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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 L >Ez-  
    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! _czbUl  
     
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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)  *Fa )\.XX  
    QvG56:M3  
    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。 S3ab0JM  
    NCowt|#t  
    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) ~_'0]P\  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. -@rxiC:Q  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of ?3`q+[:  
    %   order N and frequency M, evaluated at R.  N is a vector of sa_R$ /H  
    %   positive integers (including 0), and M is a vector with the CV s8s  
    %   same number of elements as N.  Each element k of M must be a fs&,w  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) C1V# ?03eI  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is 'nMApPl  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix O~.U:45t  
    %   with one column for every (N,M) pair, and one row for every U);OR  
    %   element in R. {T&v2u#S  
    % 6MuWlCKF8  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- 3=Z<wD s  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is (Up'$J}  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to =K :(&6f<t  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 mV<i JZh  
    %   for all [n,m]. DMB"Y,  
    % QjLji +L  
    %   The radial Zernike polynomials are the radial portion of the !(Q l)C  
    %   Zernike functions, which are an orthogonal basis on the unit \yM-O-{  
    %   circle.  The series representation of the radial Zernike v51EXf  
    %   polynomials is -&imjy<  
    % opdu=i=E  
    %          (n-m)/2 aM), M]m[  
    %            __ Yb}w;F8(  
    %    m      \       s                                          n-2s S."7+g7Ar  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r On4w/L9L5  
    %    n      s=0 N<(HPE};  
    % Id %_{),HX  
    %   The following table shows the first 12 polynomials. qbXz7s*{  
    % u yFn}y62  
    %       n    m    Zernike polynomial    Normalization Te;gVG*  
    %       --------------------------------------------- z5 Bi=~=#  
    %       0    0    1                        sqrt(2) }w@gj"\H  
    %       1    1    r                           2 rR]-RX(  
    %       2    0    2*r^2 - 1                sqrt(6) k^^:;OR  
    %       2    2    r^2                      sqrt(6) 6yI}1g  
    %       3    1    3*r^3 - 2*r              sqrt(8) X/Y#U\  
    %       3    3    r^3                      sqrt(8) 71O3O7  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) }kE87x'  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) =2,0Wo]$  
    %       4    4    r^4                      sqrt(10) j5\$[-';  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) kg3ppt  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) 0O~p7D  
    %       5    5    r^5                      sqrt(12) )Pli})   
    %       --------------------------------------------- &3Tx@XhO  
    % 8%[HYgd5)  
    %   Example: _UkmYZ/  
    % W/r^ugDV  
    %       % Display three example Zernike radial polynomials (S oo<.9~  
    %       r = 0:0.01:1; b{RqwV5P  
    %       n = [3 2 5]; !%xP}{(7  
    %       m = [1 2 1]; m"Qq{p|'  
    %       z = zernpol(n,m,r); &x0C4Kh  
    %       figure zE`R,:VI  
    %       plot(r,z) 8Mu;U3cIW  
    %       grid on YsP/p-  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') B!bsTvX  
    % /s c.C  
    %   See also ZERNFUN, ZERNFUN2. B,_`btJh  
    qX$u4I!,  
    % A note on the algorithm. Ax!Gu$K2o  
    % ------------------------ <tbZj=*O/o  
    % The radial Zernike polynomials are computed using the series kX[fy7rVt  
    % representation shown in the Help section above. For many special ~O: U|&  
    % functions, direct evaluation using the series representation can '# z]M  
    % produce poor numerical results (floating point errors), because ]` ]g@v  
    % the summation often involves computing small differences between SMoz:J*Q(  
    % large successive terms in the series. (In such cases, the functions D|_V<'  
    % are often evaluated using alternative methods such as recurrence NP/>H9Q2%  
    % relations: see the Legendre functions, for example). For the Zernike %6ub3PLw8  
    % polynomials, however, this problem does not arise, because the gLQ #4H  
    % polynomials are evaluated over the finite domain r = (0,1), and 3]U]?h  
    % because the coefficients for a given polynomial are generally all +y&d;0!  
    % of similar magnitude. 8~ #M{}  
    % @(:v_l  
    % ZERNPOL has been written using a vectorized implementation: multiple #U=;T]!'$  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] j7 d:v7+_  
    % values can be passed as inputs) for a vector of points R.  To achieve 59*M"1['Q  
    % this vectorization most efficiently, the algorithm in ZERNPOL 0xpx(T[  
    % involves pre-determining all the powers p of R that are required to M3pjXc<O  
    % compute the outputs, and then compiling the {R^p} into a single ^bUxLa[.  
