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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 j7ZxA*  
    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! O6NH  
     
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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 xx!8cvD4?  
    function z = zernfun(n,m,r,theta,nflag) v^@)&,  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. Oe;#q  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N R?iCJ5m  
    %   and angular frequency M, evaluated at positions (R,THETA) on the ,:PMS8pS  
    %   unit circle.  N is a vector of positive integers (including 0), and |:5O|m '  
    %   M is a vector with the same number of elements as N.  Each element TiI/I`A  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) <b H *f w  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, KbLSK  
    %   and THETA is a vector of angles.  R and THETA must have the same ?d3K:|g  
    %   length.  The output Z is a matrix with one column for every (N,M) *@''OyL  
    %   pair, and one row for every (R,THETA) pair. L0"|4=  
    % r{v3 XD/  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike **%&|9He  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), .4\I?  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral b_RO%L:"yL  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, +"-l~`+<es  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized FzX ;~CA  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. kVQm|frUz  
    % Lbrl CB+  
    %   The Zernike functions are an orthogonal basis on the unit circle. 4,LS08&gh  
    %   They are used in disciplines such as astronomy, optics, and FDD=I\Ic  
    %   optometry to describe functions on a circular domain. A#cFO)"  
    % THhxj)  
    %   The following table lists the first 15 Zernike functions. 5kw  K%  
    % d[9{&YnH !  
    %       n    m    Zernike function           Normalization &Tt7VYJfIV  
    %       -------------------------------------------------- YCiG~y/~  
    %       0    0    1                                 1 cEu_p2(7!B  
    %       1    1    r * cos(theta)                    2 U!q2bF<@  
    %       1   -1    r * sin(theta)                    2 [<@T%yq  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) 'Hx#DhiFz  
    %       2    0    (2*r^2 - 1)                    sqrt(3) >`UqS`YQK  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) 6 8,j~e3-i  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) yZ6WbI8n  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) 6d]4 %QT  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) k_]'?f7Z  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) Pg T3E  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) LSc^3=X  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) :bct+J}l~  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) Eh8GqFEM  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Bbs1U  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) OU%"dmSDk  
    %       -------------------------------------------------- P?V+<c{  
    % C{/U;Ie-b  
    %   Example 1: TNqL ')f  
    % k*;U?C!  
    %       % Display the Zernike function Z(n=5,m=1) ;>Z+b#C[  
    %       x = -1:0.01:1; 4A@HR  
    %       [X,Y] = meshgrid(x,x); .t\J @?Z  
    %       [theta,r] = cart2pol(X,Y); r5s{t4 ;Ch  
    %       idx = r<=1; lVT*Ev{&.  
    %       z = nan(size(X)); 2?%*UxcO  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); e=VSO!(rY  
    %       figure y`zdI_!7  
    %       pcolor(x,x,z), shading interp >bV3~m$a+  
    %       axis square, colorbar R?)Yh.vi=t  
    %       title('Zernike function Z_5^1(r,\theta)') (Z>?\iNJ  
    % 1R@G7m  
    %   Example 2: VgXT4gO!  
    % zqj|$YNC  
    %       % Display the first 10 Zernike functions _UTN4z2aTG  
    %       x = -1:0.01:1; [,Rc&7p~R  
    %       [X,Y] = meshgrid(x,x); ^Ak?2,xB#+  
    %       [theta,r] = cart2pol(X,Y); 12#yHsk  
    %       idx = r<=1; \uHC9}0  
    %       z = nan(size(X)); t8RtJ2;  
    %       n = [0  1  1  2  2  2  3  3  3  3]; <7`k[~)VB  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; %R4 \[e  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; !QVhP+l'H  
    %       y = zernfun(n,m,r(idx),theta(idx)); EgG3XhfS  
    %       figure('Units','normalized') $MDmY4\  
    %       for k = 1:10 }5PC53q  
    %           z(idx) = y(:,k); }OIe!  
    %           subplot(4,7,Nplot(k)) f`ibP6%  
    %           pcolor(x,x,z), shading interp m<j;f  
    %           set(gca,'XTick',[],'YTick',[]) l7T?Yx j  
    %           axis square  cRK Lyb  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ?a,#p  
    %       end Mo3%OR  
    % dn'|~zf.  
