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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 正序阅读 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 2:2rwH }e  
    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! i' N  
     
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    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)  4$d|}ajH  
    6"eGd"  
    数值分析方法看一下就行了。其实就是正交多项式的应用。zernike也只不过是正交多项式的一种。 ~F>oNbJIv  
    kn`KU.J.  
    07年就写过这方面的计算程序了。
    2024年6月28-30日于上海组织线下成像光学设计培训,欢迎报名参加。请关注子在川上光学公众号。详细内容请咨询13661915143(同微信号)
    离线guapiqlh
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    只看该作者 8楼 发表于: 2014-03-04
    我也一直想了解这个多项式的应用,还没用过呢
    离线li_xin_feng
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    只看该作者 7楼 发表于: 2012-09-28
    我也正在找啊
    离线niuhelen
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    只看该作者 6楼 发表于: 2011-03-12
    这三个文件,我不知道该怎样把我的面型节点的坐标及轴向位移用起来,还烦请指点一下啊,谢谢啦!
    离线niuhelen
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) 91mXvQ:u  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. V{ra,a*  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of Y@M=6G  
    %   order N and frequency M, evaluated at R.  N is a vector of [UR+G8X21m  
    %   positive integers (including 0), and M is a vector with the 5#$E4k:YV  
    %   same number of elements as N.  Each element k of M must be a ~9h6"0K!  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) %w/o#*j<;  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is NTs< ;ED  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix n_.2B$JD  
    %   with one column for every (N,M) pair, and one row for every p^5B_r:  
    %   element in R. 7{8!IcR #  
    % H6bomp"  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- <u u1e@P  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is mZ ONxR6q$  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to nH NMoA  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 g0cCw2S  
    %   for all [n,m]. c^A3|tCi  
    % <4C`^p  
    %   The radial Zernike polynomials are the radial portion of the (}gF{@sn  
    %   Zernike functions, which are an orthogonal basis on the unit o=q N+-N  
    %   circle.  The series representation of the radial Zernike @hQ+pG@s  
    %   polynomials is "EWU:9\0  
    % [WY NA-O  
    %          (n-m)/2 E I)Pfx"0  
    %            __ j=PQoEtU'<  
    %    m      \       s                                          n-2s T/)$}#w0i  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r Q(oWaG  
    %    n      s=0 ,XI,B\eNk  
    % %}+j4n  
    %   The following table shows the first 12 polynomials. ^p|@{4f]  
    % s-*8=  
    %       n    m    Zernike polynomial    Normalization $-5iwZ  
    %       --------------------------------------------- Gv?3}8Wp  
    %       0    0    1                        sqrt(2) ;G;vpl  
    %       1    1    r                           2 e_\4(4x  
    %       2    0    2*r^2 - 1                sqrt(6) vb5tyY0c  
    %       2    2    r^2                      sqrt(6) MfCu\[qOz  
    %       3    1    3*r^3 - 2*r              sqrt(8) lv&<kYWY  
    %       3    3    r^3                      sqrt(8) u2-%~Rlo  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) m-*du(  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) H.O7Y  
    %       4    4    r^4                      sqrt(10) _BHb0zeot  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) p?0 a"5Q  
    %       5    3    5*r^5 - 4*r^3            sqrt(12)  W* `2lf  
