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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 XEp{VC@=  
    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! VcYrK4  
     
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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)  cWsNr'MS*  
    Tod&&T'UW  
    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。 C.yQ=\U2  
    zuad~%D<I  
    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) *eTqVG.  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. '6iEMg&3  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of RNEp4x  
    %   order N and frequency M, evaluated at R.  N is a vector of Z*]9E^  
    %   positive integers (including 0), and M is a vector with the PB\(=  
    %   same number of elements as N.  Each element k of M must be a Q0`wt.}V2  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) ;40/yl3r3[  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is D[[|")Fn  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix H7&8\ FNa  
    %   with one column for every (N,M) pair, and one row for every 0y'H~(  
    %   element in R. :1. L}4"gg  
    % Y1W1=Uc uk  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- ,*TmIPNK  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is p SH=%u>  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to K*vt;L  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 rK6l8)o  
    %   for all [n,m]. hc(#{]].  
    %  j|DsG,  
    %   The radial Zernike polynomials are the radial portion of the E1aHKjLQ  
    %   Zernike functions, which are an orthogonal basis on the unit W dK #ZOR  
    %   circle.  The series representation of the radial Zernike Tj` ,Z5vy  
    %   polynomials is 5FPM`hLT  
    % ouvA~/5  
    %          (n-m)/2 x*\Y)9Vgy  
    %            __ +;(c:@>@,  
    %    m      \       s                                          n-2s `t>l:<@%  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r A7Cm5>Y_S  
    %    n      s=0 lV3x*4O=  
    % Fh&G;aEq  
    %   The following table shows the first 12 polynomials. y4 #>X  
    % K^)Eb(4  
    %       n    m    Zernike polynomial    Normalization Z!a =dnwHz  
    %       --------------------------------------------- 7dTkp!'X-  
    %       0    0    1                        sqrt(2) _2Zx?<] 2E  
    %       1    1    r                           2 6m/r+?'  
    %       2    0    2*r^2 - 1                sqrt(6) + /4A  
    %       2    2    r^2                      sqrt(6) ONB{_X?  
    %       3    1    3*r^3 - 2*r              sqrt(8) u OmtyX  
    %       3    3    r^3                      sqrt(8) 4$HhP, gL=  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) MIeU,KT#U  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) z3{G9Np  
    %       4    4    r^4                      sqrt(10) q"CVcLi9  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) q5J5>  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) Y!aSs3c  
    %       5    5    r^5                      sqrt(12) L:$ ,v^2  
    %       --------------------------------------------- u"r`3P`  
    % WH#1 zv  
    %   Example: bI7Vwyz  
    % !]A  
    %       % Display three example Zernike radial polynomials U|H=Y"pL  
    %       r = 0:0.01:1; b"<liGh"n-  
    %       n = [3 2 5]; TM__I\+Q  
    %       m = [1 2 1]; %vn"{3y>rF  
    %       z = zernpol(n,m,r); 6fE7W>la  
    %       figure e-})6)XgA  
    %       plot(r,z) !,_u)4  
    %       grid on K C*e/J  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') PV.X z0@R  
    % '|6]_   
    %   See also ZERNFUN, ZERNFUN2. >mbHy<<  
    jKz$@gP  
    % A note on the algorithm. wyH[x!QX  
    % ------------------------ IkL#SgY  
    % The radial Zernike polynomials are computed using the series *?@?f&E/  
    % representation shown in the Help section above. For many special NR$3%0 nC6  
    % functions, direct evaluation using the series representation can *nT<m\C6  
    % produce poor numerical results (floating point errors), because H5/6TX72N  
    % the summation often involves computing small differences between f=l rg KE  
    % large successive terms in the series. (In such cases, the functions Fk&c=V;SU  
    % are often evaluated using alternative methods such as recurrence ueogaifvB  
    % relations: see the Legendre functions, for example). For the Zernike "@^k)d$  
