切换到宽版
  • 广告投放
  • 稿件投递
  • 繁體中文
    • 11174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 倒序阅读 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 i2@VB6]?  
    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! oiF}?:7Q7  
     
    分享到
    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
    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 bHPYp5UwN  
    function z = zernfun(n,m,r,theta,nflag) ^M3~^lV  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. V `b2TS  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N W 0(_ ~  
    %   and angular frequency M, evaluated at positions (R,THETA) on the fdxLAC  
    %   unit circle.  N is a vector of positive integers (including 0), and Ky|88~}:C9  
    %   M is a vector with the same number of elements as N.  Each element Y,GU%[+  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) u}>#Eb  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, LUG;(Fko  
    %   and THETA is a vector of angles.  R and THETA must have the same XxT#X3D/,"  
    %   length.  The output Z is a matrix with one column for every (N,M) O!zV)^r  
    %   pair, and one row for every (R,THETA) pair. bBu,#Mc  
    % *-+&[P]m  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike [DJflCR&  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), <A<{,:5C  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral iocI:b <  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, pA`+hQNN  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized  :l~ I  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. Ot:CPm@  
    % q`|LRz&al  
    %   The Zernike functions are an orthogonal basis on the unit circle. *YW/_  
    %   They are used in disciplines such as astronomy, optics, and r>dwDBE  
    %   optometry to describe functions on a circular domain. &J55P]7w  
    % ZtV9&rd7  
    %   The following table lists the first 15 Zernike functions. YsG%6&zEq  
    % 3b*cU}go  
    %       n    m    Zernike function           Normalization /d0K7F  
    %       -------------------------------------------------- \qR7mI/*  
    %       0    0    1                                 1 oE<`VY|  
    %       1    1    r * cos(theta)                    2 vh"R'o  
    %       1   -1    r * sin(theta)                    2 ]p*l%(dhY  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) +~'865{  
    %       2    0    (2*r^2 - 1)                    sqrt(3) cmBB[pk\  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) wi hH?~]  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) ~Cl){8o  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) `k OD[*  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) lwHzj&/ ~  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) P#pn*L*"T  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) rJPb 3F  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) |s)Rxq){"V  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) &/mA7Vf>eR  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 09dK0H3(  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) 0FGe=$vD  
    %       -------------------------------------------------- l-K9LTd  
    % "XB[|#&  
    %   Example 1: _Bj)r}~7#  
    % SLO%7%>p  
    %       % Display the Zernike function Z(n=5,m=1) q:l>O5  
    %       x = -1:0.01:1; aki _RG>U'  
    %       [X,Y] = meshgrid(x,x); Ae mDJ8Y  
    %       [theta,r] = cart2pol(X,Y); =3|O %\  
    %       idx = r<=1; MA;1 ;uI,  
    %       z = nan(size(X)); Q&MZN);.  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); 2}YOcnB  
    %       figure zEs>b(5u  
    %       pcolor(x,x,z), shading interp |\QgX%  
    %       axis square, colorbar #rxVd 7f  
    %       title('Zernike function Z_5^1(r,\theta)') umD!2 w  
    % M9EfU  
    %   Example 2: N U|d  
    % NZ ;{t\  
    %       % Display the first 10 Zernike functions Fkvl%n  
    %       x = -1:0.01:1; ^m?KRm2  
    %       [X,Y] = meshgrid(x,x); /3A^I{e74  
    %       [theta,r] = cart2pol(X,Y); Em?d*z  
    %       idx = r<=1; :q=%1~Idla  
    %       z = nan(size(X)); +lJG(Qd  
    %       n = [0  1  1  2  2  2  3  3  3  3]; cU0s p  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; Xg<*@4RD8  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; !vX D  
    %       y = zernfun(n,m,r(idx),theta(idx)); 5V5%/FU m  
    %       figure('Units','normalized') *_R]*o!W'  
    %       for k = 1:10 ` jzTmt  
    %           z(idx) = y(:,k); I([!]z  
    %           subplot(4,7,Nplot(k)) ulu9'ch  
    %           pcolor(x,x,z), shading interp ?dD&p8{  
    %           set(gca,'XTick',[],'YTick',[]) ~7Ts_:E-  
    %           axis square C3< m7h  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) Wi[~fI8^!  
