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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 ut26sg{s(  
    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! D0T0Km/"  
     
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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 EdQ:8h  
    function z = zernfun(n,m,r,theta,nflag) S}=d74(/n  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. N[$bP)h7  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 25xpq^Zw  
    %   and angular frequency M, evaluated at positions (R,THETA) on the WfbG }%&J  
    %   unit circle.  N is a vector of positive integers (including 0), and PoyY}Ra  
    %   M is a vector with the same number of elements as N.  Each element ]y*AA58;  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) F Qtlo+3  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, j=U [V&T  
    %   and THETA is a vector of angles.  R and THETA must have the same 9f ,$JjX[  
    %   length.  The output Z is a matrix with one column for every (N,M) F,2)Udim  
    %   pair, and one row for every (R,THETA) pair. 2qEm,x'S  
    % o(~QuHOp8>  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike sflH{!;p  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), Wj2s+L7,  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral \x JGR!  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, BMlnzi  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized O*MC"%T  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 9NCo0!Fb  
    % X"V,3gDG  
    %   The Zernike functions are an orthogonal basis on the unit circle. W5a)`%H  
    %   They are used in disciplines such as astronomy, optics, and J!?hajw7N  
    %   optometry to describe functions on a circular domain. 9IFK4>&O6  
    %  $$E!u}  
    %   The following table lists the first 15 Zernike functions. v-`RX;8  
    % )4oTA@wR  
    %       n    m    Zernike function           Normalization  /[f9Z:>V  
    %       -------------------------------------------------- c(@V t&gE  
    %       0    0    1                                 1 Kyy CS>  
    %       1    1    r * cos(theta)                    2 ]Lg$p  
    %       1   -1    r * sin(theta)                    2 fp^!?u  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) _jmkAmeu  
    %       2    0    (2*r^2 - 1)                    sqrt(3) |2mm@):  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) Xy{\>}i]N  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) 3Qt-%=b&  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) V+7x_>!&)  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) N}0-L$@SL  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) _8$arjx=  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) LfD7 0r\  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) yLfb'Ba  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) {Lj]++`fB]  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) M7R.? nk  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) UR')) 1n  
    %       -------------------------------------------------- 9!hiCqA&  
    % B%95M|  
    %   Example 1: 0 rbMT`Hy  
    % ?3ldHWa  
    %       % Display the Zernike function Z(n=5,m=1)  6C6<,c   
    %       x = -1:0.01:1; yyZV/ x~  
    %       [X,Y] = meshgrid(x,x); [[(29|`]  
    %       [theta,r] = cart2pol(X,Y); Bny3j~*U  
    %       idx = r<=1; 2y6 e]D  
    %       z = nan(size(X)); 0pT?qsM2  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); a6AD`| U8  
    %       figure ^O_E T$  
    %       pcolor(x,x,z), shading interp %5|awWo_?  
    %       axis square, colorbar d(u"^NH;  
    %       title('Zernike function Z_5^1(r,\theta)') &6-udZB-  
    % m[~fT(NI  
    %   Example 2: @1_M's;  
    % KiN8N=z  
    %       % Display the first 10 Zernike functions "F nH>g-  
    %       x = -1:0.01:1; Y%AVC9(  
    %       [X,Y] = meshgrid(x,x); ,DUD4 [3  
    %       [theta,r] = cart2pol(X,Y); fi*@m,-  
    %       idx = r<=1; F91'5D,u0  
    %       z = nan(size(X)); Wr.G9zq.+  
    %       n = [0  1  1  2  2  2  3  3  3  3]; +C4UM9  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; #*QnO\.  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; X 4\  
    %       y = zernfun(n,m,r(idx),theta(idx)); b}DxD1*nsI  
    %       figure('Units','normalized') `9IG//  
    %       for k = 1:10 r(g:b ^S  
    %           z(idx) = y(:,k); "<f"r#   
    %           subplot(4,7,Nplot(k)) >OP[ qj  
    %           pcolor(x,x,z), shading interp X wvH  
    %           set(gca,'XTick',[],'YTick',[]) @edx]H1~^  
    %           axis square <Sm@ !yx  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) JHN{vB  
    %       end O,m0Xb2s]~  
    % neN #Mo'A  
    %   See also ZERNPOL, ZERNFUN2. G.CkceWRn  
    9F[k;Uw  
    %   Paul Fricker 11/13/2006 koQ\]t'*As  
    {9>LF  
    cB,O"-  
    % Check and prepare the inputs: HE>6A|rgDr  
    % ----------------------------- UVND1XV^f  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) =ELl86=CG  
        error('zernfun:NMvectors','N and M must be vectors.') 0E[&:6#Y  
    end Tw^b!74gq  
    4hRc,Vq  
    if length(n)~=length(m) |Nx7jGd:i  
        error('zernfun:NMlength','N and M must be the same length.') KxZup\\:v  
    end 0$8iWL  
    5Q $6~\  
    n = n(:); ;Mzy>*#$Q  
    m = m(:); N@Fof(T&  
    if any(mod(n-m,2)) OsQB` D  
        error('zernfun:NMmultiplesof2', ... wGRMv1|lIu  
              'All N and M must differ by multiples of 2 (including 0).') 8R G U^&  
    end 6|h~pH  
    Rn(6Fk?   
