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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 X MF? y  
    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! 2v4&'C  
     
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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)  MPsm)jqX  
    "fd'~e$S#  
    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。 + j6^g*  
    *AYjMCo  
    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) x?-kt.M  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. EN2/3~syO-  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of >U'gQS?\]  
    %   order N and frequency M, evaluated at R.  N is a vector of {FNq&)#`  
    %   positive integers (including 0), and M is a vector with the uze5u\  
    %   same number of elements as N.  Each element k of M must be a ;"D I)hd z  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) *Mr'/qp,  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is !5h@uar  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix `}&}2k  
    %   with one column for every (N,M) pair, and one row for every 1jQlwT(:  
    %   element in R. yM*< BV  
    % \dc*!Es  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- ^Dw18gqr=@  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is _8nT$!\\  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to +^@6{1  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 /kK:{  
    %   for all [n,m]. 3D"?|rd~  
    % g|V0[Hnq6  
    %   The radial Zernike polynomials are the radial portion of the .2:S0=xt<  
    %   Zernike functions, which are an orthogonal basis on the unit I=Xj;\b  
    %   circle.  The series representation of the radial Zernike |+(Hia,X  
    %   polynomials is >>HC|  
    % SB2Ij',  
    %          (n-m)/2 #`{L_n$c  
    %            __ bR;Wf5  
    %    m      \       s                                          n-2s CaqMLi%  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r qz/d6-0"  
    %    n      s=0 b&Go'C{p  
    % Y!L<& sl   
    %   The following table shows the first 12 polynomials. p*S;4+>#  
    % :yC|Q)  
    %       n    m    Zernike polynomial    Normalization 07tSXl5!  
    %       --------------------------------------------- 0}y-DCuQ  
    %       0    0    1                        sqrt(2) H g;;>  
    %       1    1    r                           2 ?e+$?8l[3  
    %       2    0    2*r^2 - 1                sqrt(6) /0I=?+QSo  
    %       2    2    r^2                      sqrt(6) /N82h`\n  
    %       3    1    3*r^3 - 2*r              sqrt(8) AT]Ty  
    %       3    3    r^3                      sqrt(8) iKN800^u  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) BY^5z<^.  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) GLIP;)h1  
    %       4    4    r^4                      sqrt(10) G@;I^_gN  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) o@g/,V $  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) Kw ^tvRt'*  
    %       5    5    r^5                      sqrt(12) 9,zM.g9Qv  
    %       --------------------------------------------- 9 ]W4o"  
    % oc3}L^aD  
    %   Example: 3teanU`  
    % z ''-AH,  
    %       % Display three example Zernike radial polynomials 5.e. BT  
    %       r = 0:0.01:1; mrz@Y0mgL  
    %       n = [3 2 5]; y?s8UEC  
    %       m = [1 2 1]; C2 ] x  
    %       z = zernpol(n,m,r); ,HM~Zs  
    %       figure $1$T2'C~+  
    %       plot(r,z) \9t6 #8  
    %       grid on 86,$ I+  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') FI$#x%A  
    % ,"W.A  
    %   See also ZERNFUN, ZERNFUN2. .}l&lj@#  
    ^  M4-O~  
    % A note on the algorithm. vAMr&[  
    % ------------------------ [5Dg%?x  
    % The radial Zernike polynomials are computed using the series AE"E($S`  
    % representation shown in the Help section above. For many special j,/t<@S>  
    % functions, direct evaluation using the series representation can 8fwM)DKS  
    % produce poor numerical results (floating point errors), because #Qp.O@e  
    % the summation often involves computing small differences between M:Aik&  
    % large successive terms in the series. (In such cases, the functions W=k%aB?p  
    % are often evaluated using alternative methods such as recurrence /Aq):T T  
    % relations: see the Legendre functions, for example). For the Zernike ?hQ,'M2  