    %       end R16'?,  
    % hc~s"Atck  
    %   See also ZERNPOL, ZERNFUN2. {S,l_d+(  
    (ohq0Y  
    %   Paul Fricker 11/13/2006 Y3r%B9~  
    wB.Nn/p  
    )ap_Z6  
    % Check and prepare the inputs: b`)){LR  
    % -----------------------------  $rz=6h  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 8#(Q_  
        error('zernfun:NMvectors','N and M must be vectors.') T?:glp[4I  
    end ojQI7 Uhw  
    1"/He ` 4  
    if length(n)~=length(m) A/s>PhxV  
        error('zernfun:NMlength','N and M must be the same length.') ,oaw0Vw  
    end e_s&L,ze  
    #[zI5)Meh  
    n = n(:); \]P!.}nX#  
    m = m(:); &8%e\W\K:/  
    if any(mod(n-m,2)) V6t,BJjS  
        error('zernfun:NMmultiplesof2', ... b8LoIY*  
              'All N and M must differ by multiples of 2 (including 0).') -:30:oq  
    end .u:81I=w(  
    N-I5X2  
    if any(m>n) 'rMN=1:iu"  
        error('zernfun:MlessthanN', ... /I)yU>o  
              'Each M must be less than or equal to its corresponding N.') } @K FB  
    end vk*=4}:  
    1QmH{jM  
    if any( r>1 | r<0 ) Y2d;E.DH8  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') p3]_}Y D[#  
    end >Y_*%QGH_  
    MS0Fl|YA  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 0KMctPT]p  
        error('zernfun:RTHvector','R and THETA must be vectors.')  `)GrwfC  
    end  PZ{Dv'C  
    0j30LXI_  
    r = r(:); [%9no B  
    theta = theta(:); /%0<p,T  
    length_r = length(r); C0S^h<iSe*  
    if length_r~=length(theta) %=?cZfFqO  
        error('zernfun:RTHlength', ... 9:`(Q3Ei  
              'The number of R- and THETA-values must be equal.') F%i^XA]a*  
    end -8r  
    TJ: ]SB  
    % Check normalization: Ku\Y'ub  
    % -------------------- ~_Lr=CD;4  
    if nargin==5 && ischar(nflag) Nluv/?<  
        isnorm = strcmpi(nflag,'norm'); ({JHZ6uZ  
        if ~isnorm @J5Jpt*IE  
            error('zernfun:normalization','Unrecognized normalization flag.') TF 'U  
        end 4'-|UPhx  
    else Si_%Rr&jW  
        isnorm = false; 'XzXZJ[uq  
    end s3]?8hXd  
    4hAl-8~Q6  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% b&=5m  
    % Compute the Zernike Polynomials EhO|~A*R  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -O&CI)`;B  
    +)j1.X  
    % Determine the required powers of r: u0#}9UKQ  
    % ----------------------------------- 'ihhoW8  
    m_abs = abs(m); AX= 1b,s  
    rpowers = []; 4O;OjUI0a  
    for j = 1:length(n) mt5KbA>nU  
        rpowers = [rpowers m_abs(j):2:n(j)]; 6ezS{Q  
    end z]2]XTmWs  
    rpowers = unique(rpowers); %I-+Ead0i  
    ;x:rZV/  
    % Pre-compute the values of r raised to the required powers, LJOr!rWi  
    % and compile them in a matrix: {_Lg tu  
    % ----------------------------- Ya;9]k8,  
    if rpowers(1)==0 =egW  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); N nk@h  
        rpowern = cat(2,rpowern{:}); Ea?XT&,  
        rpowern = [ones(length_r,1) rpowern]; *P 3V  
    else /}Lt,9  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); D K=cVpN%s  
        rpowern = cat(2,rpowern{:}); nK$X[KrV'  
    end K-f1{ 0  
    Pfm_@'8  
    % Compute the values of the polynomials: '0\@McU]  
    % -------------------------------------- K"b`#xN(t  
    y = zeros(length_r,length(n)); %e`$p=m  
    for j = 1:length(n) WBNw~|DO]  
        s = 0:(n(j)-m_abs(j))/2; +&Hr4@pgW  
        pows = n(j):-2:m_abs(j); rHf&:~   
        for k = length(s):-1:1 CBDG./  
            p = (1-2*mod(s(k),2))* ... Rb%%?*|  
                       prod(2:(n(j)-s(k)))/              ... $&"V^@  