    % polynomials, however, this problem does not arise, because the `z}?"BW|  
    % polynomials are evaluated over the finite domain r = (0,1), and +qN>.y!Y  
    % because the coefficients for a given polynomial are generally all nUaJzPl  
    % of similar magnitude. .r=4pQ@#  
    % >>4qJ%bL  
    % ZERNPOL has been written using a vectorized implementation: multiple zF`0J  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] <q58uuK  
    % values can be passed as inputs) for a vector of points R.  To achieve ~gJwW+  
    % this vectorization most efficiently, the algorithm in ZERNPOL KWbI'}_z  
    % involves pre-determining all the powers p of R that are required to  Po+.&7F  
    % compute the outputs, and then compiling the {R^p} into a single i'<[DjMDlm  
    % matrix.  This avoids any redundant computation of the R^p, and &C5_g$Ma.Z  
    % minimizes the sizes of certain intermediate variables. pHGYQ;:L  
    % RT4x\&q  
    %   Paul Fricker 11/13/2006 Uk[b|<U-`d  
    SBu"3ym  
    Ve$o}h-  
    % Check and prepare the inputs: # " 6Qj'/h  
    % ----------------------------- )EPjAv  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ^ 'MT0j  
        error('zernpol:NMvectors','N and M must be vectors.') olB.*#gA  
    end ;$,U~0  
    G{~J|{t\yz  
    if length(n)~=length(m) tn\yI!a  
        error('zernpol:NMlength','N and M must be the same length.') LG9+GszX 2  
    end oi7@s0@  
    P7bMIe  
    n = n(:); ;J( 8 L  
    m = m(:); .<0ye_S'y  
    length_n = length(n); f].h^ ~.q  
    ](]i 'fE>  
    if any(mod(n-m,2)) y%$AhRk*U  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') 4&lv6`G `  
    end q4h]o^+  
    x M/+L:_<  
    if any(m<0) /|m2WxK)  
        error('zernpol:Mpositive','All M must be positive.') 4HXo>0  
    end :1Xz4wkWS*  
    ='r!g  
    if any(m>n) JAnZdfRt  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') {GT*ZU*  
    end bn&TF3b  
    #<"~~2?  
    if any( r>1 | r<0 ) |fJ};RLI"  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') h|9L5  
    end dh\'<|\K  
    fSj5ZsO  
    if ~any(size(r)==1) Pl06:g2I  
        error('zernpol:Rvector','R must be a vector.') lN 4oW3QT  
    end J$DE"| -  
    GT.,  
    r = r(:); !x=~g"d<&  
    length_r = length(r); z]y.W`i   
    K=Z|/Kkh  
    if nargin==4 `:fZ)$sY  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); %)8}X>xq  
        if ~isnorm {%5eMyF#  
            error('zernpol:normalization','Unrecognized normalization flag.') LKB$,pR~1l  
        end CJx|?yK2  
    else Xf]d. :  
        isnorm = false; 9MJG;+B~  
    end zV37$Hb  
    ;%9|k U  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @+2=g WH  
    % Compute the Zernike Polynomials r.&Vw|*>  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% BsDn5\ q  
    ZgcMv,=  
    % Determine the required powers of r: e2TiBTbQaF  
    % ----------------------------------- '3tCH)s  
    rpowers = []; ibk6|pp  
    for j = 1:length(n) 7hcYD!DS  
        rpowers = [rpowers m(j):2:n(j)]; f|c{5$N!  
    end 9ULQrq$?  
    rpowers = unique(rpowers); ,AFu C <  
    g}{aZ$sta  
    % Pre-compute the values of r raised to the required powers, :J@ gmY:C  
    % and compile them in a matrix: R4cM%l_#W  
    % ----------------------------- nPl?K:(  
    if rpowers(1)==0 ^A/k)x6  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); |v%YQ R  
        rpowern = cat(2,rpowern{:}); 9wwqcx)3(  
        rpowern = [ones(length_r,1) rpowern];  skViMo  
    else UKvWJnz  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); sY Qk  
        rpowern = cat(2,rpowern{:}); 4N3R|  
    end lvz7#f L~  
    Y eo]]i{  
    % Compute the values of the polynomials: dn+KH+v  
    % -------------------------------------- \'D0'\:vz  
    z = zeros(length_r,length_n); *vxk@ `K~  
    for j = 1:length_n D=Gtq6jd  
        s = 0:(n(j)-m(j))/2; WX?IYQ+  
        pows = n(j):-2:m(j); *)T^Ch D,  
        for k = length(s):-1:1 b=NxUd O  
            p = (1-2*mod(s(k),2))* ... ?P`K7  
                       prod(2:(n(j)-s(k)))/          ... %T%sGDCV  
                       prod(2:s(k))/                 ... i%]EEVmN  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... 6SkaH<-&K  
                       prod(2:((n(j)+m(j))/2-s(k)));  "Og7rl  
            idx = (pows(k)==rpowers); E A1?)|}n  