    % polynomials, however, this problem does not arise, because the GxIw4m9  
    % polynomials are evaluated over the finite domain r = (0,1), and [d_sd  
    % because the coefficients for a given polynomial are generally all GI:$(<  
    % of similar magnitude. cOr@dUSL  
    % `b{.K,  
    % ZERNPOL has been written using a vectorized implementation: multiple uF=xo`=|  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] _~ipO1*  
    % values can be passed as inputs) for a vector of points R.  To achieve %g>{m2o  
    % this vectorization most efficiently, the algorithm in ZERNPOL %aszZP  
    % involves pre-determining all the powers p of R that are required to An%V>a-[  
    % compute the outputs, and then compiling the {R^p} into a single @Sl!p)  
    % matrix.  This avoids any redundant computation of the R^p, and =abth6#)  
    % minimizes the sizes of certain intermediate variables. P00pSRQHD  
    % 3[jk}2R';p  
    %   Paul Fricker 11/13/2006 D@ji1$K  
    ,T|%vqbmw  
    Y%V|M0 0`  
    % Check and prepare the inputs: HGgw<Os-k  
    % ----------------------------- m9Uoq[1  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Y{8L ~U:  
        error('zernpol:NMvectors','N and M must be vectors.') c[Mz#BWG  
    end (1vmtg.O  
    ZREAEGi{  
    if length(n)~=length(m) ^gdg0y!5~  
        error('zernpol:NMlength','N and M must be the same length.') X&<#3n  
    end afZPju"-  
    2ju1<t,8)  
    n = n(:); .F~EQ %  
    m = m(:); Lu~e^Ul   
    length_n = length(n); "Jp6EL%  
    Hf/2KYZ  
    if any(mod(n-m,2)) [\ JZpF  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') YJ5;a\QxN  
    end Z6cG<,DQ  
    rr[9sk`^H  
    if any(m<0) !HXdUAKu  
        error('zernpol:Mpositive','All M must be positive.') 7<=7RPWmD  
    end wBcDL/(>  
    (!';  
    if any(m>n) ?nFT51 t/4  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') pg~`NN  
    end N[}XLhbt  
    #oYX0wvl  
    if any( r>1 | r<0 ) VmTk4?V4  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') ubhem(p#  
    end OD"eB?  
    qR_"aQ7s2  
    if ~any(size(r)==1) !UUh7'W4u  
        error('zernpol:Rvector','R must be a vector.') l1o dkNf|  
    end U6=m4]~Z  
    $`'^&o;&f  
    r = r(:); 0EXAdRR  
    length_r = length(r); H[x9 7r  
    ?<  w +{  
    if nargin==4 U gB  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); {\t:{.F A  
        if ~isnorm 7$GP#V1r/  
            error('zernpol:normalization','Unrecognized normalization flag.') xuelo0h,  
        end a~ REFy  
    else 6x@-<{L  
        isnorm = false; ,XP9NHE  
    end CSU>nIE0  
    vS<;:3  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% hJ:Hv.{`)W  
    % Compute the Zernike Polynomials (oJ#`k:&n  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% i6m;2 UAa  
    ==(M vu`  
    % Determine the required powers of r: ;T52 aX  
    % ----------------------------------- l}AB):<Z  
    rpowers = []; xs &vgel>  
    for j = 1:length(n) n?,fF(  
        rpowers = [rpowers m(j):2:n(j)]; 9/s-|jD  
    end v2@M,xbxF:  
    rpowers = unique(rpowers); JmYi&  
    I )B2Z(<Q  
    % Pre-compute the values of r raised to the required powers, #8/Z)-G  
    % and compile them in a matrix: !#iP)"O  
    % ----------------------------- n6o}$]H  
    if rpowers(1)==0 )QZ?Bf  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ?qK:P  
        rpowern = cat(2,rpowern{:}); q>omCk%h  
        rpowern = [ones(length_r,1) rpowern]; y6j TT%  
    else 9J]LV'f7  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); h}`<pq  
        rpowern = cat(2,rpowern{:}); xD lC]loi7  
    end Nq~bO_-I  
    mI`dZ3h  
    % Compute the values of the polynomials: F37,u|  
    % -------------------------------------- xEiW]Eo  
    z = zeros(length_r,length_n); Bv=Z*"Fv  
    for j = 1:length_n AARhGx|L<  
        s = 0:(n(j)-m(j))/2; E>V8|Hz;  
        pows = n(j):-2:m(j); *smo{!0Gg  
        for k = length(s):-1:1 d7G'+B1  
            p = (1-2*mod(s(k),2))* ... \|&5eeE@  
                       prod(2:(n(j)-s(k)))/          ... Q'=!1^&  
                       prod(2:s(k))/                 ... 0^[$0]Mt[  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... :e9E#o  
                       prod(2:((n(j)+m(j))/2-s(k))); |n &6z  
            idx = (pows(k)==rpowers); ?)PcYrV  
            z(:,j) = z(:,j) + p*rpowern(:,idx); nEn2!)$  
        end Lq&xlW j  