                       prod(2:s(k))/                     ... 52b*[tZ  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... YKbaf(K )9  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); ?UK|>9y}Z  
            idx = (pows(k)==rpowers); 7lS#f1E  
            y(:,j) = y(:,j) + p*rpowern(:,idx); ovwQ2TuK  
        end f)g7 3=  
         Fe.t/amS/  
        if isnorm MB%Q WU  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); wtT}V=_  
        end N?5x9duK  
    end f+|$&p%  
    % END: Compute the Zernike Polynomials { .*y  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Z0`T\ay  
    &AlJ "N|  
    % Compute the Zernike functions: % , N<  
    % ------------------------------  f>s?4  
    idx_pos = m>0; S.Z9$k%   
    idx_neg = m<0; = pI?A^  
    2P]L9'N{Y  
    z = y; @"Z7nJX  
    if any(idx_pos) 7T"XPV|W6  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); hXb%;GL  
    end n!')wIk  
    if any(idx_neg) K9vIm4::d$  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Qj3a_p$)P  
    end r?CI)Y;  
    *26334B.R  
    % EOF zernfun
    离线niuhelen
    发帖
    19
    光币
    28
    光券
    0
    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) ~+<olss_  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. 6YuY|JD  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated peJKNX.!q  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive Z4){ 7|~a  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, DI`%zLDcY  
    %   and THETA is a vector of angles.  R and THETA must have the same saU]`w_Z*  
    %   length.  The output Z is a matrix with one column for every P-value, QZX~T|Ckv  
    %   and one row for every (R,THETA) pair. Sa"9^_.2#  
    % +n})Y  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike }Z TGi,P c  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) (~$/$%b  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) q~L^au8  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 U!XS;a)  
    %   for all p. 0wFH!s/B  
    % 3+J0!FVla  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 l;sy0S"DO]  
    %   Zernike functions (order N<=7).  In some disciplines it is .bVmqR`  
    %   traditional to label the first 36 functions using a single mode l{VSb92f  
    %   number P instead of separate numbers for the order N and azimuthal /%A;mlf{  
    %   frequency M. +,v-=~5  
    % +3i7D  
    %   Example: hUz[uyt  
    % W23]Bx  
    %       % Display the first 16 Zernike functions KOg?FmD  
    %       x = -1:0.01:1; Y F:2>w<  
    %       [X,Y] = meshgrid(x,x); [+w3J#K  
    %       [theta,r] = cart2pol(X,Y); 8F)G7 H ,  
    %       idx = r<=1; tRw@U4=y  
    %       p = 0:15; `.#@@5e  
    %       z = nan(size(X)); +%N KQ'49I  
    %       y = zernfun2(p,r(idx),theta(idx)); tn|,O.t  
    %       figure('Units','normalized') 4yaxl\2  
    %       for k = 1:length(p) )' xETA  
    %           z(idx) = y(:,k); =2y8 CgLj  
    %           subplot(4,4,k) pium$4l2#  
    %           pcolor(x,x,z), shading interp x'Pi5NRE  
    %           set(gca,'XTick',[],'YTick',[]) kCUT ^  
    %           axis square z"DkFvA  
    %           title(['Z_{' num2str(p(k)) '}']) iRG?# "  
    %       end Rq~t4sA:  
    % rOD1_X-  
    %   See also ZERNPOL, ZERNFUN. 6@tvRDeaDW  
    5)zn:$cz  
    %   Paul Fricker 11/13/2006 ^IgY d*5  
    PiLJZBUv  
    TO;.eN!sv  
    % Check and prepare the inputs: ?IX!+>.H  
    % ----------------------------- ZX b}91rzt  
    if min(size(p))~=1 [O-sVYB  
        error('zernfun2:Pvector','Input P must be vector.') /T0nLp`gi  
    end {+("C] b  
    >+cVs:  
    if any(p)>35 lf>nbvp  
        error('zernfun2:P36', ... (I{ $kB"p  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... %up}p/?  
