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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 o'V%EQ  
    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! WE!vSZ3R  
     
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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)  T~8  .9g  
    <g2_6C\j  
    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。 w Fn[9_`*  
    &lc8G  
    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) $r.U  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. ReB7vpd  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of RNJ FSD.  
    %   order N and frequency M, evaluated at R.  N is a vector of 3 pWM~(#>-  
    %   positive integers (including 0), and M is a vector with the }U 5Y=RYo  
    %   same number of elements as N.  Each element k of M must be a 5a`%)K  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) dz9Y}\2tf  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is Qc-(*}  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix [uuj?Rbd  
    %   with one column for every (N,M) pair, and one row for every z5cYyx r>  
    %   element in R. R'K/t|MC  
    % &V=7D#L  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- bi[7!VQf  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is <>&=n+i  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to ;<Qdy` T  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 D#rrW?-z  
    %   for all [n,m]. <lwuTow  
    % GlYly5F  
    %   The radial Zernike polynomials are the radial portion of the j+ ::y) $  
    %   Zernike functions, which are an orthogonal basis on the unit pK_?}~  
    %   circle.  The series representation of the radial Zernike _2Py\+$  
    %   polynomials is d.F)9h]XHO  
    % 'Z!G a.I  
    %          (n-m)/2 c$M%G)P  
    %            __ }6m?d!m  
    %    m      \       s                                          n-2s C 0C0GqN,  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r I*)VZW  
    %    n      s=0 >l1 r,/\\  
    % =]>%t]  
    %   The following table shows the first 12 polynomials. }p3b#fAr  
    % I<\ '%  
    %       n    m    Zernike polynomial    Normalization [I+9dSM1t  
    %       --------------------------------------------- Opg#*w%-  
    %       0    0    1                        sqrt(2) D,;\F,p  
    %       1    1    r                           2 dSK 0h(8  
    %       2    0    2*r^2 - 1                sqrt(6) f?UzD#50D  
    %       2    2    r^2                      sqrt(6) j~av\SCU*  
    %       3    1    3*r^3 - 2*r              sqrt(8) F@W*\3)  
    %       3    3    r^3                      sqrt(8) /p`&;/V|  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) JSVeU54T^<  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) /XpSe<3  
    %       4    4    r^4                      sqrt(10) 4MvC]_&  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) MgJ5B(c  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) ocA]M=3~k  
    %       5    5    r^5                      sqrt(12) Zr/r2  
    %       --------------------------------------------- C8b''9t.  
    % H#(<-)j0_  
    %   Example: n~r 9!m$<  
    % )iE"Tl  
    %       % Display three example Zernike radial polynomials  M[P^]J@  
    %       r = 0:0.01:1; !$xu(D.  
    %       n = [3 2 5]; dk5|@?pe  
    %       m = [1 2 1]; 1"E\C/c  
    %       z = zernpol(n,m,r); KFhG(   
    %       figure F8mC?fbK9  
    %       plot(r,z) 8C&x MA^  
    %       grid on KCqqJ}G  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') #uvJH8)D  
    % +<(a}6dt  
    %   See also ZERNFUN, ZERNFUN2. Uene=Q6>  
    /=T"=bP#/  
    % A note on the algorithm. 3:`XG2'  
    % ------------------------ TipHV;|e  
    % The radial Zernike polynomials are computed using the series (F5ttQPh  
    % representation shown in the Help section above. For many special sBW3{uK  
    % functions, direct evaluation using the series representation can -Zy)5NB-tZ  
    % produce poor numerical results (floating point errors), because JeQ[qQ  
    % the summation often involves computing small differences between 8 njuDl  
    % large successive terms in the series. (In such cases, the functions /tKGwX]y  
    % are often evaluated using alternative methods such as recurrence vA>W9OI   
    % relations: see the Legendre functions, for example). For the Zernike rw u3Nb  
    % polynomials, however, this problem does not arise, because the G}Z4g  
    % polynomials are evaluated over the finite domain r = (0,1), and _w u*M  
    % because the coefficients for a given polynomial are generally all 3wt  
    % of similar magnitude. U":"geU  
    % !#}>Hv^N  
    % ZERNPOL has been written using a vectorized implementation: multiple Q<=Y  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] uLW/f=7 L  
    % values can be passed as inputs) for a vector of points R.  To achieve Jt=>-Spj  
    % this vectorization most efficiently, the algorithm in ZERNPOL UxqWnHH.`  
    % involves pre-determining all the powers p of R that are required to TVkcDS  
    % compute the outputs, and then compiling the {R^p} into a single (V9h2g&8L  
    % matrix.  This avoids any redundant computation of the R^p, and rg)h 5G  
