切换到宽版
  • 广告投放
  • 稿件投递
  • 繁體中文
    • 11419阅读
    • 9回复

    [求助]ansys分析后面型数据如何进行zernike多项式拟合? [复制链接]

    上一主题 下一主题
    离线niuhelen
     
    发帖
    19
    光币
    28
    光券
    0
    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 U :IQWlC  
    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! @s@r5uR9B  
     
    分享到
    离线phoenixzqy
    发帖
    4352
    光币
    5478
    光券
    1
    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)  kwWDGA?zFB  
    arET2(h  
    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。 j%Usui<DL  
    PkMN@JS  
    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
    发帖
    856
    光币
    846
    光券
    0
    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
    发帖
    59
    光币
    0
    光券
    0
    只看该作者 7楼 发表于: 2012-09-28
    我也正在找啊
    离线niuhelen
    发帖
    19
    光币
    28
    光券
    0
    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
    发帖
    19
    光币
    28
    光券
    0
    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) G?kK:eV  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. 7-:R{&3Lm:  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of ja^_Lh9  
    %   order N and frequency M, evaluated at R.  N is a vector of 50_[n$tqE  
    %   positive integers (including 0), and M is a vector with the >3ax `8  
    %   same number of elements as N.  Each element k of M must be a A:y HClmn  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) i/j53towe  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is -M/j&<;LW  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix <Pzy'9  
    %   with one column for every (N,M) pair, and one row for every 'X<4";$mU  
    %   element in R. ]Hp>~Zvbb  
    % Hz\@#   
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- ?iZ2sRWR6  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is B (Ps/  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to &1(- 8z*  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 28k=@k^q  
    %   for all [n,m]. /~M H]Gh  
    % N=AHS  
    %   The radial Zernike polynomials are the radial portion of the 2n)?)w]!M  
    %   Zernike functions, which are an orthogonal basis on the unit KL3Z(  
    %   circle.  The series representation of the radial Zernike GLgf%A`5/_  
    %   polynomials is aaP_^m O  
    % {`QA.he.  
    %          (n-m)/2 )/?H]o$NU  
    %            __ c/Xg ARCO  
    %    m      \       s                                          n-2s [Ur\^wS  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r ,jOJ\WXP  
    %    n      s=0 'IG@JL'  
    % 0z'GN#mT5  
    %   The following table shows the first 12 polynomials. k,[*h-{8  
    % vfc:ok1  
    %       n    m    Zernike polynomial    Normalization &\1n=y  
    %       --------------------------------------------- Q_U.J0  
    %       0    0    1                        sqrt(2) y{S8?$dU$:  
    %       1    1    r                           2 l|=4FIMD  
    %       2    0    2*r^2 - 1                sqrt(6) %Yj%0  
    %       2    2    r^2                      sqrt(6) s bj/d~$N  
    %       3    1    3*r^3 - 2*r              sqrt(8) ;I&VpAPx  
    %       3    3    r^3                      sqrt(8) yL*]_  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) <XIIT-b[  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) ,Klv[_x7  
    %       4    4    r^4                      sqrt(10) |RFBhB/u  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) >~SS^I0  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) nq)F$@  
    %       5    5    r^5                      sqrt(12) ,;_+o]  
    %       --------------------------------------------- 0?<#!  
    % 7 !$[XD  
    %   Example: h:nybLw?  
    % 7~ PL8  
    %       % Display three example Zernike radial polynomials OvtE)u l@  
    %       r = 0:0.01:1; N ~{N Nf Y  
    %       n = [3 2 5]; @eJCr)#}  
    %       m = [1 2 1]; P.}d@qD{)  
    %       z = zernpol(n,m,r); ")T\_ME  
    %       figure yd).}@  
    %       plot(r,z) hq)1YO  
    %       grid on {%f{U"m  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') eXW|{asx  
    % g1s%x=7/  
    %   See also ZERNFUN, ZERNFUN2. TIWR[r1!  
    rW:krx9  
    % A note on the algorithm. HeOdCr-PN  
    % ------------------------ b6bs .  
