  
    %           axis square ( jU $  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) peu9B gs  
    %       end *V hEl7  
    % jz_Y|"{`v  
    %   See also ZERNPOL, ZERNFUN2. eMnK@J  
    ! DOyOTR&3  
    %   Paul Fricker 11/13/2006 _|["}M"?  
    vN^.MR+<  
    > )< ?  
    % Check and prepare the inputs: Ez~5ax7x  
    % ----------------------------- 2, )>F"R  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) m|W17LhW{  
        error('zernfun:NMvectors','N and M must be vectors.') V3ozaVk;  
    end '>t&fzD0  
    dscah0T  
    if length(n)~=length(m) \4wMv[;7  
        error('zernfun:NMlength','N and M must be the same length.') _M/N_Fm  
    end OJpfiZ@Q_  
    : wS&3:h  
    n = n(:); %4m Nk}tyH  
    m = m(:); g_cED15  
    if any(mod(n-m,2)) Zpg;hj5_  
        error('zernfun:NMmultiplesof2', ... Ht;Rz*}  
              'All N and M must differ by multiples of 2 (including 0).') cZ_)'0  
    end vQLYWRXiA  
    2pdeJ  
    if any(m>n) rb-ao\  
        error('zernfun:MlessthanN', ... g0j)k6<6(Y  
              'Each M must be less than or equal to its corresponding N.') KV$&qM.  
    end A]!0Z:{h%  
    ZwBz\jmbP  
    if any( r>1 | r<0 ) ~BuzI9~7P  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') N_bgWQY  
    end QUW`Yc  
    } doAeTZ  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) *|Vf1R]  
        error('zernfun:RTHvector','R and THETA must be vectors.') Uo >aQk  
    end %urvX$r4K  
    }R<t=):  
    r = r(:); Q&:)D7m\)S  
    theta = theta(:); :@i+yN cV  
    length_r = length(r); iSO xQ  
    if length_r~=length(theta) G^t)^iI"'  
        error('zernfun:RTHlength', ... 56z>/`=  
              'The number of R- and THETA-values must be equal.') kMCP .D45;  
    end Zq 85q  
    cxs@ph&Wk  
    % Check normalization: fE~KWLm  
    % -------------------- )). =MTk  
    if nargin==5 && ischar(nflag) `[5xncZ-  
        isnorm = strcmpi(nflag,'norm'); &zF>5@fM  
        if ~isnorm n7bVL#Sq[  
            error('zernfun:normalization','Unrecognized normalization flag.') ((A@VcX  
        end #aL.E(%  
    else `15}jTi  
        isnorm = false; HNS^:X R  
    end m8F$h-  
    MS;^:t1`  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% n{!{,s  
    % Compute the Zernike Polynomials HSNj  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% =h4u N,  
    ;)FvTm'"\.  
    % Determine the required powers of r: ^WB[uFt-  
    % ----------------------------------- f4  S:L&  
    m_abs = abs(m); K>+ v" x  
    rpowers = []; w3,KqF  
    for j = 1:length(n) E~}H,*)  
        rpowers = [rpowers m_abs(j):2:n(j)]; Y9X,2L7V  
    end m+'1c}n^7  
    rpowers = unique(rpowers); o4p5`jOG@  
    [Ix6ArY  
    % Pre-compute the values of r raised to the required powers, HD KF>S_S  
    % and compile them in a matrix: Jn{)CZ  
    % ----------------------------- 9ia&/BT7"z  
    if rpowers(1)==0 -Ct+W;2  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); tRU/[?!  
        rpowern = cat(2,rpowern{:}); dY}5Kmt  
        rpowern = [ones(length_r,1) rpowern]; A x8>  
    else 0J'^<G TL  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); |.Vgk8oTl  
        rpowern = cat(2,rpowern{:}); OE(y$+L3_I  
    end (9]1p;  
    _DSDY$Ec  
    % Compute the values of the polynomials: LAc60^t1  
    % -------------------------------------- %TFsk  
    y = zeros(length_r,length(n)); xMk>r1Ud  
    for j = 1:length(n) =Ya^PAj '}  
        s = 0:(n(j)-m_abs(j))/2; =)+^y}xb  
        pows = n(j):-2:m_abs(j); >oq\`E  
        for k = length(s):-1:1 ]zj#X\  
            p = (1-2*mod(s(k),2))* ... n>u_>2Ikkj  
                       prod(2:(n(j)-s(k)))/              ... ltNI+G  
                       prod(2:s(k))/                     ... X$;x2mz nM  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... p+iNi4y@  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); @Pc7$qD%  
            idx = (pows(k)==rpowers); -%J9!(  
            y(:,j) = y(:,j) + p*rpowern(:,idx); _"p(/H  
        end jX4$PfOhR  
         O8#]7\)  
        if isnorm :7X4VHw/  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 0@?m"|G  
        end 2gK]w$H7!  