    %   See also ZERNPOL, ZERNFUN2. ^"<Bk<b(  
    C"n!mr{srt  
    %   Paul Fricker 11/13/2006 \1<aBgK i  
    =A,T:!}'  
    1ik.|T<f0  
    % Check and prepare the inputs: kO`!!M[Oo  
    % ----------------------------- k+[oYd  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) IY.M#Q ]  
        error('zernfun:NMvectors','N and M must be vectors.')  lPz`?Hn  
    end }8 ;,2E*z  
    lGahwn:  
    if length(n)~=length(m) =4+2y '  
        error('zernfun:NMlength','N and M must be the same length.') zfDfy!\2_  
    end yqx!{8=V  
    K+ /wJ9^B  
    n = n(:); KJ/Gv#Kj  
    m = m(:); &^&0,g?To  
    if any(mod(n-m,2)) e%:vLE 9  
        error('zernfun:NMmultiplesof2', ... dCn9]cj/  
              'All N and M must differ by multiples of 2 (including 0).') U&(gNuR>J  
    end vO?sHh  
    5hEA/G  
    if any(m>n) GBZu<t/  
        error('zernfun:MlessthanN', ... j@nK6`d+1  
              'Each M must be less than or equal to its corresponding N.') jHT^I as  
    end j/oc+ M^  
    _)pOkS  
    if any( r>1 | r<0 ) <J~6Q  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') J0bcW25  
    end 4J'0k<5S  
    U43U2/^  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) (5`(H.(  
        error('zernfun:RTHvector','R and THETA must be vectors.') a"4X7 D+  
    end jK\kASwG  
    30w(uF  
    r = r(:); ~~WY?I-  
    theta = theta(:); n=DmdQ}  
    length_r = length(r); BJHWx,v  
    if length_r~=length(theta) GZ5DI+3  
        error('zernfun:RTHlength', ... )X*_oH=  
              'The number of R- and THETA-values must be equal.') (oCpQDab@  
    end ,*V%  
    rUV'DC?eE  
    % Check normalization: zO9WqP_`iR  
    % -------------------- TG?>;It&  
    if nargin==5 && ischar(nflag) $pPc}M[h  
        isnorm = strcmpi(nflag,'norm'); d+h~4'ebv  
        if ~isnorm  m5J@kE%  
            error('zernfun:normalization','Unrecognized normalization flag.') |jH Yf42Q  
        end 8:I-?z;S  
    else LD WYFOGQ  
        isnorm = false; FN26f*/  
    end Zl# ';~9W  
    `|nJAW3  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% g]MgT-C|  
    % Compute the Zernike Polynomials s 64@<oU<"  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @QpL*F  
    vfDX~_N  
    % Determine the required powers of r: 'SE5sB  
    % ----------------------------------- 5 <KBMCn  
    m_abs = abs(m); 6R3/"&P(/#  
    rpowers = []; o@$py U8  
    for j = 1:length(n) Sd I>  
        rpowers = [rpowers m_abs(j):2:n(j)]; iqX%pR~Yo  
    end %Y.@AiViz  
    rpowers = unique(rpowers); (3x2^M8  
    AKLFUk  
    % Pre-compute the values of r raised to the required powers, !*qQ 7  
    % and compile them in a matrix: /viBJ`-O  
    % ----------------------------- lUnC+w#[  
    if rpowers(1)==0 ^Kl<<pUaV  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); |1!OwQax  
        rpowern = cat(2,rpowern{:}); ^5!"[RB\  
        rpowern = [ones(length_r,1) rpowern]; Qdc#v\B  
    else -:!T@rV,d  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); %t.\J:WN;  
        rpowern = cat(2,rpowern{:}); $Vs5d= B  
    end CC`#2j  
    {9F}2 SJ  
    % Compute the values of the polynomials: ucLh|}jJ5  
    % -------------------------------------- p)Ht =~  
    y = zeros(length_r,length(n)); F CfU=4O  
    for j = 1:length(n) >"]t4]GVf  
        s = 0:(n(j)-m_abs(j))/2; 1X&scVw  
        pows = n(j):-2:m_abs(j); n#P?JyGm1g  
        for k = length(s):-1:1 &oVZ2.O#(  
            p = (1-2*mod(s(k),2))* ... 68qCY  
                       prod(2:(n(j)-s(k)))/              ... KAy uv  
                       prod(2:s(k))/                     ... ,/p .!+  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... d$MewDW UN  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); Q;z'"P   
            idx = (pows(k)==rpowers); Q^lgtb  
            y(:,j) = y(:,j) + p*rpowern(:,idx); ` gor  
        end .,p@ee$q  
         l2!ztK1^  
        if isnorm t<p4H^  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); >o"0QD  
        end Ao9=TC'v$'  
    end %LL?'&&  