    if any(m>n) kkvG=  
        error('zernfun:MlessthanN', ... [nL{n bli  
              'Each M must be less than or equal to its corresponding N.') EZICH&_  
    end ?]1_ 2\M  
    14U:.Q  
    if any( r>1 | r<0 ) FVi7gg.?  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') /)Ga<  
    end }q-*Ls~  
    NR </Jm*  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) d~ m,hCTe  
        error('zernfun:RTHvector','R and THETA must be vectors.') gK8E|f-z  
    end X:3W9`s )*  
    >P-{2 a,4  
    r = r(:);  nIWZo ~  
    theta = theta(:); J0%e6{C1  
    length_r = length(r); "9>.,nzt  
    if length_r~=length(theta) j>D[iHrH  
        error('zernfun:RTHlength', ... D\"F?>  
              'The number of R- and THETA-values must be equal.') ?+^vU5b1u  
    end ]Ak/:pu  
    YwYCXFQ|  
    % Check normalization: 3b9SyU2  
    % -------------------- qKL :#ny  
    if nargin==5 && ischar(nflag) 1$A7BP  
        isnorm = strcmpi(nflag,'norm'); |3ob1/)p0  
        if ~isnorm CAs8=N#H%  
            error('zernfun:normalization','Unrecognized normalization flag.') xna4W|-  
        end g`NJ `  
    else /b ]Yya#  
        isnorm = false; -chk\75  
    end #.Q8q  
    BAy]&q|.  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% gk-g!v&  
    % Compute the Zernike Polynomials e\Igc.  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% cV|u]ce%1  
    N,oN3mFF  
    % Determine the required powers of r: XkGS3EY  
    % ----------------------------------- @)i A V1r"  
    m_abs = abs(m); b~5Q|3P9  
    rpowers = []; 0vi)m y;!  
    for j = 1:length(n) W.|r=   
        rpowers = [rpowers m_abs(j):2:n(j)]; xD|/98  
    end ;XUiV$  
    rpowers = unique(rpowers); |mHxkd  
    7QnQ=gu  
    % Pre-compute the values of r raised to the required powers, S(&]?!  
    % and compile them in a matrix: +?&|p0  
    % ----------------------------- n"Gow/-;  
    if rpowers(1)==0 =x QLf4>  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); nKR=/5a4Y  
        rpowern = cat(2,rpowern{:}); j 1Ng[  
        rpowern = [ones(length_r,1) rpowern]; Hea76P5$P+  
    else B#Q=Fo 6  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 8dBG ZwyET  
        rpowern = cat(2,rpowern{:}); r=S6yq}  
    end .#BWu(EYV  
    Pl9Ky(Q`V  
    % Compute the values of the polynomials: 9hNHcl.  
    % -------------------------------------- JGZxNUr^  
    y = zeros(length_r,length(n)); - C  
    for j = 1:length(n) SniKC qmC]  
        s = 0:(n(j)-m_abs(j))/2; >}?4;:.=  
        pows = n(j):-2:m_abs(j); KeIk9T13O  
        for k = length(s):-1:1 |o5F%1o  
            p = (1-2*mod(s(k),2))* ... q%rfKHMA50  
                       prod(2:(n(j)-s(k)))/              ... "&@v[O)!xu  
                       prod(2:s(k))/                     ... [WAnII  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... (*XSr Q  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); DqQ+8 w  
            idx = (pows(k)==rpowers); KcW]"K>p!  
