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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 <!g]q1  
    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! /6Q]f  
     
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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 Q__CW5&'u  
    function z = zernfun(n,m,r,theta,nflag) gMI%!Y  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. EjLq&QR.  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N n#g_)\  
    %   and angular frequency M, evaluated at positions (R,THETA) on the Q"dq_8\`U  
    %   unit circle.  N is a vector of positive integers (including 0), and &Gjpc>d  
    %   M is a vector with the same number of elements as N.  Each element (p{%]M  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) gLX<> |)*  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, w\acgQ^%e  
    %   and THETA is a vector of angles.  R and THETA must have the same uK@d?u!`  
    %   length.  The output Z is a matrix with one column for every (N,M) 9$\s v5  
    %   pair, and one row for every (R,THETA) pair. p[JIH~nb  
    % 4>=M"D hB  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike M5h r0 R{  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), u9"yU:1keb  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral RG{T\9]n  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, YbU8 xq  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized (U.Go/A#wE  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ?Z 2,?G  
    % QFx3N%  
    %   The Zernike functions are an orthogonal basis on the unit circle. =$J(]KPv!?  
    %   They are used in disciplines such as astronomy, optics, and M!J7Vj?Ps  
    %   optometry to describe functions on a circular domain. aDdGhB  
    % rJ Jx8)M  
    %   The following table lists the first 15 Zernike functions. _li3cXE  
    % btuG%D{a^  
    %       n    m    Zernike function           Normalization 'IX1WS&\"  
    %       -------------------------------------------------- @e)}#kN.  
    %       0    0    1                                 1 8X7??f1;Y  
    %       1    1    r * cos(theta)                    2 ~pRgTXbz  
    %       1   -1    r * sin(theta)                    2 |T6K?:U7  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) JJd qdX;  
    %       2    0    (2*r^2 - 1)                    sqrt(3) Xj\ToO  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) @wcF#?J  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) ==[=Da~  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) n{;Q"\*Sg  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) uI-T]N:W8x  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) l1 Kv`v\  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) 77@N79lqO  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) m=01V5_  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) BX?DI-o^h  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) :/T\E\Qr  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) zL yI|%KH  
    %       -------------------------------------------------- XYo,5-  
    % 5*$yY-A  
    %   Example 1: xG/Q%A  
    % LDjtkD.r  
    %       % Display the Zernike function Z(n=5,m=1) Q~(Gll;  
    %       x = -1:0.01:1; g0grfGo2p  
    %       [X,Y] = meshgrid(x,x); bp?5GU&Uy  
    %       [theta,r] = cart2pol(X,Y); UTkPA2x  
    %       idx = r<=1; XZIapT  
    %       z = nan(size(X)); a!$kKOK  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); Y*c]C;%=  
    %       figure :oIBJ u%/  
    %       pcolor(x,x,z), shading interp !rUP&DA  
    %       axis square, colorbar jA{5)-g  
    %       title('Zernike function Z_5^1(r,\theta)') &!8 WRJ  
    % J9mK9{#q  
    %   Example 2: ~*iF`T6  
    % ;MS.ag#  
    %       % Display the first 10 Zernike functions RM|J |R  
    %       x = -1:0.01:1; 072C!F  
    %       [X,Y] = meshgrid(x,x); }emUpju<C  
    %       [theta,r] = cart2pol(X,Y); {fXkbMO|  
    %       idx = r<=1; ;R*-cm  
    %       z = nan(size(X)); 7S{qo&j'  
    %       n = [0  1  1  2  2  2  3  3  3  3]; D^6*Cwb  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; w<9rTHG8,  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; O@Aazc5K  
    %       y = zernfun(n,m,r(idx),theta(idx)); .C^P6S2oJ  
    %       figure('Units','normalized') 8o5[tl ?w  
    %       for k = 1:10 FHOw ]"#  
    %           z(idx) = y(:,k); t$!zgUJ  
    %           subplot(4,7,Nplot(k)) ]pR?/3  
    %           pcolor(x,x,z), shading interp )7 p" -  
    %           set(gca,'XTick',[],'YTick',[]) yzS^8,  
    %           axis square ETHcZ  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) N!K%aH~O  
    %       end Pm/<^z%  
    % _KH91$iW8m  
    %   See also ZERNPOL, ZERNFUN2. "h+Z[h6T  
    eI1zRoIl-  
    %   Paul Fricker 11/13/2006 ukR0E4p  
    *J-pAN  
    jR/Gd01)  
    % Check and prepare the inputs: Ugri _  
    % ----------------------------- CQWXLQED>  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) & BPYlfB1  
        error('zernfun:NMvectors','N and M must be vectors.') VIp|U{  
    end g Q\.|'%  
    A^OwT#  
    if length(n)~=length(m) |}G"^r  
        error('zernfun:NMlength','N and M must be the same length.') O=o}uB-*6  
    end W>pe-  
    J>_mDcPo  
    n = n(:); |*{*tW C1  
    m = m(:); geG0F}oC!  