    % minimizes the sizes of certain intermediate variables. PrnrXl S  
    % /H&aMk}J@y  
    %   Paul Fricker 11/13/2006 #5{sglC"|F  
    #93}E Y  
    P;GprJ`l  
    % Check and prepare the inputs: rO^xz7K^  
    % ----------------------------- FdxsU DL  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) L8P 36]>  
        error('zernpol:NMvectors','N and M must be vectors.') $c =&0yt5  
    end $9H[3OZPVv  
    1uM/2sX  
    if length(n)~=length(m) _Ex?Xk  
        error('zernpol:NMlength','N and M must be the same length.') pGkef0p@  
    end # "r kuDO  
    VkXn8J  
    n = n(:); q$>_WF#||  
    m = m(:); =] KIkS3  
    length_n = length(n); !sK#zAR2  
    6\`DlUn'*  
    if any(mod(n-m,2)) !%62Phai  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') I#c(J  
    end W-Of[X{<  
    s`vSt* ]K  
    if any(m<0) U.Hdbmix  
        error('zernpol:Mpositive','All M must be positive.') .0 X$rX=  
    end <Kp+&(l,l  
    PP4d?+;V  
    if any(m>n) B1,?{Ur  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') *`.LA@bHU  
    end ;tr)=)q &  
    Oga1u  
    if any( r>1 | r<0 ) s01$fFJgO  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') m3"c (L`B  
    end 'FxYMSZS$  
    yk#rd~2Z0  
    if ~any(size(r)==1)  8bGD  
        error('zernpol:Rvector','R must be a vector.') L8D m9}  
    end S)Mby  
    zKMv7;s?  
    r = r(:); ?o>6S EGW  
    length_r = length(r); '\'7yN'  
    Cz[5Ug'V  
    if nargin==4 )<Ob  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); @7X\tV.Z  
        if ~isnorm 2%]t3\XW  
            error('zernpol:normalization','Unrecognized normalization flag.') 8J^d7uC  
        end W U0UG$o`  
    else w= B  
        isnorm = false; 8E-Ip>{>  
    end APOea  
    U,d2DAvt  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -s33m]a;  
    % Compute the Zernike Polynomials :SdIU36  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% oE5;|x3  
    PbQE{&D#  
    % Determine the required powers of r: 'Ye]eL,I\  
    % ----------------------------------- 8(uw0~GO  
    rpowers = []; (I!1sE!?1  
    for j = 1:length(n) 8z0Hx  
        rpowers = [rpowers m(j):2:n(j)]; +>^[W~[2  
    end Ltl]j*yei  
    rpowers = unique(rpowers); \CDAFu#  
    DbQBVy  
    % Pre-compute the values of r raised to the required powers, Sn0Xl3yr  
    % and compile them in a matrix: 'l8eH$  
    % ----------------------------- eoC<a"bJ>  
    if rpowers(1)==0 k=FcPF"  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); QdirE4W  
        rpowern = cat(2,rpowern{:}); (w}r7`n  
        rpowern = [ones(length_r,1) rpowern]; R'r|E_  
    else [ns&Y0Y`t  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); '&/(oJ ;O~  
        rpowern = cat(2,rpowern{:}); % hNn%Oy:E  
    end (+@faP   
    *:(1K%g  
    % Compute the values of the polynomials: {.cB>L  
    % -------------------------------------- [KD}U-(Wg  
    z = zeros(length_r,length_n); d{?)q  
    for j = 1:length_n 0:HC;J  
        s = 0:(n(j)-m(j))/2; ;g6 nHek  
        pows = n(j):-2:m(j); Hc>([?P%t  
        for k = length(s):-1:1 F61 +n!%8  
            p = (1-2*mod(s(k),2))* ... dPRtN@3  
                       prod(2:(n(j)-s(k)))/          ... YBR)s\*  
                       prod(2:s(k))/                 ... fO0- N>W'P  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... FU^Y{sbDg  
                       prod(2:((n(j)+m(j))/2-s(k))); #T Z!#,q  
            idx = (pows(k)==rpowers); =":@Foa  
            z(:,j) = z(:,j) + p*rpowern(:,idx); rffVfw  
        end ER/\ +Z#Z  
         T3 =)F%  
        if isnorm W&Y4Dq^  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); Wnb)*pPP  
        end {E3;r7  
    end p0"BO4({{  
    $&bU2]  
    % EOF zernpol
    离线niuhelen
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) JiG8jB7%}  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. Kv(Y }  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated D 86 K$IT  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive y[TaM9<  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, A)=X?x  
    %   and THETA is a vector of angles.  R and THETA must have the same <t% Ao,"  
    %   length.  The output Z is a matrix with one column for every P-value, a g|9$  
    %   and one row for every (R,THETA) pair. #Lu4OSM+  
    % e,PQ)1  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike b=6ZdN1  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) >?H_A  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) 3 ATN?V@  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 `PXoJl  
    %   for all p. @`#OC#  
    % DK2c]i^|=  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 <e@I1iL37y  
    %   Zernike functions (order N<=7).  In some disciplines it is 0b!fWS?,k0  
    %   traditional to label the first 36 functions using a single mode 1',+&2)oj  
    %   number P instead of separate numbers for the order N and azimuthal 37GHt9l  
    %   frequency M. cj,&&3sbV  
    % H J2O@e  
    %   Example: p?EEox  
    % _p3WE9T  
    %       % Display the first 16 Zernike functions  ."$=  
    %       x = -1:0.01:1; M$DwQ}Z  
    %       [X,Y] = meshgrid(x,x); I_{9eG1w?  