            y(:,j) = y(:,j) + p*rpowern(:,idx); Uiz#QGt  
        end  n}f*>Mn  
         p%?VW  
        if isnorm }}cS-p  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); uFXu9f+  
        end (mvzGXNz4  
    end l+V#`S*q  
    % END: Compute the Zernike Polynomials `g~T #U\>d  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% DjK  
    c!2j+ORz  
    % Compute the Zernike functions: Qgel^"t]i  
    % ------------------------------ ?F!='6D}b  
    idx_pos = m>0; =ghN)[AZV  
    idx_neg = m<0; #xlT,:_:)  
    f(}AdW}?  
    z = y; ar!`8"  
    if any(idx_pos) o`EL)K{  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); A=+ |&+? t  
    end QE b ^'y  
    if any(idx_neg) `'gadCTb=  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); K9@F1ccQ/  
    end ^Hplrwj}  
    /Ayo78Pi  
    % 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) E/Ng   
    %ZERNFUN2 Single-index Zernike functions on the unit circle. lls-Nir%  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated L]_1z  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive o2J-&   
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, YgFmJ.1  
    %   and THETA is a vector of angles.  R and THETA must have the same  oRbG6Vv/  
    %   length.  The output Z is a matrix with one column for every P-value, <Y9 L3O`[  
    %   and one row for every (R,THETA) pair. %xH2jf  
    % <691pk X  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike Uql|32j  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) '%} k"&t$i  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) h\@\*Xz<v  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 y!,Ly_x$@  
    %   for all p. 4J"S?HsW|  
    % e=yQFzQT)  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 c[h{C!d1  
    %   Zernike functions (order N<=7).  In some disciplines it is B_u1FWc  
    %   traditional to label the first 36 functions using a single mode GW[g!6 6^  
    %   number P instead of separate numbers for the order N and azimuthal uq~Z  
    %   frequency M. YV p sf8R  
    % ioZ{2kK  
    %   Example: s_j ?L  
    % ^/H9`z;  
    %       % Display the first 16 Zernike functions 8^8fUN4<=  
    %       x = -1:0.01:1; YaWZOuxm  
    %       [X,Y] = meshgrid(x,x); #KOr-Yg|U  
    %       [theta,r] = cart2pol(X,Y); .h0@Vs  
    %       idx = r<=1; ^V1iOf:  
    %       p = 0:15; i 1{Lx)  
    %       z = nan(size(X)); &:3uK`  
    %       y = zernfun2(p,r(idx),theta(idx)); )e1&[0  
    %       figure('Units','normalized') ]V 4Fm{]  
    %       for k = 1:length(p) XlPi)3m4/S  
    %           z(idx) = y(:,k); >3v j<v}m  
    %           subplot(4,4,k) iFypKpHg~  
    %           pcolor(x,x,z), shading interp 3kc.U  
    %           set(gca,'XTick',[],'YTick',[]) @`,~d{ziF  
    %           axis square 3/j^Ao\fw  
    %           title(['Z_{' num2str(p(k)) '}']) sX :)g>b   
    %       end _~=qByD   
    % d[p-zn.  
    %   See also ZERNPOL, ZERNFUN. .d4L@{V  
    D #`o  
    %   Paul Fricker 11/13/2006 "k0bj>  
    s\CZ os&  
     ./iC  
    % Check and prepare the inputs: 5vfzSJ  
    % ----------------------------- WPN4mEow  
    if min(size(p))~=1 >l!#_a  
        error('zernfun2:Pvector','Input P must be vector.') h.~:UR*   
    end IO*}N"  
    qir/Sa' [  
    if any(p)>35 wOlnDQs  
        error('zernfun2:P36', ... 323zR*\m  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... .:`+4n  
               '(P = 0 to 35).']) #DqVh!t"  
    end T&ECGF;Y/  
    6ojEEM  
    % Get the order and frequency corresonding to the function number: hhqSfafUX  
    % ---------------------------------------------------------------- EGY'a*]cU  
    p = p(:); d,c8ks(  
    n = ceil((-3+sqrt(9+8*p))/2); hJ>Kfm  
    m = 2*p - n.*(n+2); [b=l'e/  
    ;`{PA !>  
    % Pass the inputs to the function ZERNFUN: ;?*`WB  
    % ---------------------------------------- >E9:3&[F  
    switch nargin "X.JD  
        case 3 na5:)j4<  
            z = zernfun(n,m,r,theta); `2B,+ytW8  
        case 4 |2YkZ nJn  
            z = zernfun(n,m,r,theta,nflag); O]XdPH20  
        otherwise ?tf/#5t}  
            error('zernfun2:nargin','Incorrect number of inputs.') w6PKr^  
    end o)(N*tC  
    6<uJ}3  
