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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 Az.Y-O<$\  
    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! l"cO@.T3  
     
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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 d>aZpJ[.  
    function z = zernfun(n,m,r,theta,nflag) T2Vj &EA@  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. 4+W}TKw  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N U ^,ld`  
    %   and angular frequency M, evaluated at positions (R,THETA) on the {#;6$dU;(  
    %   unit circle.  N is a vector of positive integers (including 0), and SOUA,4  
    %   M is a vector with the same number of elements as N.  Each element J*;t{M5  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) jAJkCCG  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, R]r~TJ o  
    %   and THETA is a vector of angles.  R and THETA must have the same 2N]y)S_<V  
    %   length.  The output Z is a matrix with one column for every (N,M) =_UPZ]  
    %   pair, and one row for every (R,THETA) pair. -~aVt~{k/  
    % #A))#sT'R  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike M9N|Ql  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 2+^#<Uok  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral |4'E&(BU-  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, tl4;2m3w  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized z^oi15D|{  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. LD6fi  
    % Z@h]dU5%a  
    %   The Zernike functions are an orthogonal basis on the unit circle. 4s"HO/  
    %   They are used in disciplines such as astronomy, optics, and QHQj6]  
    %   optometry to describe functions on a circular domain. g =%W"v  
    % '2 )d9_ w  
    %   The following table lists the first 15 Zernike functions. 62zlO{ >rJ  
    % 3oIoQj+D  
    %       n    m    Zernike function           Normalization b"zq3$6*  
    %       -------------------------------------------------- J L Z  
    %       0    0    1                                 1 .58 AXg  
    %       1    1    r * cos(theta)                    2 MdyH/.Te  
    %       1   -1    r * sin(theta)                    2 pkT a^I  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) Y# lE  
    %       2    0    (2*r^2 - 1)                    sqrt(3) oFsMQ Py  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) U "}Kth  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) 6F<L4*4U  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) z;3}GxE-si  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) ~pw_*AN  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) ,fN iZ  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) lz>5bR'  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) G)putk@   
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) ^6`R:SV4Gx  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) x7/2e{p uu  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) # ._!.P  
    %       -------------------------------------------------- dk.da&P  
    % 9XoKOR(  
    %   Example 1: [&39Yv.k,7  
    % 8"4`W~ 3  
    %       % Display the Zernike function Z(n=5,m=1) ``Nj Nd  
    %       x = -1:0.01:1; PEBQ|k8g&  
    %       [X,Y] = meshgrid(x,x);  CZuxH  
    %       [theta,r] = cart2pol(X,Y); $Qm;F% >  
    %       idx = r<=1; ^*0;Z<_  
    %       z = nan(size(X)); aE;!mod  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); m\VJ=  
    %       figure w S;(u[W  
    %       pcolor(x,x,z), shading interp qS7*.E~j|]  
    %       axis square, colorbar sX=!o})0  
    %       title('Zernike function Z_5^1(r,\theta)') crmnh4-  
    % q|j;dI&  
    %   Example 2: `t8e2?GH  
    % Pjx9@i  
    %       % Display the first 10 Zernike functions m t*v@'l.  
    %       x = -1:0.01:1; 0W>O,%z&P#  
    %       [X,Y] = meshgrid(x,x); GY4yZa  
    %       [theta,r] = cart2pol(X,Y); 7kb`o y;(^  
    %       idx = r<=1; Onk~1ks:  
    %       z = nan(size(X)); U} g%`<  
    %       n = [0  1  1  2  2  2  3  3  3  3]; rKjQEO$yi  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; J=Jw"? f  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; F:H76O`8  
    %       y = zernfun(n,m,r(idx),theta(idx)); |Rl|Th  
    %       figure('Units','normalized') 7'<4'BGzl]  
    %       for k = 1:10 Mr&]RTEE  
    %           z(idx) = y(:,k); /wK7l-S  
    %           subplot(4,7,Nplot(k)) V*/))n?  