               '(P = 0 to 35).']) 1'6cGpZY  
    end *i$ePVU  
    %@ mGK8  
    % Get the order and frequency corresonding to the function number: Jx-wO/  
    % ---------------------------------------------------------------- TTI81:fku  
    p = p(:); 0)uYizJce  
    n = ceil((-3+sqrt(9+8*p))/2); (L6Cy% KgV  
    m = 2*p - n.*(n+2); }0=<6\+:`  
    =Pe><k  
    % Pass the inputs to the function ZERNFUN: h`MdKX$  
    % ---------------------------------------- RE46k`44  
    switch nargin KA]*ox6j;  
        case 3 S+x_c4 T  
            z = zernfun(n,m,r,theta); sCH)gr@gJ^  
        case 4 Zu<]bv  
            z = zernfun(n,m,r,theta,nflag); !UV5zmS  
        otherwise fW0$s`  
            error('zernfun2:nargin','Incorrect number of inputs.') ^E>CGGS4  
    end d-!<C7O}  
    !ZtSbOC'  
    % EOF zernfun2
    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) X:q_c=X  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. cqL(^R.  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of `6w#8}  
    %   order N and frequency M, evaluated at R.  N is a vector of jMpa?Jp1  
    %   positive integers (including 0), and M is a vector with the delf ]  
    %   same number of elements as N.  Each element k of M must be a /8CY0Ey  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) `<vxG4=62\  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is 9R]](g#  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix sB7" 0M  
    %   with one column for every (N,M) pair, and one row for every {sc[RRN~C  
    %   element in R. ubGs/Vzye  
    % D\rmaF+  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- `1'5j "v  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is l)@:T|)c  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to #\FT EY!  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 q.oLmX  
    %   for all [n,m]. n9}RW;N+u  
    % h`?k.{})M  
    %   The radial Zernike polynomials are the radial portion of the E <@\>y.[  
    %   Zernike functions, which are an orthogonal basis on the unit uW[3G  
    %   circle.  The series representation of the radial Zernike , {<Fz%  
    %   polynomials is Di.;<v#FL  
    % z}APR@?`n8  
    %          (n-m)/2 CIQwl 6H9  
    %            __ "3y}F  
    %    m      \       s                                          n-2s 84`rbL!M  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r L+R >%d s  
    %    n      s=0 x;U|3{I o  
    % jH0Bo;  
    %   The following table shows the first 12 polynomials. yh!B!v'  
    % ^ KAG|r9  
    %       n    m    Zernike polynomial    Normalization !J# .!}3  
    %       --------------------------------------------- `p|[rS>  
    %       0    0    1                        sqrt(2) #]zhZW4  
    %       1    1    r                           2 +qE']yzm!  
    %       2    0    2*r^2 - 1                sqrt(6) >l2w::l%  
    %       2    2    r^2                      sqrt(6) |cu`f{E2]  
    %       3    1    3*r^3 - 2*r              sqrt(8) dQ6GhS ~  
    %       3    3    r^3                      sqrt(8) ~07RFR  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) 8A/>JD3^  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) oFyeH )!  
    %       4    4    r^4                      sqrt(10) qy9i9$8  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) -A;w$j6*  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) gb_X?j%p7  
    %       5    5    r^5                      sqrt(12) JN^bo(kb  
    %       --------------------------------------------- cHEz{'1m  
    % Z3`2-r_=  
    %   Example: \3j)>u,r  
    % 6+ANAk  
    %       % Display three example Zernike radial polynomials )Pa*+ew7  
    %       r = 0:0.01:1; n_;S2KM  
    %       n = [3 2 5]; Oxj(g;}  
    %       m = [1 2 1]; A;t zRe  
    %       z = zernpol(n,m,r); V/&o]b   
    %       figure 5G oK"F0i  
    %       plot(r,z) w`-$-4i  
    %       grid on `:3&@.{T(  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') T0?uC/7H  
    % oek #^:pF  
    %   See also ZERNFUN, ZERNFUN2. _/ Tlqzp  
    \bRy(Z)  
    % A note on the algorithm. "cQvd(kug  
    % ------------------------ z+Z%H#9e  
    % The radial Zernike polynomials are computed using the series sSy$(%  
    % representation shown in the Help section above. For many special uZ<%kV1B  
    % functions, direct evaluation using the series representation can 9nO(xJ"e4  
    % produce poor numerical results (floating point errors), because r hZQQOQ  
    % the summation often involves computing small differences between F'ENq6  
    % large successive terms in the series. (In such cases, the functions G V=OKf#  
    % are often evaluated using alternative methods such as recurrence q /:T1a7!  
