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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 Sqj'2<~W  
    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! pC,MiV$c"  
     
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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 ]CyWL6 z  
    function z = zernfun(n,m,r,theta,nflag) SPKGbp&  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. ?H8w/{J   
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ?2hoY  
    %   and angular frequency M, evaluated at positions (R,THETA) on the HU ]Yv+3   
    %   unit circle.  N is a vector of positive integers (including 0), and tWL3F?wd  
    %   M is a vector with the same number of elements as N.  Each element cA%70Y:AV  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) +r[u4?  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, :9H=D^J  
    %   and THETA is a vector of angles.  R and THETA must have the same L?!*HS7 m  
    %   length.  The output Z is a matrix with one column for every (N,M) t4)~A5s  
    %   pair, and one row for every (R,THETA) pair. qPsf`nI7  
    % r@L19d)J  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike HNN,1MN  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ^n#6CW*n  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral {8D`A;KD  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, uP bvN[~t  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized 82#7TX4  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. mk?&`_X1  
    % 4Z>KrFO  
    %   The Zernike functions are an orthogonal basis on the unit circle. ju#/ {V;D  
    %   They are used in disciplines such as astronomy, optics, and ~oO>6  
    %   optometry to describe functions on a circular domain. 8Z{&b,Y4L  
    % c6gRXp'ID  
    %   The following table lists the first 15 Zernike functions. 9%aBW7@SK  
    % B-`d7c5  
    %       n    m    Zernike function           Normalization & Ji!*~sE  
    %       -------------------------------------------------- d`9% :2qE  
    %       0    0    1                                 1 @,0W(  
    %       1    1    r * cos(theta)                    2 _r+2o-ZR  
    %       1   -1    r * sin(theta)                    2 \C;cs&\Q  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) K#q1/2  
    %       2    0    (2*r^2 - 1)                    sqrt(3) <PL94  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) &r s+x<  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) 7+wy`xi  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) ^g/    
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) 3u+A/  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) 0NlC|5ma)  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) 2(V;OWY(@  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) `l+{jrRb<  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) ^hPREbD+f  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ?,8|K B  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) aOZSX3;wg  
    %       -------------------------------------------------- =\mAvVe  
    % l1*qDzb  
    %   Example 1: _7df(+.{<A  
    % V<&x+?>S  
    %       % Display the Zernike function Z(n=5,m=1) OxGKtnAjf  
    %       x = -1:0.01:1; =+24jHs  
    %       [X,Y] = meshgrid(x,x); -Lh\]  
    %       [theta,r] = cart2pol(X,Y); aH7i$U&  
    %       idx = r<=1; NN(ZH73  
    %       z = nan(size(X)); i\E}!Rwl+  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); 9K~0:c  
    %       figure >2Z0XEe  
    %       pcolor(x,x,z), shading interp 6lW\-h`N G  
    %       axis square, colorbar iZsZSW \  
    %       title('Zernike function Z_5^1(r,\theta)') MR$R#  
    % |C;8GSw>|F  
    %   Example 2: 8:*   
    % qfJi[8".  
    %       % Display the first 10 Zernike functions ]Rah,4?9f  
    %       x = -1:0.01:1; aU&p7y4C@  
    %       [X,Y] = meshgrid(x,x); l|WdJn o  
    %       [theta,r] = cart2pol(X,Y); X/< zxM  
    %       idx = r<=1; awYnlE/Z1  
    %       z = nan(size(X)); r 7 dwj  
    %       n = [0  1  1  2  2  2  3  3  3  3]; %x}iEqkU  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; (<bYoWrK#  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; <Wd#HKIG>l  
    %       y = zernfun(n,m,r(idx),theta(idx)); 80(Olf@PE  
    %       figure('Units','normalized') S($8_u$U  
    %       for k = 1:10 ]iI2  
    %           z(idx) = y(:,k); /\7E&n:)2  
    %           subplot(4,7,Nplot(k)) nZtMF%j'  
    %           pcolor(x,x,z), shading interp q4y P\B  
    %           set(gca,'XTick',[],'YTick',[]) "Zh3,  
    %           axis square :mYVHLmea  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 'dXGd.V7u  
    %       end -hd@<+;E  
    % %P8*Az&]T  
    %   See also ZERNPOL, ZERNFUN2. .ei5+?V<i  
    9~7s*3zI  
    %   Paul Fricker 11/13/2006 [SKN}:D  
    _^%DfMP3i\  
    *JD-|m K  
    % Check and prepare the inputs: #fGI#]SG?  