    end SN"Y@y)=  
    % END: Compute the Zernike Polynomials W>!:K^8]  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% !)oQ9,N  
    rEp\ld  
    % Compute the Zernike functions: VOj7Tz9UD  
    % ------------------------------ Yz2N(g[  
    idx_pos = m>0; ,1 H|{<  
    idx_neg = m<0; rYt|[Pk  
    wclj9&k  
    z = y; 2|?U%YrHWs  
    if any(idx_pos) N}/V2K]Q  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); /Zs_G=\>  
    end d1.@v;  
    if any(idx_neg) 56YqYu.  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); j9c:SP5  
    end Y*9vR~#H  
    Z L0Vx6Ph  
    % EOF zernfun
    离线niuhelen
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) %Qj$@.*:  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. v" #8^q  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated zE8_3UC  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive IX 6 jb"  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, ?2o+x D2  
    %   and THETA is a vector of angles.  R and THETA must have the same ^3vI NF  
    %   length.  The output Z is a matrix with one column for every P-value, Jon3ywd1Y  
    %   and one row for every (R,THETA) pair. dL'oIBp  
    % 30w(uF  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike ~~WY?I-  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) n=DmdQ}  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) g}6M+QNj  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 lhE]KdE3  
    %   for all p. i\ 7JQZ  
    % 'p!&&.%  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 Yt_tAm  
    %   Zernike functions (order N<=7).  In some disciplines it is !j  #8zN  
    %   traditional to label the first 36 functions using a single mode MsIaMW_  
    %   number P instead of separate numbers for the order N and azimuthal k=d _{2 ~  
    %   frequency M. 6Yva4Lv  
    % iX2exJto  
    %   Example: e GAto  
    % ?Nt m5(R  
    %       % Display the first 16 Zernike functions OP(om$xm  
    %       x = -1:0.01:1; ;x_T*} CH  
    %       [X,Y] = meshgrid(x,x); ~|~2B$JeV  
    %       [theta,r] = cart2pol(X,Y); u9q#L.Ij  
    %       idx = r<=1; :zIB3nT^  
    %       p = 0:15; YJz06E1 -9  
    %       z = nan(size(X)); 7/]Ra  
    %       y = zernfun2(p,r(idx),theta(idx)); " 5Pqvi  
    %       figure('Units','normalized') @~=d4Wj6  
    %       for k = 1:length(p) :Eg4^,QX  
    %           z(idx) = y(:,k); ooa"Th<  
    %           subplot(4,4,k) NU.4_cixb  
    %           pcolor(x,x,z), shading interp ae0Mf0<#)  
    %           set(gca,'XTick',[],'YTick',[]) NP\/9 8|1  
    %           axis square /&=y_%VR  
    %           title(['Z_{' num2str(p(k)) '}']) bB/fU7<{)u  
    %       end ~t*_  
    % k'"R;^~xg  
    %   See also ZERNPOL, ZERNFUN. A] 'XC"lS  
    ?` ebi|6  
    %   Paul Fricker 11/13/2006 [ p0_I7  
    E_D@ 7a  
    xOxyz6B\  
    % Check and prepare the inputs: m=iKu(2xRq  
    % ----------------------------- *g'%5i1ed  
    if min(size(p))~=1 ki `ur%h  
        error('zernfun2:Pvector','Input P must be vector.') 5 r<cna  
    end ?6\A$?  