    % END: Compute the Zernike Polynomials h&Q-QU  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% b]Lp_t  
    _>{"vY  
    % Compute the Zernike functions: &xFs0R i(  
    % ------------------------------ c<)O#i@3/  
    idx_pos = m>0; 2+\@0j[q  
    idx_neg = m<0; \xk8+=/A  
    j4D`Xq2 X  
    z = y; l2 #^}-  
    if any(idx_pos) \T`iq[+6  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ^12}#I  
    end `v Ebm Xb  
    if any(idx_neg) u |ru$cIo  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); AT^MQvn  
    end ]<o^Q[OL  
    v kW2&  
    % 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) 9Or  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. ulY<4MN  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated 'miY"L:| O  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive C@FX[:l@-  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, E1rxuV|9  
    %   and THETA is a vector of angles.  R and THETA must have the same l*4_  
    %   length.  The output Z is a matrix with one column for every P-value, [- x]%  
    %   and one row for every (R,THETA) pair. k3B]u.Lo  
    % \kksZ4,  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike cvv(OkC  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) m"8Gh `Fo  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) Eh?,-!SUQn  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 \_zp4Xb2  
    %   for all p. 1ml{oqNj  
    % ,~xX[uB  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 h*X u/aOg  
    %   Zernike functions (order N<=7).  In some disciplines it is ePwoza  
    %   traditional to label the first 36 functions using a single mode JlN<w  
    %   number P instead of separate numbers for the order N and azimuthal -D30(g{O  
    %   frequency M. &H@OLyC  
    % 9^1.nE(R&  
    %   Example: oSqkAAGz\  
    % 7<3eB)S  
    %       % Display the first 16 Zernike functions /N(Ol WEp  
    %       x = -1:0.01:1; R4g% $}  
    %       [X,Y] = meshgrid(x,x); LIDYKKDJ^  
    %       [theta,r] = cart2pol(X,Y); n g?kl|VG  
    %       idx = r<=1; niP/i  
    %       p = 0:15; hiA%Tq?  
    %       z = nan(size(X)); ZA/:\6gm  
    %       y = zernfun2(p,r(idx),theta(idx)); $P%b?Y/  
    %       figure('Units','normalized') k'$UA$2d  
    %       for k = 1:length(p) $X:r&7t+Q[  
    %           z(idx) = y(:,k); h$y0>eMWs  
    %           subplot(4,4,k) YF<;s^&@u  
    %           pcolor(x,x,z), shading interp /MQI5Djg  
    %           set(gca,'XTick',[],'YTick',[]) a6fqtkZ x  
    %           axis square 2OJ=Xb1  
    %           title(['Z_{' num2str(p(k)) '}']) 7IH^5r  
    %       end 8'X:}O/  
    % A~UDtXN*4  
    %   See also ZERNPOL, ZERNFUN. h,C?%H+/0Q  
    <[mvfw  
    %   Paul Fricker 11/13/2006 %4rPkPAtrp  
    hJ1:#%Qe.  
    LxC"j1wfl  
    % Check and prepare the inputs: m"Y|xvIA  
    % ----------------------------- KD5}Nk)t  
    if min(size(p))~=1 l^ aUN  
        error('zernfun2:Pvector','Input P must be vector.') H6PS7g"  
    end j4G?=oDb  
    /*8Ms`  
    if any(p)>35 w;p!~o &  
        error('zernfun2:P36', ... m!-,K8  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... s&7,gWy}BE  
               '(P = 0 to 35).']) 7VQk$im399  
    end ;0f?-W?1  
    gM<*(=x'  
    % Get the order and frequency corresonding to the function number: pK~K>8\  
    % ---------------------------------------------------------------- AK*F,H9  
    p = p(:); O1_dA%m  
    n = ceil((-3+sqrt(9+8*p))/2); i; 3^vhbQ  
    m = 2*p - n.*(n+2); aN5w  
    5-ju5z?=  
    % Pass the inputs to the function ZERNFUN: $`&uu  
    % ---------------------------------------- C4jq T  
    switch nargin YQI&8~z  
        case 3 ,^UNQO*{GI  
            z = zernfun(n,m,r,theta); +EWfsKz  
        case 4 Iw0Q1bK(  
            z = zernfun(n,m,r,theta,nflag); !?7c2QRN  
        otherwise _lE0_X|d  
            error('zernfun2:nargin','Incorrect number of inputs.') 7EKQE>xj  
    end /Af:{|'$%  
    .WR+)^&zz  
    % EOF zernfun2