    % The radial Zernike polynomials are computed using the series Ax;=Zh<DAv  
    % representation shown in the Help section above. For many special :OG I|[  
    % functions, direct evaluation using the series representation can $KK~KEZ2  
    % produce poor numerical results (floating point errors), because O`B,mgT(  
    % the summation often involves computing small differences between {_QdB;VwH  
    % large successive terms in the series. (In such cases, the functions FQ]/c#J  
    % are often evaluated using alternative methods such as recurrence jN\u}!\O  
    % relations: see the Legendre functions, for example). For the Zernike TmsIyDcD~  
    % polynomials, however, this problem does not arise, because the ;]u9o}[ 2  
    % polynomials are evaluated over the finite domain r = (0,1), and +(W1x C0  
    % because the coefficients for a given polynomial are generally all U ? +_\  
    % of similar magnitude. DN*5q9.  
    % UFG_ZoD+  
    % ZERNPOL has been written using a vectorized implementation: multiple {KG6#/%;  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] q,vWu(.  
    % values can be passed as inputs) for a vector of points R.  To achieve kAki 9a(=!  
    % this vectorization most efficiently, the algorithm in ZERNPOL j3gDGw;  
    % involves pre-determining all the powers p of R that are required to ^7-zwl(>?N  
    % compute the outputs, and then compiling the {R^p} into a single |eqBCZn  
    % matrix.  This avoids any redundant computation of the R^p, and *m~-8_ >;  
    % minimizes the sizes of certain intermediate variables. X@rA2);6  
    % TSlB.pw%v  
    %   Paul Fricker 11/13/2006 [9 W@<p  
    eTiTS*`u  
    -8Jw_  
    % Check and prepare the inputs: zLpCKndj  
    % ----------------------------- P {TJ$  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) )  =<HDek  
        error('zernpol:NMvectors','N and M must be vectors.') O>~,RI!  
    end /yOx=V  
    1( pHC  
    if length(n)~=length(m) g !'R}y  
        error('zernpol:NMlength','N and M must be the same length.') Ri.tA  
    end Zh"m;l/]  
    mdj%zJ8/  
    n = n(:); ?=VvFfv%  
    m = m(:); T5S4,.o9W  
    length_n = length(n); J2YQdCL  
    B5b:znW2@  
    if any(mod(n-m,2)) ]&cnc8tC  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') 0MG>77  
    end I;(3)^QH#  
    L:z0cvn"  
    if any(m<0) xa>| k>I  
        error('zernpol:Mpositive','All M must be positive.') D|]BFu)F  
    end eqbN_$>  
    dY*q[N/pO  
    if any(m>n) x:Y9z_)O  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') (WM3(US|  
    end C]`uC^6g  
    *{g3ia  
    if any( r>1 | r<0 ) YR%iZ"`*+O  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') O$Rz/&  
    end 3/G^V'Yu  
    ,YYEn^:>  
    if ~any(size(r)==1) GG} %  
        error('zernpol:Rvector','R must be a vector.') _?{7%(C  
    end }A#IBqf5  
    _P>YG<*"kQ  
    r = r(:); "yWw3(V2>  
    length_r = length(r); @:lM|2:  
    TdtV (  
    if nargin==4 *ByHTd  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); SQKhht`M  
        if ~isnorm Syk)S<  
            error('zernpol:normalization','Unrecognized normalization flag.') i "8mrWb  
        end  T]#V  
    else :^;c(>u{  
        isnorm = false; e+ xQ\LH  
    end +#O+%!  
    s|[>@~gXk  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% v+c>iI  
    % Compute the Zernike Polynomials 3EoCEPb#  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 9T,/R1N8  
    Dg&84,bv^  
    % Determine the required powers of r: LQ+/|_(.  