    %       5    5    r^5                      sqrt(12) 7KuTC%7  
    %       --------------------------------------------- g9GE0DbT`  
    % wEKm3mY;  
    %   Example: *2=:(OK  
    % r}D`15IHJ  
    %       % Display three example Zernike radial polynomials  ]c[80F-  
    %       r = 0:0.01:1; S"5</*  
    %       n = [3 2 5]; AM'-(x|  
    %       m = [1 2 1]; e|"`W`"-  
    %       z = zernpol(n,m,r); )h2wwq0]  
    %       figure 'S@h._q  
    %       plot(r,z) +)L 'qbCSM  
    %       grid on y5|`B(  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') W O|2x0K  
    % ]/bf#&@g`k  
    %   See also ZERNFUN, ZERNFUN2. y?CEV-3+  
    c<pr1g  
    % A note on the algorithm. A5y?|q>5  
    % ------------------------ #*}4=  
    % The radial Zernike polynomials are computed using the series 'WxcA)z0cQ  
    % representation shown in the Help section above. For many special {j ${i  
    % functions, direct evaluation using the series representation can &0Wv+2l @  
    % produce poor numerical results (floating point errors), because WP2|0ib  
    % the summation often involves computing small differences between HMrS::  
    % large successive terms in the series. (In such cases, the functions 3~a!h3.f  
    % are often evaluated using alternative methods such as recurrence 42ttmN1F  
    % relations: see the Legendre functions, for example). For the Zernike i/-Xpj]Zf  
    % polynomials, however, this problem does not arise, because the 7=Ew[MOmM  
    % polynomials are evaluated over the finite domain r = (0,1), and `<b 3e(A  
    % because the coefficients for a given polynomial are generally all M:Xswwq  
    % of similar magnitude. #f\U3p  
    % yZUB8erb.  
    % ZERNPOL has been written using a vectorized implementation: multiple cl^wLC'o  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] o_b j@X  
    % values can be passed as inputs) for a vector of points R.  To achieve L*D-RYW  
    % this vectorization most efficiently, the algorithm in ZERNPOL )/Ee#)z*  
    % involves pre-determining all the powers p of R that are required to ,]y)Dy  
    % compute the outputs, and then compiling the {R^p} into a single 1i$9x$4~E  
    % matrix.  This avoids any redundant computation of the R^p, and w[~$.FM/  
    % minimizes the sizes of certain intermediate variables. m`I6gnLj  
    % BqCBH!^x  
    %   Paul Fricker 11/13/2006 #wk'&XsC#z  
    -81usu&NH  
    jiC;*]n  
    % Check and prepare the inputs: 8e[kE>tS._  
    % ----------------------------- 1EyM,$On  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Beiz*2-}a  
        error('zernpol:NMvectors','N and M must be vectors.') z )a8 ^]`  
    end %_KNAuM  
    CmY'[rI  
    if length(n)~=length(m) `:}GE@]  
        error('zernpol:NMlength','N and M must be the same length.') Ip4CC'  
    end f,)[f M4  
    kQsyvE  
    n = n(:);  [^8*9?i4  
    m = m(:); UStZ3A'  
    length_n = length(n); 5ok3q@1_]{  
    5d*k[fZ  
    if any(mod(n-m,2)) a4 O  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') vz#rbBY*;  
    end uG${`4  
    #J\ 2/~  
    if any(m<0) q/XZb@rt  
        error('zernpol:Mpositive','All M must be positive.') Nye Ga  
    end <&t^&6k  
    cCw?%qq,L  
    if any(m>n) |9?67-  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') D?) "Z$  
    end fY}e.lD  
    :@`Ll;G  
    if any( r>1 | r<0 ) v,KH2 (N  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') vaxNF%^~yN  
    end zq8 z#FN  
    =L#tSa=M"  
    if ~any(size(r)==1) o/CSIvz1  
        error('zernpol:Rvector','R must be a vector.') vMRM/.  