    %           pcolor(x,x,z), shading interp Mc\lzq8\ 1  
    %           set(gca,'XTick',[],'YTick',[]) ]f-e/8$`@  
    %           axis square CBvBBt*  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) "=RB #  
    %       end {=(4  
    % }x8fXdd  
    %   See also ZERNPOL, ZERNFUN2. z=u4&x|xA  
    =CJs&Qa2  
    %   Paul Fricker 11/13/2006 ;1y\!f3#V~  
    q`{.2yV  
    )XNcy"   
    % Check and prepare the inputs: $iB(N ZV  
    % ----------------------------- } M1<a4~  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 9R E;50h  
        error('zernfun:NMvectors','N and M must be vectors.') Oc8+an1m  
    end 3b_#xr-  
    ROfmAc  
    if length(n)~=length(m) 1n5&PNu  
        error('zernfun:NMlength','N and M must be the same length.') jALo;PDJ  
    end & v`kyc  
    : Z.mM5  
    n = n(:); y"]> Rr  
    m = m(:); n^A=ar.  
    if any(mod(n-m,2)) Pgo5&SQb  
        error('zernfun:NMmultiplesof2', ... kBT cN D|  
              'All N and M must differ by multiples of 2 (including 0).') H11Wb(6Wu  
    end LRmO6>y  
    jG/kT5S  
    if any(m>n) Rp|:$5&nE  
        error('zernfun:MlessthanN', ... vuK 5DG4  
              'Each M must be less than or equal to its corresponding N.') PK~okz4b  
    end X(1.Hjh  
    SrKF\h%/+  
    if any( r>1 | r<0 ) 5-g02g  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') k{;:KW|  
    end j9,X.?Xvx  
    Zaj<*?\  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Fb*;5VNU.  
        error('zernfun:RTHvector','R and THETA must be vectors.') [;b9'7j'  
    end 'R$~U?i8  
    /)G9w]|T  
    r = r(:); J d`NS3;*p  
    theta = theta(:); c9& 8kq5  
    length_r = length(r); >s>5k O  
    if length_r~=length(theta) }%}eyLm(  
        error('zernfun:RTHlength', ... HsXFglQ  
              'The number of R- and THETA-values must be equal.') ="4jk=on  
    end }Jc^p  
    6yR7RF}  
    % Check normalization: Oll\T GXP!  
    % -------------------- v14[G@V~\  
    if nargin==5 && ischar(nflag) bv] ZUF0  
        isnorm = strcmpi(nflag,'norm'); cEN^H  
        if ~isnorm I 7TMv.  
            error('zernfun:normalization','Unrecognized normalization flag.') Rbl(oj#  
        end 9*x9sfCv9  
    else 1k7E[G~G|  
        isnorm = false; \ pq]q  
    end }skXh_Vu4  
    UOwj"#  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% EEaFi 8  
    % Compute the Zernike Polynomials B>'\g O\2  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ]l\J"*"aB  
    +uH1rF_&@  
    % Determine the required powers of r: lyT~>.?{  
    % ----------------------------------- 8Ej2JMc  
    m_abs = abs(m); -V+fQGZe  
    rpowers = []; [~;9Mi.XL  
    for j = 1:length(n) rN*4Y  
        rpowers = [rpowers m_abs(j):2:n(j)]; yb]a p  
    end  [g/g(RL  
    rpowers = unique(rpowers); mT9TSW}  
    c1Hv^*Y  
    % Pre-compute the values of r raised to the required powers, +Gjy%JFp  
    % and compile them in a matrix: 5=$D~>-#  
    % ----------------------------- 4RK^efnp  
    if rpowers(1)==0 \;sUJr"$  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); xOt|j4  
        rpowern = cat(2,rpowern{:}); m/{rmtA4  
        rpowern = [ones(length_r,1) rpowern]; |5W u0T  
    else c~Ha68  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); Lkb?,j5  
        rpowern = cat(2,rpowern{:}); `yf#(YP  
    end *AJW8tIP  
    )D@ NX/}  
    % Compute the values of the polynomials: YS/DIH{9e  
    % -------------------------------------- 2#rF/!`^  
    y = zeros(length_r,length(n)); VMNihx0FJ  
    for j = 1:length(n) 7N:,F9V<  
        s = 0:(n(j)-m_abs(j))/2; 7y60-6r  
        pows = n(j):-2:m_abs(j); -yC},tK  
        for k = length(s):-1:1 hxv/285B  
            p = (1-2*mod(s(k),2))* ... .NPai4V'  
                       prod(2:(n(j)-s(k)))/              ... jKtbGVZ 7r  
                       prod(2:s(k))/                     ... 9\dC8  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ;MO %))  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); Vdjca:`  
            idx = (pows(k)==rpowers); *l5/q\D  
            y(:,j) = y(:,j) + p*rpowern(:,idx); 8J@REP4  
        end jfI|( P  
         FkRrW^?5G  
        if isnorm tewC *%3V  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); )Z]8SED  
        end :*\JJ w  
    end 1_F2{n:yp  
    % END: Compute the Zernike Polynomials yDHH05Yl  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% l.&6|   
    "d{ |_Cf  
    % Compute the Zernike functions: U/TF,JUI  
    % ------------------------------ QYg2'`(  
    idx_pos = m>0; O* 7" Q&  
    idx_neg = m<0; O8M;q!)y  
    D V C};  
    z = y; a*o=,!  