    % ----------------------------- !B*l'OJw  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ^-^ii 3G`  
        error('zernfun:NMvectors','N and M must be vectors.') mb\"qD5  
    end =&K8~   
    7g7[a/Bts  
    if length(n)~=length(m) z7X,5[P  
        error('zernfun:NMlength','N and M must be the same length.') Of}dsav   
    end j LM}hwJ8  
    "@^^niSFl  
    n = n(:); GBY-WN4sc[  
    m = m(:); H$ZLtPv5  
    if any(mod(n-m,2)) 6 h%,%  
        error('zernfun:NMmultiplesof2', ... |!q,J  
              'All N and M must differ by multiples of 2 (including 0).') :Tcvj5  
    end fW w+'xF!  
    G' '9eV$  
    if any(m>n) /O}lSXo6E  
        error('zernfun:MlessthanN', ... wnU-5r&!]  
              'Each M must be less than or equal to its corresponding N.') HcQ)XJPK  
    end Bra}HjHO  
    v[<x>?i D_  
    if any( r>1 | r<0 ) xj[v$HP  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') 5AK@e|G$w  
    end qi*Dd[OG  
    0j :u.x  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) YU`}T<;bg  
        error('zernfun:RTHvector','R and THETA must be vectors.') cE^Ljk  
    end 2YQ;Kh"S   
    ,R\ex =c  
    r = r(:); t"Bp # U1  
    theta = theta(:); Iw4[D#o  
    length_r = length(r); j+PLtE   
    if length_r~=length(theta) t$&'mJ_-w  
        error('zernfun:RTHlength', ... \fsNI T/  
              'The number of R- and THETA-values must be equal.') S.Q:O{]  
    end B7wzF"  
    Zu*7t<W  
    % Check normalization: 'YJ~~o  
    % -------------------- YwS/O N  
    if nargin==5 && ischar(nflag) bcUSjG>  
        isnorm = strcmpi(nflag,'norm'); h$#PboLd  
        if ~isnorm rd;E /:`5  
            error('zernfun:normalization','Unrecognized normalization flag.') `An p;el  
        end }qJ`nN8  
    else Y4+ ]5;B8  
        isnorm = false; <J>k%,:B  
    end =te4p@  
    )gD2wk(  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% lvRTy|%[  
    % Compute the Zernike Polynomials Y_6 v@SiO  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 4s9.")G  
    (Y*9 [hm  
    % Determine the required powers of r: A<^X P-Nrp  
    % ----------------------------------- M?=I{}!@Q  
    m_abs = abs(m); >oN Wf  
    rpowers = []; r/@Wn  
    for j = 1:length(n) &^w "  
        rpowers = [rpowers m_abs(j):2:n(j)]; RUq[HxF) 6  
    end E#3tkFF0Z[  
    rpowers = unique(rpowers); L]H' ]wpn=  
    bPif"dhHe  
    % Pre-compute the values of r raised to the required powers, #C'E'g0  
    % and compile them in a matrix: ^QjkZ^<dD  
    % ----------------------------- o bN8+ j  
    if rpowers(1)==0 y\4/M6  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); TyA1Qk\  
        rpowern = cat(2,rpowern{:}); X Ai0lN{,  
        rpowern = [ones(length_r,1) rpowern]; ]jPP]Z:y  
    else ;y{(#X#  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); H|7XfM  
        rpowern = cat(2,rpowern{:}); }#; .b'`  
    end 2#00<t\  
    '/9q7?[E!  
    % Compute the values of the polynomials: uJlW$Oc:.  
    % -------------------------------------- pr1bsrMuL  
    y = zeros(length_r,length(n)); I<XYLe[_S  
    for j = 1:length(n) +S+=lu _  
        s = 0:(n(j)-m_abs(j))/2; e2K9CE.O  
        pows = n(j):-2:m_abs(j); 3TZ:  
        for k = length(s):-1:1 +V9xKhR;x  
            p = (1-2*mod(s(k),2))* ... : 2$*'{mM  
                       prod(2:(n(j)-s(k)))/              ... q9PjQ%  
                       prod(2:s(k))/                     ... [k(b<'  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... e*}GQ  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); Fq!_VF^r  
            idx = (pows(k)==rpowers); {IvA 5^  
            y(:,j) = y(:,j) + p*rpowern(:,idx); cH4 PrMm&  
        end ( ;^>G[  
         g oyQ',+  
        if isnorm bBXLW}W  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi);  UBj&T^j  
        end Y)$%-'=b+  
    end ~Hv>^u Mh  
    % END: Compute the Zernike Polynomials S-ZN}N{,6  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ,!GoFu  
    3&5b!Y  
    % Compute the Zernike functions: C2\WvE%!  
