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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 }#2I/dn  
    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! tHV+#3h  
     
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    离线phility
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    只看该作者 1楼 发表于: 2011-03-12
    可以用matlab编程,用zernike多项式进行波面拟合,求出zernike多项式的系数,拟合的算法有很多种,最简单的是最小二乘法,你可以查下相关资料,挺简单的
    离线phility
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    只看该作者 2楼 发表于: 2011-03-12
    泽尼克多项式的前9项对应象差的
    离线niuhelen
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    只看该作者 3楼 发表于: 2011-03-12
    回 2楼(phility) 的帖子
    非常感谢啊,我手上也有zernike多项式的拟合的源程序,也不知道对不对,不怎么会有 D!mx&O9  
    function z = zernfun(n,m,r,theta,nflag) \7G.anY  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. ~0rvrDDg  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N MH=Ld=i  
    %   and angular frequency M, evaluated at positions (R,THETA) on the 9yp'-RKjw  
    %   unit circle.  N is a vector of positive integers (including 0), and JZ/T:Hsh4  
    %   M is a vector with the same number of elements as N.  Each element 5C-XQS1  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) $V;0z~&!'  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, q^6l`JJ  
    %   and THETA is a vector of angles.  R and THETA must have the same x5b .^75p$  
    %   length.  The output Z is a matrix with one column for every (N,M) :XB^IyO-A  
    %   pair, and one row for every (R,THETA) pair. aa}U87]k  
    % a~Yq0d?`D  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ~)*uJ wW/a  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ?N&s .  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral !ezy  v`  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 4jW <*jM  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized pzb`M'Z?C  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. *RFBLCt  
    % =nv/ r  
    %   The Zernike functions are an orthogonal basis on the unit circle. ne%(`XY{Q]  
    %   They are used in disciplines such as astronomy, optics, and NtkZ\3  
    %   optometry to describe functions on a circular domain. [0lO0ik>G  
    % 0P;\ :-&p  
    %   The following table lists the first 15 Zernike functions. Wm/0Pi  
    % 7#C3E$gn?  
    %       n    m    Zernike function           Normalization av~kF  
    %       -------------------------------------------------- ~R~eQ=8  
    %       0    0    1                                 1 o_&Qb^W  
    %       1    1    r * cos(theta)                    2 WTu!/J<\  
    %       1   -1    r * sin(theta)                    2 { }P~nP  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) 3\K;y>NK  
    %       2    0    (2*r^2 - 1)                    sqrt(3) D[` ~=y(  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) vJ e c+a  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) } wx(P3BHD  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) )\ J~KB4  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) f& Vx`oj  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) oG c9 6B%  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) eQ<G Nvm  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) nGxG!  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) R,8T t!n  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) o0TB>DX$`  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) %`lLX/4~  
    %       -------------------------------------------------- 3e1%G#fu  
    % w@H@[x  
    %   Example 1: 6uxF<  
    % f{h2>nEj \  
    %       % Display the Zernike function Z(n=5,m=1) e^UUR-K%  
    %       x = -1:0.01:1; py6O\` \  
    %       [X,Y] = meshgrid(x,x); 5m\)82s  
    %       [theta,r] = cart2pol(X,Y); %2'Y@AX`  
    %       idx = r<=1; i:H]Sb)<b  
    %       z = nan(size(X)); X39%O'  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); ~Xc1y!"9*  
    %       figure |Rz}bsrZ  
    %       pcolor(x,x,z), shading interp {Rn*)D9  
    %       axis square, colorbar `bWc<4T  
    %       title('Zernike function Z_5^1(r,\theta)') er<_;"`1  
    % MHS|gR.c  
    %   Example 2: 'N`x@(  
    % =)J )xH!N  
    %       % Display the first 10 Zernike functions Ss:'H H4  
    %       x = -1:0.01:1; N!<X% Ym  
    %       [X,Y] = meshgrid(x,x); ,nJCqX~ /G  
    %       [theta,r] = cart2pol(X,Y); \W|ymV_Ki  
    %       idx = r<=1; +pe\9F  
    %       z = nan(size(X)); K6,d{n  
    %       n = [0  1  1  2  2  2  3  3  3  3]; ;rl61d}NH#  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; .e\PCf9v  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; WLH ;{  
