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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 i#I+   
    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! ?b?`(JTR  
     
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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)  6'y+Ev$9  
    /G$8j$  
    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。 ,]@K6  
    I}/o`oc  
    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) (V @g?|LZ  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. I_.(&hMn  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of E&V"z^qs_  
    %   order N and frequency M, evaluated at R.  N is a vector of 2D`@$)KL  
    %   positive integers (including 0), and M is a vector with the SQ5SvYH  
    %   same number of elements as N.  Each element k of M must be a @PuJre4!;L  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) $s.:wc^  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is )0`;leli  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix |J2_2a/"  
    %   with one column for every (N,M) pair, and one row for every !>b>"\b  
    %   element in R. W3* BdpTw  
    % 7a0ZI  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- [CBA Lj5  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is c#nFm&}dm  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to `;WiTE)&)  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 >i~W$; t  
    %   for all [n,m]. /S1EQ%_  
    % E-_)w  
    %   The radial Zernike polynomials are the radial portion of the /,$;xt-J35  
    %   Zernike functions, which are an orthogonal basis on the unit o%X_V!B{V  
    %   circle.  The series representation of the radial Zernike 7CYu"+Ea  
    %   polynomials is Qi2yaEB  
    % <ro0}%-z>M  
    %          (n-m)/2 1i#uKKwE  
    %            __ NUiZ!&  
    %    m      \       s                                          n-2s cyA|6Ltg%  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r @gENv~m<OI  
    %    n      s=0 g 'c4&Do  
    % YQ X+lE  
    %   The following table shows the first 12 polynomials. K+Q81<X~  
    % VJBVk8P  
    %       n    m    Zernike polynomial    Normalization xB3;%Lc  
    %       --------------------------------------------- rZ 9bz}K  
    %       0    0    1                        sqrt(2) sp0& " &5  
    %       1    1    r                           2 7!w@u6Q  
    %       2    0    2*r^2 - 1                sqrt(6) </tiNc  
    %       2    2    r^2                      sqrt(6) fL ng[&  
    %       3    1    3*r^3 - 2*r              sqrt(8) K`* 8 *k{  
    %       3    3    r^3                      sqrt(8) &+6XdhX  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) #rMMOu9r2  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) i0{pm q  
    %       4    4    r^4                      sqrt(10) !1+L0,I6  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) mu)?SGpyE  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) u /JEQz1  
    %       5    5    r^5                      sqrt(12) UoPd>q4Uj  
    %       --------------------------------------------- "UKX~}8T  
    % SPOg'  
    %   Example: ur={+0 y  
    % X<\^*{  
    %       % Display three example Zernike radial polynomials #Bj{ 4OeV  
    %       r = 0:0.01:1; U`K5 DZ~  
    %       n = [3 2 5]; &WN4/=QW-J  
    %       m = [1 2 1]; O^G/(  
    %       z = zernpol(n,m,r); -'BJhi\Y]~  
    %       figure }cgEC-  
    %       plot(r,z) WqqrfzlM  
    %       grid on H)t YxW  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') uljd)kLy4O  
    % M|?qSFv:  
    %   See also ZERNFUN, ZERNFUN2. g[*+R9'  
    | ctGxS9  
    % A note on the algorithm. RO'MFU<g  
    % ------------------------ cZ \#074u/  
    % The radial Zernike polynomials are computed using the series l*HONl&j  
    % representation shown in the Help section above. For many special N\=pH{  
    % functions, direct evaluation using the series representation can sn_]7d+ Q  
    % produce poor numerical results (floating point errors), because 6hXL`A&},  
    % the summation often involves computing small differences between 1lfkb1BM  
    % large successive terms in the series. (In such cases, the functions af\>+7x93  
    % are often evaluated using alternative methods such as recurrence X/lLM`  
    % relations: see the Legendre functions, for example). For the Zernike ?(Dkh${@  
    % polynomials, however, this problem does not arise, because the \E9Z H3;  
    % polynomials are evaluated over the finite domain r = (0,1), and @cAv8i K  
    % because the coefficients for a given polynomial are generally all D^=_408\  
    % of similar magnitude. epCU(d*b  
    % ~-GgVi*I  
    % ZERNPOL has been written using a vectorized implementation: multiple r^ S 4 I&  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] ;WJ}zjo >  
    % values can be passed as inputs) for a vector of points R.  To achieve )s,L:{<  
    % this vectorization most efficiently, the algorithm in ZERNPOL ~l}rYi>g%  
    % involves pre-determining all the powers p of R that are required to 9@./=5N~3  
    % compute the outputs, and then compiling the {R^p} into a single zHG KPuk'  
    % matrix.  This avoids any redundant computation of the R^p, and 2al%J%  
    % minimizes the sizes of certain intermediate variables. Vky~yTL)\  
    % # ,u7lAz  
    %   Paul Fricker 11/13/2006 upQ:C>S  
    L-^vlP)Vu  
    m;WUp{'  
    % Check and prepare the inputs: ]7O)iq%  
    % ----------------------------- +Q If7=  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Yb%H9A  
        error('zernpol:NMvectors','N and M must be vectors.') w3PE.A"Q  
    end /S$p_7N  
    I.Co8is  
    if length(n)~=length(m) bRJYw6oA<  
        error('zernpol:NMlength','N and M must be the same length.') 8;P8CKe  
    end zwN;CD1  
    IQMk:  
    n = n(:); ,]i ^/fT  
    m = m(:); JHwkLAuz  
    length_n = length(n); :7D&=n)  
    9b@L^]Kg  
    if any(mod(n-m,2)) /YR*KxIx  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') [^A.$,  
    end {0q;:7Bt  
    ElZ'/l*\  
    if any(m<0) F}DdErd!f  
        error('zernpol:Mpositive','All M must be positive.') "V&2 g?  