    % ------------------------------ [^bq?w  
    idx_pos = m>0; O^ui+44wp  
    idx_neg = m<0; ;m>/tD%  
    )G?\{n-  
    z = y; K/*"U*9Kv  
    if any(idx_pos) O6 s3#iu  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); k~K;r8D/  
    end [hpkE lE  
    if any(idx_neg) )ZBNw{nh  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); >;Vfs{Z(q  
    end cQ1Axs TO  
    )"hd"  
    % 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) (Cq 38~mR  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. l;A'^  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated rr\9HA  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive n 3&h1-  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, $IX(a4'  
    %   and THETA is a vector of angles.  R and THETA must have the same <T)0I1S  
    %   length.  The output Z is a matrix with one column for every P-value, Rx"+i0  
    %   and one row for every (R,THETA) pair. Gf9sexn]l  
    % 50Gu~No6  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike ^W%F?#ELN2  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) )B&`<1Oie  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) (G} }h  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 O43"-  
    %   for all p. Cg~1<J?2  
    % 60n P'xfR  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 a3w6&e`  
    %   Zernike functions (order N<=7).  In some disciplines it is x0{B7/FN  
    %   traditional to label the first 36 functions using a single mode $|&<cenMT  
    %   number P instead of separate numbers for the order N and azimuthal |.yS~XFJS  
    %   frequency M. %g5#q64  
    % <A|z   
    %   Example: 1BW9,Xr  
    % ,ZrR*W?iF  
    %       % Display the first 16 Zernike functions jck(cc= R  
    %       x = -1:0.01:1; RyukQY~<W  
    %       [X,Y] = meshgrid(x,x); na9YlJ\  
    %       [theta,r] = cart2pol(X,Y); L aA<`  
    %       idx = r<=1; &J|I&p   
    %       p = 0:15; TJW8l[M  
    %       z = nan(size(X)); pp1Kor  
    %       y = zernfun2(p,r(idx),theta(idx)); qc)+T_m  
    %       figure('Units','normalized') o$%KbfXO]  
    %       for k = 1:length(p) g@M5_I(W  
    %           z(idx) = y(:,k); ZBx,'ph}4  
    %           subplot(4,4,k) 4G;KT~Cgb  
    %           pcolor(x,x,z), shading interp i?@7>Ca  
    %           set(gca,'XTick',[],'YTick',[]) VyYrL]OrA  
    %           axis square mAW(j@5sp  
    %           title(['Z_{' num2str(p(k)) '}']) 33eOM(`D[  
    %       end dxZn| Y  
    % 0LzS #J+  
    %   See also ZERNPOL, ZERNFUN. *KiY+_8>  
    Pj g#  
    %   Paul Fricker 11/13/2006 5?Q5cD2]\6  
    U Rq9:{  
    ]c 'EJu  
    % Check and prepare the inputs: 0;=]MEk?  
    % ----------------------------- ?*kB>U9e  
    if min(size(p))~=1 +"WNG  
        error('zernfun2:Pvector','Input P must be vector.') BonjK#  
    end F2Ny=H &G  
    zT/woiyB`  
    if any(p)>35 R\5fl[  
        error('zernfun2:P36', ... Pd d(1K*  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... .),9a,  
               '(P = 0 to 35).']) |na9I6  
    end Oo(xYy  
    j1K~zG  
    % Get the order and frequency corresonding to the function number: ;BWWafZ  
    % ---------------------------------------------------------------- (o{x*';i4  
    p = p(:); LSXsq}  
    n = ceil((-3+sqrt(9+8*p))/2); &n83>Q  
    m = 2*p - n.*(n+2); MDd 2B9cy[  
    :c )R6=v  
    % Pass the inputs to the function ZERNFUN: ,5t_}d|3C=  
    % ---------------------------------------- 94*MRn1E  
    switch nargin 7pep\  
        case 3 `\3RFr  
            z = zernfun(n,m,r,theta); /9P7;1?  