    if any(idx_pos) QupCr/Hs  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); $L3UDX+F  
    end G"C'/  
    if any(idx_neg) &L;0%  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); -l^u1z  
    end ]r|X[9  
    _57i[U r  
    % 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) YQ;?N66  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. s<aJ pi{n4  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated )]?sCNb  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive r 5:DIA!  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, IL&Mf9m  
    %   and THETA is a vector of angles.  R and THETA must have the same H71LJfH  
    %   length.  The output Z is a matrix with one column for every P-value, YiB^m   
    %   and one row for every (R,THETA) pair. *i&ks> 4N  
    % ([^1gG+>J  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike IrM Ws86;  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) ]w"r4HlCx  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) ' FF@I^O  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 3N[t2Y1r  
    %   for all p. R>yoMk/u  
    % ,A4v|]kq]  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 1o V\QK&  
    %   Zernike functions (order N<=7).  In some disciplines it is %?^IS&]Z  
    %   traditional to label the first 36 functions using a single mode DFcgUEq  
    %   number P instead of separate numbers for the order N and azimuthal n~@;[=o?5  
    %   frequency M. :!n_a*.{  
    % Jt  ^a  
    %   Example: Mnc9l ^  
    % 4v_<<l  
    %       % Display the first 16 Zernike functions r ".*l?=  
    %       x = -1:0.01:1; .]JGCTB3  
    %       [X,Y] = meshgrid(x,x); krFuEaO  
    %       [theta,r] = cart2pol(X,Y); M2l0x @|  
    %       idx = r<=1; //xK v{3fI  
    %       p = 0:15; C|*U)#3:F  
    %       z = nan(size(X)); tZB" (\  
    %       y = zernfun2(p,r(idx),theta(idx)); -XRn%4EX?  
    %       figure('Units','normalized') ;p)RMRMg  
    %       for k = 1:length(p) B<%cqz@  
    %           z(idx) = y(:,k); Yw7txp`i  
    %           subplot(4,4,k) +`}QIp0  
    %           pcolor(x,x,z), shading interp 5_!s\5  
    %           set(gca,'XTick',[],'YTick',[]) xf% _HMKc  
    %           axis square 3"FvYv{  
    %           title(['Z_{' num2str(p(k)) '}']) W US[hx,  
    %       end ]?rVram;z  
    % IOhJL'r  
    %   See also ZERNPOL, ZERNFUN. F>RL&i  
    $KHw=<:)/  
    %   Paul Fricker 11/13/2006 mE\sD<b  
    6?i]oy^X]p  
    +N n $  
    % Check and prepare the inputs: l!qhK'']V"  
    % ----------------------------- jlXzfD T  
    if min(size(p))~=1 `ECY:3"$KA  
        error('zernfun2:Pvector','Input P must be vector.') RUco3fZ   
    end t/%{R.1MN  
    5 nF46c  
    if any(p)>35 F#-mseKhc  
        error('zernfun2:P36', ... amvD5  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... -Y>,\VEK  
               '(P = 0 to 35).']) xP/?E  
    end IyK^` y  
    EdcbWf7  
    % Get the order and frequency corresonding to the function number: /o L& <e  
    % ---------------------------------------------------------------- wr5ScsNS  
    p = p(:); r ]s7a?O  
    n = ceil((-3+sqrt(9+8*p))/2); 7qsu0 .[d  
    m = 2*p - n.*(n+2); ``mnk>/  
    iq1HA.X(  
    % Pass the inputs to the function ZERNFUN: 7M8oI.?C|  
    % ---------------------------------------- fab. %$  
    switch nargin H1Jk_@b  
        case 3 <$z6:4uN_  
            z = zernfun(n,m,r,theta);  3-~*  
        case 4 aQ.QkM Z  
            z = zernfun(n,m,r,theta,nflag);  m#K)%0  
        otherwise u:]c  
            error('zernfun2:nargin','Incorrect number of inputs.') "%peYNZ&%  
    end 5}C.^J`  
    e'I/}J  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) v*";A  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. yopC <k  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of LUs)"ZAi|  
    %   order N and frequency M, evaluated at R.  N is a vector of |`|#-xu  
    %   positive integers (including 0), and M is a vector with the HVdy!J  
    %   same number of elements as N.  Each element k of M must be a m>{a<N  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) VQ"Z3L3-4  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is gU&y5s~  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix `- HI)-A97  
    %   with one column for every (N,M) pair, and one row for every '@/1e\-y  
    %   element in R. oBO4a^D  
    % 5^ck$af  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- @ D,]v:  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is LD*XNcE  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to N_^PoX935O  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 ;FGS(.mjlC  
    %   for all [n,m]. gc\/A\F<  
    % ,&~-Sq) ~  
    %   The radial Zernike polynomials are the radial portion of the mv,5Q6!  