    if any(mod(n-m,2)) 1bV G%N  
        error('zernfun:NMmultiplesof2', ... Orq/38:4G  
              'All N and M must differ by multiples of 2 (including 0).') 'NtI bS  
    end CPJ<A,V  
    UXct+l  
    if any(m>n) UdO8KD#r3  
        error('zernfun:MlessthanN', ... d7V/#34  
              'Each M must be less than or equal to its corresponding N.') KtQs uL%  
    end ^OY$ W  
    ~}_^$l8#-Q  
    if any( r>1 | r<0 ) /]U$OP*0  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') 5sY $  
    end eHgr"f*7   
    ?IGp?R^j"  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) mE<_oRM)  
        error('zernfun:RTHvector','R and THETA must be vectors.') TZgtu+&  
    end ;dzy 5o3  
    P#A,(Bke3  
    r = r(:); MRpMmu  
    theta = theta(:); @D9O<x  
    length_r = length(r); M XG>|  
    if length_r~=length(theta) $>/d)o  
        error('zernfun:RTHlength', ... 8>C4w 5kF  
              'The number of R- and THETA-values must be equal.') ,Q"'q0hM=  
    end 0fqcPi  
    =IL\T8y09  
    % Check normalization: +-!3ruwSn  
    % -------------------- Z|qI[uiO  
    if nargin==5 && ischar(nflag) ,buX|  
        isnorm = strcmpi(nflag,'norm'); )?jFz'<r  
        if ~isnorm Y(,RJ&7  
            error('zernfun:normalization','Unrecognized normalization flag.') B!&5*f}*  
        end I=L[ "]  
    else V6merT79  
        isnorm = false; q{9vY:`[  
    end ROkwjw  
    'dj3y/ k%  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% I "x'  
    % Compute the Zernike Polynomials ika*w  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ,ojJ;w5D  
    oywPPVxj  
    % Determine the required powers of r: +`F(wk["m  
    % ----------------------------------- "r6qFxY  
    m_abs = abs(m); |Y"XxM9  
    rpowers = []; ?c8~VQaQ  
    for j = 1:length(n) | lLe^FM  
        rpowers = [rpowers m_abs(j):2:n(j)]; IgbuMEfL  
    end Z.${WZW  
    rpowers = unique(rpowers);  m}yu4  
    va@;V+cD  
    % Pre-compute the values of r raised to the required powers, l4RqQ+[KA;  
    % and compile them in a matrix: @JSWqi>  
    % ----------------------------- qK'mF#n0#  
    if rpowers(1)==0 j"jssbu}  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ewcFzlA@  
        rpowern = cat(2,rpowern{:}); 0j$=KA  
        rpowern = [ones(length_r,1) rpowern]; ]:f.="  
    else 4<s;xSCL  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); w^L`"  
        rpowern = cat(2,rpowern{:}); ~;(\a@ _  
    end 'Ft81e)/  
    wQ(DX!   