    end "F7g8vu  
    q\x*@KQgM  
    r = r(:); DHaSBk  
    length_r = length(r); g%4-QCZ,  
    CTD{!I(  
    if nargin==4 E;@` { v  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); G!ty@ Fx  
        if ~isnorm Y@Lv>p  
            error('zernpol:normalization','Unrecognized normalization flag.') 0N;Pb(%7UU  
        end INyreoMp  
    else $83TA> <a  
        isnorm = false; ullq}}  
    end TlYeYN5V  
    51*o&:eim  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 3G~ T_J&  
    % Compute the Zernike Polynomials _WVeb}  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% >Yl?i&3n  
    9} :n  
    % Determine the required powers of r: ;4z6="<Y  
    % ----------------------------------- _Su? VxU  
    rpowers = []; $Dxz21|P7  
    for j = 1:length(n) ]>b.oI/  
        rpowers = [rpowers m(j):2:n(j)]; LR@rn2Z  
    end 2ZNTj u7h  
    rpowers = unique(rpowers); _SJ#k|vcq  
    |dsd5Vdr  
    % Pre-compute the values of r raised to the required powers, 5%rD7/7N  
    % and compile them in a matrix: g7EJyA  
    % ----------------------------- +Tf,2?O  
    if rpowers(1)==0 ac6L3=u\  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); q=M!YWz  
        rpowern = cat(2,rpowern{:}); (%rO'X  
        rpowern = [ones(length_r,1) rpowern]; :D-My28'  
    else DB We>Ef(  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); frWw-<HoI  
        rpowern = cat(2,rpowern{:}); npkE [JE:  
    end f\nF2rlu  
    L%# #U'e3  
    % Compute the values of the polynomials: \S{ise/U  
    % -------------------------------------- }oIA*:5  
    z = zeros(length_r,length_n); ryy".'v  
    for j = 1:length_n -fI-d1@  
        s = 0:(n(j)-m(j))/2; vrXUS9i.  
        pows = n(j):-2:m(j); E?l_ *[G  
        for k = length(s):-1:1 Qr6[h!  
            p = (1-2*mod(s(k),2))* ... g""1f%U_p  
                       prod(2:(n(j)-s(k)))/          ... P3jDx{F  
                       prod(2:s(k))/                 ... qgbp-A!2zF  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... oyZ}JTl( Q  
                       prod(2:((n(j)+m(j))/2-s(k))); f }PT3  
            idx = (pows(k)==rpowers); cT'D2Yeq  
            z(:,j) = z(:,j) + p*rpowern(:,idx); 8%S5Fc #am  
        end I'{-T=R-q  
         .E-)R  
        if isnorm Q&}`( ]k  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); .uG|Vq1v  
        end ~5<-&Dyp7  
    end 9^h0D}#@  
    `rzgC \  
    % EOF zernpol
    离线niuhelen
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) B~K@o.%  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. |B yw]\3v  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated r8x<- u4  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive ^iAOz-H  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, 6K501!70g6  
    %   and THETA is a vector of angles.  R and THETA must have the same s 4uZ;  
    %   length.  The output Z is a matrix with one column for every P-value, 'yd<<BM`  
    %   and one row for every (R,THETA) pair. {XAm3's  
    % eT* )r~  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike c@!%.# |y  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) qOAK`{b  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) VX0q!Q  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 ?UCK  
    %   for all p. \6~(# y  
    % @(Q 'J`  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 / qp)n">  
    %   Zernike functions (order N<=7).  In some disciplines it is O}5mDx  
    %   traditional to label the first 36 functions using a single mode r!A1Sfo4P  
    %   number P instead of separate numbers for the order N and azimuthal R+ #(\  
    %   frequency M. vDl6TKXcu  
    % 8D7 = ]  
    %   Example: Nr 5h%<` I  
    % X&R ,-^  
    %       % Display the first 16 Zernike functions AG/?LPJ  
    %       x = -1:0.01:1; <d!_.f}v  
    %       [X,Y] = meshgrid(x,x); RS'!>9I  
    %       [theta,r] = cart2pol(X,Y); d/oxRzk'L  
    %       idx = r<=1; vZ3/t8$*  
    %       p = 0:15; JtA tG%  
    %       z = nan(size(X)); ]@YBa4}w  
    %       y = zernfun2(p,r(idx),theta(idx)); $KDH"J  
    %       figure('Units','normalized') P(B:tg  
    %       for k = 1:length(p) uXD?s3Wv  
    %           z(idx) = y(:,k); [AgS@^"sf5  
    %           subplot(4,4,k) /sHWJ?`&/,  
    %           pcolor(x,x,z), shading interp AY3nQH   
    %           set(gca,'XTick',[],'YTick',[]) hS(}<B{x!  