    %       y = zernfun(n,m,r(idx),theta(idx)); 57EL&V%j  
    %       figure('Units','normalized') f'Rq#b@  
    %       for k = 1:10 lYU?j|n  
    %           z(idx) = y(:,k); XII',&  
    %           subplot(4,7,Nplot(k)) 7wHd*{^9N  
    %           pcolor(x,x,z), shading interp ~xcU6@/  
    %           set(gca,'XTick',[],'YTick',[]) KBA& s  
    %           axis square \"d\b><R  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) rr2^sQ;_  
    %       end oo1h"[  
    % D8`SI2 1P  
    %   See also ZERNPOL, ZERNFUN2. 4^!%>V"d/  
    %K0Wm#)  
    %   Paul Fricker 11/13/2006 e@PY(#ru  
    h]}DMVV]  
    #;h> x  
    % Check and prepare the inputs: bL|$\'S  
    % ----------------------------- .-1'#Z1T  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Gsy'':u  
        error('zernfun:NMvectors','N and M must be vectors.') c=re(  
    end lInf,Q7W  
    o bGvd6\  
    if length(n)~=length(m) 9ZDbZc  
        error('zernfun:NMlength','N and M must be the same length.') azG"Mt |7Z  
    end J 2k4k  
    gI/(hp3ob  
    n = n(:); ]Mvpec_B  
    m = m(:); Su<>UsdUC  
    if any(mod(n-m,2)) pz"}o#R"x  
        error('zernfun:NMmultiplesof2', ... 3teP6|K'g  
              'All N and M must differ by multiples of 2 (including 0).') $Qxy@vU  
    end <:!:7  
    uW4.Q_O!H  
    if any(m>n) 'Jd*r(2d  
        error('zernfun:MlessthanN', ... +mYK  
              'Each M must be less than or equal to its corresponding N.') /$9We8  
    end Q~`{^fo1  
    "oh ;?gQ.  
    if any( r>1 | r<0 ) s\ Ln  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') &,* ILz  
    end 2_TFc2d  
    Nl^u A  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) xRZ/[1f!  
        error('zernfun:RTHvector','R and THETA must be vectors.') rx@2Dmt6  
    end 7@&kPh}PG  
    &d]@$4u$;  
    r = r(:); 'f8'|o)  
    theta = theta(:); gOMy8w4>  
    length_r = length(r); `chD*@76I  
    if length_r~=length(theta) At&kW3(  
        error('zernfun:RTHlength', ... D$VRE^k  
              'The number of R- and THETA-values must be equal.') *DvQnj  
    end r(rT.D&  
    H;#C NB<e  
    % Check normalization: 2I 7|hZ,  
    % -------------------- %q6I-  
    if nargin==5 && ischar(nflag) U#{(*)qr  
        isnorm = strcmpi(nflag,'norm'); g*!1S  
        if ~isnorm ,o}CBB! k  
            error('zernfun:normalization','Unrecognized normalization flag.') TgKSE1  
        end 2SlI5+u  
    else o ^ 08<  
        isnorm = false; V5gr-^E  
    end 4~2 9,  
    M^G9t*I  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% )_}xK={  
    % Compute the Zernike Polynomials 5uJ!)Q  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% .R^ R|<x  
    "*:?m{w5  
    % Determine the required powers of r: l nJ  
    % ----------------------------------- 0qm CIcg  
    m_abs = abs(m); c=aZ[  
    rpowers = []; iDdR-T|  
    for j = 1:length(n) Y Azj>c&  
        rpowers = [rpowers m_abs(j):2:n(j)]; y2R\SL,  
    end l< HnPR/  
    rpowers = unique(rpowers); OHv9|&Tpl  
    gUYTVp Vf  
    % Pre-compute the values of r raised to the required powers, 8t |?b  
    % and compile them in a matrix: Pfd%[C/vdm  
    % ----------------------------- >)k[085t  
    if rpowers(1)==0 D`U,T& @  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); X.ZG-TC  
        rpowern = cat(2,rpowern{:}); n6 wx/:  
        rpowern = [ones(length_r,1) rpowern]; s.a@uR^  
    else ->Fsmb+R  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 5?|y%YH;R\  
        rpowern = cat(2,rpowern{:}); mRN[l j  
    end w }8=sw  
    t{`uN  
    % Compute the values of the polynomials: rl-#Ez  
    % -------------------------------------- j$4lyDfD  
    y = zeros(length_r,length(n)); !j3Xzn9  
    for j = 1:length(n) "V5_B^Gzb]  
        s = 0:(n(j)-m_abs(j))/2; UG]x CkDS  
        pows = n(j):-2:m_abs(j); ZgmK~iJ  
        for k = length(s):-1:1 Q |hBGH9:B  
            p = (1-2*mod(s(k),2))* ... b#n  
                       prod(2:(n(j)-s(k)))/              ... Z% ]LZ/O8  
                       prod(2:s(k))/                     ... {mLv?"M]  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... %VE FruM  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); QBA{*@ A-  
            idx = (pows(k)==rpowers); +e#(p<  
            y(:,j) = y(:,j) + p*rpowern(:,idx); OaY]}4tI$  
        end Z1p%6f`  
         L!fIAd`  
        if isnorm nYO$ |/e  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); Fxn=+Xgg  
        end I<" UQ\)  
    end ^ '_Fd  
    % END: Compute the Zernike Polynomials h]4qJ  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %D7'7E8.  
