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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 n\Lb.}]1~  
    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! `Ry]y"K  
     
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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)  ,5*eX  
    }I2@%tt?  
    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。 ])!o5`ltZ  
    o$Z6zmxO  
    07年就写过这方面的计算程序了。
    2024年6月28-30日于上海组织线下成像光学设计培训,欢迎报名参加。请关注子在川上光学公众号。详细内容请咨询13661915143(同微信号)
    在线guapiqlh
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    只看该作者 8楼 发表于: 2014-03-04
    我也一直想了解这个多项式的应用,还没用过呢
    离线li_xin_feng
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    只看该作者 7楼 发表于: 2012-09-28
    我也正在找啊
    离线niuhelen
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    只看该作者 6楼 发表于: 2011-03-12
    这三个文件,我不知道该怎样把我的面型节点的坐标及轴向位移用起来,还烦请指点一下啊,谢谢啦!
    离线niuhelen
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) /q`xCS  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. `6]%P(#a  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of &-B^~M*??  
    %   order N and frequency M, evaluated at R.  N is a vector of ]X ?7ZI^  
    %   positive integers (including 0), and M is a vector with the zIu E9l  
    %   same number of elements as N.  Each element k of M must be a s pp f  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) zM(vr"U   
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is NP/Gn6fr  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix 2h1vVF3  
    %   with one column for every (N,M) pair, and one row for every sWc*5Rt  
    %   element in R. Yd=>K HVD  
    % t? yz  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- E(8* pI  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is L"4mL,  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to [k;\SXDZo  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 + |#O@k  
    %   for all [n,m]. 9vGu0Um  
    % Ne[7gxpu  
    %   The radial Zernike polynomials are the radial portion of the G(G{RAk>  
    %   Zernike functions, which are an orthogonal basis on the unit 3 +G$-ru  
    %   circle.  The series representation of the radial Zernike Z:sg}  
    %   polynomials is 4hTMbS_;  
    % K k-S}.E  
    %          (n-m)/2 x"gd8j]s  
    %            __ JS CZ{v J$  
    %    m      \       s                                          n-2s uP~@U"!  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r =IQ5<;U3  
    %    n      s=0 ; Q3n  
    % , P70J b  
    %   The following table shows the first 12 polynomials. pxCK;]  
    % e} P I^bc  
    %       n    m    Zernike polynomial    Normalization mUdOX7$c>  
    %       --------------------------------------------- H1QJ k_RL  
    %       0    0    1                        sqrt(2) $ us]35Z3  
    %       1    1    r                           2 Rld!,t  
    %       2    0    2*r^2 - 1                sqrt(6) XF;ES3 d  
    %       2    2    r^2                      sqrt(6) 34%RZG_o'  
    %       3    1    3*r^3 - 2*r              sqrt(8) =E.t`x=  
    %       3    3    r^3                      sqrt(8) |yQZt/*SOZ  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) z8SmkL  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) Z\ja  
    %       4    4    r^4                      sqrt(10) X[&Wkr8x '  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) ^h ~x)@=  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) )ttUWy$w  
    %       5    5    r^5                      sqrt(12) _/6!yyl  
    %       --------------------------------------------- Py@wJEo  
    % j}JrE,|  
    %   Example: hRrn$BdLX  
    % X.f>'0i  
    %       % Display three example Zernike radial polynomials ,!Z *5  
    %       r = 0:0.01:1; V-Sd[  