    % -----------------------------------  Z>[7#;;  
    rpowers = []; vOQ% f?%G\  
    for j = 1:length(n) 80xr zv  
        rpowers = [rpowers m(j):2:n(j)]; =L6#=7hcl  
    end Bo 35L:r|  
    rpowers = unique(rpowers); fgLjF,Y  
    )>volP  
    % Pre-compute the values of r raised to the required powers, ,:_c-d#  
    % and compile them in a matrix: +y7z>Fwl  
    % ----------------------------- )uPJ? 2S9  
    if rpowers(1)==0 tne_]+  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 0[;2dc  
        rpowern = cat(2,rpowern{:}); 9shf y4?k  
        rpowern = [ones(length_r,1) rpowern]; 2 $>DX\h  
    else 12$0-@U  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 8@3K, [Mo  
        rpowern = cat(2,rpowern{:}); Z;0~f<e%  
    end U& ?hG>  
    <izQ]\kL  
    % Compute the values of the polynomials: #&3,T1i`  
    % -------------------------------------- @[GV0*yz$  
    z = zeros(length_r,length_n); p/H.bG!z  
    for j = 1:length_n /y$Omc^  
        s = 0:(n(j)-m(j))/2; %#6@PQ[R.  
        pows = n(j):-2:m(j); "wUIsuG/p  
        for k = length(s):-1:1 GES}o9?#  
            p = (1-2*mod(s(k),2))* ... }@DCcf$<  
                       prod(2:(n(j)-s(k)))/          ... te_2"Z  
                       prod(2:s(k))/                 ... ((y|?Z$  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... eP{srP3 9  
                       prod(2:((n(j)+m(j))/2-s(k))); X.]I4O&_  
            idx = (pows(k)==rpowers); 2q f|+[X  
            z(:,j) = z(:,j) + p*rpowern(:,idx); }nmlN  
        end yR}. Xq/  
         `Sod]bO +U  
        if isnorm t],a1I.gk  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); 2u{~35  
        end kY0HP a  
    end [%W'd9`>  
    7 qKz_O  
    % EOF zernpol
    离线niuhelen
    发帖
    19
    光币
    28
    光券
    0
    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) ('J/Ww<  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. ^2+Ex+  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated woI5aee|  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive 8R~<$ xz  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, I&PJ[U#~a  
    %   and THETA is a vector of angles.  R and THETA must have the same r>mBe;[TX  
    %   length.  The output Z is a matrix with one column for every P-value, _,3ljf?WQM  
    %   and one row for every (R,THETA) pair. 2+]5}'M  
    % "Ih3  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike !2!~_*sGe  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) Y,]Lk<Hm3  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) a@}.96lStD  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 0KA*6]h t  
    %   for all p. s 6Wp"V(  
    % MT6p@b5  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 "8za'@D"f  
    %   Zernike functions (order N<=7).  In some disciplines it is .1QGNW  
    %   traditional to label the first 36 functions using a single mode iFIGJS  
    %   number P instead of separate numbers for the order N and azimuthal Iu'9yb  
    %   frequency M. manw;`Q  
    % `IHP_IfR  
    %   Example: X'A`" }=_  
    % )k<cd.MX  
    %       % Display the first 16 Zernike functions pyEQb#  
    %       x = -1:0.01:1; EEe$A?a;  
    %       [X,Y] = meshgrid(x,x); %0\@\fC41  
    %       [theta,r] = cart2pol(X,Y); )@]%:m!ER  
    %       idx = r<=1; iSfRJ:_&6  
    %       p = 0:15; (Tx_`rO4VY  
    %       z = nan(size(X)); |mT%IR  
    %       y = zernfun2(p,r(idx),theta(idx)); ammi4k/  
    %       figure('Units','normalized') ~!uX"F8Xl  
    %       for k = 1:length(p) kD#T _d  
    %           z(idx) = y(:,k); If'q8G3]-  
    %           subplot(4,4,k) KpN]9d   
    %           pcolor(x,x,z), shading interp @52#ZWy  
    %           set(gca,'XTick',[],'YTick',[]) ` w;Wud'*<  
    %           axis square Lg4|6.Ez|P  
    %           title(['Z_{' num2str(p(k)) '}']) *F$@!ByV  
    %       end i0M6;W1T  
    % O:BdZ5 b  
    %   See also ZERNPOL, ZERNFUN. 74^v('-2  
    ~cU1 /CW8  
    %   Paul Fricker 11/13/2006 'O a3 6@  
    @&T' h}|:  
    wd:Yy  
    % Check and prepare the inputs: nD i^s{  
    % ----------------------------- zC50 @S3|  
    if min(size(p))~=1 , ['}9:f9  
        error('zernfun2:Pvector','Input P must be vector.') ?K$&|w%{3  
    end iXWzIb}CJ-  
    8W3zrnc  
    if any(p)>35 B*/!s7c.  