    %   Zernike functions, which are an orthogonal basis on the unit {^D; ($lm  
    %   circle.  The series representation of the radial Zernike Qz"+M+~%&  
    %   polynomials is R-Ys<;  
    % `# U<'$  
    %          (n-m)/2 Cnr=1E=  
    %            __ < z#.J]  
    %    m      \       s                                          n-2s a&Qr7tT Y"  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r /3#h]5Y"T  
    %    n      s=0 C$0rl74Wi  
    % /a*8z,x  
    %   The following table shows the first 12 polynomials. &S=Qu?H  
    % GrUpATIx  
    %       n    m    Zernike polynomial    Normalization )K8 ^}L,  
    %       --------------------------------------------- 4_D *xW  
    %       0    0    1                        sqrt(2) .-'_At4g  
    %       1    1    r                           2 +zwS[P@  
    %       2    0    2*r^2 - 1                sqrt(6) j0=F__H#@  
    %       2    2    r^2                      sqrt(6) ZZw2m@T>  
    %       3    1    3*r^3 - 2*r              sqrt(8) 97[wz C,  
    %       3    3    r^3                      sqrt(8) TMGYNb%<bX  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) v%r/PHw  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) ^7uXpqQBr  
    %       4    4    r^4                      sqrt(10) w&@zJ[  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) 8BrC@L2E0  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) 3@nIoN'z  
    %       5    5    r^5                      sqrt(12) MDQ:6Ri  
    %       --------------------------------------------- ?S;z!) H)P  
    % 2D(sA  
    %   Example: Ee_?aG e&  
    % =0L%<@yA  
    %       % Display three example Zernike radial polynomials <FX ]n<  
    %       r = 0:0.01:1; sSf;j,7V  
    %       n = [3 2 5]; T6b~uE  
    %       m = [1 2 1]; lN&+<>a  
    %       z = zernpol(n,m,r); ,PoG=W  
    %       figure EKO~\d  
    %       plot(r,z) ;GE6S{~-  
    %       grid on )Tieef*Q~  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') KWxTN|>  
    % qzNXz_#+u  
    %   See also ZERNFUN, ZERNFUN2. WJxcJE  
    S|xwYaoy%  
    % A note on the algorithm. :f:&B8  
    % ------------------------ HE{UgU:tY  
    % The radial Zernike polynomials are computed using the series rizjH+  
    % representation shown in the Help section above. For many special CDF;cM"td  
    % functions, direct evaluation using the series representation can eIy:5/s  
    % produce poor numerical results (floating point errors), because o~9sO=-O  
    % the summation often involves computing small differences between EXF]y}n  
    % large successive terms in the series. (In such cases, the functions >0[:uu,'>  
    % are often evaluated using alternative methods such as recurrence TQ:h[6v  
    % relations: see the Legendre functions, for example). For the Zernike [m4M#Lg\0  
    % polynomials, however, this problem does not arise, because the =E$bZe8  
    % polynomials are evaluated over the finite domain r = (0,1), and Qn|8Ic` *  
    % because the coefficients for a given polynomial are generally all "N ">RjJ"  
    % of similar magnitude. sPb}A$'  
    % /NjBC[P  
    % ZERNPOL has been written using a vectorized implementation: multiple </Y(4Xwf=  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] :Jf</uP_  
    % values can be passed as inputs) for a vector of points R.  To achieve *- ~GVe  
    % this vectorization most efficiently, the algorithm in ZERNPOL am !ssF5s  
    % involves pre-determining all the powers p of R that are required to s?s ,wdp  
    % compute the outputs, and then compiling the {R^p} into a single .%dGSDru  
    % matrix.  This avoids any redundant computation of the R^p, and P3YM4&6XA  
    % minimizes the sizes of certain intermediate variables. 4s~X  
    % x4PzP  
    %   Paul Fricker 11/13/2006 }A]e C  
     j'Jb+@W?  