    %       n = [3 2 5]; xp }hev^@$  
    %       m = [1 2 1]; _m gHJ0v'  
    %       z = zernpol(n,m,r); \eT5flC  
    %       figure 'rO!AcdLU  
    %       plot(r,z) d%RC  
    %       grid on *n 6s.$p)%  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') S+atn]eU@  
    % BGD8w2  
    %   See also ZERNFUN, ZERNFUN2. $Q96,rb}k;  
    [z`31F  
    % A note on the algorithm. ||hb~%JK6  
    % ------------------------ El[)?+;D  
    % The radial Zernike polynomials are computed using the series G~2jUyv  
    % representation shown in the Help section above. For many special 1 u| wMO  
    % functions, direct evaluation using the series representation can Crho=RJPR  
    % produce poor numerical results (floating point errors), because 3=FZ9>by  
    % the summation often involves computing small differences between X(]WVCu  
    % large successive terms in the series. (In such cases, the functions zF8dKFE~  
    % are often evaluated using alternative methods such as recurrence AX;8^6.F3  
    % relations: see the Legendre functions, for example). For the Zernike sk,ox~0R  
    % polynomials, however, this problem does not arise, because the vq^f}id  
    % polynomials are evaluated over the finite domain r = (0,1), and wVicyiY]  
    % because the coefficients for a given polynomial are generally all *W0y: 3dB3  
    % of similar magnitude. 6K-_pg]  
    % s.N7qO^:E  
    % ZERNPOL has been written using a vectorized implementation: multiple m#PY,y  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] tD(7^GuR  
    % values can be passed as inputs) for a vector of points R.  To achieve =vDEfO/T  
    % this vectorization most efficiently, the algorithm in ZERNPOL !`g~F\l  
    % involves pre-determining all the powers p of R that are required to 1zm ulj%&  
    % compute the outputs, and then compiling the {R^p} into a single \>:CvTzF  
    % matrix.  This avoids any redundant computation of the R^p, and 6r"eN%m  
    % minimizes the sizes of certain intermediate variables. B$ajK`x&I  
    % >/kc dWl  
    %   Paul Fricker 11/13/2006 Ljxz.2LGr  
    ~]pE'\D7Ad  
    CFzNwgv]z  
    % Check and prepare the inputs: Rot@x r7Hc  
    % ----------------------------- ~$:|VHl  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) q>$ev)W  
        error('zernpol:NMvectors','N and M must be vectors.') L+Xc-uv["p  
    end (l-tvk4Ln  
    NdtB1b  
    if length(n)~=length(m) !sDh4jQ`  
        error('zernpol:NMlength','N and M must be the same length.') { QHVo#  
    end 3L833zL  
    *.sVr7=j  
    n = n(:); 6x h:/j3  
    m = m(:); kbTm^y"  
    length_n = length(n); -fwoTGlX  
    0E,8R{e  
    if any(mod(n-m,2)) 4]G?G]lS>  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') e,F1Xi #d  
    end 5R'TcWf#W  
    U1DXe h~V  
    if any(m<0) _LMM,!f  
        error('zernpol:Mpositive','All M must be positive.') 8YZbP5'  
    end u.d).da  
    ]h>_\9qO  
    if any(m>n) T&%ux=Jt  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') A?CcHw rT  
    end Bt> }rYz1  
    r"``QmM  
    if any( r>1 | r<0 ) ,TXTS*V?  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') eqP&8^HP  
    end GNXHM*~  
    Gb8D[1=u=  
    if ~any(size(r)==1) 0Fk5kGD,&K  
        error('zernpol:Rvector','R must be a vector.') 1<BX]-/tP  
    end }4Tc  
    xIxn"^'  
    r = r(:); FME3sa$  
    length_r = length(r); : >6F+XZ  
    v1BDP<qU2  
    if nargin==4 ap&?r`Tu  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); 0'V5/W  
        if ~isnorm RIb4!!',c  
            error('zernpol:normalization','Unrecognized normalization flag.') f|h|q_<;  
        end }`W){]{k O  
    else (8Bk;bd  
        isnorm = false; kSR\RuY*  
    end LV\DBDM  
    .q `Hjmg<  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% gCioq.  
