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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 }v$=mLy  
    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! J)yy}[Fx  
     
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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)  +siNU#!  
    [%,=0P}  
    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。 d?y\~<  
    =LY^3TlDj  
    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) EoW zHa  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. ;QD;5 <1  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of P,=J"%a-  
    %   order N and frequency M, evaluated at R.  N is a vector of =C1Qo#QQ%  
    %   positive integers (including 0), and M is a vector with the }mZ*f y0t  
    %   same number of elements as N.  Each element k of M must be a jt?%03iuk  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) ,?~,"IQyi[  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is |sM#g1D@  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix GhA~PjZS  
    %   with one column for every (N,M) pair, and one row for every Vzm7xl [  
    %   element in R. 2DdLqZY#  
    % 9gayu<J  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- ~x|Sv4M  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is )WJI=jl  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to 4>`w9   
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 *X5LyO3-gP  
    %   for all [n,m]. 3PeJPw  
    % 4zbV' ]  
    %   The radial Zernike polynomials are the radial portion of the uW_ /7ex  
    %   Zernike functions, which are an orthogonal basis on the unit S^=/}PT'  
    %   circle.  The series representation of the radial Zernike $& gidz/w  
    %   polynomials is <QLj6#d7Y  
    % NH6!|T  
    %          (n-m)/2 ~3]8f0^%m  
    %            __ n:z>l,`C]  
    %    m      \       s                                          n-2s !gQ(1u|r  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r I65GUX#DV  
    %    n      s=0 :b)@h|4  
    % iAhRlQ{Qu  
    %   The following table shows the first 12 polynomials. 1H@F>}DP  
    % 3e1"5~?'<  
    %       n    m    Zernike polynomial    Normalization dU n#'<g5  
    %       --------------------------------------------- o62gLO]z@  
    %       0    0    1                        sqrt(2) <-7Ha_#  
    %       1    1    r                           2 .AS,]*?Zn%  
    %       2    0    2*r^2 - 1                sqrt(6) )A;<'{t #L  
    %       2    2    r^2                      sqrt(6) =J\7(0Dz4t  
    %       3    1    3*r^3 - 2*r              sqrt(8) -W vAmi  
    %       3    3    r^3                      sqrt(8) U?yXTMD  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) n&&y\?n  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) ?q`mr_x%?  
    %       4    4    r^4                      sqrt(10) M!@[lJ  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) uS.a9 Q(  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) rMloj8O*  
    %       5    5    r^5                      sqrt(12) l).Ijl}AH;  
    %       --------------------------------------------- %&GQ]pmcY  
    % l>q.BG  
    %   Example: kp"cHJNx  
    % FiL JF!  
    %       % Display three example Zernike radial polynomials /m:}rD  
    %       r = 0:0.01:1; VQ`O;n6/`  
    %       n = [3 2 5]; oaE3Aa  
    %       m = [1 2 1]; !{\c`Z<#  
    %       z = zernpol(n,m,r); U {v_0\ES  
    %       figure " WL  
    %       plot(r,z) vS<e/e+  
    %       grid on % VZ\4+8S  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') L.[2l Q  
    % ' 'N@ <|  
    %   See also ZERNFUN, ZERNFUN2. @^@-A\7[KO  
    E ..[F<5  
    % A note on the algorithm. c8MNo'h  
    % ------------------------ \GP c_m:qL  
    % The radial Zernike polynomials are computed using the series Atw^C+"vW&  
    % representation shown in the Help section above. For many special =r8(9:F!  
    % functions, direct evaluation using the series representation can 54&2SU$kx  
    % produce poor numerical results (floating point errors), because Joj8'  
    % the summation often involves computing small differences between j>zVC;Sj*  
    % large successive terms in the series. (In such cases, the functions '@bA_F(  
    % are often evaluated using alternative methods such as recurrence 2{\Y<%.  
    % relations: see the Legendre functions, for example). For the Zernike 2(|V1]6D?  