    %           axis square #J&45  
    %           title(['Z_{' num2str(p(k)) '}']) 5>{  
    %       end <Sw>5M!j  
    % 8:s" ^YLN  
    %   See also ZERNPOL, ZERNFUN. |oCE7'BaP  
    ?}<4LK]  
    %   Paul Fricker 11/13/2006 (<y~]igy  
    ~@g7b`t=la  
    hbfTv;=z  
    % Check and prepare the inputs: c~j")o  
    % ----------------------------- )y8 u+5^  
    if min(size(p))~=1 yn&+ >{  
        error('zernfun2:Pvector','Input P must be vector.') 0V:7pSC{P  
    end s'/b&Idf8  
    6R_G{AWLL  
    if any(p)>35 H#yBWvj*H  
        error('zernfun2:P36', ... a W1y0  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... :mOHR&2xR%  
               '(P = 0 to 35).']) ca~nfo  
    end doeYc  
    j p g$5jZ  
    % Get the order and frequency corresonding to the function number: w,uyN  
    % ---------------------------------------------------------------- 6KT]3*B   
    p = p(:); g~,"C8-H  
    n = ceil((-3+sqrt(9+8*p))/2); xz9x t  
    m = 2*p - n.*(n+2); +v$,/~$tI  
    7&ty!PpD  
    % Pass the inputs to the function ZERNFUN: 3eOwy~  
    % ---------------------------------------- ZY N HVR  
    switch nargin !cblmF;0  
        case 3 l]:nncpns  
            z = zernfun(n,m,r,theta); vd0;33$L  
        case 4 zB,Vi-)vH  
            z = zernfun(n,m,r,theta,nflag); u7L!&/6On  
        otherwise T&@xgj|!)  
            error('zernfun2:nargin','Incorrect number of inputs.') j A/xe  
    end d"h*yH@  
     Z1@E  
    % EOF zernfun2
    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 gmFCjs  
    function z = zernfun(n,m,r,theta,nflag) km%c0:  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. /Mac:;W`  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N @jXdQY%{  
    %   and angular frequency M, evaluated at positions (R,THETA) on the 6R.%I{x'  
    %   unit circle.  N is a vector of positive integers (including 0), and >a6{y   
    %   M is a vector with the same number of elements as N.  Each element ^T^l3B[  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) +`y{r^xD  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, U^AywE]  
    %   and THETA is a vector of angles.  R and THETA must have the same 0Yh Mwg?  
    %   length.  The output Z is a matrix with one column for every (N,M) uv&??F]/  
    %   pair, and one row for every (R,THETA) pair. g>L4N.ZH_v  
    % y;'yob  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike UG@9X/l}  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), }8joltf  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral HUP~  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, +JDQ`Qk  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized {>x6SVF  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. J(0E'o{ug  
    % o8PK,!Pl  
    %   The Zernike functions are an orthogonal basis on the unit circle. 9FGe (t <  
    %   They are used in disciplines such as astronomy, optics, and >#9 f{  
    %   optometry to describe functions on a circular domain. FR bmeq3c  
    % o#p{0y  
    %   The following table lists the first 15 Zernike functions. bSG}I|  
    % 8Uv2p{ <#  
    %       n    m    Zernike function           Normalization yniXb2iM  
    %       -------------------------------------------------- T +a\dgd  
    %       0    0    1                                 1  BVJ6U[h`  
    %       1    1    r * cos(theta)                    2 fN!ci']  
    %       1   -1    r * sin(theta)                    2 <./r%3$;7  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) IdHyd Y1  
    %       2    0    (2*r^2 - 1)                    sqrt(3) 5c 8tH=  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) *h <_gn  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) FrKI=8  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) w<qn@f  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) rAv)k&l  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) ?j'Nx_RoX  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) PU& v{gn  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Qru iQ/t  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) [Yi;k,F:  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 7I#<w[l>k  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) ls;!Og9  
    %       -------------------------------------------------- 5{PT  
    % 5.IX  
    %   Example 1: lR<1x  
    % r bfIH":  
    %       % Display the Zernike function Z(n=5,m=1) 3QD+&9{D  
    %       x = -1:0.01:1; k=^~\$e  
    %       [X,Y] = meshgrid(x,x); L  `\>_  
    %       [theta,r] = cart2pol(X,Y); 2#i*'.  