            z(:,j) = z(:,j) + p*rpowern(:,idx); .j0$J\:i  
        end P@Oo$ o  
         IY\5@PVZ  
        if isnorm *C*U5~Zq7:  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); UECK:61Me  
        end u0c1:Uv#~e  
    end  w``ST  
    6Y?|w3f   
    % 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) A[{yCn`tM  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. h]}wp;Z  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated 8^1 Te m  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive YZ8>OwQz2  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, EUX\^c]n  
    %   and THETA is a vector of angles.  R and THETA must have the same )g%d:xI  
    %   length.  The output Z is a matrix with one column for every P-value, Flm%T-Dl  
    %   and one row for every (R,THETA) pair. @:vwb\azVD  
    % |3"KK  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike ,<P vovg_  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) 0znR0%~  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) Js?]$V"  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 MH\dC9%p  
    %   for all p. 16(QR-  
    % hD!7Cl Q  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 J<h $ wM  
    %   Zernike functions (order N<=7).  In some disciplines it is E4/Dr}4  
    %   traditional to label the first 36 functions using a single mode 2eY_%Y0  
    %   number P instead of separate numbers for the order N and azimuthal qqY"*uJ'  
    %   frequency M. Wt-GjxGi  
    % ^k">A:E2  
    %   Example: 3bH'H*2  
    % Y\8)OBZ  
    %       % Display the first 16 Zernike functions n 0L^e  
    %       x = -1:0.01:1; Cnh \%OW  
    %       [X,Y] = meshgrid(x,x); vXZOy%$o  
    %       [theta,r] = cart2pol(X,Y); )F]]m#`  
    %       idx = r<=1; E]-/Zbvdv  
    %       p = 0:15; QlU8uI[dk  
    %       z = nan(size(X)); :':s@gqr  
    %       y = zernfun2(p,r(idx),theta(idx)); e6$WQd`O  
    %       figure('Units','normalized') HQhM'x  
    %       for k = 1:length(p) ;[OH(!  
    %           z(idx) = y(:,k); I1M%J@Cz  
    %           subplot(4,4,k) BW*rIn<?G  
    %           pcolor(x,x,z), shading interp ~=l;=7 T  
    %           set(gca,'XTick',[],'YTick',[]) S_UIO.K  
    %           axis square v PG},m~-  
    %           title(['Z_{' num2str(p(k)) '}']) UySZbmP48  
    %       end :*9Wh  
    % &d^m 1  
    %   See also ZERNPOL, ZERNFUN. 8'io$ 6d=  
    uz jU2  
    %   Paul Fricker 11/13/2006 <R=Zs[9M1  
    z<XtS[ki  
    )1`0PJoHE  
    % Check and prepare the inputs: fJ!R6D  
    % ----------------------------- }Oq5tC@$G  
    if min(size(p))~=1 r52gn(,  
        error('zernfun2:Pvector','Input P must be vector.') Pw"-S?`(  
    end Z,Dl` w  
    I:1C8*/  
    if any(p)>35 ` 7V]y -  
        error('zernfun2:P36', ... .Vvx,>>D  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... Ean5b>\  
               '(P = 0 to 35).']) ],Do6 @M-  
    end 4O!ikmY:t  
    srrgvG,  
    % Get the order and frequency corresonding to the function number: v;D~Pa  
    % ---------------------------------------------------------------- H8}oIA"b  
    p = p(:); M@v.c; Lt  
    n = ceil((-3+sqrt(9+8*p))/2); ')<hON44EX  
    m = 2*p - n.*(n+2); PIS2Ed]  
    K_Eux rPn  
    % Pass the inputs to the function ZERNFUN: *#+An<iT ;  
    % ---------------------------------------- 7Kxp=-k  
    switch nargin Yufc{M00  
        case 3 59;KQ  
            z = zernfun(n,m,r,theta); T%*D~=fQ'  
        case 4 ":QZy8f9%  
            z = zernfun(n,m,r,theta,nflag); tJ$_lk ~6q  
        otherwise 07{)?1cod4  
            error('zernfun2:nargin','Incorrect number of inputs.') t!7-DF|N  
    end ~6LN6}~|.  
    N6i Q8P -  
    % 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 qF;|bF  
    function z = zernfun(n,m,r,theta,nflag) > /caXvS  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. i?^L/b`H  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N J<jy2@"tXo  
    %   and angular frequency M, evaluated at positions (R,THETA) on the |Ds1  
    %   unit circle.  N is a vector of positive integers (including 0), and fVpMx4&F   
    %   M is a vector with the same number of elements as N.  Each element D2~*&'4y  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) >}6%#CAf  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, 4 "'~NvO  
    %   and THETA is a vector of angles.  R and THETA must have the same a<bwzX|.  