    %       [theta,r] = cart2pol(X,Y); 3?-V>-[G_  
    %       idx = r<=1; KyVe0>{_u  
    %       p = 0:15; IFHgD}kp%#  
    %       z = nan(size(X)); l%vhV&  
    %       y = zernfun2(p,r(idx),theta(idx)); n=<q3}1Jej  
    %       figure('Units','normalized') 9}7oKlyk  
    %       for k = 1:length(p) 6"#Tvj~-8  
    %           z(idx) = y(:,k); B)LXxdkOn  
    %           subplot(4,4,k) *GY,h$Ul  
    %           pcolor(x,x,z), shading interp y"{UN M|R  
    %           set(gca,'XTick',[],'YTick',[]) dW] Ej"W  
    %           axis square 9 u6 g  
    %           title(['Z_{' num2str(p(k)) '}']) -0[>}!l=G  
    %       end c;A ew!  
    % J{v6DYhi  
    %   See also ZERNPOL, ZERNFUN. 4.$hHFqS^5  
    ^$^Vd@t>a  
    %   Paul Fricker 11/13/2006 dvH67 x  
    vM$#m1L?  
    #EwRb<'Em  
    % Check and prepare the inputs: s?z=q%-p  
    % ----------------------------- pD)/- Dgdm  
    if min(size(p))~=1 OmQuAG ^\x  
        error('zernfun2:Pvector','Input P must be vector.') 7i%P&oB  
    end 8I|1P l  
    {;wK,dU  
    if any(p)>35 0Mzc1dG:  
        error('zernfun2:P36', ... "1\RdTw  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... A*?/F:E  
               '(P = 0 to 35).']) &vGEz*F  
    end KH CdO  
    vFkyfX(   
    % Get the order and frequency corresonding to the function number: %QlBFl0a  
    % ---------------------------------------------------------------- |R|U z`  
    p = p(:); Y=#mx3.  
    n = ceil((-3+sqrt(9+8*p))/2); LP-KD  
    m = 2*p - n.*(n+2); uc{Qhw!;:  
    m/"=5*pA  
    % Pass the inputs to the function ZERNFUN: [~&:`I1  
    % ---------------------------------------- pu m9x)y1  
    switch nargin 7{6cLYl  
        case 3 ~P.-3  
            z = zernfun(n,m,r,theta); pR^Y|NG!  