    % polynomials, however, this problem does not arise, because the [g_@<?zg  
    % polynomials are evaluated over the finite domain r = (0,1), and g!UM8I-$  
    % because the coefficients for a given polynomial are generally all c$;enAf@  
    % of similar magnitude. - Zh+5;8g  
    % ap!<8N  
    % ZERNPOL has been written using a vectorized implementation: multiple !bg3  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] C+j+q648>  
    % values can be passed as inputs) for a vector of points R.  To achieve up?8Pq*  
    % this vectorization most efficiently, the algorithm in ZERNPOL |v&&%>A2  
    % involves pre-determining all the powers p of R that are required to xPv&(XZR  
    % compute the outputs, and then compiling the {R^p} into a single ?a}~yz#B(  
    % matrix.  This avoids any redundant computation of the R^p, and czzV2P/t}  
    % minimizes the sizes of certain intermediate variables. 3z<t#  
    % Oh: -Y]m=  
    %   Paul Fricker 11/13/2006 ohl%<FqS  
    LWE !+(n  
    -XBNtM_ "  
    % Check and prepare the inputs: 2ou?:5i  
    % -----------------------------  %JZIg!  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 7E'C o|  
        error('zernpol:NMvectors','N and M must be vectors.') I{ $|Ed1  
    end *`W82V  
    f&|SGD*  
    if length(n)~=length(m) f$L5=V  
        error('zernpol:NMlength','N and M must be the same length.') lbY>R@5  
    end |(N4x(xl  
    _)Ms9RN  
    n = n(:); Z3d&I]Tf  
    m = m(:); {*m?t 7  
    length_n = length(n); Q=[&~^ Y)  
    mAMKCxz,  
    if any(mod(n-m,2)) lF<(yF5  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') j:#[voo7  
    end +^.(3Aw  
    Tm `CA0@  
    if any(m<0) 03xQ%"TU<  
        error('zernpol:Mpositive','All M must be positive.') %K%z<R8  
    end %`~8j H@  
    <8Ad\MU  
    if any(m>n) bm^ou#]|  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') "6ZatRUd  
    end cX2b:  
    '*pq@|q;t  
    if any( r>1 | r<0 ) P*}Oi7Z  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') :V [vE h  
    end D 6(w}W  
    D_{J:Hb  
    if ~any(size(r)==1) pD{Li\LY  
        error('zernpol:Rvector','R must be a vector.') n\QG-?%Pi  
    end C$_H)I  
    .R1)i-^  
    r = r(:); zr,jaR;  
    length_r = length(r); /{lls2ycW%  
    +um; eL7  
    if nargin==4 jooh`| `P  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); |Q{l ]D  
        if ~isnorm 0-@waK  
            error('zernpol:normalization','Unrecognized normalization flag.') 49CMRO,T  
        end r6A7}v  
    else iU &V}p  
        isnorm = false; OS3J,f}<=  
    end P iN3t]2  
    tqHXzmsjW  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |YH1q1l  
    % Compute the Zernike Polynomials sbRg=k&Ns  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Yd@9P 2C  
    <1"6`24  
    % Determine the required powers of r: 6ik6JL$AI  
    % ----------------------------------- GovGh? X#x  
    rpowers = []; 8!1o,=I$  
    for j = 1:length(n) 7|2:;5:U  
        rpowers = [rpowers m(j):2:n(j)]; 1vobfZ-w9  
    end X/@Gx 4  
    rpowers = unique(rpowers); hM;EUWv  
    wc;5tb#  
    % Pre-compute the values of r raised to the required powers, <4Ak$ E %"  
    % and compile them in a matrix: XVY^m}pMe  
    % ----------------------------- i22R3&C  
    if rpowers(1)==0 Ouj5NL  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ct/I85c@P  
        rpowern = cat(2,rpowern{:}); __zsrIUJ  
        rpowern = [ones(length_r,1) rpowern]; R (6Jvub"I  
    else #0weN%  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 7UMsKE-  
        rpowern = cat(2,rpowern{:});  p.zU9rID  
    end ?g9CeeH*  
    &vLZj  
    % Compute the values of the polynomials: `P'{HT  
    % -------------------------------------- P afmHXx  
    z = zeros(length_r,length_n); aFhsRE?YC=  
    for j = 1:length_n 5t0$nKah]  
        s = 0:(n(j)-m(j))/2; }=wSfr9g  
        pows = n(j):-2:m(j);  ;v.l<AOE  
        for k = length(s):-1:1 ZM<1;!i  
            p = (1-2*mod(s(k),2))* ... r&^4L  
                       prod(2:(n(j)-s(k)))/          ... 3B>!9:w~f  
                       prod(2:s(k))/                 ... |gT$M _}  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... 1,4kw~tA  
                       prod(2:((n(j)+m(j))/2-s(k))); ~jJu*s$?  