    % matrix.  This avoids any redundant computation of the R^p, and '{ f=hE_/  
    % minimizes the sizes of certain intermediate variables. Y ?'tUV  
    % 8LJ{i%  
    %   Paul Fricker 11/13/2006 aMK~1]Cx  
    l#bAl/c`  
    IfV  3fJ7  
    % Check and prepare the inputs: b |7ja_  
    % ----------------------------- lIf(6nm@  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ?4[H]BK  
        error('zernpol:NMvectors','N and M must be vectors.') 4v dNMV~  
    end dDtFx2(R  
    R1ktj  
    if length(n)~=length(m) (~s|=Hxq|-  
        error('zernpol:NMlength','N and M must be the same length.') $h28(K%  
    end 5j^NV&/_  
    !gP0ndRJ=  
    n = n(:); Zb''mf\  
    m = m(:); z`$J_CjY  
    length_n = length(n); ;(6P6@+o  
    'C?NJ~MN  
    if any(mod(n-m,2)) XU-m"_t  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') ml u 3K  
    end SH8/0g?  
    fgF;&(b  
    if any(m<0) 27 GhE  
        error('zernpol:Mpositive','All M must be positive.') I@ \#up}  
    end Jx|I6 y  
    RuAlB*  
    if any(m>n) .ve *Vp  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') zAScRg$:?  
    end l@~LV}BI  
    \#dl6:"  
    if any( r>1 | r<0 ) =AOWeLk*G  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') <I=$ry6 8  
    end I&]G   
    GAEO$e:  
    if ~any(size(r)==1) MZv\ C  
        error('zernpol:Rvector','R must be a vector.') S~F`  
    end p!W[X%`)  
    )\ 0F7Z  
    r = r(:); 9dKul,c  
    length_r = length(r); 8SmjZpQ?  
    >HTbegi  
    if nargin==4 ?IYY'fS"  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); B 0)]s<<  
        if ~isnorm g]$ 4~"|.  
            error('zernpol:normalization','Unrecognized normalization flag.') |)U|:F/{@  
        end '$m7ft}  
    else LSd*| 3E}n  
        isnorm = false; p1O6+hRio  
    end ?S#\K^  
    ]=&L_(34  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% H\G{3.T.9  
    % Compute the Zernike Polynomials 83iCL;GS=  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *l>0t]5YH  
    Z(Q2Ue;}&  
    % Determine the required powers of r: JW+*d`8Z[  
    % ----------------------------------- J|QiH<  
    rpowers = []; <94G  
    for j = 1:length(n) uJow7-FD  
        rpowers = [rpowers m(j):2:n(j)]; U;^[$Aq  
    end f7<pEGb  
    rpowers = unique(rpowers); "{BqtU*.  
    Ax<\jW<  
    % Pre-compute the values of r raised to the required powers, z,Lzgh  
    % and compile them in a matrix: N{joXHCu  
    % ----------------------------- 'K@0Wp  
    if rpowers(1)==0 [*Ju3  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); FJS'G^  
        rpowern = cat(2,rpowern{:}); XGs^rIf  
        rpowern = [ones(length_r,1) rpowern]; KSqTY>%fnv  
    else UpUp8%fCU  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); t#b0H)  
        rpowern = cat(2,rpowern{:}); <,8l *1C  
    end >Hwc,j q  
    \8b6\qF/\  
    % Compute the values of the polynomials: lAASV{s{  
    % -------------------------------------- 'jaoO9KY K  
    z = zeros(length_r,length_n); 0Xl%uF+w  
    for j = 1:length_n 'Z8aPHD  
        s = 0:(n(j)-m(j))/2; IF_DZ   
        pows = n(j):-2:m(j); :#X[%"g.  