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    function z = zernpol(n,m,r,nflag) uV:;q>XM'%  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. 3UIR^Rh+  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of ]KS|r+  
    %   order N and frequency M, evaluated at R.  N is a vector of (\ze T5  
    %   positive integers (including 0), and M is a vector with the :Qg3B ';  
    %   same number of elements as N.  Each element k of M must be a 1R1DK$^c  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) h] (BTb#-  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is R~$W  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix lkWID  
    %   with one column for every (N,M) pair, and one row for every /\S1p3EW*  
    %   element in R. '= _}&  
    % +@Oo)#V|.  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- L+}q !'8S  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is wsyG~^>  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to S0_#h)  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 6O^'J~wiI  
    %   for all [n,m]. \@6nRs8b|N  
    % |Go?A/'  
    %   The radial Zernike polynomials are the radial portion of the %d5;JEgA:g  
    %   Zernike functions, which are an orthogonal basis on the unit &J)q_Z8  
    %   circle.  The series representation of the radial Zernike idLysxN  
    %   polynomials is F j_r n  
    % \(PC#H%  
    %          (n-m)/2 8#gS{   
    %            __ 0ivlKe%  
    %    m      \       s                                          n-2s GUJaeFe  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r \4RVJ[2  
    %    n      s=0 FF"6~  
    % zW`$T 88~  
    %   The following table shows the first 12 polynomials. *RQkL'tRf  
    % ps#+i  
    %       n    m    Zernike polynomial    Normalization gHLBtl/  
    %       --------------------------------------------- :>U2yI  
    %       0    0    1                        sqrt(2) YlW~  
    %       1    1    r                           2 c$)Y$@D  
    %       2    0    2*r^2 - 1                sqrt(6) 6t0!a@t  
    %       2    2    r^2                      sqrt(6) }E5oa\ 1u  
    %       3    1    3*r^3 - 2*r              sqrt(8) SCClD6k=V  
    %       3    3    r^3                      sqrt(8) gWo`i  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) W|K"0ab  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) '/^bO#G:  
    %       4    4    r^4                      sqrt(10) b_&;i4[  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) ?*}^xXI/  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) x/NR_~Rnk  
    %       5    5    r^5                      sqrt(12) yJx{6  
    %       --------------------------------------------- i2ap]  
    % jXEuK:exQ  
    %   Example: ({#9gTP2b  
    % 6N}>@Y5  
    %       % Display three example Zernike radial polynomials ~+1t3M e  
    %       r = 0:0.01:1; *xEcX6ZHX  
    %       n = [3 2 5]; 6&p I{  
    %       m = [1 2 1]; ~c~$2Xo  
    %       z = zernpol(n,m,r); _pSCv:3T  
    %       figure ~429sT(   
    %       plot(r,z)  rO]7 g  
    %       grid on |=C&JA  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') 3/P# 2&jt  
    % ?dJ-g~  
    %   See also ZERNFUN, ZERNFUN2. KdT1Nb=  
    ~n:dHK`  
    % A note on the algorithm. j?&Rf,,%  
    % ------------------------ `6KTQk'  
    % The radial Zernike polynomials are computed using the series i5  x[1  
    % representation shown in the Help section above. For many special {EKzPr/  
    % functions, direct evaluation using the series representation can d\Xi1&&  
    % produce poor numerical results (floating point errors), because 60KhwD1  
    % the summation often involves computing small differences between j9zK=eG  
    % large successive terms in the series. (In such cases, the functions H6ff b)&  
    % are often evaluated using alternative methods such as recurrence K1rF;7Y6  
    % relations: see the Legendre functions, for example). For the Zernike 'J R2@W`]]  
    % polynomials, however, this problem does not arise, because the 2ZMYA=[!  