    % Compute the Zernike Polynomials o*DN4oa)  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Y%PwktQm  
    zA$k0p  
    % Determine the required powers of r: v%"|WV[N  
    % ----------------------------------- P%{^i]  
    rpowers = []; E"+QJ~!  
    for j = 1:length(n) O-LO/*5MI  
        rpowers = [rpowers m(j):2:n(j)]; K[ (NTp$E  
    end -j73Wz  
    rpowers = unique(rpowers); |K.mP4CKY  
    2.%.Z_k)  
    % Pre-compute the values of r raised to the required powers, V'kX)$  
    % and compile them in a matrix: [x9KVd ^d  
    % ----------------------------- c lNkph  
    if rpowers(1)==0 8-BflejX  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); kPVO?uO  
        rpowern = cat(2,rpowern{:}); k 9L? +PD  
        rpowern = [ones(length_r,1) rpowern]; +pR[U4$  
    else !q9+9 *6  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); |2abmuR0  
        rpowern = cat(2,rpowern{:}); ~xD ={9BL  
    end ,wIONDnLZ  
    byT h/H  
    % Compute the values of the polynomials: TMig-y*[  
    % -------------------------------------- 73xAG1D$r  
    z = zeros(length_r,length_n); o| #Qu8Lk  
    for j = 1:length_n JKGc3j,+#  
        s = 0:(n(j)-m(j))/2; SzjkI+-$:  
        pows = n(j):-2:m(j); huJ&]"C  
        for k = length(s):-1:1 .u4 W /  
            p = (1-2*mod(s(k),2))* ... f ` R/ i  
                       prod(2:(n(j)-s(k)))/          ... 7cTV?nc  
                       prod(2:s(k))/                 ... Jh ]i]7r  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... G5C I<KRK#  
                       prod(2:((n(j)+m(j))/2-s(k))); [/Rf\T(,jn  
            idx = (pows(k)==rpowers); ,6om\9.E@  
            z(:,j) = z(:,j) + p*rpowern(:,idx); C}_ ojcR  
        end ynE)Xdh  
         Q aS\(_  
        if isnorm MO n  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); a>GyO&+Dkg  
        end P/Q!<I  
    end k~jP'aD  
    9D7+[`r(-  
    % 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) J/[=p<I)  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. ~v6OsH%vx  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated k3) dEH1z  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive WJ4li@T7V  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, qI~xlW  
    %   and THETA is a vector of angles.  R and THETA must have the same x "^Xj]-  
    %   length.  The output Z is a matrix with one column for every P-value, 0V'nK V"|  
    %   and one row for every (R,THETA) pair. {TX]\ufG  
    % vTlwRG=5  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike K95p>E`9e  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) 2vAQ  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) ?TU}~}  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 `C$:Yf]%nG  
    %   for all p. L$IQuy  
    % Q\ U:~g3  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 E|6VX4`+  
    %   Zernike functions (order N<=7).  In some disciplines it is gx\&_) w N  
    %   traditional to label the first 36 functions using a single mode W9D86]3Y  
    %   number P instead of separate numbers for the order N and azimuthal E ,|xJjh  
    %   frequency M. dIRm q+d^  
    % 2JJ"O|Ibz  
    %   Example: mR}6r2O2\Q  
    % l i0i"  
    %       % Display the first 16 Zernike functions &?*V0luP)  
    %       x = -1:0.01:1; c@/(B:@  
    %       [X,Y] = meshgrid(x,x); 3b+d"`Y^S  
    %       [theta,r] = cart2pol(X,Y); Hhari!R XC  
    %       idx = r<=1; dt`{!lts'  
    %       p = 0:15; ^(|vsFzn  
    %       z = nan(size(X)); 2\7`/,U6  
    %       y = zernfun2(p,r(idx),theta(idx)); .zn;:M#T  
    %       figure('Units','normalized') nij!1z|M  
    %       for k = 1:length(p) `<\1[HJ\  
    %           z(idx) = y(:,k); +(C6#R<LI  
    %           subplot(4,4,k) G|( ]bvJ?  