            idx = (pows(k)==rpowers); }Za[<t BWS  
            z(:,j) = z(:,j) + p*rpowern(:,idx); "ibKi=  
        end X\M0Q%8  
         N!hp^V<7  
        if isnorm IUwY/R9Q  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); rGx1>xd(k  
        end eqXW|,zUm  
    end $.v5G>- )3  
    pS51fF9  
    % 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) OfIml.  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. CI ~+(+q  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated XYf;72*  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive Ktg6*L/  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, !Il<'+ ^  
    %   and THETA is a vector of angles.  R and THETA must have the same n&k1'KL&  
    %   length.  The output Z is a matrix with one column for every P-value, 5q@o,d  
    %   and one row for every (R,THETA) pair. i $#bg^  
    % 3]/w3|y  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike ,f4Hl%T;  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) ?2QssfB  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) gy,B+~p  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 Z-Zox-I1}-  
    %   for all p. 9^>nZ6  
    % "rBo?%:  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 -k")#1  
    %   Zernike functions (order N<=7).  In some disciplines it is XP~4jOL]  
    %   traditional to label the first 36 functions using a single mode n`;=^^B  
    %   number P instead of separate numbers for the order N and azimuthal #*XuU8q?  
    %   frequency M. |Kh#\d  
    % `UGHk*DL)  
    %   Example: NkA|T1w7  
    % PudwcP {  
    %       % Display the first 16 Zernike functions @<r  ;>G  
    %       x = -1:0.01:1; yLG`tU1  
    %       [X,Y] = meshgrid(x,x); HS>Z6|uLY  
    %       [theta,r] = cart2pol(X,Y); Q(>89*b&  
    %       idx = r<=1; gtqgf<mS  
    %       p = 0:15; 5o'V}  
    %       z = nan(size(X)); j8_WEjG  
    %       y = zernfun2(p,r(idx),theta(idx)); ney6N@  
    %       figure('Units','normalized') /5EM;Mx  
    %       for k = 1:length(p) j)]mN$Sa:  
    %           z(idx) = y(:,k);  UcKpid  
    %           subplot(4,4,k) c5nl!0XX  
    %           pcolor(x,x,z), shading interp tFO86 !ln  
    %           set(gca,'XTick',[],'YTick',[]) hZU @35~BN  
    %           axis square gfR B  
    %           title(['Z_{' num2str(p(k)) '}']) ZQZ>{K  
    %       end ":tQYo]d  
    % "~> # ;x{  
    %   See also ZERNPOL, ZERNFUN. ix [aS  
    ^dM,K p  
    %   Paul Fricker 11/13/2006 ej4xW~_  
    @OV\raUO&V  
    +Gg6h=u  
    % Check and prepare the inputs: M\ B A+  
    % ----------------------------- &>XIK8*  
    if min(size(p))~=1 [yJcM [p\  
        error('zernfun2:Pvector','Input P must be vector.') i*_T\_=  
    end f4@>7K]9TA  
    g/'CX}g`  
    if any(p)>35 NffZttN  
        error('zernfun2:P36', ... Zx@/5!_n.  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... @rB!47!  
               '(P = 0 to 35).']) z GhJ  
    end E %FCOKw_  
    Xb@lKX5Re  
    % Get the order and frequency corresonding to the function number: |kB1>$  
    % ---------------------------------------------------------------- gf$5pp-  
    p = p(:); 07:CcT  
    n = ceil((-3+sqrt(9+8*p))/2); G];5'd~C;d  
    m = 2*p - n.*(n+2); WPPz/c|j  
    A'^y+42jY  
    % Pass the inputs to the function ZERNFUN: .v?Ir)  
    % ---------------------------------------- 8!(4;fN$j.  
    switch nargin c*sK| U7)  
        case 3 Vcm9:,Xlw  
            z = zernfun(n,m,r,theta); S:"R/EE(  
        case 4 +l+8Z:i<  
            z = zernfun(n,m,r,theta,nflag); vN=e1\  
        otherwise .'.#bH9K  
            error('zernfun2:nargin','Incorrect number of inputs.') qq9fZZb  
    end 2y s'q !  