    % Compute the values of the polynomials: )nHMXZ>Td  
    % -------------------------------------- 7b1 yF,N  
    y = zeros(length_r,length(n)); w (HVC  
    for j = 1:length(n) E:rJi]  
        s = 0:(n(j)-m_abs(j))/2; ;*5z&1O  
        pows = n(j):-2:m_abs(j); u4lM>(3Y}  
        for k = length(s):-1:1 kg Bkwp  
            p = (1-2*mod(s(k),2))* ... pRfKlTU\  
                       prod(2:(n(j)-s(k)))/              ... vT5GUO{5  
                       prod(2:s(k))/                     ... Cnpl0rV~5  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... JSg=9p$  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); ;FlDRDZ%  
            idx = (pows(k)==rpowers); 7NEOaX(J9  
            y(:,j) = y(:,j) + p*rpowern(:,idx); igOX0  
        end 9ZOQNN<ex  
         B)/&xQu  
        if isnorm -~.+3rcZ]  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); =)y$&Ydj  
        end ;R >>,&g  
    end ]>)}xfL &,  
    % END: Compute the Zernike Polynomials .NdsKhg b  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% CMC p7- v  
    ->|eMV'd  
    % Compute the Zernike functions: =0e>'Iw2  
    % ------------------------------ tDAX pi(  
    idx_pos = m>0; []\-*{^r  
    idx_neg = m<0; pe[huYE  
    6+sz4  
    z = y; h9S f  
    if any(idx_pos) qw4wg9w5p  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); o ^w^dgJ  
    end L^^f.w#m  
    if any(idx_neg) Z+R-}<   
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); \R&ZWJKh  
    end d >M0:  
    Q]/g=Nn ^~  
    % EOF zernfun
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) :@~3wD[y  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. ^CwS'/fdN  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated };(2 na  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive 09Oe-Bg  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, #r&yH^-  
    %   and THETA is a vector of angles.  R and THETA must have the same N}bZdE9F  
    %   length.  The output Z is a matrix with one column for every P-value, K(mzt[n(  
    %   and one row for every (R,THETA) pair. f$nZogaQ  
    % i/N68  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike aLevml2:T  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) ,J*#Ixe}  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) #z}IW(u<  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 0/+TQD!L  
    %   for all p. YcaLc_pUx  
    % :fG9p`  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 wu0J XB%&^  
    %   Zernike functions (order N<=7).  In some disciplines it is "^pF2JI  
    %   traditional to label the first 36 functions using a single mode +Y~,1ai 5^  
    %   number P instead of separate numbers for the order N and azimuthal 9&c *%mm  
    %   frequency M. Re\V<\$J  
    % t;%MSedn  
    %   Example: UJX5}36  
    % xI=[=;L  
    %       % Display the first 16 Zernike functions xJzO?a'  
    %       x = -1:0.01:1; \]/ 6>yT  
    %       [X,Y] = meshgrid(x,x); YF");itH  
    %       [theta,r] = cart2pol(X,Y); .9bP8u2B{  
    %       idx = r<=1; `4]-B@ 7_  
    %       p = 0:15; /Bnh%6#ab  
    %       z = nan(size(X)); fl9J  
    %       y = zernfun2(p,r(idx),theta(idx)); !P:~oo =  
    %       figure('Units','normalized') BYDOTy/%nJ  
    %       for k = 1:length(p) $uNYus^vS  
    %           z(idx) = y(:,k); 9%Ftln6  
    %           subplot(4,4,k) bO{wQ1)Z_  
    %           pcolor(x,x,z), shading interp .!Q[kn0a  
    %           set(gca,'XTick',[],'YTick',[]) rUF= uO(  
    %           axis square 9%uJ:c?  
    %           title(['Z_{' num2str(p(k)) '}']) my3W[3#  
    %       end {,m W7  
    % T;I>5aQ:q4  
    %   See also ZERNPOL, ZERNFUN. tO QY./I  
    a U*cwR  
    %   Paul Fricker 11/13/2006 Yg7C"3;Vt  
    (OK;*ZH+T@  
    W[W}:@KZ  
    % Check and prepare the inputs: *$ 7c||J7  
    % ----------------------------- I%G6V a@  
    if min(size(p))~=1 au1(.(  
        error('zernfun2:Pvector','Input P must be vector.') 3m`y?Dd  
    end A =k{Rl{LA  
    5SY(:!  
    if any(p)>35 {@[z-)N7\,  
        error('zernfun2:P36', ... H]W59-{a  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... aV8]?E5G  
               '(P = 0 to 35).']) '.DFyHsq  
    end aTY\mKk  
    ygp NMq#?X  
    % Get the order and frequency corresonding to the function number: tm}0kWx  
    % ---------------------------------------------------------------- Go_~8w0<  
    p = p(:); /v:+ vh*mS  
    n = ceil((-3+sqrt(9+8*p))/2); 6jtnH'E/  
    m = 2*p - n.*(n+2); . ),m7"u|  
    :AB$d~${M>  
    % Pass the inputs to the function ZERNFUN: Dos`lh  
    % ---------------------------------------- h=~ TgTv  
    switch nargin h{cJ S9e}  
        case 3 Lm}:`  
            z = zernfun(n,m,r,theta); ZjXpMx,  
        case 4 )W/ mt[;  
            z = zernfun(n,m,r,theta,nflag); ] T! >]  
        otherwise x, ^j=n  
            error('zernfun2:nargin','Incorrect number of inputs.') ceR zHq=  
    end g k[8'  
    v5GV"qY  
    % 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^b2H !~  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. vo[Zuv?<h  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of RS~jHwIh  
    %   order N and frequency M, evaluated at R.  N is a vector of iC`K$LY4W  
    %   positive integers (including 0), and M is a vector with the afcI5w;>}  
    %   same number of elements as N.  Each element k of M must be a (JHL0Z/  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) :.IVf Zw  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is 7N / v  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix v'*#P7%Kf  
    %   with one column for every (N,M) pair, and one row for every  IR,`-  
    %   element in R. wYJ.F  
    % uf (`I  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- shOQ/  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is T,,,+gPx  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to "3A.x1uQ  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 `2y2Bk  
    %   for all [n,m]. ([='LyH];z  
    % >v9 ("  
    %   The radial Zernike polynomials are the radial portion of the AAE8j.  