         be@MQ}6>  
        if isnorm ):[[Ch_  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); 7PvuKAv?k  
        end FP\[7?ZLn  
    end r-V./M@L  
    QWP_8$Q  
    % 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) PQs9@]w[  
    %ZERNFUN2 Single-index Zernike functions on the unit circle.  )Ob{]  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated I'j? T.  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive w])Sz*J  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, nAYjSE  
    %   and THETA is a vector of angles.  R and THETA must have the same 8_ LDS  
    %   length.  The output Z is a matrix with one column for every P-value, >ylVES/V  
    %   and one row for every (R,THETA) pair. c*W$wr  
    % qjFgy)qV  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike f ;Dz(~ hw  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) 2,fB$5+  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) :`|,a (  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 aG ,uF  
    %   for all p. ])JJ`Z8Bk  
    % =osj}(  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 +(<f(]bG  
    %   Zernike functions (order N<=7).  In some disciplines it is -_}EQ9Q  
    %   traditional to label the first 36 functions using a single mode BOG )JaDW  
    %   number P instead of separate numbers for the order N and azimuthal K XGs'D  
    %   frequency M. ppYz~ {"r  
    % }0TY  
    %   Example: ~Mx fud  
    % A4^+p0@  
    %       % Display the first 16 Zernike functions Mc6y'w  
    %       x = -1:0.01:1; jL8zH  
    %       [X,Y] = meshgrid(x,x); 4j*}|@x  
    %       [theta,r] = cart2pol(X,Y); I5~DC  
    %       idx = r<=1; : ?V;  
    %       p = 0:15; 6IX!9I\sT  
    %       z = nan(size(X)); We ->d |=  
    %       y = zernfun2(p,r(idx),theta(idx)); Dn[1BWM/7  
    %       figure('Units','normalized') Dz{e@+>M  
    %       for k = 1:length(p) anvj{1  
    %           z(idx) = y(:,k); YJy*OS_&  
    %           subplot(4,4,k) A"'MRYT`  
    %           pcolor(x,x,z), shading interp i_MI!o  
    %           set(gca,'XTick',[],'YTick',[]) aI<~+]  
    %           axis square ,jn?s^X6Dj  
    %           title(['Z_{' num2str(p(k)) '}']) VtO+=mZV  
    %       end JVkawkeX  
    % 0D'Wr(U(  
    %   See also ZERNPOL, ZERNFUN. W)#`4a^xj7  
    --9mTqx  
    %   Paul Fricker 11/13/2006 $o/>wgQY-  
    VPHCPGrk  
    WdT|xf.Q&  
    % Check and prepare the inputs:  W 6~=?C  
    % ----------------------------- d}ZH Y[  
    if min(size(p))~=1 B4}XK =)  
        error('zernfun2:Pvector','Input P must be vector.') j#&  
    end Trrh`@R  
    0 OBkd  
    if any(p)>35 ,B2 -'O  
        error('zernfun2:P36', ... %gaKnT(|r  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... \p$0  
               '(P = 0 to 35).']) $c}0L0  
    end ]c)SVn$6  
    o >Lk`\  
    % Get the order and frequency corresonding to the function number: Xo`1#6xsE  
    % ---------------------------------------------------------------- ca =e_sg  
    p = p(:); OL rD4 e  
    n = ceil((-3+sqrt(9+8*p))/2); ju#6 3  
    m = 2*p - n.*(n+2); 4i,SiFKB  
    lQ/XJw  
    % Pass the inputs to the function ZERNFUN: Db=gS=Qm  
    % ---------------------------------------- jO 55<s94  
    switch nargin ]lUu%<-;  
        case 3 ))`Zv=y"  
            z = zernfun(n,m,r,theta); Nj0)/)<r+  
        case 4 MxRU6+a  
            z = zernfun(n,m,r,theta,nflag); vNC0M:p,  
        otherwise MbfzGYA2~  
            error('zernfun2:nargin','Incorrect number of inputs.') Wv"tAseu  
    end E: GJ$I  
    (5~C _Y  
    % 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 @@IA35'tc  
    function z = zernfun(n,m,r,theta,nflag) n#Roz5/U  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. R`2A-c  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N AxlFU~E4  
    %   and angular frequency M, evaluated at positions (R,THETA) on the M"^Vf{X^  
    %   unit circle.  N is a vector of positive integers (including 0), and N-`;\  
    %   M is a vector with the same number of elements as N.  Each element Eap/7U1Q  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) 6;cY!  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, n=? 0g;1!  
    %   and THETA is a vector of angles.  R and THETA must have the same A Vm{#^p[(  
    %   length.  The output Z is a matrix with one column for every (N,M) 0j(jJAE.  