        case 4 jmwQc&  
            z = zernfun(n,m,r,theta,nflag); =iQ`F$M  
        otherwise Toa#>Z*+Rb  
            error('zernfun2:nargin','Incorrect number of inputs.') % /wP2O<  
    end V-o`L`(F`  
    <]SS gQ9/"  
    % EOF zernfun2
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    只看该作者 3楼 发表于: 2011-03-12
    回 2楼(phility) 的帖子
    非常感谢啊,我手上也有zernike多项式的拟合的源程序,也不知道对不对,不怎么会有 ():?FJ M  
    function z = zernfun(n,m,r,theta,nflag) 8f`b=r(a>  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. vd X~E97  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 1*Fvx-U'  
    %   and angular frequency M, evaluated at positions (R,THETA) on the 8=_| qy}l/  
    %   unit circle.  N is a vector of positive integers (including 0), and kl<B*:RqH  
    %   M is a vector with the same number of elements as N.  Each element UHDI9>G~,  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) ,h(+\^ ?,  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, U=<.P;+f9  
    %   and THETA is a vector of angles.  R and THETA must have the same uL{~(?U$  
    %   length.  The output Z is a matrix with one column for every (N,M) |$-d, ] V  
    %   pair, and one row for every (R,THETA) pair. IgnY* 2FT  
    % ^T J   
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike V5^b6$R@  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), &_x/Dzu!z  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral y5tAp  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, vrEaNT$J-  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized 'f<_SKd  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. k=/|?%  
    % )jZ=/ xG  
    %   The Zernike functions are an orthogonal basis on the unit circle. E3C[o! 5  
    %   They are used in disciplines such as astronomy, optics, and H_r'q9@<>  
    %   optometry to describe functions on a circular domain. .2-JV0  
    % &@Gu~)^(  
    %   The following table lists the first 15 Zernike functions. L5P}%1 _  
    % mZJzBYM)  
    %       n    m    Zernike function           Normalization FH5bC6  
    %       -------------------------------------------------- \36;csu  
    %       0    0    1                                 1 [";5s&)q  
    %       1    1    r * cos(theta)                    2 .F$AmVTN  
    %       1   -1    r * sin(theta)                    2 uTloj .  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) FwzA_ nn  
    %       2    0    (2*r^2 - 1)                    sqrt(3) &1C9K>  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) ZUI\0qh+  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) sWCm[HpG  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) Q]'!FmXf  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) JF\viMfR  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) 9<r}s  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) <R8Z[H:bV  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) PKs%-Uk  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) *M<=K.*\G  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) aw~EK0yU   
    %       4    4    r^4 * sin(4*theta)             sqrt(10) bHT@]`@@  
    %       -------------------------------------------------- .qPfi] ty  
    % ASU\O3%%  
    %   Example 1: y$Noo)Z  
    % I*R$*/)  
    %       % Display the Zernike function Z(n=5,m=1) Qg.:w  
    %       x = -1:0.01:1; PGhZ`nl  
    %       [X,Y] = meshgrid(x,x); e[dRHl  
    %       [theta,r] = cart2pol(X,Y); */e5lRO\  
    %       idx = r<=1; y5D?Bg|M  
    %       z = nan(size(X)); RUtS_Z&  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); J0! E@   
    %       figure M\6v}kUY  
    %       pcolor(x,x,z), shading interp i */U.'#  
    %       axis square, colorbar YYh_lAS>  
    %       title('Zernike function Z_5^1(r,\theta)') L2$L.@  
    % A:J{  
    %   Example 2: Y--8v#t  
    % bD-Em#>  
    %       % Display the first 10 Zernike functions ?0%TE\I8  
    %       x = -1:0.01:1; LkB!:+v |B  
    %       [X,Y] = meshgrid(x,x); }]?G"f t K  
    %       [theta,r] = cart2pol(X,Y); Y"%o\DS*  
    %       idx = r<=1; *?"{T;4u~O  
    %       z = nan(size(X)); e;[8 GE.   
    %       n = [0  1  1  2  2  2  3  3  3  3]; 3) 0~:  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; AAY UXY!  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; lhj2u]yU0S  
    %       y = zernfun(n,m,r(idx),theta(idx)); e !Okc*,  
    %       figure('Units','normalized') u.FDe2|[)  
    %       for k = 1:10 5/ju it  
    %           z(idx) = y(:,k); wO%:WL$5  
    %           subplot(4,7,Nplot(k)) p00AcUTq  
    %           pcolor(x,x,z), shading interp `{_PSzM  
    %           set(gca,'XTick',[],'YTick',[]) (W!$6+GT  