        for k = length(s):-1:1 lF4u{B9DM  
            p = (1-2*mod(s(k),2))* ... ;!u;!F!i  
                       prod(2:(n(j)-s(k)))/          ... I4e+$bU3  
                       prod(2:s(k))/                 ... ^PqF<d6  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... l?E|R Kp  
                       prod(2:((n(j)+m(j))/2-s(k))); hKe30#:v  
            idx = (pows(k)==rpowers); j F5Blc  
            z(:,j) = z(:,j) + p*rpowern(:,idx); xAdq+$><  
        end &=q! Wdw~  
         v%91k  
        if isnorm }vh Za p^  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); q~Jq/E"f  
        end Px;Cg 6  
    end l[Z)@bC1   
    v 1.*IV5Y  
    % 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) AXW!]=?X  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. *|RS*ABte  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated Sp?NfJ\Ie  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive W|R-J  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, wGLF%;rRe4  
    %   and THETA is a vector of angles.  R and THETA must have the same N6/T#UVns  
    %   length.  The output Z is a matrix with one column for every P-value, ltA/  
    %   and one row for every (R,THETA) pair. tYe:z:7l?<  
    % U}AX0*S  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike ?]!vRmZ;  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) t_c?Wp~tH  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) 49h0^;xlo:  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 `E>vG-9  
    %   for all p. <^><3U`  
    % .[j%sGdKl  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 bhg}-dto  
    %   Zernike functions (order N<=7).  In some disciplines it is =>LZm+P  
    %   traditional to label the first 36 functions using a single mode |4>:M\h  
    %   number P instead of separate numbers for the order N and azimuthal 8T5k-HwE  
    %   frequency M. e!0OW7 kV  
    % w~@[ r4W  
    %   Example: `gguip-C  
    % G$=-,6kZO  
    %       % Display the first 16 Zernike functions i0Pn Z J  
    %       x = -1:0.01:1; Mg? L-C  
    %       [X,Y] = meshgrid(x,x); aiux^V  
    %       [theta,r] = cart2pol(X,Y); N}Q FGX  
    %       idx = r<=1; O|z%DkH[  
    %       p = 0:15; x)viY5vjH  
    %       z = nan(size(X)); =ApY9`  
    %       y = zernfun2(p,r(idx),theta(idx)); `,#!C`E 9  
    %       figure('Units','normalized') +{-]P\oc  
    %       for k = 1:length(p) 8wFn}lw&  
    %           z(idx) = y(:,k); dgc&[  
    %           subplot(4,4,k) _z"o1`{w  
    %           pcolor(x,x,z), shading interp -!dQ)UEP  
    %           set(gca,'XTick',[],'YTick',[]) +R\~3uj[7  
    %           axis square uMiyq<  
    %           title(['Z_{' num2str(p(k)) '}']) BKb<2  
    %       end eyB_l.U7  
    % nNR:cG fG  
    %   See also ZERNPOL, ZERNFUN. )f*Iomp]@  
    dY'Y5Th~  
    %   Paul Fricker 11/13/2006 WU\m^!`w=F  
    #7W.s!#}Dd  
    - 9&g[  
    % Check and prepare the inputs: pVG>A&4  
    % ----------------------------- p24.bLr  
    if min(size(p))~=1 H1T~u{8j}  
        error('zernfun2:Pvector','Input P must be vector.') Pj!%ym3A  
    end hyu}}0:  
    n;OHH{E{  
    if any(p)>35 ?5~!i9pY  
        error('zernfun2:P36', ... v5 Y)al@  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... _rjBc ;a  
               '(P = 0 to 35).']) 'Y)/~\FI  
    end e_.~n<=  
    x[Q&k[xV  
    % Get the order and frequency corresonding to the function number: SIv[9G6  
    % ---------------------------------------------------------------- kI+b <$:D  
    p = p(:); V9{B}5KC  
    n = ceil((-3+sqrt(9+8*p))/2); sU%" azc  
    m = 2*p - n.*(n+2); AM/lbMr  
    \+]O*Bm&`8  
    % Pass the inputs to the function ZERNFUN: -\,VGudM}  
    % ---------------------------------------- 065A?KyD  
    switch nargin 9 z*(8d  
        case 3 <^sAY P|  
            z = zernfun(n,m,r,theta); B;c=eMw  
        case 4 X[{\ 3Av  
            z = zernfun(n,m,r,theta,nflag); Pz {Ig  
        otherwise rC rr"O#j  
            error('zernfun2:nargin','Incorrect number of inputs.') %zQ2:iT5@=  
    end %kW3hQ<$  
    Y_lCcu#OA  
    % 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 _<Vg[ -:1  
    function z = zernfun(n,m,r,theta,nflag) %\_h7:  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. :z124Zf  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N U%Ol^xl  
    %   and angular frequency M, evaluated at positions (R,THETA) on the lmp R>@o"  
    %   unit circle.  N is a vector of positive integers (including 0), and qIk )'!Vk  