    %   Zernike functions, which are an orthogonal basis on the unit 9GuG"^08  
    %   circle.  The series representation of the radial Zernike `)FSJV1  
    %   polynomials is POQRq%w  
    % p*8LS7UT  
    %          (n-m)/2 Lmx95[#@a  
    %            __ y8uB>z+#+;  
    %    m      \       s                                          n-2s T<"Hh.h  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r #=@( m.k:s  
    %    n      s=0 @54D<Lj  
    % `g&<7~\=A  
    %   The following table shows the first 12 polynomials. A=/|f$s+  
    % 8o-*s+EY"&  
    %       n    m    Zernike polynomial    Normalization q"@Y2lhD!  
    %       --------------------------------------------- Re**)3#gn  
    %       0    0    1                        sqrt(2) eDR4 c%  
    %       1    1    r                           2 ]?p&sI4  
    %       2    0    2*r^2 - 1                sqrt(6) =l TV2C<  
    %       2    2    r^2                      sqrt(6)  g-MaP  
    %       3    1    3*r^3 - 2*r              sqrt(8) G/Nb@pAy[  
    %       3    3    r^3                      sqrt(8) (-tF=wR,W  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) 1CFTQB>  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) 8*;88vW"2  
    %       4    4    r^4                      sqrt(10) TOp|Qtn  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) mdW8RsR  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) 2B{~"<  
    %       5    5    r^5                      sqrt(12) FOxMt;|M  
    %       --------------------------------------------- L, L>cmpM  
    % !fXwX3B  
    %   Example: )54;YK  
    % #;?j]npg]  
    %       % Display three example Zernike radial polynomials ^fT|Wm<  
    %       r = 0:0.01:1; h=uwOi6}  
    %       n = [3 2 5]; mrR~[533j  
    %       m = [1 2 1]; TQyi -Dc  
    %       z = zernpol(n,m,r); #I jG[a-  
    %       figure Q%6 1_l  
    %       plot(r,z) 6\ g-KO  
    %       grid on !sA[A>  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') ,*SoV~  
    % _Gv[ D  
    %   See also ZERNFUN, ZERNFUN2. bLyU;  
    \M-}(>Pfk  
    % A note on the algorithm. rnvKfTpZDU  
    % ------------------------ iO)FZ%?"  
    % The radial Zernike polynomials are computed using the series w,fA-*bZ 0  
    % representation shown in the Help section above. For many special 5(0f"zY  
    % functions, direct evaluation using the series representation can ]03+8 #J  
    % produce poor numerical results (floating point errors), because Ww&~ZZZ {  
    % the summation often involves computing small differences between T2-n;8t  
    % large successive terms in the series. (In such cases, the functions WV]%llj^  
    % are often evaluated using alternative methods such as recurrence ##~";j  
    % relations: see the Legendre functions, for example). For the Zernike cs[_5r&:  
    % polynomials, however, this problem does not arise, because the @giJ&3S,  
    % polynomials are evaluated over the finite domain r = (0,1), and GMqeC  
    % because the coefficients for a given polynomial are generally all MYgh^%w:  
    % of similar magnitude. f$Fa*O-  
    % ;fLYO6  
    % ZERNPOL has been written using a vectorized implementation: multiple i`-,=RJ  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] #p@8m_g  
    % values can be passed as inputs) for a vector of points R.  To achieve "L'0"  
    % this vectorization most efficiently, the algorithm in ZERNPOL VPG+]> *  
    % involves pre-determining all the powers p of R that are required to xxWrSl`fB  
    % compute the outputs, and then compiling the {R^p} into a single dLb9p"EE#  
    % matrix.  This avoids any redundant computation of the R^p, and n\8;4]n  
    % minimizes the sizes of certain intermediate variables. =SJwCT0;  
    % GRV#f06  
    %   Paul Fricker 11/13/2006 <T?H H$es)  
    = WFn+#&^  
    Ig9yd S-.  