    ? R[GSS1  
    if any(p)>35 XGnC8Be{4  
        error('zernfun2:P36', ... 5}9rpN{y  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... C?g*c  
               '(P = 0 to 35).']) >"]t4]GVf  
    end [--] ?Dr  
    C91'dM  
    % Get the order and frequency corresonding to the function number: rc{F17~vX  
    % ---------------------------------------------------------------- KAT^vbR  
    p = p(:); IQ~EL';<w  
    n = ceil((-3+sqrt(9+8*p))/2); f0{ tBD!%  
    m = 2*p - n.*(n+2); 4kNSF  
    i94)DWZ^  
    % Pass the inputs to the function ZERNFUN: i#U_g:~wC  
    % ---------------------------------------- '<C#"2  
    switch nargin O~Dm|hP  
        case 3 :G<~x8]k0  
            z = zernfun(n,m,r,theta); 2-duzc  
        case 4 `LTD|0;  
            z = zernfun(n,m,r,theta,nflag); DT]3q4__Q  
        otherwise S: g 2V  
            error('zernfun2:nargin','Incorrect number of inputs.') bwjLMWEVq  
    end t .XuH#  
    ,UT :wpc^i  
    % 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) N2EX`@_2  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. Y]}>he1/5  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of _7-P8"m  
    %   order N and frequency M, evaluated at R.  N is a vector of W^9=z~-h  
    %   positive integers (including 0), and M is a vector with the Z#_VxA>]v  
    %   same number of elements as N.  Each element k of M must be a R8u9tTW  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) .|J-(J<>[.  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is c~Z\|Y`#B  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix URxy*)  
    %   with one column for every (N,M) pair, and one row for every upF^k%<y:  
    %   element in R. p~t5PU*(  
    % ].*I Z  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- X;p4/ *U  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is HNL;s5gq  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to jE, oEt O;  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 kD4J{\  
    %   for all [n,m]. ~VO?PfxZ  
    % \]p[DYBY#  
    %   The radial Zernike polynomials are the radial portion of the ^Ea^t.c}_  
    %   Zernike functions, which are an orthogonal basis on the unit q+Qrc]>-f  
    %   circle.  The series representation of the radial Zernike \kksZ4,  
    %   polynomials is lJXihr  
    % P'CDV3+  
    %          (n-m)/2 %y2 i1^  
    %            __ 1ml{oqNj  
    %    m      \       s                                          n-2s ,~xX[uB  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r h*X u/aOg  
    %    n      s=0 ePwoza  
    % JlN<w  
    %   The following table shows the first 12 polynomials. b+ v!3|  
    % y@Ga9bI7  
    %       n    m    Zernike polynomial    Normalization >_um-w#C  
    %       --------------------------------------------- nQ^ <h.  
    %       0    0    1                        sqrt(2) K9N\E"6ZP  
    %       1    1    r                           2 [H2"z\\u  
    %       2    0    2*r^2 - 1                sqrt(6) zJa,kN|m  
    %       2    2    r^2                      sqrt(6) J |TA12s  
    %       3    1    3*r^3 - 2*r              sqrt(8) l?LP:;S  
    %       3    3    r^3                      sqrt(8) !8i[.EAT  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) G8voqP  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) C Ejf&n  
    %       4    4    r^4                      sqrt(10) /\1MG>#K  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) :%vD hMHa  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) LTc= D  
    %       5    5    r^5                      sqrt(12) zkRL'-  
    %       --------------------------------------------- :raYt5n1,y  
    % Qh. : N  
    %   Example: ZSg["`  
    % N=P+b%%:Z  
    %       % Display three example Zernike radial polynomials C~aNOe WR  
    %       r = 0:0.01:1; |LNAd:0  
    %       n = [3 2 5]; /SDDCZ`;|c  
    %       m = [1 2 1]; ^l"  
    %       z = zernpol(n,m,r); Q:~>$5Em5  
    %       figure %.*?i9}  
    %       plot(r,z) 6S2v3  
    %       grid on F)g.xQ  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') 89{@2TXR  
    % 6~j.S "  
    %   See also ZERNFUN, ZERNFUN2. K1K3s< y+  
    O*7Gl G  
    % A note on the algorithm. zf>r@>S!L  
    % ------------------------ hhVyz{u  
    % The radial Zernike polynomials are computed using the series HC*V\vz  
    % representation shown in the Help section above. For many special %SJ9Jr,  
    % functions, direct evaluation using the series representation can GGR hM1II  
    % produce poor numerical results (floating point errors), because E1e#E3Yq}s  
    % the summation often involves computing small differences between Q]}aZ4L  
    % large successive terms in the series. (In such cases, the functions zT5@wm  
    % are often evaluated using alternative methods such as recurrence T]tG,W1>i  