        error('zernfun2:P36', ... :'h$]p%  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... . 0dGS  
               '(P = 0 to 35).']) . !1[I{KU  
    end &l6@C3N$  
    z ]f(lwo{  
    % Get the order and frequency corresonding to the function number: _'D(>e?  
    % ---------------------------------------------------------------- `%YMUBaI  
    p = p(:); Ry95a%&/s  
    n = ceil((-3+sqrt(9+8*p))/2); wx-\@{E  
    m = 2*p - n.*(n+2); }u#3hYa  
    'Agw~ &$  
    % Pass the inputs to the function ZERNFUN: EPE_2a}  
    % ---------------------------------------- @x `X|>&  
    switch nargin e&sH<hWR  
        case 3 c0wLc,)G  
            z = zernfun(n,m,r,theta); l]G iz&  
        case 4 Zk`y"[J  
            z = zernfun(n,m,r,theta,nflag); 8#!g;`~ D  
        otherwise ?j&hG|W9<z  
            error('zernfun2:nargin','Incorrect number of inputs.') tR51Pw  
    end -9vNV:c  
    B=Kr J{&!  
    % EOF zernfun2
    离线niuhelen
    发帖
    19
    光币
    28
    光券
    0
    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 RB4n>&Y  
    function z = zernfun(n,m,r,theta,nflag) :G>w MMv&z  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. "R5G^-<h p  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 0 s+X:*C~  
    %   and angular frequency M, evaluated at positions (R,THETA) on the LZ wCe$1  
    %   unit circle.  N is a vector of positive integers (including 0), and g}!{_z  
    %   M is a vector with the same number of elements as N.  Each element JDf>Qg{  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) t U}6^yc  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, ECt<\h7}  
    %   and THETA is a vector of angles.  R and THETA must have the same m 3UK`~ji  
    %   length.  The output Z is a matrix with one column for every (N,M) D?#l8  
    %   pair, and one row for every (R,THETA) pair. CHTK.%AQH!  