    %   pair, and one row for every (R,THETA) pair. rxj@NwAno  
    % nKB&|!  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike _-]!;0E IV  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), T[-c|  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral *O> aqu  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, -fJ@R1]  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized 1?|6odc  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. O}_a3>1DY  
    % cmhN(==  
    %   The Zernike functions are an orthogonal basis on the unit circle. j yRSEk$  
    %   They are used in disciplines such as astronomy, optics, and *frJ^ Ws{  
    %   optometry to describe functions on a circular domain. bz0P49%  
    % `QdQ?9x{F  
    %   The following table lists the first 15 Zernike functions. M~Qj'VVL  
    % tRnW%F5  
    %       n    m    Zernike function           Normalization :KSor}t  
    %       -------------------------------------------------- u\R`IZ&O  
    %       0    0    1                                 1 GrR0RwnH)?  
    %       1    1    r * cos(theta)                    2 l(,;wAH  
    %       1   -1    r * sin(theta)                    2 pP* ~ =?  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) P%sO(_PuT  
    %       2    0    (2*r^2 - 1)                    sqrt(3) tIb21c q  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) VS|( "**  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) 8A^jD(|  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) ^mueFw}\  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) BwJ^_:(p~  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) c5E#QV0&v~  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) $i:||L^8p  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) +Y)#yGUn  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) /J.\p/%\  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) EeJqszmH  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) 5 n+ e  
    %       -------------------------------------------------- =+`j?1  
    % 7grt4k  
    %   Example 1: r1oku0o  
    % w,Zx5bBg%  
    %       % Display the Zernike function Z(n=5,m=1) cZr G:\A  
    %       x = -1:0.01:1; 7q!yCU  
    %       [X,Y] = meshgrid(x,x); a3UPbl3^  
    %       [theta,r] = cart2pol(X,Y); %gu$_S  
    %       idx = r<=1; sQ}%7BMK  
    %       z = nan(size(X)); ?#m<\]S<  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); 8.CKH4h  
    %       figure ?orhJS  
    %       pcolor(x,x,z), shading interp a,~D+s;^  
    %       axis square, colorbar }B"|z'u  
    %       title('Zernike function Z_5^1(r,\theta)') +z|UpI  
    % hA*Z'.[  
    %   Example 2: z0 2}&^Zzk  
    % 4e@&QOo`Cu  
    %       % Display the first 10 Zernike functions .vN%UNu  
    %       x = -1:0.01:1; 6!+X.+  
    %       [X,Y] = meshgrid(x,x); LgP>u?]n  
    %       [theta,r] = cart2pol(X,Y); `M?v!]o  
    %       idx = r<=1; }2ql?K  
    %       z = nan(size(X)); W""*hJ  
    %       n = [0  1  1  2  2  2  3  3  3  3]; {b'}:aMc  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; EK?@Z.q+  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; RQ^m6)BTo  
    %       y = zernfun(n,m,r(idx),theta(idx)); _k_>aG23  
    %       figure('Units','normalized') 4L=$K2R2r  
    %       for k = 1:10 @%OPy|=,{  
    %           z(idx) = y(:,k); jj!N39f   
    %           subplot(4,7,Nplot(k)) QSHJmk 6L  
    %           pcolor(x,x,z), shading interp 4<T*i{[  
    %           set(gca,'XTick',[],'YTick',[]) 9DOkQnnc  
    %           axis square Ak5[PBbW  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) >-5td=:Z  
    %       end 1mHwYT+  
    % |5=~(-I>@  
    %   See also ZERNPOL, ZERNFUN2. K`Bq(z?/  
    -RG8<bI,  
    %   Paul Fricker 11/13/2006 Z}8k[*.  
    @s%X  
    /!=U +X  
    % Check and prepare the inputs: M=5d95*-}  
    % ----------------------------- [)#u<lZ<~  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) D:wnO|:  
        error('zernfun:NMvectors','N and M must be vectors.') t_dcV%=  
    end WI1T?.Gc   
    U~uwm/h  
    if length(n)~=length(m) fav5e'[$  
        error('zernfun:NMlength','N and M must be the same length.') l`@0zw+  
    end 6exI_3A4jh  
    "jL1. 9%"  
    n = n(:); q&zny2])  
    m = m(:); C=N! z  
    if any(mod(n-m,2)) m`hGDp3  
        error('zernfun:NMmultiplesof2', ... o]Z _@VI  
              'All N and M must differ by multiples of 2 (including 0).') -xJX_6}A  
    end wgY6D!Y   
    TC qkm^xv  
    if any(m>n) 7:n?PN(p6a  
        error('zernfun:MlessthanN', ... In f9wq\  
              'Each M must be less than or equal to its corresponding N.') ,*/Pg 52?  