    %           pcolor(x,x,z), shading interp 8;Yx<woR  
    %           set(gca,'XTick',[],'YTick',[]) ds?v'|  
    %           axis square o[cV1G  
    %           title(['Z_{' num2str(p(k)) '}']) N1|$$9G+  
    %       end X!m9lV<  
    % S%yd5<%_  
    %   See also ZERNPOL, ZERNFUN. IFDZfx  
    Y@b.sMg{  
    %   Paul Fricker 11/13/2006 :&:JTa1cv  
    mw='dFt  
    Mi/&f   
    % Check and prepare the inputs: )tl.s)"N  
    % ----------------------------- ,:Lb7bFv>  
    if min(size(p))~=1 ad:&$  
        error('zernfun2:Pvector','Input P must be vector.') k[HAkB \{  
    end .8P.)%  
    Er+nk`UR_  
    if any(p)>35 Kwg4sr5"D  
        error('zernfun2:P36', ... s;64N'HH  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... Z| V`B `  
               '(P = 0 to 35).']) QoG cWJ  
    end `kU/NKq  
    D?0zhU  
    % Get the order and frequency corresonding to the function number: []A%<EI7  
    % ---------------------------------------------------------------- nSkPM 5\TI  
    p = p(:); D;_ MPN[  
    n = ceil((-3+sqrt(9+8*p))/2); %7gkNa  
    m = 2*p - n.*(n+2); uU:CR>=AKW  
    FKT1fv[H  
    % Pass the inputs to the function ZERNFUN: [&nh5 |f  
    % ---------------------------------------- Hrzf'a|^  
    switch nargin qHP78&wUx  
        case 3 'ul~7h;n  
            z = zernfun(n,m,r,theta); :@!ic<p  
        case 4 T+<A`k: -  
            z = zernfun(n,m,r,theta,nflag); Fm # w2o  
        otherwise tWoh''@#  
            error('zernfun2:nargin','Incorrect number of inputs.') /l<<_uk$  
    end |"9 #bU  
    $.GOZqMs  
    % EOF zernfun2
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    只看该作者 3楼 发表于: 2011-03-12
    回 2楼(phility) 的帖子
    非常感谢啊,我手上也有zernike多项式的拟合的源程序,也不知道对不对,不怎么会有 o`q_wdy?  
    function z = zernfun(n,m,r,theta,nflag) hweaGL t0  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. '^FGc  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N _Jt 2YZdA  
    %   and angular frequency M, evaluated at positions (R,THETA) on the `p7&> BOA  
    %   unit circle.  N is a vector of positive integers (including 0), and _!?Hu/zo  
    %   M is a vector with the same number of elements as N.  Each element LI6hE cM=  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) V]vc(rH  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, !\,kZ|#>  
    %   and THETA is a vector of angles.  R and THETA must have the same ?w+Ix~k  
    %   length.  The output Z is a matrix with one column for every (N,M) 't9hXzAfW  
    %   pair, and one row for every (R,THETA) pair. -~QHqU.  
    % pKjoi{ Z  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike p!<$vE  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), nYt/U\n!  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral fz3 lV  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, "n=vN<8(o  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized Xe^Cn R  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. d'|, [p  
    % ]wWPXx[>/  
    %   The Zernike functions are an orthogonal basis on the unit circle. )5.C]4jol  
    %   They are used in disciplines such as astronomy, optics, and LT,?$I  
    %   optometry to describe functions on a circular domain. A,) VM9M_l  
    % T1r3=Y4  
    %   The following table lists the first 15 Zernike functions. A?oXqb  
    % u]ZqOJXxu  
    %       n    m    Zernike function           Normalization  =Mb1o[  
    %       -------------------------------------------------- f*24)Wn<  
    %       0    0    1                                 1 fVM`-8ZTq  
    %       1    1    r * cos(theta)                    2 ]l(wg]  
    %       1   -1    r * sin(theta)                    2 6Vbzd0dk  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) 6Kj'Zy VL  
    %       2    0    (2*r^2 - 1)                    sqrt(3) Cua%1]"4w  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) U7DCx=B  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) ;_(PVo  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) ad_`x  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) s-7RW  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) u^j {U}  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) 3w!c`;c%  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) _"%B7FK  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) hG_?8:W8HT  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) .y&QqxiE  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) atW'  
    %       -------------------------------------------------- k3t78Qg  
    % 6y5A"-  
    %   Example 1: pW]4bx@E  
    % x+@&(NMP5  
    %       % Display the Zernike function Z(n=5,m=1) Fbp{,V@F2  
    %       x = -1:0.01:1; fof2 xcH!  