    % 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) $s*nh>@7  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. udEJo~u  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of /uh?F  
    %   order N and frequency M, evaluated at R.  N is a vector of L7gZ4Hu=`  
    %   positive integers (including 0), and M is a vector with the !zu YO3:  
    %   same number of elements as N.  Each element k of M must be a 015 ;'V#we  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) )@IDmz>  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is xb N)z  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix sULCYiT|Hn  
    %   with one column for every (N,M) pair, and one row for every 4;rt|X77  
    %   element in R. xla64Qld  
    % CJDnHuozc  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- \z~wm&  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is q{fgsc8v\  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to e%Sw(=a  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 L,(H(GeX  
    %   for all [n,m]. ^PNE6  
    % .nN>Ipv  
    %   The radial Zernike polynomials are the radial portion of the d4 Hpe>  
    %   Zernike functions, which are an orthogonal basis on the unit  1\[En/6  
    %   circle.  The series representation of the radial Zernike lj U|9|v  
    %   polynomials is N=JZtf/i  
    % [SJ)4e|)  
    %          (n-m)/2 n^a&@?(+  
    %            __ 8)NQt$lWp  
    %    m      \       s                                          n-2s C1x"q9| \`  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r cl`!A2F1G#  
    %    n      s=0 BA5b;+o-  
    % 6t,_Xqg*  
    %   The following table shows the first 12 polynomials. xT]|78h$   
    % *VbB'u:  
    %       n    m    Zernike polynomial    Normalization +1te8P*  
    %       --------------------------------------------- c2tf7fkH  
    %       0    0    1                        sqrt(2) fcim4dfP  
    %       1    1    r                           2 Hv>16W$_  
    %       2    0    2*r^2 - 1                sqrt(6) ']x`d  
    %       2    2    r^2                      sqrt(6) r?:zKj8/u  
    %       3    1    3*r^3 - 2*r              sqrt(8) (=T%eJ61  
    %       3    3    r^3                      sqrt(8) =SY`Xkj[  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) Wubvvm8U  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) }.L\O]~{  
    %       4    4    r^4                      sqrt(10) "%mu~&Ga  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) }#b[@3/T  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) VbwB<nQl  
    %       5    5    r^5                      sqrt(12) hB !>*AsG  
    %       --------------------------------------------- Xcy Xju#"p  
    % 6JCq?:#ab  
    %   Example: :vsF4  
    % oZ /z{`  
    %       % Display three example Zernike radial polynomials snH9@!cG8  
    %       r = 0:0.01:1; LE'8R~4.<  
    %       n = [3 2 5]; $GMva}@G`  
    %       m = [1 2 1]; 3YFbT Z  
    %       z = zernpol(n,m,r); k)a3j{{  
    %       figure f3p)Q<H>`(  
    %       plot(r,z) 2i4&*& A  
    %       grid on S5,y!K]C~  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') ~8j4IO(  
    % =!~6RwwwY  
    %   See also ZERNFUN, ZERNFUN2. C{5bG=Sg~  
    kdam]L:9  
    % A note on the algorithm. w]% |^:  
    % ------------------------ mF6 U{=  
    % The radial Zernike polynomials are computed using the series TTfU(w%&P  
    % representation shown in the Help section above. For many special wH<'*>/  
    % functions, direct evaluation using the series representation can Jn+k$'6 %#  
    % produce poor numerical results (floating point errors), because >$g+Gx\v4  
    % the summation often involves computing small differences between /Cl=;^)  
    % large successive terms in the series. (In such cases, the functions ag7(nn0!  
    % are often evaluated using alternative methods such as recurrence Y\e8oIYu7  
    % relations: see the Legendre functions, for example). For the Zernike H[u[3  
    % polynomials, however, this problem does not arise, because the /Tc I  
    % polynomials are evaluated over the finite domain r = (0,1), and 8M_p'AR\,y  
    % because the coefficients for a given polynomial are generally all l)d(N7HME  
    % of similar magnitude. K,$Ro@!  