    ob/HO (h3  
    % Compute the Zernike functions: ;KG}Yr72  
    % ------------------------------ d <zD@ z  
    idx_pos = m>0; .ts XQf  
    idx_neg = m<0; DLO#_t^v.  
    fT=ZiHJ3Gu  
    z = y; AP9\]qZ(7  
    if any(idx_pos) $?9u;+jIR  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); H~:g =Zw  
    end ;a[3RqmKW  
    if any(idx_neg) swMR+F#u*  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ncpA\E;ff^  
    end ) }k"7"  
    Vkqfs4t  
    % 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) MN>U jFA  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. GF0Utp:Zf;  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated z] |Y   
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive $:?=A5ttuo  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, 4Cvo^k/I  
    %   and THETA is a vector of angles.  R and THETA must have the same W\e!rq  
    %   length.  The output Z is a matrix with one column for every P-value, ])WIw'L!  
    %   and one row for every (R,THETA) pair. + Cq&~<B  
    % L)Da1<O  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike `$/M\aM%  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) -Q1~lN m:  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) *C> N  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 , - _ReL  
    %   for all p. 1i Q(q\%  
    % k7z{q/]M  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 x6|QTO  
    %   Zernike functions (order N<=7).  In some disciplines it is XsMETl"Av4  
    %   traditional to label the first 36 functions using a single mode y|WOw(#  
    %   number P instead of separate numbers for the order N and azimuthal ,UD5>Ai  
    %   frequency M. m h|HEkM  
    % ?[lKft  
    %   Example: p+{*w7?8"[  
    % k59.O~0V  
    %       % Display the first 16 Zernike functions O[ ^zQA  
    %       x = -1:0.01:1; >JN[5aus  
    %       [X,Y] = meshgrid(x,x); oXqx]@7  
    %       [theta,r] = cart2pol(X,Y); ?=?9a  
    %       idx = r<=1; ]8opI\  
    %       p = 0:15; );^{;fLy%  
    %       z = nan(size(X)); PmDar<m  
    %       y = zernfun2(p,r(idx),theta(idx)); 1{wOjq(4  
    %       figure('Units','normalized') KYFkO~N  
    %       for k = 1:length(p) T9gQq 7(l  
    %           z(idx) = y(:,k); Oin:5K)4-  
    %           subplot(4,4,k) *Rj*%S  
    %           pcolor(x,x,z), shading interp y;HJ"5.Mw  
    %           set(gca,'XTick',[],'YTick',[]) @wXo{p@W  
    %           axis square x 0L,$Ol  
    %           title(['Z_{' num2str(p(k)) '}']) f&Meiu+  
    %       end r$Y% 15JV  
    % }5ONDg(I~  
    %   See also ZERNPOL, ZERNFUN. [m]O^Hp{{  
    U,$^| Iz  
    %   Paul Fricker 11/13/2006 i(<do "Am<  
    q.RW_t~  
    |7G=f9V  
    % Check and prepare the inputs: =7U 8`]WA  
    % ----------------------------- 7ZgFCK,8m,  
    if min(size(p))~=1 F}#=qBa[  
        error('zernfun2:Pvector','Input P must be vector.') <1E* wPm8  
    end f.u[!T  
    {I"d"'h  
    if any(p)>35 SyR[G*djl  
        error('zernfun2:P36', ... )TgjaR9G  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... *:V"C\`^n  
               '(P = 0 to 35).']) O7T wM Yh  
    end -"3<Ll  
    *AN2&>Y  
    % Get the order and frequency corresonding to the function number: y]E ?\03"  
    % ---------------------------------------------------------------- )`]} D[j  
    p = p(:); gUxJ>~  
    n = ceil((-3+sqrt(9+8*p))/2); ?gOZY\[ma  
    m = 2*p - n.*(n+2); 1)wzSEV@  
    <lVW; l7  
    % Pass the inputs to the function ZERNFUN: w.H\j9E l  
    % ---------------------------------------- K)t+lJ  
    switch nargin B (dq$+4  
        case 3 p[-bu B]  
            z = zernfun(n,m,r,theta); D'+kzb@  
        case 4 lO0 PZnW9  
            z = zernfun(n,m,r,theta,nflag); d/:zO4v3  
        otherwise @~<M_63  
            error('zernfun2:nargin','Incorrect number of inputs.') ySwvjP7f  
    end AW:WDNQh8n  
    {sL(PS.z  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) #b4Pn`[   
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. Y8 a![  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of niV=Ijt{5  
    %   order N and frequency M, evaluated at R.  N is a vector of sX,oJIt  
    %   positive integers (including 0), and M is a vector with the bqAv)2  
    %   same number of elements as N.  Each element k of M must be a +LM /< l  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) G6(U\VFqO  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is Ue <Y ~A  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix %vO b"K$X  
    %   with one column for every (N,M) pair, and one row for every Dh|8$(Jt  