    end Ow wH 45  
    jx!)N>  
    if any(m>n) /4YXx|V  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') ^5qX+!3r{  
    end L=iaL[zdJ  
    e7t).s)b{  
    if any( r>1 | r<0 ) 8U/q3@EC  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') HD IB GG~  
    end [Oxmg?W  
    H;k;%Zg;  
    if ~any(size(r)==1) 7fLLV2  
        error('zernpol:Rvector','R must be a vector.') Dp6]!;kx  
    end 3q R@$pm  
    5znLpBX<N  
    r = r(:); xH; qJRHa  
    length_r = length(r); ME[Wg\  
    o\@1\#a  
    if nargin==4 ~cz] Rhq  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); ^b~&}uU  
        if ~isnorm }pbyC  
            error('zernpol:normalization','Unrecognized normalization flag.') W'E!5T^  
        end t LdBnf  
    else Cc0`Ylx~(  
        isnorm = false; 6`]R)i]  
    end df nmUE  
    LG [ 2u  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% $lOx 6rL  
    % Compute the Zernike Polynomials @/lLL GrZ"  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% /R^HRzTO  
    F 71  
    % Determine the required powers of r: #^gn,^QQ  
    % -----------------------------------  .LEQ r)  
    rpowers = []; ,ZJI]Q=!  
    for j = 1:length(n) j?jEWreq]~  
        rpowers = [rpowers m(j):2:n(j)]; g (33h2"  
    end i/Q*AG>b  
    rpowers = unique(rpowers); /R8>f  
    I--WS[  
    % Pre-compute the values of r raised to the required powers, {p|OKf  
    % and compile them in a matrix: aa_&WHXkt  
    % ----------------------------- q#pBlJ.LK  
    if rpowers(1)==0 yc+#LZ~(a  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); /_rQ>PgSZW  
        rpowern = cat(2,rpowern{:}); 7$z")JB  
        rpowern = [ones(length_r,1) rpowern]; !w[<?+%%n  
    else }H?8~S =  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); QBR9BR  
        rpowern = cat(2,rpowern{:}); qRJg/~_h{  
    end %;!@\5$  
    i 6kW"5t  
    % Compute the values of the polynomials: {DI_i +2  
    % -------------------------------------- y+(<Is0w  
    z = zeros(length_r,length_n); F4k<YU  
    for j = 1:length_n vPR1 TMi>  
        s = 0:(n(j)-m(j))/2; 25l6@7q.  