    %       idx = r<=1; .kl.awT  
    %       z = nan(size(X)); VB}4#-dG?  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); $ ;J:kd;<  
    %       figure t.s;dlx[@  
    %       pcolor(x,x,z), shading interp l KdY!j"  
    %       axis square, colorbar _nn\O3TB  
    %       title('Zernike function Z_5^1(r,\theta)') z1AYXW6F  
    % 2HX#:y{\l  
    %   Example 2: *XCgl*% *  
    % ;YfKG8(0  
    %       % Display the first 10 Zernike functions ,E._A(Z  
    %       x = -1:0.01:1; "p"M9P'  
    %       [X,Y] = meshgrid(x,x); \nzaF4+$  
    %       [theta,r] = cart2pol(X,Y); i&di}x  
    %       idx = r<=1; MEI.wJZ  
    %       z = nan(size(X)); aioN)V  
    %       n = [0  1  1  2  2  2  3  3  3  3]; KAFx^JLo  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; bTd94  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; pm4'2B|)g  
    %       y = zernfun(n,m,r(idx),theta(idx)); r8wip\[  
    %       figure('Units','normalized') N!Q~?/!d  
    %       for k = 1:10 4nz$J a)  
    %           z(idx) = y(:,k); Vlf=gP  
    %           subplot(4,7,Nplot(k)) |eu:qn8  
    %           pcolor(x,x,z), shading interp tK0Ksnl^  
    %           set(gca,'XTick',[],'YTick',[]) \'>8 (i~  
    %           axis square (c\i.z  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) wBJP8wES=  
    %       end U4.- {.  
    % 8o7%qWX  
    %   See also ZERNPOL, ZERNFUN2. HX`>" ?{  
    .wPu #*  
    %   Paul Fricker 11/13/2006 !xRboPg  
    jTh^#Q  
    T1_qAz+  
    % Check and prepare the inputs: +gh*n,:|  
    % ----------------------------- -]-?>gkN5  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) R)Y*<Na  
        error('zernfun:NMvectors','N and M must be vectors.') ? 3t]9z  
    end kKHGcm^r  
    |%tI!RN):  
    if length(n)~=length(m) g-NfZj?  
        error('zernfun:NMlength','N and M must be the same length.') Y2 oN.{IH  
    end |EpL~ G_  
    `9vCl@"IV  
    n = n(:); BIn7<.&  
    m = m(:); km=d'VvnI  
    if any(mod(n-m,2)) #^zUaPV 7r  
        error('zernfun:NMmultiplesof2', ... L>X39R~  
              'All N and M must differ by multiples of 2 (including 0).') 0,M1Q~u%.  
    end q)F@f /  
    lD]/Kx  
    if any(m>n) [7+dZL[  
        error('zernfun:MlessthanN', ... s6HfN'  
              'Each M must be less than or equal to its corresponding N.') :L&d>Ii|'  
    end \*r]v;NcP  
    ?c0@A*:o  
    if any( r>1 | r<0 ) QP={b+8  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') i4g99Kvl  
    end ,Srj38p  
    JZom#A. dt  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Rct=v DU  
        error('zernfun:RTHvector','R and THETA must be vectors.') ?]Wg{\NC6  
    end .0ExHcr  
    x/]]~@:  
    r = r(:); ,2/y(JX}*!  