    %   element in R. ApYri|^r  
    % :n&n"`D~  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- n1xN:A  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is L{\au5-4  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to @^$Xy<x  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 *a7&v3X  
    %   for all [n,m]. S 5Q$dAL  
    % tc@([XqH  
    %   The radial Zernike polynomials are the radial portion of the T.zU erbO  
    %   Zernike functions, which are an orthogonal basis on the unit ` AA[k  
    %   circle.  The series representation of the radial Zernike 9ci=]C5o3K  
    %   polynomials is T&=1IoOg  
    % D@(Y.&_  
    %          (n-m)/2 FXPw 5  
    %            __ n^;:V8k  
    %    m      \       s                                          n-2s W|@/<K$V  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r %<q l  
    %    n      s=0 ||&EmH  
    % yU~OfwQ  
    %   The following table shows the first 12 polynomials. WX} "Pj/6  
    % 4#Fz!Km  
    %       n    m    Zernike polynomial    Normalization v(\kSlJ  
    %       --------------------------------------------- 6t|FuTC  
    %       0    0    1                        sqrt(2) ZgL4$%  
    %       1    1    r                           2 [*O#6Xu  
    %       2    0    2*r^2 - 1                sqrt(6) &41=YnC6  
    %       2    2    r^2                      sqrt(6) f#a ~av9rC  
    %       3    1    3*r^3 - 2*r              sqrt(8) (ROurq"  
    %       3    3    r^3                      sqrt(8) f_r1(o 5:Y  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) RbJ,J)C>  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) 9i8D_[  
    %       4    4    r^4                      sqrt(10) `n]y"rj'  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) SR#X\AWM  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) $Blo`'  
    %       5    5    r^5                      sqrt(12) )A a98Eu?2  
    %       --------------------------------------------- ->x+ p"  
    % 3 [lF  
    %   Example: -bK#&o,  
    % sY=fS2b#)  
    %       % Display three example Zernike radial polynomials f$mfY6v  
    %       r = 0:0.01:1; C G~ )`  
    %       n = [3 2 5]; 1q*85 [Y  
    %       m = [1 2 1]; Y%b 5{1  
    %       z = zernpol(n,m,r); ` RUr/|S  
    %       figure W :PGj0?  
    %       plot(r,z) #_}lF<k  
    %       grid on SnRTC<DDh  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') q79)nhC F  
    % &_ Ewu@4  
    %   See also ZERNFUN, ZERNFUN2. n`T 4aDm  
    W xyQA:3s  
    % A note on the algorithm. 7'_zJI^  
    % ------------------------ O^I~d{M 5I  
    % The radial Zernike polynomials are computed using the series wxARD3%  
    % representation shown in the Help section above. For many special $WvI%r  
    % functions, direct evaluation using the series representation can 5@"&%8oeq0  
    % produce poor numerical results (floating point errors), because L=Q- r[  
    % the summation often involves computing small differences between ,8g~,tMr+  
    % large successive terms in the series. (In such cases, the functions y@J]busU  
    % are often evaluated using alternative methods such as recurrence _cx}e!BK#  
    % relations: see the Legendre functions, for example). For the Zernike Xi_>hL+R(  
    % polynomials, however, this problem does not arise, because the W{l+_a{/9  
    % polynomials are evaluated over the finite domain r = (0,1), and ;8;nY6Ie  
    % because the coefficients for a given polynomial are generally all W|3XD-v@  
    % of similar magnitude. *A`hKx  
    % E27wxMU  
    % ZERNPOL has been written using a vectorized implementation: multiple HH3WZ^0>  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] 2i#wJ8vrF  
    % values can be passed as inputs) for a vector of points R.  To achieve zr?%k]A%UO  
    % this vectorization most efficiently, the algorithm in ZERNPOL 0O?B!Jr]RM  
    % involves pre-determining all the powers p of R that are required to )'xTDi  
    % compute the outputs, and then compiling the {R^p} into a single |]~tX zY  
    % matrix.  This avoids any redundant computation of the R^p, and PIn'tV  
    % minimizes the sizes of certain intermediate variables. `~[zIq:}7  
    % VcT(n7  
    %   Paul Fricker 11/13/2006 H1f){L97wR  
    s,N%sO;  
    ; )O)\__"-  
    % Check and prepare the inputs: ;VM/Cxgep  
    % ----------------------------- JQ'NFl9<  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) `f}c 1  
        error('zernpol:NMvectors','N and M must be vectors.') @0?Mwy!  