    YD@Z}NE v"  
    % Check and prepare the inputs: `mW~{)x  
    % ----------------------------- 5~Ek_B  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 00'SceL=`  
        error('zernpol:NMvectors','N and M must be vectors.') pouXt-%2X  
    end wx`.  
    ES!e/l  
    if length(n)~=length(m) w N`Nj m9!  
        error('zernpol:NMlength','N and M must be the same length.') ',!jYh}Uxk  
    end pH.&C 5kA  
    ?{}P#sn  
    n = n(:); PNd'21N  
    m = m(:); >itNa.K  
    length_n = length(n);  Qe7=6<  
    oemN$g&7  
    if any(mod(n-m,2)) *!TQC6b$  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') y$bY 8L  
    end Q"U%]2@=  
    fVgN8b|&'  
    if any(m<0) YlUh|sK7m  
        error('zernpol:Mpositive','All M must be positive.') QZG<sZ0"  
    end wkPjMmW+!  
    XN6$TNsD$  
    if any(m>n) s3m \  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') UCjx   
    end /;w(sU  
    :*h1ik4t  
    if any( r>1 | r<0 ) J)y g<*/3  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') d (x'\4(K  
    end 1ig*Xp[  
    ?>{u@tYL  
    if ~any(size(r)==1) #"~\/sb   
        error('zernpol:Rvector','R must be a vector.') U?Dr0wD;[  
    end < `"  
    u?fM.=/N  
    r = r(:); JFO,Q -y\  
    length_r = length(r); rv(N0p/  
    EI*~VFx  
    if nargin==4 N>YSXh`W`y  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); 3) d }3w {  
        if ~isnorm #:N#i  
            error('zernpol:normalization','Unrecognized normalization flag.') 5,oLl {S'  
        end _ q1\8y  
    else ZklpnL*!  
        isnorm = false; *P9"1K +  
    end $0K@= 7ms  
    T [xIn+w  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {]8|\CcY?  
    % Compute the Zernike Polynomials P(Rl/eyRM  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LQr!0p.i"  
    "_LqIW1   
    % Determine the required powers of r: L7aVj&xM  
    % ----------------------------------- Li|~%E1  
    rpowers = []; )D#}/3s  
    for j = 1:length(n) 4H,c;g=!  
        rpowers = [rpowers m(j):2:n(j)]; :L+ xEL  
    end #9r}Kr=P  
    rpowers = unique(rpowers); r5UV BV8T  
    z9OpMA  
    % Pre-compute the values of r raised to the required powers, jQ'g'c!  
    % and compile them in a matrix: EV Z1Z  
    % ----------------------------- s@"|o3BX  
    if rpowers(1)==0 fap]`P~#L  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ](Wa:U}Xs  
        rpowern = cat(2,rpowern{:}); |>xuH#Q  
        rpowern = [ones(length_r,1) rpowern]; g.di3GGi  
    else *S.FM.r  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); gCPH>8JwS0  
        rpowern = cat(2,rpowern{:}); [pp|*@1T  
    end r,.j^a  
    ,aUbB8  
    % Compute the values of the polynomials: f42F@M(:  
    % -------------------------------------- /;Hqv`X7  
    z = zeros(length_r,length_n); KMkD6g  
    for j = 1:length_n QN$s %&O  
        s = 0:(n(j)-m(j))/2; 7]1a3Jk  
        pows = n(j):-2:m(j); t bR  
        for k = length(s):-1:1 (M1YOK)I  
            p = (1-2*mod(s(k),2))* ... a?nK|Q=e  
                       prod(2:(n(j)-s(k)))/          ... +aEm]=3  
                       prod(2:s(k))/                 ... k'|yUJ,  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... k) Lhzr[  
                       prod(2:((n(j)+m(j))/2-s(k))); S .x>w/  
            idx = (pows(k)==rpowers); i~v[3e9y7  
            z(:,j) = z(:,j) + p*rpowern(:,idx); LXxQI(RO  
        end )V>OND  
         W?aP%D"(i  
        if isnorm \db=]L=|  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); T-ST M"~%  
        end ]nebL{}5  
    end 56c[$ q  
    yv]|Ce@8A  
    % 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)  Y?K?*`Pkc1  
    _ Dz*%  
    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。 $39TP@?:Z)  
    *w ^!\  
    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)