        pows = n(j):-2:m(j); J{@gp,&e  
        for k = length(s):-1:1 .i"v([eQ  
            p = (1-2*mod(s(k),2))* ... Z9i,#/  
                       prod(2:(n(j)-s(k)))/          ... uwj/]#`  
                       prod(2:s(k))/                 ... \_!FOUPz(  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... G`R Ed-Z[  
                       prod(2:((n(j)+m(j))/2-s(k))); a)(j68c  
            idx = (pows(k)==rpowers); M`FsKK`  
            z(:,j) = z(:,j) + p*rpowern(:,idx); 5w gtc~  
        end la8se=^  
         H#E   
        if isnorm R# 8D}5[&  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); ,vrdtL  
        end ,YSQog  
    end }Tu_?b`RUm  
    @!Il!+^3  
    % 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) 6O"Vy  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. !#rZ eDmw  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated F@ZG| &  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive +=$\7z>s  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, Y5-X)f  
    %   and THETA is a vector of angles.  R and THETA must have the same nv{ou [vQ  
    %   length.  The output Z is a matrix with one column for every P-value, jn^i4f>N  
    %   and one row for every (R,THETA) pair. 9$~D4T  
    % ' hO+b  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike XZV)4=5iSO  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) %N/I;`  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) [dk|lkj@u\  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 /Or76kE  
    %   for all p. J%aW^+O  
    % 3 cT  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 Yl&eeM  
    %   Zernike functions (order N<=7).  In some disciplines it is UldKlQ8  
    %   traditional to label the first 36 functions using a single mode IqfR`iAix  
    %   number P instead of separate numbers for the order N and azimuthal *7ap[YXZ\w  
    %   frequency M. a gBKp!  
    % A!Ng@r  
    %   Example: xE9^4-Px*  
    % *#N%3:@T  
    %       % Display the first 16 Zernike functions ~SV;"e2N.  
    %       x = -1:0.01:1; S^I38gJd  
    %       [X,Y] = meshgrid(x,x); ) OZDq]mV  
    %       [theta,r] = cart2pol(X,Y); 'V4.umj1~  
    %       idx = r<=1; 0K7-i+\#  
    %       p = 0:15; %T}{rU~X  
    %       z = nan(size(X)); K.o?g?&<  
    %       y = zernfun2(p,r(idx),theta(idx)); 6du"^g  
    %       figure('Units','normalized') y|.wL=;  
    %       for k = 1:length(p) q<oA%yR  
    %           z(idx) = y(:,k); HZ[&ZNTa  
    %           subplot(4,4,k) "y "C#:5  
    %           pcolor(x,x,z), shading interp 66:|)  
    %           set(gca,'XTick',[],'YTick',[]) 8}xU]N#EV  
    %           axis square JR 2v}b  
    %           title(['Z_{' num2str(p(k)) '}']) DQ9 <N~l  
    %       end @a%,0Wn  
    % %04>R'mN  
    %   See also ZERNPOL, ZERNFUN. T1n GBl\(  
    :eHh }  
    %   Paul Fricker 11/13/2006 8uyVx9C0  
    "9LPq  
    " 8;D^  
    % Check and prepare the inputs: p"q-sMYl  
    % ----------------------------- 0<nKB}9  
    if min(size(p))~=1 {:4); .  
        error('zernfun2:Pvector','Input P must be vector.') oWs&W  
    end t,k9:p  
    um&N|5lHb  
    if any(p)>35 SkvKzV.R;  
        error('zernfun2:P36', ... ; wW6x  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... Y]Su<t gX?  
               '(P = 0 to 35).']) '? yZ,t  
    end %{zM> le9  
    {[r'+=}l\S  
    % Get the order and frequency corresonding to the function number: "q#(}1Zd  
    % ---------------------------------------------------------------- _AVCh)Zb  
    p = p(:); C$ZY=UXz!T  
    n = ceil((-3+sqrt(9+8*p))/2); 8f8+3  
    m = 2*p - n.*(n+2); IEC:zmkn  
    (c(?s`;  
    % Pass the inputs to the function ZERNFUN: ip1jY!   