    end e|+U7=CK  
    e~c;wP~cO  
    if length(n)~=length(m) &}VGC=F;d  
        error('zernpol:NMlength','N and M must be the same length.') bV&/)eqv  
    end H^p ?t=Y  
    ZebXcT ,41  
    n = n(:); )MLOYX  
    m = m(:); xA#B1qbw  
    length_n = length(n); BV$lMLD{r  
    m>$+sMZE  
    if any(mod(n-m,2)) KP[ax2!x  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') ~qLbyzHaB  
    end  |X`xJL  
    {QTfD~z^K  
    if any(m<0) *nluK  
        error('zernpol:Mpositive','All M must be positive.') |Rw0$he  
    end :^71,An >E  
    Z&@X4X"q  
    if any(m>n) ||cG/I&,  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') Wu<  
    end aG8}R~wH&  
    \CM(  
    if any( r>1 | r<0 ) K0yTHX?(.  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') ]nhLv!Co  
    end 1[C,*\X8v  
    }XWic88!~  
    if ~any(size(r)==1) GptJQ=pV  
        error('zernpol:Rvector','R must be a vector.') 3_B .W  
    end aAF:nyV~~0  
    'N)&;ADx-G  
    r = r(:); Cq?l>  
    length_r = length(r); NwoBM6 #  
    2I(0EBW  
    if nargin==4 w#U3h]>,  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); dY68wW>d|  
        if ~isnorm ;FQAL@"Yj  
            error('zernpol:normalization','Unrecognized normalization flag.') R8F[ 7&(  
        end ]T}G-  
    else W`^'hka  
        isnorm = false; <33[qt~  
    end cBBc^SR  
    PQ<""_S||  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 49^;T;'v  
    % Compute the Zernike Polynomials FF6[qSV  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% rXuhd [!(P  
    DGj:qd(  
    % Determine the required powers of r: m:d P,  
    % ----------------------------------- Yvs)H'n=  
    rpowers = []; VmHok  
    for j = 1:length(n) {3lsDU4  
        rpowers = [rpowers m(j):2:n(j)]; 28C/^4  
    end [!,&A{.!  
    rpowers = unique(rpowers); >B U 0B  
    Mi,yg=V  
    % Pre-compute the values of r raised to the required powers, ]C3{ _?=  
    % and compile them in a matrix: Oj\lg2Ck  
    % ----------------------------- q|b#=Af]g  
    if rpowers(1)==0 QUVwO m  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); c^><^LGb  
        rpowern = cat(2,rpowern{:}); M9HM:  
        rpowern = [ones(length_r,1) rpowern]; !fZ\GOx  
    else U8-#W(tRR  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); *?Nrx=O*  
        rpowern = cat(2,rpowern{:}); fchsn*R%-  
    end EeG7 %S 5(  
    QxH%4 )?  
    % Compute the values of the polynomials: ]@vX4G/  
    % -------------------------------------- X#'DS&{  
    z = zeros(length_r,length_n); ' 7+x,TszI  
    for j = 1:length_n  gPh;  
        s = 0:(n(j)-m(j))/2; [5e}A&  
        pows = n(j):-2:m(j); Urj8v2k  
        for k = length(s):-1:1 jB!p,fqcb  
            p = (1-2*mod(s(k),2))* ... aTuD|s  
                       prod(2:(n(j)-s(k)))/          ... zUXQl{  
                       prod(2:s(k))/                 ... &YGd!Q  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... $~!%Px)  
                       prod(2:((n(j)+m(j))/2-s(k))); N9tH0  
            idx = (pows(k)==rpowers); m~'!  
            z(:,j) = z(:,j) + p*rpowern(:,idx); KV9'ew+M  
        end #(F/P!qk  
         ,Md8A`7x~  
        if isnorm SE.r 'J0  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); .T 6 NMIp*  
        end f6vhW66:?x  
    end ayfR{RYi  
    O;z:?  
    % 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)   nU4to  
    aFaioE#h(  
    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^jlb  
    F r!FV4  
    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)