    % (F^R9G|  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike /"J 6``MV  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), j7)mC4o:%  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral a/ uo)']B  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ZBDF>u@  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized  2d*bF.  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. e1g3a1tnWl  
    % tN<X3$aN  
    %   The Zernike functions are an orthogonal basis on the unit circle. *%/O (ohs@  
    %   They are used in disciplines such as astronomy, optics, and # bHkI~  
    %   optometry to describe functions on a circular domain. L ~'98C  
    % Gtaa^mnxD  
    %   The following table lists the first 15 Zernike functions. d<d3j9u(#  
    % ,KJHYm=Q  
    %       n    m    Zernike function           Normalization 8#;=>m%  
    %       -------------------------------------------------- zg3kU65PJE  
    %       0    0    1                                 1 g"748LY>=p  
    %       1    1    r * cos(theta)                    2 |!] "y<  
    %       1   -1    r * sin(theta)                    2 vyDxX  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) keC'/\e  
    %       2    0    (2*r^2 - 1)                    sqrt(3) |K_%]1*riC  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) i{m!v6j:  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) |kK5:\H  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) |dQz(z&6{5  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) m"rht:v5  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) yZ{yzv'D&  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) O|sk "YXF  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) PwW$=M{\.  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) !#pc@(rE  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10)  FkrXM!mJ  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) Mv%Qze,\V^  
    %       -------------------------------------------------- k 6M D3c  
    % <=p>0L  
    %   Example 1: L9O;K$[s  
    % nHm29{G0  
    %       % Display the Zernike function Z(n=5,m=1) k Nc- @B  
    %       x = -1:0.01:1; Hy4;i^Ik <  
    %       [X,Y] = meshgrid(x,x); Bc.de&Bxz_  
    %       [theta,r] = cart2pol(X,Y); (=uT*Cb  
    %       idx = r<=1; P!Fy kg  
    %       z = nan(size(X)); _^Q!cB'~/`  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); 7zCJ3p  
    %       figure b5H}0<  
    %       pcolor(x,x,z), shading interp ?:3hp2k<  
    %       axis square, colorbar {!D(3~MI  
    %       title('Zernike function Z_5^1(r,\theta)') nE u:& 4  
    % O6NH  
    %   Example 2: 5@+?{Cl  
    % - (WH+  
    %       % Display the first 10 Zernike functions ('J@GTe@xj  
    %       x = -1:0.01:1; -_nQn  
    %       [X,Y] = meshgrid(x,x); F/ZFO5C%  
    %       [theta,r] = cart2pol(X,Y); 4ams~  
    %       idx = r<=1; _!1LV[x!s  
    %       z = nan(size(X)); D(ItNMc Ku  
    %       n = [0  1  1  2  2  2  3  3  3  3]; <c[\\ :Hh*  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; BW)-F (v   
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; "'bl)^+?,  
    %       y = zernfun(n,m,r(idx),theta(idx)); ,93Uji[l  
    %       figure('Units','normalized') :+DrV\)  
    %       for k = 1:10 z |llf7:  
    %           z(idx) = y(:,k); Xi%Og\vm5  
    %           subplot(4,7,Nplot(k)) cy.r/Z}  
    %           pcolor(x,x,z), shading interp z(A[xN@/W<  
    %           set(gca,'XTick',[],'YTick',[]) [-*&ZYp  
    %           axis square %\ i&g$  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ]UUa/ep-  
    %       end ]O@iT= *3  
    % V5(_7b#z``  
    %   See also ZERNPOL, ZERNFUN2. avq$aq(3&  
    _M/N_Fm  
    %   Paul Fricker 11/13/2006 OJpfiZ@Q_  
    : wS&3:h  
    sR1_L/.  
    % Check and prepare the inputs: ]uox ^HC  
    % ----------------------------- vcdVck@  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 0]bt}rh  
        error('zernfun:NMvectors','N and M must be vectors.') e:Y+-C5  
    end (*$F7oO<  
    H9)n<r  
    if length(n)~=length(m) y/\b0&  
        error('zernfun:NMlength','N and M must be the same length.') I9 zs  
    end '(@q"`n  
    K1hkOj;S  
    n = n(:); ,e43m=KhK  
    m = m(:); $h p UI  