    % ---------------------------------------- (O?z6g  
    switch nargin U> q&+:+  
        case 3 _QXo4z!a8  
            z = zernfun(n,m,r,theta); Ta9;;B?$  
        case 4 7yQ r  
            z = zernfun(n,m,r,theta,nflag); YI%S)$  
        otherwise ;R 2(Gb  
            error('zernfun2:nargin','Incorrect number of inputs.') >z[d ~  
    end m1daOeZ]P  
    :^l*_v{  
    % 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 ZO}V}3  
    function z = zernfun(n,m,r,theta,nflag) 4SG[_:+!  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. 3%cNePlr  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N vo0[Z,aH5  
    %   and angular frequency M, evaluated at positions (R,THETA) on the v- {kPc=:#  
    %   unit circle.  N is a vector of positive integers (including 0), and gO$!_!@LM  
    %   M is a vector with the same number of elements as N.  Each element !w C4ei`  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) Y61E|:fV!  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, uQ8]j.0  
    %   and THETA is a vector of angles.  R and THETA must have the same 8,['q~z  
    %   length.  The output Z is a matrix with one column for every (N,M) BA-n+WCWJ  
    %   pair, and one row for every (R,THETA) pair. \!w7 N :m  
    % WX?|iw I~  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike K0+J!- a]7  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ` $zi?A:j  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral ]?<uf40Mm  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, g$]9xn#_[  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized pl7!O9bo  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 7L]fCw p[  
    % DtZkrj)D/  
    %   The Zernike functions are an orthogonal basis on the unit circle. TF{ xFb)  
    %   They are used in disciplines such as astronomy, optics, and d[O.UzQ  
    %   optometry to describe functions on a circular domain. Zu+Z7@$}/  
    % @Z|cUHo  
    %   The following table lists the first 15 Zernike functions. qbT].,?!U  
    % "` 9W"A=  
    %       n    m    Zernike function           Normalization RrRCT.+E  
    %       -------------------------------------------------- <X;y 4lPZ  
    %       0    0    1                                 1 hVR=g!e#X  
    %       1    1    r * cos(theta)                    2 V qYe0-^=P  
    %       1   -1    r * sin(theta)                    2 nE/T)[1|  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) g(<@r2p  
    %       2    0    (2*r^2 - 1)                    sqrt(3) _{ ?1+  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) sRYFu%  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) 0}w>8L7i{  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) .|o7YTcR:  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) dc:|)bK M  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) o3uv"# C  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) P/ug'  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) "lUw{3  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) ? ZN8Ku  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ,AM6E63  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) *;4r|# LG  
    %       -------------------------------------------------- *8MU,6  
    % M6g!bK2l  
    %   Example 1: Dj %jrtT  
    % dIK!xOStA  
    %       % Display the Zernike function Z(n=5,m=1) @AWKEo<7.I  
    %       x = -1:0.01:1; H!vvdp?Z  
    %       [X,Y] = meshgrid(x,x); B8C"i%8V)  
    %       [theta,r] = cart2pol(X,Y); #V~r@,  
    %       idx = r<=1;  |\,e9U>  
    %       z = nan(size(X)); '2# O{  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); /Nxy?g|,  
    %       figure sLB{R#Pt  
    %       pcolor(x,x,z), shading interp Q=>@:1=  
    %       axis square, colorbar O5e9vQH  
    %       title('Zernike function Z_5^1(r,\theta)') Jzfz y0$  
    % LQR9S/?Ld  
    %   Example 2: XhTp'2,]  
    % K uFDkT!  
    %       % Display the first 10 Zernike functions 8)M . W  
    %       x = -1:0.01:1; +:oHI[1HG  
    %       [X,Y] = meshgrid(x,x); /FB'  
    %       [theta,r] = cart2pol(X,Y); N/^r9Nu  
    %       idx = r<=1; [}+ MZ  
    %       z = nan(size(X)); X $cW!a  
    %       n = [0  1  1  2  2  2  3  3  3  3]; *4WOmsj  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; \N7 E!82  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; 9 ?h)U|J?G  
    %       y = zernfun(n,m,r(idx),theta(idx)); ?p6+?\H  
    %       figure('Units','normalized') jJg 'Y:K9q  
    %       for k = 1:10 <A!v'Y  
    %           z(idx) = y(:,k); |J~;yO SD  
    %           subplot(4,7,Nplot(k)) ^<ayPV)+  
    %           pcolor(x,x,z), shading interp 4qiG>^h9  
    %           set(gca,'XTick',[],'YTick',[]) GHH1jJ_[7  
    %           axis square I~#'76L[  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) %*:-4K  
    %       end g+)T\_#u  
    % py@5]n%  
    %   See also ZERNPOL, ZERNFUN2. ^[d)Hk}L  
    qhiQ!fMQ  
    %   Paul Fricker 11/13/2006 v{&cgod  
    0*,r  
    <7u*OYjA  
    % Check and prepare the inputs: $t[`}I }  
    % ----------------------------- E!jM&\Zj  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) RqH"+/wR  
        error('zernfun:NMvectors','N and M must be vectors.') K4A=lD+  
    end vek9. 4! ]  
    ])T*T$u  
    if length(n)~=length(m) eK4\v:oG1  
        error('zernfun:NMlength','N and M must be the same length.') l[rIjyL@  
    end 4,T S1H  
    @P[Tu; 4  
    n = n(:); +9,"ne1'e  
    m = m(:); &LHQ) ?  