    %       [X,Y] = meshgrid(x,x); \i[BP  
    %       [theta,r] = cart2pol(X,Y); c0Dmq)HK?  
    %       idx = r<=1; Dr9 ?2  
    %       z = nan(size(X)); 1H,g=Y4f%  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); q,2]5 '  
    %       figure oiH|uIsqR  
    %       pcolor(x,x,z), shading interp 8V-\e?&^  
    %       axis square, colorbar cFagz* !  
    %       title('Zernike function Z_5^1(r,\theta)') BvU"4d;x  
    % lI/0:|l  
    %   Example 2: Z.wA@ ~e  
    % &|<xqt  
    %       % Display the first 10 Zernike functions G3G6IP  
    %       x = -1:0.01:1; vwr74A.g0  
    %       [X,Y] = meshgrid(x,x); "|m|E/Z-9  
    %       [theta,r] = cart2pol(X,Y); =D^TK-H  
    %       idx = r<=1; 3},Zlu  
    %       z = nan(size(X)); 3[XQR8o  
    %       n = [0  1  1  2  2  2  3  3  3  3]; poJg"R4  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; vLO&Lpv  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; !%Y~~'5 h  
    %       y = zernfun(n,m,r(idx),theta(idx)); C`'W#xnp1  
    %       figure('Units','normalized') ?'r9"M>  
    %       for k = 1:10 ?Mp1~{8  
    %           z(idx) = y(:,k); `E\imL  
    %           subplot(4,7,Nplot(k)) %k0EpJE%  
    %           pcolor(x,x,z), shading interp R1-k3;v^  
    %           set(gca,'XTick',[],'YTick',[]) $iM=4 3W  
    %           axis square L;QY<b  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ?_`0G/xl  
    %       end &)pK%SAM  
    % wG8Wez%  
    %   See also ZERNPOL, ZERNFUN2. *wV[TKaN  
    L "<B;u5pM  
    %   Paul Fricker 11/13/2006 o/,NGU  
    Z7jX9e"L  
    A7P`lJgv  
    % Check and prepare the inputs: 2BzqY`O  
    % ----------------------------- [^~7]2i  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) A.Bk/N1G  
        error('zernfun:NMvectors','N and M must be vectors.') &gc `<kLu  
    end +@VYs*&&  
    r?l;I3~  
    if length(n)~=length(m) P=H+ #  
        error('zernfun:NMlength','N and M must be the same length.') MF[z -7  
    end 1'G8o=~  
    J Lb6C 52  
    n = n(:); Ewo*yY>  
    m = m(:); y7<&vIEC  
    if any(mod(n-m,2)) 0p fnV%  
        error('zernfun:NMmultiplesof2', ... &14W vAU  
              'All N and M must differ by multiples of 2 (including 0).') A6ewdT?>,  
    end F3ZxhkF  
    J< JBdk  
    if any(m>n) J  fcMca  
        error('zernfun:MlessthanN', ... eSl-9 ^  
              'Each M must be less than or equal to its corresponding N.') -cNx1et  
    end FoPginZ]J  
    G5Q!L;3HZ  
    if any( r>1 | r<0 ) ~_!ts{[E  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') )%du@a8  
    end ke/_k/  
    ;^l_i4A  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) fo\\o4Qyh  
        error('zernfun:RTHvector','R and THETA must be vectors.') yZSvn[f  
    end FQf #*  
    bdV3v`  
    r = r(:); [V@yRWI  
    theta = theta(:); b"8FlZ$  
    length_r = length(r); H?}wl%  
    if length_r~=length(theta) Fc0jQ@4=  
        error('zernfun:RTHlength', ... !Y;<:zx5  
              'The number of R- and THETA-values must be equal.') U  5`y  
    end ~SV Q;U)-  
    =LZ>s u  
    % Check normalization: # bX~=`  
    % -------------------- \iMyo  
    if nargin==5 && ischar(nflag) Q?;C4n4]l  
        isnorm = strcmpi(nflag,'norm'); 7dD.G/'  
        if ~isnorm Ku3!*n_\  
            error('zernfun:normalization','Unrecognized normalization flag.') ;.Zh,cU  
        end jXEGSn  
    else =aow d4 t  
        isnorm = false; ) Ypz!  