    7O84R^!|2  
    % 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 O{:_-eI&d  
    function z = zernfun(n,m,r,theta,nflag) u2%/</]h  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. -L<''2t  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N !|}(tqt  
    %   and angular frequency M, evaluated at positions (R,THETA) on the /G[; kR"  
    %   unit circle.  N is a vector of positive integers (including 0), and %P05k  
    %   M is a vector with the same number of elements as N.  Each element YaI8hj@}  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) VbQ9o  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, tai  
    %   and THETA is a vector of angles.  R and THETA must have the same WtlPgT;wE  
    %   length.  The output Z is a matrix with one column for every (N,M) t F^|,9_<  
    %   pair, and one row for every (R,THETA) pair. |a/1mUxQ&  
    % zfAHE {c  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 1xAZ0X#  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), aM/sD=}  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral G4DuqN~2m  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ^,F8 ha  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized T <J%|d .'  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. (NPDgR/  
    % pI*/ - !I  
    %   The Zernike functions are an orthogonal basis on the unit circle. @w:6m&KL9  
    %   They are used in disciplines such as astronomy, optics, and 0NKo)HT  
    %   optometry to describe functions on a circular domain. g_{hB5N](7  
    % U)mg]o-VE  
    %   The following table lists the first 15 Zernike functions. cEzWIS?pp\  
    % =pHWqGOD  
    %       n    m    Zernike function           Normalization 2Hltgt,  
    %       -------------------------------------------------- v}w=I}<x  
    %       0    0    1                                 1 {p#[.E8  
    %       1    1    r * cos(theta)                    2 } ti+tM*  
    %       1   -1    r * sin(theta)                    2 M`{x*qR  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) zSs5F_  
    %       2    0    (2*r^2 - 1)                    sqrt(3) ODE9@]a  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) k8]=5C?k  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) ~xz3- a/  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) eq>E<X#<  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) E*rnk4Y  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) %*4Gx +b  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) %) A-zzj  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) /y2upu*!  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) G}.t!"  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) p_z_d6?  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) Gp6|0:2,L~  
    %       -------------------------------------------------- =l%"Om*A  
    % GUUVE@Z  
    %   Example 1: >C|/%$kk:f  
    % )dFTH?Mpo  
    %       % Display the Zernike function Z(n=5,m=1) QC+oSb!!?  
    %       x = -1:0.01:1; |UbwPL_L  
    %       [X,Y] = meshgrid(x,x); 3)SO-Bz\  
    %       [theta,r] = cart2pol(X,Y); ,]ALyWGuX  
    %       idx = r<=1; gm;6v30e  
    %       z = nan(size(X)); B5%N@g$`j  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); DFvLCGkDk  
    %       figure WgBV,{ C  
    %       pcolor(x,x,z), shading interp e [D'0L  
    %       axis square, colorbar O/;$0`~hY  
    %       title('Zernike function Z_5^1(r,\theta)') MguH)r` uT  
    % 9p,<<5{  
    %   Example 2: DkO>?n:-C  
    % 0>jo+b\D$  
    %       % Display the first 10 Zernike functions cB5|% @$I  
    %       x = -1:0.01:1; \qPgQsy4  
    %       [X,Y] = meshgrid(x,x); (+g!~MP  
    %       [theta,r] = cart2pol(X,Y); 7.O1 ~-  
    %       idx = r<=1; ,$ICv+7]  
    %       z = nan(size(X)); 5x/q\p-{/  
    %       n = [0  1  1  2  2  2  3  3  3  3]; @C),-TM  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; n1Ag o3NM  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; #^IEQZgH  
    %       y = zernfun(n,m,r(idx),theta(idx)); /?b<}am  
    %       figure('Units','normalized') ^:0NKq\  
    %       for k = 1:10 7 R1;'/;  
    %           z(idx) = y(:,k); , O=@I  
    %           subplot(4,7,Nplot(k)) |qra.\  
    %           pcolor(x,x,z), shading interp M5OH-'  
    %           set(gca,'XTick',[],'YTick',[]) m .2)P~a  
    %           axis square w}Q|*!?_  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) F8 4LMk?U  
    %       end m9^ ? p  
    % Zxw>|eKI>D  
    %   See also ZERNPOL, ZERNFUN2. h#bpog  
    IQK__)  
    %   Paul Fricker 11/13/2006 -CW$p=y}  
    p-U'5<n  
    7Kx3G{5ja  
    % Check and prepare the inputs: >M7e'}0 ;  
    % ----------------------------- Hl&]r'bK  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) LyA=(h6  
        error('zernfun:NMvectors','N and M must be vectors.') 5gq3 >qo  
    end ?9q{b\=l  
    qIQvix$8  
    if length(n)~=length(m) o{\@7'G  
        error('zernfun:NMlength','N and M must be the same length.') %^RlE@l9  
    end 1 sCF -r  
    UP:+1Sp9  
    n = n(:); }#@P+T:b  
    m = m(:); Jrlc%,pZ  
    if any(mod(n-m,2)) 2S^xqvh  
        error('zernfun:NMmultiplesof2', ... n }lav  
              'All N and M must differ by multiples of 2 (including 0).') s#sr1[9}G  
    end 1N< )lZl)  
    7I4G:-V:^  
    if any(m>n) {: EQ  
        error('zernfun:MlessthanN', ... fw^mjD  
              'Each M must be less than or equal to its corresponding N.') _-g:T&#  
    end `xbk)oW#  
    Ki-CJ y  
    if any( r>1 | r<0 ) }vO^%Gd  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') ,<zZKR_  
    end r2QC$V:0  
    "z^Ysvw&~  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) d; @Kz^  
        error('zernfun:RTHvector','R and THETA must be vectors.') ;D]TPBE  
    end :i*JlKHJ d  
    n[WXIE<  
    r = r(:); #v-)Ie\F?  