    if any(mod(n-m,2)) NDCZc_  
        error('zernfun:NMmultiplesof2', ... >qCUs3}C{*  
              'All N and M must differ by multiples of 2 (including 0).') S}ZM;M  
    end e9"<.:&  
    ; l+3l ez  
    if any(m>n) U0UOubA  
        error('zernfun:MlessthanN', ... Hg]Q.SeJ(  
              'Each M must be less than or equal to its corresponding N.') tAc[r)xFw  
    end "4}{Z)&R2  
    Dj #G{X".  
    if any( r>1 | r<0 ) 7BdvJ"  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') uiM*!ge  
    end 4k<4=E  
    -=O9D- x=  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) _T8#36iR  
        error('zernfun:RTHvector','R and THETA must be vectors.') EZE/~$`3   
    end !,INrl[  
    WX]kez{<uP  
    r = r(:); /dh w~|  
    theta = theta(:); l`fjz-eE  
    length_r = length(r); Y }Rx`%X  
    if length_r~=length(theta) fMI4'.Od  
        error('zernfun:RTHlength', ...  :v8j3=  
              'The number of R- and THETA-values must be equal.') X^r HugQ  
    end q-X)tH_+w@  
    lLyMm8E%pZ  
    % Check normalization: jQC6N#L  
    % -------------------- ZGe+w](  
    if nargin==5 && ischar(nflag) Cddw\|'3  
        isnorm = strcmpi(nflag,'norm'); Cf J@|Rh  
        if ~isnorm [:TOU^  
            error('zernfun:normalization','Unrecognized normalization flag.') buG0#:  
        end Vb|DNl@  
    else =H3 JRRS  
        isnorm = false; F=$2Gz 'RT  
    end uXNJ{]o  
    K zKHC  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ID E3>D  
    % Compute the Zernike Polynomials Z?O aY4  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 3RZP 12x  
    p!|ok #sW  
    % Determine the required powers of r: 03iO4yOu  
    % ----------------------------------- Z"] ben  
    m_abs = abs(m); B&+V%~/  
    rpowers = []; xaAJ>0IM  
    for j = 1:length(n) MjQKcL4%7  
        rpowers = [rpowers m_abs(j):2:n(j)]; HBV~`0O$  
    end o5 @ l!NQ  
    rpowers = unique(rpowers); `GUj.+u  
    o[!]xmj  
    % Pre-compute the values of r raised to the required powers, `BdZqXKG  
    % and compile them in a matrix: W5<1@  
    % ----------------------------- n,bZj<3t  
    if rpowers(1)==0 ]ta]OK{s"  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); \[% [`m  
        rpowern = cat(2,rpowern{:}); 6Z\[{S];  
        rpowern = [ones(length_r,1) rpowern]; ~^w;`~L  
    else S%R:GZEf_  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); VSc;}LH  
        rpowern = cat(2,rpowern{:}); "=MRzSke3  
    end My&h{Qk  
    r8pTtf#Q  
    % Compute the values of the polynomials: *ukE"Aj  
    % --------------------------------------  M#IGq  
    y = zeros(length_r,length(n)); /<\>j+SC  
    for j = 1:length(n) Xv|~1v%s7  
        s = 0:(n(j)-m_abs(j))/2; JLp.bxx  
        pows = n(j):-2:m_abs(j); #(?EL@5  
        for k = length(s):-1:1 j$4Tot  
            p = (1-2*mod(s(k),2))* ... /NNe/7'l  
                       prod(2:(n(j)-s(k)))/              ... 9\0  
                       prod(2:s(k))/                     ... (B.J8`h }  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... q UY;CEf  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); w3a`G|  
            idx = (pows(k)==rpowers); jt@k< #h~  
            y(:,j) = y(:,j) + p*rpowern(:,idx); J'sVT{@GS  
        end >t.2!Z_RQ  
         ]/XNfb  
        if isnorm vClD)Ar  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); _Y40a+hk]  
        end =mxmJFA  
    end rg]eSP3 W  
    % END: Compute the Zernike Polynomials <*<7p{x  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% }Gqx2 )H  
    (x2I*<7P  
    % Compute the Zernike functions: QHUoAa`6v  
    % ------------------------------ DI'wZySS^  
    idx_pos = m>0; Vf`n>  
    idx_neg = m<0; -5l74f!i  
    ?_3K]i1IS  
    z = y; X<9jBj/t  
    if any(idx_pos) {a-p/\U  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); P ^R224R  
    end n{J<7I e"*  
    if any(idx_neg) r5xu#%hgp;  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); #G:~6^A  
    end 4nzUDeI3MG  
    U{ gJn#e/.  
    % 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