    if any(mod(n-m,2)) j7Fb4;o{  
        error('zernfun:NMmultiplesof2', ... r\Y,*e  
              'All N and M must differ by multiples of 2 (including 0).') 0\XWdTj{  
    end :ZY%-]u7  
    (0.oE%B",1  
    if any(m>n) \85%d0@3  
        error('zernfun:MlessthanN', ... t9U6\ru  
              'Each M must be less than or equal to its corresponding N.') rQ{|0+l  
    end ~'%d]s+q  
    aI&~aezmN  
    if any( r>1 | r<0 ) # &.syD#  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.')  B`e/ /  
    end 7JBs7LG  
    */h(4Hz  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Oq~{HJ{  
        error('zernfun:RTHvector','R and THETA must be vectors.') m@XX2l9:9  
    end xR0*w7YE  
    S'34](9n6  
    r = r(:); tV(iC~/  
    theta = theta(:); ]%D!-[C%1  
    length_r = length(r); X1(ds*'Kv  
    if length_r~=length(theta) Ob]\t/:%P  
        error('zernfun:RTHlength', ... ]:Ep1DIMl  
              'The number of R- and THETA-values must be equal.') U\lbh;9G  
    end \)/qCeiZ  
    CWkWW/ZI  
    % Check normalization: 1rZ E2  
    % -------------------- @>O7/d?O  
    if nargin==5 && ischar(nflag) +pqbl*W;1  
        isnorm = strcmpi(nflag,'norm'); 6"G(Iq'2t3  
        if ~isnorm "qq$i35x  
            error('zernfun:normalization','Unrecognized normalization flag.') 8*u'D@0  
        end %U{sn\V  
    else I%r7L  
        isnorm = false; C{/U;Ie-b  
    end TNqL ')f  
    k*;U?C!  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ;>Z+b#C[  
    % Compute the Zernike Polynomials s U`#hL6;  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% RL4|!HzR  
    Z0Sqw  
    % Determine the required powers of r: B0b|+5WhR  
    % ----------------------------------- _m?i$5  
    m_abs = abs(m); d~QKZ&jf  
    rpowers = []; esTL3 l{[  
    for j = 1:length(n) Q.$8>)  
        rpowers = [rpowers m_abs(j):2:n(j)]; L-E &m*%  
    end 8i] S[$Fc  
    rpowers = unique(rpowers); Vwp>:'Pu  
    h81giY]  
    % Pre-compute the values of r raised to the required powers, *Hn=)q  
    % and compile them in a matrix: F.y_H#h  
    % ----------------------------- c\ZI 5&4jT  
    if rpowers(1)==0 i}8OaX3x  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); R-zS7Jyox  
        rpowern = cat(2,rpowern{:}); ]zj#X\  
        rpowern = [ones(length_r,1) rpowern]; n>u_>2Ikkj  
    else ltNI+G  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); )8^E{w^D}  
        rpowern = cat(2,rpowern{:}); m&=Dy5  
    end I@m(}  
    VvIUAn  
    % Compute the values of the polynomials: %TI3Eb  
    % -------------------------------------- vh.8m $,  
    y = zeros(length_r,length(n)); tF,`v{-up  
    for j = 1:length(n) *^@b0f~vj  
        s = 0:(n(j)-m_abs(j))/2; OH>Gc-V  
        pows = n(j):-2:m_abs(j); ;V~x[J|x  
        for k = length(s):-1:1 u^SInanw  
            p = (1-2*mod(s(k),2))* ... [gUD +  
                       prod(2:(n(j)-s(k)))/              ... VM5'd  
                       prod(2:s(k))/                     ... R(0[bMr3Q  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... \1<aBgK i  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); ]/h$6mrL  
            idx = (pows(k)==rpowers); yH:p*|%:  
            y(:,j) = y(:,j) + p*rpowern(:,idx); _}47U7s8  
        end 2|?U%YrHWs  
         N}/V2K]Q  
        if isnorm +vJ}'uR3P  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); rCqwJoC`v  
        end lmcgOTT):  
    end ,k.")  
    % END: Compute the Zernike Polynomials +(x(Ybl#  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Z L0Vx6Ph  
    V"8Go;[  
    % Compute the Zernike functions: yD \Kn{  
    % ------------------------------ !lg_zAV  
    idx_pos = m>0; M3UC9t9]  
    idx_neg = m<0; ?r|iZKa  
    ;C=d( pY  
    z = y; 8)iI=,T*  
    if any(idx_pos) ._p2"<  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); >P(.yQ8&kL  
    end z+oy#p6+F.  
    if any(idx_neg) 19R~&E's  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ~b *|V  
    end ,]JIp~=nsh  
    !ckluj  
    % EOF zernfun
    离线phility
    发帖
    69
    光币
    11
    光券
    0
    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
    离线phility
    发帖
    69
    光币
    11
    光券
    0
    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