    % _'.YC<;  
    % ZERNPOL has been written using a vectorized implementation: multiple ?kF_C,k/>N  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] PdkS3Hz  
    % values can be passed as inputs) for a vector of points R.  To achieve ,~TV/l<  
    % this vectorization most efficiently, the algorithm in ZERNPOL )M: pg%  
    % involves pre-determining all the powers p of R that are required to qGYru1  
    % compute the outputs, and then compiling the {R^p} into a single @j{n V@|  
    % matrix.  This avoids any redundant computation of the R^p, and .O1Kwu  
    % minimizes the sizes of certain intermediate variables. x3QQ`w-  
    % &y~~Z [.F,  
    %   Paul Fricker 11/13/2006 mT3'kUZ}]  
    "lT>V)NB'  
    Ibbpy++d[  
    % Check and prepare the inputs: jW!x!8=  
    % ----------------------------- ]6*+i $  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Yqz B="  
        error('zernpol:NMvectors','N and M must be vectors.') 50?5xSEM0_  
    end 4kr! Af  
    PIthv [F  
    if length(n)~=length(m) vr$zYdV>  
        error('zernpol:NMlength','N and M must be the same length.') ,Qw\w,  
    end RPh8n4&("  
    W3h{5\d!  
    n = n(:); O\5q_>]  
    m = m(:); IuW5LS  
    length_n = length(n); ).8i*Ys,:  
    4Up3x+bg  
    if any(mod(n-m,2)) Wb7z&vj  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') "+BNas^rF  
    end D$vP&7pOr4  
    yJMHm8OB7  
    if any(m<0) t)62_nu  
        error('zernpol:Mpositive','All M must be positive.') B|zVq=l~  
    end yClbM5,  
    A:JW Ux  
    if any(m>n) mKh <M)Bz  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') &# w~S~  
    end /Sn>{ &  
    3v:c".O2O  
    if any( r>1 | r<0 ) 4pw:O^v  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') .15^c+j  
    end a+_F^   
    [h=[@jiB  
    if ~any(size(r)==1) D_(K{? KU  
        error('zernpol:Rvector','R must be a vector.') f|cF [&wo  
    end d$O)k+j  
    NU#rv%p  
    r = r(:); D,)^l@UP  
    length_r = length(r); xdV $dDCT  
    {R{Io|   
    if nargin==4 LqOjVQxz  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); \~{b;$N}  
        if ~isnorm 1 xu2$x.b  
            error('zernpol:normalization','Unrecognized normalization flag.') DP-euz  
        end w*-1*XNA  
    else : ~R:[T2P  
        isnorm = false; YYiT,Xp<A  
    end tG"lI/  
    NW=tZVQ<X  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% "p[FFg  
    % Compute the Zernike Polynomials ,2y " \_  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% A[mm_+D>  
    ?\J.Tv $$$  
    % Determine the required powers of r: }ippi6b:r  
    % ----------------------------------- 0s%rd>3  
    rpowers = []; fmv8)$W#U  
    for j = 1:length(n) GA.4'W^&a  
        rpowers = [rpowers m(j):2:n(j)];  &9*MO  
    end {k#RWDespy  
    rpowers = unique(rpowers); 9" RGf 1]  
    U if61)+!i  
    % Pre-compute the values of r raised to the required powers, )b_ GKA `  
    % and compile them in a matrix: u3XQ<N{Gj  
    % ----------------------------- "k%B;!We)  
    if rpowers(1)==0 /t<C_lLM  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); m@W\Pic,j.  
        rpowern = cat(2,rpowern{:}); j& x=?jX  
        rpowern = [ones(length_r,1) rpowern]; ncy?w e  
    else A` iZ"?  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); )ZP-t!).G#  
        rpowern = cat(2,rpowern{:}); .!&S{;Vv?W  
    end "~uo4n~H  
    5{@Hpj/B  
    % Compute the values of the polynomials: IUJRP  
    % -------------------------------------- sJHN4  
    z = zeros(length_r,length_n); '+Gy)@c  
    for j = 1:length_n NxyrP**j  
        s = 0:(n(j)-m(j))/2; UJX=lh.o  
        pows = n(j):-2:m(j); ]F]!>dKA  
        for k = length(s):-1:1 '/@] V  
            p = (1-2*mod(s(k),2))* ... Wxxnc#;lv  
                       prod(2:(n(j)-s(k)))/          ... 8UANB]@Y}  
                       prod(2:s(k))/                 ... &d8z`amP  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... @}^eyS$|!  
                       prod(2:((n(j)+m(j))/2-s(k))); 2\: z   
            idx = (pows(k)==rpowers); *YI>Q@F9  
            z(:,j) = z(:,j) + p*rpowern(:,idx); 3X,SCG  
        end OGjeE4  
         :<}.3Q?&  
        if isnorm Y8fahQ#  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); '[6o(~ *  
        end h{sY5d'D  
    end q[}[w!to  
    ;~>E^0M  
    % 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)  -{>Nrx|  
    WV5gH*uUa  
    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。 @WX]K0 $;  
    X6mY#T'fQ  
    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)