    % relations: see the Legendre functions, for example). For the Zernike ]p\u$VY9  
    % polynomials, however, this problem does not arise, because the )5'rw<:="  
    % polynomials are evaluated over the finite domain r = (0,1), and hw|t8 ShW  
    % because the coefficients for a given polynomial are generally all gu!](yEgl  
    % of similar magnitude. XUf7yD  
    % S_j1=6 #^  
    % ZERNPOL has been written using a vectorized implementation: multiple b.@H1L  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] u(8~4P0w  
    % values can be passed as inputs) for a vector of points R.  To achieve v10mDr  
    % this vectorization most efficiently, the algorithm in ZERNPOL Jxf}b}^T  
    % involves pre-determining all the powers p of R that are required to A ^-Z)0 :  
    % compute the outputs, and then compiling the {R^p} into a single 5'62ulwMP=  
    % matrix.  This avoids any redundant computation of the R^p, and zF=#6  
    % minimizes the sizes of certain intermediate variables. 3]!h{_:u  
    % 2*Va9HP!q  
    %   Paul Fricker 11/13/2006 =#b4c>  
    prqT(1  
    89db5Dx  
    % Check and prepare the inputs: 4p?+LdL  
    % ----------------------------- -ywX5B  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) cB_pyX9Z  
        error('zernpol:NMvectors','N and M must be vectors.') iF8@9m  
    end XRtyC4f  
    gj[z ka0_  
    if length(n)~=length(m) pmoGudaRF  
        error('zernpol:NMlength','N and M must be the same length.') T\)dt?Tv#\  
    end HE@-uh  
    6W]OpM  
    n = n(:); >,kL p|gA  
    m = m(:); 3>H2xh3Y  
    length_n = length(n); %7iUlO}}V  
    0 -!?W  
    if any(mod(n-m,2)) 3,%nkW  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') E7k-pquvE  
    end 4P`PmQ=GQh  
    YfV"_G.ad|  
    if any(m<0) ]"C| qR*  
        error('zernpol:Mpositive','All M must be positive.') r^fxyN2V  
    end E1^aAlVSD  
    !NIL pimi  
    if any(m>n) YomwjKyuP  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') P_kaIPP  
    end >-UD]?>  
    %uh R'8"  
    if any( r>1 | r<0 ) 8y-e+  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') $,}Qf0(S  
    end &"sX^6t  
    -6AOK<kfI  
    if ~any(size(r)==1) 2;dM:FHLhO  
        error('zernpol:Rvector','R must be a vector.') "9)1K!tH  
    end n'! -Pv  
    <m~T>Ql1  
    r = r(:); u4go*#  
    length_r = length(r);  =|^X$H  
    32M6EEmPG  
    if nargin==4 JJ^iy*v  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); Ngn\nkf  
        if ~isnorm C<zx'lw!  
            error('zernpol:normalization','Unrecognized normalization flag.') j7QBU  
        end |3s.;w K  
    else LBW.*PHW  
        isnorm = false; E6,`Ld;c[  
    end Kh>?!` lL  
    Vww@eK%5Q  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% _K 4eD.  
    % Compute the Zernike Polynomials }".\ 4B$n  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% zfk'>_'  
    6lB{Ao?|  
    % Determine the required powers of r: HY*l4QK  
    % ----------------------------------- ~,(0h:8  
    rpowers = []; gn-=##fT:i  
    for j = 1:length(n) h<PYE]?l  
        rpowers = [rpowers m(j):2:n(j)]; \9dz&H  
    end 6^7)GCq [  
    rpowers = unique(rpowers); c.|sW2/  
    -0=}|$H.  
    % Pre-compute the values of r raised to the required powers, M>`?m L  
    % and compile them in a matrix: v1`bDS?*Q  
    % ----------------------------- uYG #c(lc  
    if rpowers(1)==0 3MS3O.0]/  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); f]@[4<Ny  
        rpowern = cat(2,rpowern{:}); >WGX|"!"  
        rpowern = [ones(length_r,1) rpowern]; &xPOp$Sx~  
    else GM34-GH+  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ;g;,%jdCS  
        rpowern = cat(2,rpowern{:}); |[lxV&SD .  
    end yb@X*PW/z  
    mafAC73  
    % Compute the values of the polynomials: BDv|~NHs  
    % -------------------------------------- bAA'=z<  
    z = zeros(length_r,length_n);  e B9m4  
    for j = 1:length_n ZwY`x')  
        s = 0:(n(j)-m(j))/2; 6/p]jN  
        pows = n(j):-2:m(j); -ucz+{  
        for k = length(s):-1:1 v.~Nv@+kR  
            p = (1-2*mod(s(k),2))* ... *@b~f&Lx6  
                       prod(2:(n(j)-s(k)))/          ... g7E`;&f  
                       prod(2:s(k))/                 ... g4BwKENM  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... Z7K!"I  
                       prod(2:((n(j)+m(j))/2-s(k))); LbtlcpF*~5  
            idx = (pows(k)==rpowers); pn ~/!y  
            z(:,j) = z(:,j) + p*rpowern(:,idx); BP7<^`i&  
        end >X@.f1/5X  
         [4V|UvKz  
        if isnorm HnH2u;  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); M8 ^ziZY  
        end @E&X &F%  
    end 8yJk81 gY  
    3RTB~K8:{  
    % EOF zernpol
    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)  S| ?--vai_  
    uTF EI.N  
    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。 h3 ZL0Fi*  
    XsCbJ[Z_?q  
    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)