        case 4 P^lRJB<$Q  
            z = zernfun(n,m,r,theta,nflag); ~e){2_J&n  
        otherwise *8%uXkMm  
            error('zernfun2:nargin','Incorrect number of inputs.') QOJ5  
    end vZ=dlu_t  
    kqC7^x  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) kH62#[J)yM  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. #CB`7 }jq  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of w~e$ul(IQM  
    %   order N and frequency M, evaluated at R.  N is a vector of !T]bz+  
    %   positive integers (including 0), and M is a vector with the U4._a  
    %   same number of elements as N.  Each element k of M must be a !OR %AdxB  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) :&O6Y-/B  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is P"+K'B7K3  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix KKwJ=za  
    %   with one column for every (N,M) pair, and one row for every TaTw,K|/  
    %   element in R. 3 DZ8-N S  
    % 0"c(n0L  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- .W0;Vhw"  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is a0B,[i  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to +H3;{ h9,  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 =>)4>WT8A  
    %   for all [n,m]. *W1:AGpz  
    % OjY#xO+'  
    %   The radial Zernike polynomials are the radial portion of the 2?9gf,U  
    %   Zernike functions, which are an orthogonal basis on the unit 1\if XJ  
    %   circle.  The series representation of the radial Zernike 7|pF (sb0  
    %   polynomials is }!5+G:JAh  
    % /Os;,g  
    %          (n-m)/2 K<D=QweOon  
    %            __ sQr |3}I(  
    %    m      \       s                                          n-2s Qj/.x#T  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r #F|w_P  
    %    n      s=0 m<e-XT  
    % <S`N9a  
    %   The following table shows the first 12 polynomials. ugMf pT)  
    % hG%J:}  
    %       n    m    Zernike polynomial    Normalization 3/?{= {  
    %       --------------------------------------------- !&8HA   
    %       0    0    1                        sqrt(2) :7Smsc"B!  
    %       1    1    r                           2 j?cE0 hz  
    %       2    0    2*r^2 - 1                sqrt(6) ;chz};zY  
    %       2    2    r^2                      sqrt(6) h(wu5G0C#u  
    %       3    1    3*r^3 - 2*r              sqrt(8) >`!Lh`n7_  
    %       3    3    r^3                      sqrt(8) W_P&;)E  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) ~xaPq=AH  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) @{y[2M} %]  
    %       4    4    r^4                      sqrt(10) ZpTT9{PT=:  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) )=c/{  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) Yr9!</;T  
    %       5    5    r^5                      sqrt(12) j* \gD  
    %       --------------------------------------------- Y$ To)qo  
    % v*k}{M  
    %   Example: BVk&TGa;[$  
    % tE]0 #B)D<  
    %       % Display three example Zernike radial polynomials :CJ]^v   
    %       r = 0:0.01:1; TUV&9wKXo  
    %       n = [3 2 5]; XFU['BI  
    %       m = [1 2 1]; K~B@8az  
    %       z = zernpol(n,m,r); Lupug"p0   
    %       figure tEXY>=  
    %       plot(r,z) ui\yY3?  
    %       grid on sF`ELrR \  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') )LP=IT  
    % j~@Hj$APa`  
    %   See also ZERNFUN, ZERNFUN2. U:n3V  
    ,xeJf6es  
    % A note on the algorithm. L6-zQztn  
    % ------------------------ O-D${==  
    % The radial Zernike polynomials are computed using the series E0Y>2HOuL  
    % representation shown in the Help section above. For many special L(bYG0ZI5C  
    % functions, direct evaluation using the series representation can L93&.d@m9  
    % produce poor numerical results (floating point errors), because \<bar ~  
    % the summation often involves computing small differences between XDWR ]  
    % large successive terms in the series. (In such cases, the functions XsVp7zk\  
    % are often evaluated using alternative methods such as recurrence l,pI~A`w_  
    % relations: see the Legendre functions, for example). For the Zernike yp}a&Dg  
    % polynomials, however, this problem does not arise, because the v7ShXX:  
    % polynomials are evaluated over the finite domain r = (0,1), and :Rq>a@Rp  
    % because the coefficients for a given polynomial are generally all ,o3{?o]s  
    % of similar magnitude. E mUA38  
    % cM= ? {W7~  
    % ZERNPOL has been written using a vectorized implementation: multiple M :4N'#`  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] [-*1M4D9  
    % values can be passed as inputs) for a vector of points R.  To achieve ^` 96L  
    % this vectorization most efficiently, the algorithm in ZERNPOL V@QWJZ"  
    % involves pre-determining all the powers p of R that are required to eiNF?](3O  
    % compute the outputs, and then compiling the {R^p} into a single ]CFh0N|(L  
    % matrix.  This avoids any redundant computation of the R^p, and l Wa4X#~.  