    %   length.  The output Z is a matrix with one column for every (N,M) u.xnOcOH!  
    %   pair, and one row for every (R,THETA) pair. 'm kLCS  
    % 1#+S+g@#  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 40m-ch6Q  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 9CD_ os\h  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral 0YDR1dO(*  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, C!bUI8x z  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized 1/J=uH  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. t;\Y{`  
    % }:)&u|d_  
    %   The Zernike functions are an orthogonal basis on the unit circle. &0JI!bR(  
    %   They are used in disciplines such as astronomy, optics, and f(MO_Sj]  
    %   optometry to describe functions on a circular domain. ]~3V}z,T*  
    % 61'XgkacDS  
    %   The following table lists the first 15 Zernike functions. =Jb>x#Y  
    % H"WprHe  
    %       n    m    Zernike function           Normalization P\k# >}}  
    %       -------------------------------------------------- 6(ol1 (U  
    %       0    0    1                                 1 E hMNap}5"  
    %       1    1    r * cos(theta)                    2 1bX<$>x9u  
    %       1   -1    r * sin(theta)                    2 \ }G> 8^  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) #S"nF@   
    %       2    0    (2*r^2 - 1)                    sqrt(3) c yz3,3\e  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) [.wYdv35  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) c5GuM|*7  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) vy I!]p  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) _.8S&  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) R8'RA%O9J  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) -nV9:opD  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) h~zT ydnH  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) j&qub_j"xX  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) /9fR'EO{x  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) C;^X[x%h7$  
    %       -------------------------------------------------- [d ]9Oa4  
    % {R `[kt  
    %   Example 1: i=2N;sAl  
    % FU4L6n  
    %       % Display the Zernike function Z(n=5,m=1) nAdf=D'P  
    %       x = -1:0.01:1; qUb&   
    %       [X,Y] = meshgrid(x,x); 'TB2:W3  
    %       [theta,r] = cart2pol(X,Y); }@d@3  
    %       idx = r<=1; M9%$lCl   
    %       z = nan(size(X)); `VguQl_,gA  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); '6%2.[ o  
    %       figure ?4T-@~~*`=  
    %       pcolor(x,x,z), shading interp ' S/gmn  
    %       axis square, colorbar ey$&;1x#5  
    %       title('Zernike function Z_5^1(r,\theta)') \qJXF|z<K  
    % ]:J$w]\  
    %   Example 2: "f OV^B  
    % .(k|wX[Fu~  
    %       % Display the first 10 Zernike functions 63IM]J  
    %       x = -1:0.01:1; Pa: |_IXA  
    %       [X,Y] = meshgrid(x,x); {E|$8)58i  
    %       [theta,r] = cart2pol(X,Y); '!B&:X)  
    %       idx = r<=1; f]sr RYSR  
    %       z = nan(size(X)); "E4a=YH_  
    %       n = [0  1  1  2  2  2  3  3  3  3]; {]4LULq  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; 8Z=R)asGS  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; 7WzxA=*#  
    %       y = zernfun(n,m,r(idx),theta(idx)); 8zW2zkv2|#  
    %       figure('Units','normalized') FGBbO\< /  
    %       for k = 1:10 H3-hcx54T  
    %           z(idx) = y(:,k); sc#qwQ#  
    %           subplot(4,7,Nplot(k)) 5*u+q2\F  
    %           pcolor(x,x,z), shading interp \1M4Dl5!  
    %           set(gca,'XTick',[],'YTick',[]) 'PW5ux@`<  
    %           axis square W ]8 QM1$  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ('+d.F[109  
    %       end 44j*KsBf  
    % &s>Jb?_5Mx  
    %   See also ZERNPOL, ZERNFUN2. nKj7.,>;:<  
    1<aP92/N&  
    %   Paul Fricker 11/13/2006 YKK*ER0  
    ~WF\  
    W=+ Y|R!  