    %   M is a vector with the same number of elements as N.  Each element GiFf0c 9  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) h%|9]5(=  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, (ai72#nFtb  
    %   and THETA is a vector of angles.  R and THETA must have the same cnYYs d{  
    %   length.  The output Z is a matrix with one column for every (N,M) E =  ^-Z  
    %   pair, and one row for every (R,THETA) pair. ; ?j~8  
    % B8>FCF&}E  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike +E `063  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), YFAnlqC  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral 3XBp6`  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, Wd1 IX^7C%  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized ?({PcF/  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. f`bIQ9R  
    % LsUFz_  
    %   The Zernike functions are an orthogonal basis on the unit circle. 2 /UI>@By  
    %   They are used in disciplines such as astronomy, optics, and w7Pe  
    %   optometry to describe functions on a circular domain. Hv+:fr"  
    % ^>t-v  
    %   The following table lists the first 15 Zernike functions. v3 !byN^  
    % }v,W-gA  
    %       n    m    Zernike function           Normalization 5Bzuj`  
    %       -------------------------------------------------- bmSpbX\  
    %       0    0    1                                 1 YDdLDE  
    %       1    1    r * cos(theta)                    2 ` 3vN R"  
    %       1   -1    r * sin(theta)                    2 *%z<P~}  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) J>/Ci\OB  
    %       2    0    (2*r^2 - 1)                    sqrt(3) cRjL3  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) )m oo?Q  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) +q 4W0  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) {lTR/  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) #r-j.f}yx  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) @m }rQT  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) ysQEJm^|-u  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10)   zd.1  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) wV]sGHuF}  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 2OA8 R}  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) 'JJ1#kKa  
    %       -------------------------------------------------- %kaTQ"PB  
    % tOu90gu  
    %   Example 1: k QB 1=c  
    % *#3voJjV(  
    %       % Display the Zernike function Z(n=5,m=1) qT&S  
    %       x = -1:0.01:1; -zkW\O[  
    %       [X,Y] = meshgrid(x,x); zDKLo 3:  
    %       [theta,r] = cart2pol(X,Y); O1l4gduN|i  
    %       idx = r<=1; ,dGFX]P  
    %       z = nan(size(X)); l;"ub^AH  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); W ??;4  
    %       figure }A)^XZ/  
    %       pcolor(x,x,z), shading interp }7f 1(#{7  
    %       axis square, colorbar v3iDh8.__  
    %       title('Zernike function Z_5^1(r,\theta)') ,APGPE}I[  
    % z{7,.S u  
    %   Example 2: 7"h=MB_  
    % UEx(~>  
    %       % Display the first 10 Zernike functions >' BU*  
    %       x = -1:0.01:1; i2`.#YJ&v  
    %       [X,Y] = meshgrid(x,x); 6i*p +S?U"  
    %       [theta,r] = cart2pol(X,Y); !nZI? z;  
    %       idx = r<=1; /zDSlj<c  
    %       z = nan(size(X)); N9fUlXhR  
    %       n = [0  1  1  2  2  2  3  3  3  3]; vV\/pu8  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; N6-2*ES  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; u|:UFz^p  
    %       y = zernfun(n,m,r(idx),theta(idx)); VO\S>kw  
    %       figure('Units','normalized') SF78 s:_!_  
    %       for k = 1:10 #8WR{  
    %           z(idx) = y(:,k); A3<P li  
    %           subplot(4,7,Nplot(k)) * wQZ '  
    %           pcolor(x,x,z), shading interp .q~,.yI&j  
    %           set(gca,'XTick',[],'YTick',[]) Yg]FF`{p=  
    %           axis square 'T #<OR  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) bUZ&}(/  
    %       end *$*nY [/5  
    % &B{Jxc`VA  
    %   See also ZERNPOL, ZERNFUN2. sf|_2sI  
    &~D.")Dz  
    %   Paul Fricker 11/13/2006 h}c6+@w&-  
    10QNV=yK7s  
    T`(;;%  
    % Check and prepare the inputs: yF [@W<  
    % ----------------------------- &Pn%zfmMN  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ,<Do ^HB/  
        error('zernfun:NMvectors','N and M must be vectors.') U:T5o]P<  
    end Z_hBd['!  
    fmT3Afl5c  
    if length(n)~=length(m) <_FF~lj  
        error('zernfun:NMlength','N and M must be the same length.') k(w9vt0?  
    end cl9;2D"Zm!  