    % minimizes the sizes of certain intermediate variables. phT|w H  
    % ^,?dk![1Cv  
    %   Paul Fricker 11/13/2006 k|Hxd^^I  
    "o=*f/M  
    x2^Yvgc-  
    % Check and prepare the inputs: IwE{Zvr  
    % ----------------------------- lO Rym:P  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) yK$.wd 2,  
        error('zernpol:NMvectors','N and M must be vectors.') DQ<{FN  
    end A[/_}bI|  
    cZ!%#A z  
    if length(n)~=length(m) v-V#?+#  
        error('zernpol:NMlength','N and M must be the same length.') XzX-Q'i=n0  
    end )4[Yplo  
    0X`sQNx  
    n = n(:); R::0.*FF  
    m = m(:); ),G?f {`!  
    length_n = length(n); qhKW6v  
    @AXRKYQ{t  
    if any(mod(n-m,2)) "Y9PS_u(~  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') (BB&ZUdyv  
    end {Hv/|.),hu  
    -P+@n)?T6  
    if any(m<0) :YZqrcr}  
        error('zernpol:Mpositive','All M must be positive.') l>KkK|!T^i  
    end 8{m5P8w'  
    ?!N@%R>5rN  
    if any(m>n) /M:R|91:_  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') #HMJBQ4v#  
    end G_fP%ovh  
    Js`xTH'  
    if any( r>1 | r<0 ) 9WXJz;  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') `N+A8  
    end &"D *  
     aOS:rC  
    if ~any(size(r)==1) oKl^Ttr  
        error('zernpol:Rvector','R must be a vector.') 3DNw=Ic0k  
    end r<Il;?S6  
    eR'Df" +  
    r = r(:); .qjVw?E  
    length_r = length(r); )2@_V %  
    *z2G(Uac  
    if nargin==4 0K\Xxo.=  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); 6g/ <FM  
        if ~isnorm J3oEN'8S  
            error('zernpol:normalization','Unrecognized normalization flag.') &kR+7  
        end fS08q9,S/  
    else 6mP s;I  
        isnorm = false; KO`ftz3 +  
    end $T\W'W R>  
    #~A(%a  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% g7($lt>  
    % Compute the Zernike Polynomials p0 @ ,-  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% gWt}q-@nRR  
    l(-We.:(  
    % Determine the required powers of r: ]*I:N  
    % ----------------------------------- z 6~cm6j  
    rpowers = []; r-M:YB  
    for j = 1:length(n) +mM=`[Z`??  
        rpowers = [rpowers m(j):2:n(j)]; d~bZOy  
    end s@USJ4#  
    rpowers = unique(rpowers); +_.k\CRms  
    ;7id![KI4  
    % Pre-compute the values of r raised to the required powers, (G:A^z  
    % and compile them in a matrix: |Ai/q6u  
    % ----------------------------- \rXmWzl{  
    if rpowers(1)==0 QucDIZ  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); DtXQLL*fl(  
        rpowern = cat(2,rpowern{:}); )gOVnA/M  
        rpowern = [ones(length_r,1) rpowern]; ?hqHTH:PU  
    else _?j66-( Q  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 4@2<dw|*h  
        rpowern = cat(2,rpowern{:}); Z XCq>  
    end MQs!+Z"m>  
    T1$fu(f  
    % Compute the values of the polynomials: #@ClhpLD  
    % -------------------------------------- st- z>}  
    z = zeros(length_r,length_n); `SFA`B)[5@  
    for j = 1:length_n A -C.Bi;/  
        s = 0:(n(j)-m(j))/2; '}4z=f`}  
        pows = n(j):-2:m(j); VI xGD#m  
        for k = length(s):-1:1 2C6o?*RjyY  
            p = (1-2*mod(s(k),2))* ... z7vc|Z|  
                       prod(2:(n(j)-s(k)))/          ... <z'Pj7c[  
                       prod(2:s(k))/                 ... PK C}!>2  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... /"(`oe<  
                       prod(2:((n(j)+m(j))/2-s(k))); U yb-feG  
            idx = (pows(k)==rpowers); +9^V9]{Vo  
            z(:,j) = z(:,j) + p*rpowern(:,idx); Y ~TR`y  
        end L3/SIoqd  
         @AB}r1E2  
        if isnorm dd+hX$,  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); f|v5i tO2  
        end OO?;??  
    end c-`'`L^J  
    ;\'d9C  
    % 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)  2>g!+p Ox  
    U2 Cmf  
    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。 I:M15  
    I7+yu>  
    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)