    % relations: see the Legendre functions, for example). For the Zernike Kqt,sJ  
    % polynomials, however, this problem does not arise, because the ^"!j m  
    % polynomials are evaluated over the finite domain r = (0,1), and a:(.{z?nM  
    % because the coefficients for a given polynomial are generally all !@x'?+   
    % of similar magnitude. ]7`)|PJ  
    % S%7^7MSqA  
    % ZERNPOL has been written using a vectorized implementation: multiple ?u9JRXj%  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] _XqD3?yH4  
    % values can be passed as inputs) for a vector of points R.  To achieve =fZ)2q  
    % this vectorization most efficiently, the algorithm in ZERNPOL MQv2C@K9F  
    % involves pre-determining all the powers p of R that are required to 'y?(s+  
    % compute the outputs, and then compiling the {R^p} into a single u~9gR@e2{  
    % matrix.  This avoids any redundant computation of the R^p, and /J"U`/ {4  
    % minimizes the sizes of certain intermediate variables. 6(`Bl$M9  
    % )`ZTu -|  
    %   Paul Fricker 11/13/2006 clZ jb  
    u-a*fT  
    mGmkeD'  
    % Check and prepare the inputs: Nuw_,-h  
    % ----------------------------- 2Rp5 E^s  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) iS{8cN3R  
        error('zernpol:NMvectors','N and M must be vectors.') Y!6,ty'  
    end 9(>l trA  
    *$ihNX]YG  
    if length(n)~=length(m) <XV\8Y+n  
        error('zernpol:NMlength','N and M must be the same length.') V+mTo^  
    end rN'')n/F  
    _[,oP s:+  
    n = n(:); NiwJ$Ah~X  
    m = m(:);  ?vgHu  
    length_n = length(n); zd+_ BPT  
    Z[DiLXHL  
    if any(mod(n-m,2)) Ed%8| M3  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') "]dNN{Wka  
    end RQZ|:SvV  
    mE"?{~XVL  
    if any(m<0) ?=%Q$|]-  
        error('zernpol:Mpositive','All M must be positive.') Q-X<zn  
    end 4&Uq\,nx  
    z@nJ-*'U8  
    if any(m>n) y~JCSzpU  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') ^&'&Y>  
    end \k{UqU+s  
    s:j"8ZH  
    if any( r>1 | r<0 ) `JGV3nN  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') 38wt=0br  
    end h]rF2 B  
    )19As8rL/o  
    if ~any(size(r)==1) cC.=,n  
        error('zernpol:Rvector','R must be a vector.') mr+J#  
    end K0#kW \4`  
    2l)J,z  
    r = r(:); Cz2OGM*mz?  
    length_r = length(r); !H`Q^Xf}  
    Kp +Lk  
    if nargin==4 (GVH#}uB  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); kMGK 8y  
        if ~isnorm nwk66o:|  
            error('zernpol:normalization','Unrecognized normalization flag.') y`?{ 2#1H  
        end gHLBtl/  
    else :>U2yI  
        isnorm = false; YlW~  
    end c$)Y$@D  
    6t0!a@t  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% }E5oa\ 1u  
    % Compute the Zernike Polynomials sE4= 2p`x  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 47R4gs#W  
    !%' 1 x2?  
    % Determine the required powers of r: 8]6u]3q#  
    % ----------------------------------- ESk<*-  
    rpowers = []; pSQ)DqW  
    for j = 1:length(n) biCX: m+_?  
        rpowers = [rpowers m(j):2:n(j)]; LFsrqdzJ  
    end 7Vf2Qx1_  
    rpowers = unique(rpowers); Ex'6 WN~kD  
    \bze-|C  
    % Pre-compute the values of r raised to the required powers, CKShz]1  
    % and compile them in a matrix: as1ZLfN.  
    % ----------------------------- z z@;UbD"  
    if rpowers(1)==0 C3n_'O  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); $2uZdl8Rvj  
        rpowern = cat(2,rpowern{:}); }QszOi\fV1  
        rpowern = [ones(length_r,1) rpowern]; K-&&%Id6R  
    else OW (45  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); L lmdydC%  
        rpowern = cat(2,rpowern{:}); sh []OSM  
    end vc.:du  
    tZS-e6*S  
    % Compute the values of the polynomials: ;P9P2&c8c  
    % -------------------------------------- )J?Nfi%  
    z = zeros(length_r,length_n); ~$1Zw&X  
    for j = 1:length_n {{b&l!  
        s = 0:(n(j)-m(j))/2; L-}>;M$Y)  
        pows = n(j):-2:m(j); 5"4O_JQ  
        for k = length(s):-1:1 u~#QvA~]  
            p = (1-2*mod(s(k),2))* ... cd8~y  
                       prod(2:(n(j)-s(k)))/          ... W+3ZuAP\n  
                       prod(2:s(k))/                 ... 9Foo8e  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... G3{t{XkV  
                       prod(2:((n(j)+m(j))/2-s(k))); 'J R2@W`]]  
            idx = (pows(k)==rpowers); @1#QbNp#  
            z(:,j) = z(:,j) + p*rpowern(:,idx); 2BLcun  
        end +@]b}W  
         <qD/ #$   
        if isnorm DPuz'e*  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); W+ tI(JZ  
        end (?SK< 4!  
    end x0^O?UR  
    h+e Oe}  
    % 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)  lej-,HX  
    `.3!  
    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。 oej5bAi  
    C[f'1O7  
    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)