    AyI}LQm]u  
    n = n(:); !:!@dC%8_  
    m = m(:); BGk<NEzH  
    if any(mod(n-m,2)) &*?!*+!,i  
        error('zernfun:NMmultiplesof2', ... B'[3kJ'  
              'All N and M must differ by multiples of 2 (including 0).') )H=[NB6J8  
    end B@~eBU,$  
    Y/Gswcz  
    if any(m>n) /Va&k4  
        error('zernfun:MlessthanN', ... RQ$o'U9A  
              'Each M must be less than or equal to its corresponding N.') dwsy(g7  
    end +{l3#Y  
    "}y3@ M^  
    if any( r>1 | r<0 ) /=O+/)l`  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') Dv\:b*  
    end P\G C8KV]  
    &VBD2_T  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) D9pxe qf+=  
        error('zernfun:RTHvector','R and THETA must be vectors.') 9zM4D  
    end V'BZ=.=  
    @"#gO:|[i0  
    r = r(:); SBB bniK-  
    theta = theta(:); Fw8X$SE"  
    length_r = length(r); ef1N#z%gt  
    if length_r~=length(theta) +'6ea+$  
        error('zernfun:RTHlength', ... :_b =Km<  
              'The number of R- and THETA-values must be equal.') L"zgBB?K6  
    end D;;o  
    oXZ@*   
    % Check normalization: '-1jWw:8  
    % -------------------- ,`B>}  
    if nargin==5 && ischar(nflag) WFc[F`b  
        isnorm = strcmpi(nflag,'norm'); H]n0JG9K  
        if ~isnorm &>^Ympr  
            error('zernfun:normalization','Unrecognized normalization flag.') =dw*B  
        end ,-NLUS "w  
    else RSVN(-wIi)  
        isnorm = false; _xZb;PbFE  
    end sN \}Q#:8  
    W*WH .1&  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %:8q7PN|  
    % Compute the Zernike Polynomials +^3L~?  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0:(dl@I)@  
    ,EJ [I^  
    % Determine the required powers of r: wQ/@+$>  
    % ----------------------------------- d'@H@  
    m_abs = abs(m); [T?6~^m=  
    rpowers = []; )-Sl/ G  
    for j = 1:length(n) EO!cv,[a  
        rpowers = [rpowers m_abs(j):2:n(j)]; FYE9&{]h  
    end b}{9 :n/SC  
    rpowers = unique(rpowers); sT T455h)  
    n[p9$W`  
    % Pre-compute the values of r raised to the required powers, T!eh?^E  
    % and compile them in a matrix: 0$dNrq  
    % ----------------------------- P`Wf'C^h  
    if rpowers(1)==0 L\'qAfRZ  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); -<^Q2]PE;  
        rpowern = cat(2,rpowern{:}); (DaP~*c3cC  
        rpowern = [ones(length_r,1) rpowern]; FXwK9 %  
    else *gMP_I  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); \\pyu]z  
        rpowern = cat(2,rpowern{:}); u2Z^iY  
    end [tw<TV"\  
    ENF@6]  
    % Compute the values of the polynomials: 9%'HB\A  
    % -------------------------------------- f$*9J  
    y = zeros(length_r,length(n)); k |aOUW  
    for j = 1:length(n) 4!RI2?4V  
        s = 0:(n(j)-m_abs(j))/2; 8Nq Iz  
        pows = n(j):-2:m_abs(j); Am^O{`r41  
        for k = length(s):-1:1 H;8]GE2n  
            p = (1-2*mod(s(k),2))* ... JM4`k8mM  
                       prod(2:(n(j)-s(k)))/              ...  G6ES]  
                       prod(2:s(k))/                     ... cO?"  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... hp%Pg &  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); ~G^doj3|+  
            idx = (pows(k)==rpowers); ${:$jX[  
            y(:,j) = y(:,j) + p*rpowern(:,idx); >knR>96  
        end [ESs?v$  
         yX Q;LQ;  
        if isnorm _BA_lkN+D  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); ^i k|l=  
        end 5'[X&r %#  
    end 1s\hJATfz  
    % END: Compute the Zernike Polynomials L|'ME| '  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% a^1c _  
    <^Nj~+G'  
    % Compute the Zernike functions: a;6\T*iJ!  
    % ------------------------------ Ln -?/[E  
    idx_pos = m>0; HWAqJb [  
    idx_neg = m<0; 8WQ%rN={8  
    M!i5StGC  
    z = y; r6_a%A*  
    if any(idx_pos) FPFYH?;$  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 7VBw@Rh  
    end tB?S0;yXjd  
    if any(idx_neg) 6242qb  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); c324@o^V  
    end *F&&rsb  
    a<36`#N  
    % 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