    end 7MY)\aH  
    t]s94 R q  
    if any( r>1 | r<0 ) i=oTg  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') \V]t!mZ-}l  
    end gaQ[3g  
    O\6vVM[  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) A-Mj|V  
        error('zernfun:RTHvector','R and THETA must be vectors.') B@ -|b  
    end ?4^} ;wDb2  
    N99[.mErU  
    r = r(:); p-.Ri^p   
    theta = theta(:); 4~!Eje!  
    length_r = length(r); 6\NvG,8  
    if length_r~=length(theta) "tqnx?pM  
        error('zernfun:RTHlength', ... 'X9AG6K1  
              'The number of R- and THETA-values must be equal.') Te# ]Cn|  
    end >-!r9"8@  
    Q4RpK(N  
    % Check normalization: d$pYo)8o({  
    % -------------------- `M&P[ .9Pz  
    if nargin==5 && ischar(nflag) 9I85EcT^4"  
        isnorm = strcmpi(nflag,'norm'); Us'Cs+5XcG  
        if ~isnorm # Mu<8`T-  
            error('zernfun:normalization','Unrecognized normalization flag.') Q|?'(J+  
        end ~p:?QB>1]  
    else <PX.l%  
        isnorm = false; $]C=qM28-  
    end Tr~sieL  
    j1/+\8Y  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% IroPx#s:i  
    % Compute the Zernike Polynomials )i;un.  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% V\0E=M*P  
    sm0fAL  
    % Determine the required powers of r: vv+km+  
    % ----------------------------------- g0PT8]8  
    m_abs = abs(m); }`9jH:q-Z  
    rpowers = []; n+2%tW  
    for j = 1:length(n) Lbcy:E*g  
        rpowers = [rpowers m_abs(j):2:n(j)]; %,0%NjK  
    end I7~|~<  
    rpowers = unique(rpowers); ?-f,8Z|h  
    oe9lF*$/  
    % Pre-compute the values of r raised to the required powers, !}_b|  
    % and compile them in a matrix: |jsb@  
    % ----------------------------- eIH$"f;L  
    if rpowers(1)==0 Fk{J@Y  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); sf$o(^P9\A  
        rpowern = cat(2,rpowern{:}); \8{\;L C  
        rpowern = [ones(length_r,1) rpowern]; zEj#arSE4  
    else {{\ce;hN  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); eNbpwne  
        rpowern = cat(2,rpowern{:}); +"dv7  
    end Jd_;@(Eg=  
    J? .F\`N)  
    % Compute the values of the polynomials: Ke!'gohv  
    % -------------------------------------- c+g@Z"es  
    y = zeros(length_r,length(n)); ##cnFQCB  
    for j = 1:length(n) (,B#t7ka  
        s = 0:(n(j)-m_abs(j))/2; 4 j X3lq|  
        pows = n(j):-2:m_abs(j); 2Q@Y^t   
        for k = length(s):-1:1 $5NKFJc  
            p = (1-2*mod(s(k),2))* ... gv|"OlB  
                       prod(2:(n(j)-s(k)))/              ... Xh F _]  
                       prod(2:s(k))/                     ... ! \sMR  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... u#@RM^738d  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); 5wv fF.v  
            idx = (pows(k)==rpowers); MLr-, "gs  
            y(:,j) = y(:,j) + p*rpowern(:,idx); -R b{^/  
        end U\zD,<I9  
         ]A^4}CK^<  
        if isnorm $,ikv?"L  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); po7>IQS]  
        end ((bTwx  
    end 6~xBi(m`  
    % END: Compute the Zernike Polynomials UG](go't  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% y t5H oy  
    .UQE{.?  
    % Compute the Zernike functions: 0^3+P%(o@  
    % ------------------------------ v-Qmx-N  
    idx_pos = m>0; $!B}$I;cd  
    idx_neg = m<0; ,[e\cnq[  
    E=$p^s  
    z = y; 3I  $>uR  
    if any(idx_pos) V 1/p_)A  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); xr%#dVk  
    end >/=> B7  
    if any(idx_neg) $n!K6fkX%  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); KOh A)  
    end VUwC-)  
    E\U`2{^.  
    % 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