    % Check and prepare the inputs: b4Ekqas  
    % ----------------------------- BDQsP$'6QT  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 4 s9LB  
        error('zernfun:NMvectors','N and M must be vectors.') nQ3A~ ()  
    end n|yO9:Uw<  
    ]7c=PC  
    if length(n)~=length(m) zX i 'kB  
        error('zernfun:NMlength','N and M must be the same length.') gf\oC> N  
    end B&"Q\'c  
    Pr C{'XDlU  
    n = n(:); 6j|{`Zd)G  
    m = m(:); 6Q5^>\Y  
    if any(mod(n-m,2)) +:/%3}`  
        error('zernfun:NMmultiplesof2', ... 2y1Sne=<Kb  
              'All N and M must differ by multiples of 2 (including 0).') k4zZ7H  
    end {?7Uj  
    %E;'ln4h&,  
    if any(m>n) %mgE;~"&  
        error('zernfun:MlessthanN', ... YtLt*Ig%  
              'Each M must be less than or equal to its corresponding N.') M X]n&  
    end 9} .z;prz  
    */S_Icf  
    if any( r>1 | r<0 ) [{/jI\?v  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') )0k53-h&  
    end )D%~` ,#pQ  
    |u p  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) bpa?C  
        error('zernfun:RTHvector','R and THETA must be vectors.') .*Qx\,  
    end F,CT Z~  
    ;q>ah!"k  
    r = r(:); -$Ih@2"6  
    theta = theta(:); 3o/[t  
    length_r = length(r); +LJ73 !  
    if length_r~=length(theta) ML p9y#  
        error('zernfun:RTHlength', ... WTiD[u  
              'The number of R- and THETA-values must be equal.') KqP#6^ _  
    end 9;If&uM  
    l;E(I_ i)  
    % Check normalization: 9W);rL|5  
    % -------------------- -trkA'ewZ  
    if nargin==5 && ischar(nflag) 2st3  
        isnorm = strcmpi(nflag,'norm'); #4;wjcGWw  
        if ~isnorm tX~w{|k  
            error('zernfun:normalization','Unrecognized normalization flag.') EKN~H$.  
        end (^>J&[=  
    else K:WDl;8 (d  
        isnorm = false; sa8Vvzvo.  
    end ue>D 7\8  
    :rP=t ,  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \GU<43J2uo  
    % Compute the Zernike Polynomials f%8C!W]Dm  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% $<OD31T  
    V28M lP  
    % Determine the required powers of r: z0 Z%m@  
    % ----------------------------------- MWh6]gGs  
    m_abs = abs(m); l}P=/#</T  
    rpowers = []; _tycgq#  
    for j = 1:length(n) Rk8P ax/JK  
        rpowers = [rpowers m_abs(j):2:n(j)]; EiaW1Cs  
    end Ni7nq8B<  
    rpowers = unique(rpowers); bhs _9ivw  
    J9 I:Q<;  
    % Pre-compute the values of r raised to the required powers, (w zQ2Dk  
    % and compile them in a matrix: )YI(/*+]  
    % ----------------------------- DW3G  
    if rpowers(1)==0 '0,^6'VWOV  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); %]7d`/  
        rpowern = cat(2,rpowern{:}); BL4-7  
        rpowern = [ones(length_r,1) rpowern]; IvNT6]6 P  
    else |&4/n6;P$0  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); .eC1qWZJpd  
        rpowern = cat(2,rpowern{:}); fd9k?,zM  
    end o,wUc"CE  
    T6kdS]4-  
    % Compute the values of the polynomials: lr$zHI7_`  
    % -------------------------------------- `QY)!$mUIF  
    y = zeros(length_r,length(n)); #,v {Ihn  
    for j = 1:length(n) B|X!>Q<g  
        s = 0:(n(j)-m_abs(j))/2; |+"(L#wk  
        pows = n(j):-2:m_abs(j); a09<!0Rp  
        for k = length(s):-1:1 3 8`<:{^Y  
            p = (1-2*mod(s(k),2))* ... Xlt|nX~#;  
                       prod(2:(n(j)-s(k)))/              ... XB5DPx  
                       prod(2:s(k))/                     ... {fp[BF  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... )=-szJjXZ  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); 7>*vI7O0l  
            idx = (pows(k)==rpowers); ,"0 :3+(8;  
            y(:,j) = y(:,j) + p*rpowern(:,idx); Yz93'HDB  
        end @|T'0_'  
         yaV|AB$v  
        if isnorm v(%*b,^  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); Jfl!#UAD|n  
        end (C)p9-,  
    end S0W||#Pr  
    % END: Compute the Zernike Polynomials 3irl (;v  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% )BfAw  
    /2VJX@h  
    % Compute the Zernike functions: 2I{"XB  
    % ------------------------------ W=4FFl[  
    idx_pos = m>0; 0Wp|1)ljA  
    idx_neg = m<0; Z<{QaY$"  
    , 9 a  
    z = y; |(^PS8wG  
    if any(idx_pos) <ZR9GlIr  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); UkGCyGyZ[  
    end Y\'}a+:@Ph  
    if any(idx_neg) Y`wSv NU  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); .6 ?U@2  
    end Ilm^G}GB  
    UJ6v(:z <  
    % 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