    % polynomials are evaluated over the finite domain r = (0,1), and Oj<.3U[C  
    % because the coefficients for a given polynomial are generally all 8}m bfu o1  
    % of similar magnitude. kG:,Ff>  
    % @SREyqC4  
    % ZERNPOL has been written using a vectorized implementation: multiple Mp:/[%9Fi  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] \%nFCK0  
    % values can be passed as inputs) for a vector of points R.  To achieve [#y/`  
    % this vectorization most efficiently, the algorithm in ZERNPOL Hl"qLrb4  
    % involves pre-determining all the powers p of R that are required to (fmcWHs  
    % compute the outputs, and then compiling the {R^p} into a single tETT\y|'  
    % matrix.  This avoids any redundant computation of the R^p, and aRBTuLa)fo  
    % minimizes the sizes of certain intermediate variables. 2|vArRKt  
    % w ^ v*1KA&  
    %   Paul Fricker 11/13/2006 OhmKjY/}  
    W2L:  
    t^HQ=*c  
    % Check and prepare the inputs: 7XKPC+)1ya  
    % ----------------------------- c\i`=>%b@  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) e0O2 >w  
        error('zernpol:NMvectors','N and M must be vectors.') 1O bxQ_x  
    end Txkmt$h  
    & 2MI(9v  
    if length(n)~=length(m) {HKd="%VG  
        error('zernpol:NMlength','N and M must be the same length.') `UFRv   
    end (0s7<&Iu  
    l4+!H\2  
    n = n(:); QJc3@  
    m = m(:); 1JIL6w_  
    length_n = length(n); %(a<(3r  
    =5NrkCk#V  
    if any(mod(n-m,2)) c"OBm#  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') +g_+JLQ  
    end `Tt}:9/3  
     %Gp%l  
    if any(m<0) 1iq,Gd-G.  
        error('zernpol:Mpositive','All M must be positive.') &fJ92v?%^S  
    end {9sA'5  
    Dm=t`_DL8  
    if any(m>n) ]>fAV(ix  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') tx}} Kd  
    end %4#,y(dO  
    NvH9?Ek"  
    if any( r>1 | r<0 ) wjk-$p  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') w=EUwt  
    end zx"'WM*  
    DA)+)PhY7K  
    if ~any(size(r)==1) * z|i{=W F  
        error('zernpol:Rvector','R must be a vector.') 5b X*8H D  
    end "dfq  
    ^UP!y!&N  
    r = r(:); jR-`ee}y2  
    length_r = length(r); *Dr-{\9  
    y6.}h9~  
    if nargin==4 lqFDX d  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); %VG;vW\V  
        if ~isnorm Le3H!9lbc  
            error('zernpol:normalization','Unrecognized normalization flag.') ,4oYKJ$+h  
        end Az4+([  
    else `ER">@&  
        isnorm = false; WAPN,WuW  
    end VXt8y)?a  
    fl| 8#\r  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ;V(- ;O  
    % Compute the Zernike Polynomials T^LpoN/T  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% X|+o4R?  
    n< UuVu  
    % Determine the required powers of r: p2T%Zl_  
    % ----------------------------------- WP,Ll\K)7  
    rpowers = []; s%h|>l[lKT  
    for j = 1:length(n) 5/j7C>  
        rpowers = [rpowers m(j):2:n(j)]; 4|Z;EAFx  
    end ;J|sH>i  
    rpowers = unique(rpowers); tins.D  
    ConXP\M-  
    % Pre-compute the values of r raised to the required powers, A/n-.ci  
    % and compile them in a matrix: qzk/P1{-  
    % ----------------------------- Q 6djfEN>  
    if rpowers(1)==0 0TA{E-A   
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); Kx.'^y  
        rpowern = cat(2,rpowern{:}); hE>ux"_2/  
        rpowern = [ones(length_r,1) rpowern]; j)4:*R.Z]  
    else xWk:7,/  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); z3!j>X_w  
        rpowern = cat(2,rpowern{:}); +a$'<GvP  
    end m0xL'g6F  
    r':wq   
    % Compute the values of the polynomials: 'n`+R~Kkh  
    % -------------------------------------- mQ 1)d5  
    z = zeros(length_r,length_n); r* #ApM"L  
    for j = 1:length_n (XtN3FTY  
        s = 0:(n(j)-m(j))/2; C!KxY/*Px  
        pows = n(j):-2:m(j); +X[+SF)!  
        for k = length(s):-1:1 b~;gj^  
            p = (1-2*mod(s(k),2))* ... I&9_F% rX  
                       prod(2:(n(j)-s(k)))/          ... F?!P7 zW  
                       prod(2:s(k))/                 ... H&K(,4u^  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... `#O%ZZ+  
                       prod(2:((n(j)+m(j))/2-s(k))); O <;Au|>*  
            idx = (pows(k)==rpowers); qYD$_a  
            z(:,j) = z(:,j) + p*rpowern(:,idx);  lJaR,,  
        end HUF],[N  
         u{#}Lo>B #  
        if isnorm 0V*B3V<  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); 2'O2n]{  
        end 3m RP.<=  
    end *|)a@V L  
    <9zzjgzG{c  
    % 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)  d/8p?Km  
    \DiAfx<Ub  
    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。 eS`ZC!W   
    bcR";cE  
    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)