    end J0Four#MD  
    R @r{  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ?^3B3qqh9  
    % Compute the Zernike Polynomials "2h5m4  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *d l"wH&  
    J>v$2?w`w  
    % Determine the required powers of r: ;]h.m)~|  
    % ----------------------------------- MOV =n75  
    m_abs = abs(m); J1-):3A  
    rpowers = []; X^in};&d  
    for j = 1:length(n) U5 rxt^  
        rpowers = [rpowers m_abs(j):2:n(j)]; k.Zll,s  
    end $T*KaX\{B  
    rpowers = unique(rpowers); -Uf4v6A  
    g)M#{"H  
    % Pre-compute the values of r raised to the required powers, 9kd.j@C  
    % and compile them in a matrix: +-HE '4mo  
    % ----------------------------- y@9Y,ZR*  
    if rpowers(1)==0 3@X|Gs'_S  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); |' @[N,  
        rpowern = cat(2,rpowern{:}); sM9- 0A  
        rpowern = [ones(length_r,1) rpowern]; -~'kP /E^  
    else G4yUC<TqBP  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); pSrsp r  
        rpowern = cat(2,rpowern{:}); UQdyv(jXq  
    end xL&PJ /'  
    ~}%&p& p  
    % Compute the values of the polynomials: ,%='>A  
    % -------------------------------------- x=3I)}J(kn  
    y = zeros(length_r,length(n)); N K"%DU<  
    for j = 1:length(n) IuWX*b`v  
        s = 0:(n(j)-m_abs(j))/2; SbJh(V-pr  
        pows = n(j):-2:m_abs(j); F25<+ 1kr  
        for k = length(s):-1:1 3qcpf:  
            p = (1-2*mod(s(k),2))* ... 9R:(^8P8  
                       prod(2:(n(j)-s(k)))/              ... hE5G!@1F  
                       prod(2:s(k))/                     ... 2e\Kw+(>{  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... lDU#7\5.  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); c V(H<"I  
            idx = (pows(k)==rpowers); 7n>|D^  
            y(:,j) = y(:,j) + p*rpowern(:,idx); mE_iS?1  
        end GsRt5?X/*  
         ]h!*T{:  
        if isnorm #?5VsD8  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); Dzm qR0)  
        end Vdy\4 nu(  
    end c8tP+O9  
    % END: Compute the Zernike Polynomials T@>6 3  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% kpY%&  
    =KW|#]RB^  
    % Compute the Zernike functions: |>[X<>m  
    % ------------------------------ ~{Ua92zV9  
    idx_pos = m>0; C0f[eA  
    idx_neg = m<0; v5gQ9  
    L`JY4JM"  
    z = y; 0Sz/c+ 6  
    if any(idx_pos) tpd|y|  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); T)O]:v  
    end aH9L|BN*  
    if any(idx_neg) aEZJNWv  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); _BCT.ual  
    end ~CJYQFt  
    `C ?a  
    % 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