    theta = theta(:); iI@m e=  
    length_r = length(r); 3w!,@=.q  
    if length_r~=length(theta) j%TcW!D-_  
        error('zernfun:RTHlength', ... okSCM#&:[2  
              'The number of R- and THETA-values must be equal.') =zX A0%  
    end kA/V=xO<  
    <}z, !w8  
    % Check normalization: KU5|~1t 4  
    % -------------------- l99{eD  
    if nargin==5 && ischar(nflag) z&W5@6")`  
        isnorm = strcmpi(nflag,'norm'); mq!_/3  
        if ~isnorm xZ.c@u6:  
            error('zernfun:normalization','Unrecognized normalization flag.') 5IfyD ]<  
        end ]$xN`O4W{  
    else pU)g93  
        isnorm = false; r[votdFo  
    end xJ[Xmre  
    Ix1[ $9  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 7$/%c{o  
    % Compute the Zernike Polynomials A3cW8 OClz  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Q ,6[  
    -)dS`hM  
    % Determine the required powers of r: j+-+<h/(  
    % ----------------------------------- H6! <y-  
    m_abs = abs(m); C?h`i ^ >2  
    rpowers = []; "JBTsQDj!  
    for j = 1:length(n) tc4"huG  
        rpowers = [rpowers m_abs(j):2:n(j)]; xZpGSlA  
    end W%.ou\GN^t  
    rpowers = unique(rpowers); Btu=MUS  
    *LZ^0c:r  
    % Pre-compute the values of r raised to the required powers, mok%TK  
    % and compile them in a matrix: SeX:A)*ez%  
    % ----------------------------- tM&;b?bJ[  
    if rpowers(1)==0 gXThdNU4G  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 1p]Z9$Y  
        rpowern = cat(2,rpowern{:}); $Afw]F$  
        rpowern = [ones(length_r,1) rpowern]; DD(K@M  
    else 6*Y>Y&sea  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); o7B }~;L  
        rpowern = cat(2,rpowern{:}); 6;^ e  
    end [WxRwE  
    `E4OgO  
    % Compute the values of the polynomials: jh3X G  
    % -------------------------------------- !(L\X'jH  
    y = zeros(length_r,length(n)); JRT,%;*,  
    for j = 1:length(n) -g`3;1EV^  
        s = 0:(n(j)-m_abs(j))/2; 5lp};  
        pows = n(j):-2:m_abs(j); JLZ=$d  
        for k = length(s):-1:1 RxZ#`$F  
            p = (1-2*mod(s(k),2))* ... SF#Rc>v  
                       prod(2:(n(j)-s(k)))/              ... {6uhUb  
                       prod(2:s(k))/                     ... 7HkQ|~zGT  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... WI+ 5x  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); .gS x`|!  
            idx = (pows(k)==rpowers); ,O[Maj/ch  
            y(:,j) = y(:,j) + p*rpowern(:,idx); V`;$Ua;y  
        end +&:?*(?Q  
         us,1:@a)a  
        if isnorm wWU5]v  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 5PXo1"n8T  
        end ~BJ~]~0P`  
    end xU5+"t~  
    % END: Compute the Zernike Polynomials _/iw=-T  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% jj&4Sv#>  
    1FO T  
    % Compute the Zernike functions: J|D$  
    % ------------------------------ [Q+qu>&HB7  
    idx_pos = m>0; iH#b"h{w  
    idx_neg = m<0; QxjX:O  
    ag \d4y6  
    z = y; `AO<r  
    if any(idx_pos) QaMB=wVr  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); QV@NA@;XZ  
    end i$Sq.NU  
    if any(idx_neg) dU4G!  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); xO<$xx  
    end #ErIot  
    OSsxO(;g  
    % EOF zernfun
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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
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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