    theta = theta(:); ,%d n)gt7  
    length_r = length(r); +u _mT$|T  
    if length_r~=length(theta) _*Vq1D]C  
        error('zernfun:RTHlength', ... Z<y +D-/  
              'The number of R- and THETA-values must be equal.') =fBJQK2sk  
    end C%#C|X193  
    fx.FHhVu  
    % Check normalization: ' 7>}I{Lq  
    % -------------------- LnZz=  
    if nargin==5 && ischar(nflag) D]b5*_CT  
        isnorm = strcmpi(nflag,'norm'); r3ZY` zf  
        if ~isnorm Q}]:lmqH  
            error('zernfun:normalization','Unrecognized normalization flag.') O\OG~`HBN  
        end 2ok>z$Y  
    else {b/60xl?  
        isnorm = false; @]*z!>1  
    end aqs']  
    @R}L 4  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% z!Jce}mx  
    % Compute the Zernike Polynomials OAw/  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% e5fJN)+a  
    S%&l(=0X  
    % Determine the required powers of r: :'GTCo$3  
    % ----------------------------------- |c8p{)  
    m_abs = abs(m); 3{CGYd]_u  
    rpowers = []; wrsETB c  
    for j = 1:length(n) 1[3"|  
        rpowers = [rpowers m_abs(j):2:n(j)]; WF-imI:EK  
    end ,O a)  
    rpowers = unique(rpowers); pSq\3Hp]Q  
    @zfeCxVOA  
    % Pre-compute the values of r raised to the required powers,  Mw'd<{  
    % and compile them in a matrix: )IZ$R*Y{  
    % ----------------------------- O";r\Z  
    if rpowers(1)==0 "cJ5Fd:*  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); shn`>=0.&  
        rpowern = cat(2,rpowern{:}); L/nz95  
        rpowern = [ones(length_r,1) rpowern]; lt0(Kf g  
    else m/Yi;>I(  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); D>*%zz|  
        rpowern = cat(2,rpowern{:}); , {z$M  
    end >47,Hq:2  
    NX)7g}S  
    % Compute the values of the polynomials: E?Q=#+}U  
    % -------------------------------------- NqqLRgMOR'  
    y = zeros(length_r,length(n)); 9lTA/-  
    for j = 1:length(n) Bfw>2  
        s = 0:(n(j)-m_abs(j))/2; jIdhmd* $z  
        pows = n(j):-2:m_abs(j); HTx7._b  
        for k = length(s):-1:1 j?z(fs-  
            p = (1-2*mod(s(k),2))* ... !JYDg  
                       prod(2:(n(j)-s(k)))/              ... 9@D,ZSi  
                       prod(2:s(k))/                     ... ?Cu#(  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... sMO3eNLn  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); PVEEKKJP]J  
            idx = (pows(k)==rpowers); O gHWmb  
            y(:,j) = y(:,j) + p*rpowern(:,idx); yMz@-B  
        end ~q|^z[7  
         8CEy#%7]}  
        if isnorm +oQ@E<)H  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 3'&]v6|  
        end uF(- h~  
    end yDd&*;9%Qg  
    % END: Compute the Zernike Polynomials O~aS&g/sf  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% QG9 2^  
    $/wr?  
    % Compute the Zernike functions: dwx1 EdJ{  
    % ------------------------------ tO~H/0  
    idx_pos = m>0; P$4?-AZ  
    idx_neg = m<0; >656if O  
    SZwfYY!ft0  
    z = y; K{|;'N-1  
    if any(idx_pos) xOu cZ+  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); >) S a#w;  
    end RPp_L>&~<  
    if any(idx_neg) Y}f%/vus  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ]m}>/2oSs  
    end ^jCkM29eu  
    ]i$CE|~  
    % 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