    %           axis square LS$82UB&  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ,?/<fxIY  
    %       end ybO,~TQ  
    % O3Mv"Py%  
    %   See also ZERNPOL, ZERNFUN2. w5jZI|  
    p2(_YN;s  
    %   Paul Fricker 11/13/2006 Af<>O$$6  
    n82Q.M-H  
    *)I1gR~  
    % Check and prepare the inputs: W2N7  
    % ----------------------------- .&xNJdsY  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) f|0QN#$  
        error('zernfun:NMvectors','N and M must be vectors.') #Q7$I.O]  
    end sdD[`#  
    ,+9r/}K]/  
    if length(n)~=length(m) >#|Yoc  
        error('zernfun:NMlength','N and M must be the same length.') #{,IY03  
    end $SR]7GZ  
    3>Snd9Q  
    n = n(:); @~3c;9LkY  
    m = m(:); %Ege^4PE  
    if any(mod(n-m,2)) |hoZ:  
        error('zernfun:NMmultiplesof2', ... :5J6rj;_  
              'All N and M must differ by multiples of 2 (including 0).') eov-"SJB  
    end ~\,6 C1M  
    ![^h<Om  
    if any(m>n) {Z.@-Tl_  
        error('zernfun:MlessthanN', ... Am4(WXVQ  
              'Each M must be less than or equal to its corresponding N.') +r_[Tj|Er  
    end &J:)*EjVl5  
    $uhDBmb  
    if any( r>1 | r<0 ) Bx4GFCdifC  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') A o$z )<d'  
    end G - WJlu  
    |vzWSm  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) <vDm(-i3  
        error('zernfun:RTHvector','R and THETA must be vectors.') NhX.yLb$   
    end q/79'>`|ai  
    caht4N{T  
    r = r(:); [hbp#I~*[  
    theta = theta(:); d?Cl04  
    length_r = length(r); Iq \oB  
    if length_r~=length(theta) ; bE6Y]"Rz  
        error('zernfun:RTHlength', ... wP?q5r5  
              'The number of R- and THETA-values must be equal.') "@$STptkc  
    end [{$0E=&0  
    n^#LB*q  
    % Check normalization: %WR"85  
    % -------------------- IoOnS)  
    if nargin==5 && ischar(nflag) /GGu` f  
        isnorm = strcmpi(nflag,'norm'); BwD1}1jp  
        if ~isnorm \-ws[  
            error('zernfun:normalization','Unrecognized normalization flag.') <t{AY^:r  
        end 5AU3s  
    else n4y6Ua9m{  
        isnorm = false; wkA!Jv%  
    end B)8Hj).@B  
    }* JMc+!9@  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% zAJUL  
    % Compute the Zernike Polynomials @8yFM%  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% hATy 3*4  
    4+,Z'J%\[7  
    % Determine the required powers of r: v*'\w#  
    % ----------------------------------- ,5*xE\9G  
    m_abs = abs(m); :exuTn  
    rpowers = []; x~tQYK   
    for j = 1:length(n) REBDr;tv  
        rpowers = [rpowers m_abs(j):2:n(j)]; j],.`Y  
    end +Q0-jS#d  
    rpowers = unique(rpowers); { ][7Np!y  
    d2yHfl]3  
    % Pre-compute the values of r raised to the required powers, >Fk `h=Wd  
    % and compile them in a matrix: vK`h;  
    % ----------------------------- "e<. n  
    if rpowers(1)==0 SJ^?D8  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); B?Sfcq-  
        rpowern = cat(2,rpowern{:}); 6*33k'=;F  
        rpowern = [ones(length_r,1) rpowern]; CT%m_lN  
    else ^|(4j_.(e  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ~ O=|v/]  
        rpowern = cat(2,rpowern{:}); bKZ#>%|:o  
    end fhx:EZ:~  
    =c^=Yvc7U  
    % Compute the values of the polynomials: kA=~ 8N  
    % -------------------------------------- E?U]w0g  
    y = zeros(length_r,length(n)); E9 q;>)}  
    for j = 1:length(n) 8lSn*;S,  
        s = 0:(n(j)-m_abs(j))/2; aZGDtzNG5h  
        pows = n(j):-2:m_abs(j); g_c)Ts(  
        for k = length(s):-1:1 \&)W#8V  
            p = (1-2*mod(s(k),2))* ... [ c[MQA0  
                       prod(2:(n(j)-s(k)))/              ... BG0M j2  
                       prod(2:s(k))/                     ... NVWeJ+w  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... .|`=mx  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); h^$}1[  
            idx = (pows(k)==rpowers); f,inQ2f}d  
            y(:,j) = y(:,j) + p*rpowern(:,idx); 4@iJ|l  
        end G2{M#H  
         AeCG2!8^0  
        if isnorm T&"dBoUq>G  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); e - ]c  
        end `R52{B#&/  
    end Mq lo:7 ^F  
    % END: Compute the Zernike Polynomials 5po' (r|U  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% :_,]?n  
    aX'g9E  
    % Compute the Zernike functions: zQ %z "tQ  
    % ------------------------------ ;=\5$J9  
    idx_pos = m>0; 'qF3,Rw  
    idx_neg = m<0; 7r[ %| :  
    tDHHQ  
    z = y; }>X\"  
    if any(idx_pos)  |iUfM3  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); [^}>AC*im  
    end Bx : So6:  
    if any(idx_neg) pkN:D+g S  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); u$=ogp =0  
    end Y!1^@;)^  
    <kXV1@>  
    % 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