    % Check and prepare the inputs: %Q9 iR5?  
    % ----------------------------- kigq(a  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) [J4 Aig  
        error('zernpol:NMvectors','N and M must be vectors.') 6%  +s`  
    end ts BPQ 8Ne  
    \LX!n!@  
    if length(n)~=length(m) N|cWTbi  
        error('zernpol:NMlength','N and M must be the same length.') ^B[%|{cO  
    end {k.Dy92  
    @]$qJFXx  
    n = n(:);  g wM~W  
    m = m(:); 6*3J3Lc_<  
    length_n = length(n); ~ KNdV  
    )aY^k|I  
    if any(mod(n-m,2)) Ul@ Jg    
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') .yp"6S^b  
    end fAMJFHW  
    WR'm<u  
    if any(m<0) c5~d^  
        error('zernpol:Mpositive','All M must be positive.') fNz*E|]8&  
    end P} =eR  
    4 j9  
    if any(m>n) %si5cc?  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') >Xz P'h  
    end rG7S^,5o  
    WU oGIT'  
    if any( r>1 | r<0 ) {4u8~whLp  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') X ?p_O2#k  
    end hVQ TW[  
    -]=-IiC#  
    if ~any(size(r)==1) WH Zz?|^  
        error('zernpol:Rvector','R must be a vector.') XRU^7@Ylks  
    end Efo,5  
    E8]PV,#xY  
    r = r(:); UPtWj8h  
    length_r = length(r); 8ut:cCrmg  
    u@!iByVAg  
    if nargin==4 @)0-oa,u+  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); `V N $ S  
        if ~isnorm ZSL:q%:.  
            error('zernpol:normalization','Unrecognized normalization flag.') A/kRw'6  
        end cI#2MjL  
    else v@OyB7}  
        isnorm = false; K*&?+_v :  
    end "zJGYBen  
    b"Ep?=*5  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% :v/6k  
    % Compute the Zernike Polynomials Bsm>^zZ`YU  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% k:(i sKIA  
    ;1Kxqp z_i  
    % Determine the required powers of r: G$\2@RT9[  
    % ----------------------------------- S30@|@fTz  
    rpowers = []; :sT\-MpQvn  
    for j = 1:length(n) !oXA^7Th6]  
        rpowers = [rpowers m(j):2:n(j)]; pa0'\  
    end A=X2zm>9  
    rpowers = unique(rpowers); C#]%  
    xJ"CAg|B  
    % Pre-compute the values of r raised to the required powers, -"L)<J@gQ?  
    % and compile them in a matrix: ?L|Jc_E  
    % ----------------------------- \-c8/=  
    if rpowers(1)==0 B"O5P>  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); PPCZT3c=  
        rpowern = cat(2,rpowern{:}); q9n0bw^N  
        rpowern = [ones(length_r,1) rpowern]; wEK@B&DV  
    else .ON+ ( #n  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); *qcL(] Yq  
        rpowern = cat(2,rpowern{:}); U:]b&I  
    end ],r?]>  
    b@ J&jE~d  
    % Compute the values of the polynomials: *b" (r|Ko  
    % -------------------------------------- cK.T=7T  
    z = zeros(length_r,length_n); Ty]/F+{  
    for j = 1:length_n &k3'UN!&Ix  
        s = 0:(n(j)-m(j))/2; I>b-w;cC  
        pows = n(j):-2:m(j); )2X ng_,  
        for k = length(s):-1:1 g{8 R+  
            p = (1-2*mod(s(k),2))* ... 7lpd$Y  
                       prod(2:(n(j)-s(k)))/          ... }z&P^p)R  
                       prod(2:s(k))/                 ... s(7'*`G"h  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... n3{m "h3  
                       prod(2:((n(j)+m(j))/2-s(k))); P=& Je?  
            idx = (pows(k)==rpowers); 0|X!Uw-Q%_  
            z(:,j) = z(:,j) + p*rpowern(:,idx); H[%F o  
        end t" 1'B!4  
         @]f"X>  
        if isnorm ]?F05!$*  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); o"^}2^)_SR  
        end zx\N^R;Jq  
    end )@Yp;=l  
    qR<  
    % 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)  'oT}jI  
    5)nv  
    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。 /t>o -  
    izC>-  
    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)