切换到宽版
  • 广告投放
  • 稿件投递
  • 繁體中文
    • 11504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 nC`#Hm.V%  
    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! I+O !<S B  
     
    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 F>*w)6 4~  
    function z = zernfun(n,m,r,theta,nflag) }aX).u  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. ={maCYlE.  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N !2&h=;i~V  
    %   and angular frequency M, evaluated at positions (R,THETA) on the `m'2RNSc+#  
    %   unit circle.  N is a vector of positive integers (including 0), and 8-8= \  
    %   M is a vector with the same number of elements as N.  Each element -JwH^*Ad  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) M;Vx[s,#,  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, XTW/3pB  
    %   and THETA is a vector of angles.  R and THETA must have the same e`}|*^-  
    %   length.  The output Z is a matrix with one column for every (N,M) 8CEy#%7]}  
    %   pair, and one row for every (R,THETA) pair. cW&OVNj  
    % 5&94VQ$d  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike yx/:<^"-$  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), p3x(:=   
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral Pi*,&D>{7  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, &a:>P>\  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized @~gz-l^$  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. |Z2_1( ku  
    % t]vX9vv+D  
    %   The Zernike functions are an orthogonal basis on the unit circle. k7W8$8 v  
    %   They are used in disciplines such as astronomy, optics, and R~Xl(O  
    %   optometry to describe functions on a circular domain. pbm4C0W}  
    % 'w9tZO\2  
    %   The following table lists the first 15 Zernike functions. kV<VhBql!  
    % };zF&  
    %       n    m    Zernike function           Normalization PwDQ<   
    %       -------------------------------------------------- e[e2X<&0RT  
    %       0    0    1                                 1 @&M $`b ^  
    %       1    1    r * cos(theta)                    2 g]d"d  
    %       1   -1    r * sin(theta)                    2 L YH9P-5H  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) * rs_k/2(  
    %       2    0    (2*r^2 - 1)                    sqrt(3) 'Y"q=@Ei9  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) `C!Pe84(  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) o-)E_X  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) Z.R^@@RqJ  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) "sHD8TUX  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) {h@R\bU  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) $ \P!P.  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) rqa;MPl  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) msoE8YK&tg  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10)  R6AZIN:  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) + qS$t  
    %       -------------------------------------------------- fYCAwS{  
    % +pjD{S~Y  
    %   Example 1: fwl RwH(  
    % zSq+#O1#  
    %       % Display the Zernike function Z(n=5,m=1) 9'4cqR  
    %       x = -1:0.01:1; fk",YtS*  
    %       [X,Y] = meshgrid(x,x); Bq$bxuhV  
    %       [theta,r] = cart2pol(X,Y); +F0M?,  
    %       idx = r<=1; &2) mpY8xQ  
    %       z = nan(size(X)); +w}5-8mH&>  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); u(REEc~nj  
    %       figure MOOL=Um3  
    %       pcolor(x,x,z), shading interp >)VrbPRuA  
    %       axis square, colorbar m Y%PG  
    %       title('Zernike function Z_5^1(r,\theta)') Pp.X Du  
    % ^R2:Z&Iv%  
    %   Example 2: !J6k\$r  
    % -i;#4@^t  
    %       % Display the first 10 Zernike functions Wxg|jP$~   
    %       x = -1:0.01:1; #D}NT*w/  
    %       [X,Y] = meshgrid(x,x); n ~ =]/  
    %       [theta,r] = cart2pol(X,Y); #~ >0Dr  
    %       idx = r<=1; &t6L8[#yd  
    %       z = nan(size(X));  kU#$  
    %       n = [0  1  1  2  2  2  3  3  3  3]; &i!.6M2  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; AalyEn&>  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; I/'jRM  
    %       y = zernfun(n,m,r(idx),theta(idx)); KD#ip3  
    %       figure('Units','normalized') rN>f"/J |  
    %       for k = 1:10 fC81(5   
    %           z(idx) = y(:,k); 9vVYZ}HC  
    %           subplot(4,7,Nplot(k)) jN B-FVaT  
    %           pcolor(x,x,z), shading interp ,':?3| $c  
    %           set(gca,'XTick',[],'YTick',[]) cZHlW|$R  
    %           axis square GadD*psD2  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) a Fl(K\  
    %       end wRWN]Vo  
    % E7 7Au;TL  
    %   See also ZERNPOL, ZERNFUN2. [zY9"B<3  
    i*F^;-q)  
    %   Paul Fricker 11/13/2006 L%=u&9DmU  
    ThFI=K  
    Q+#, VuM  
    % Check and prepare the inputs: i  #8)ad  
    % ----------------------------- ZgzrA&6  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) /dtFB5Z"w  
        error('zernfun:NMvectors','N and M must be vectors.') 9v/1>rziE  
    end Aw >DZ2  
    ^#_@Kq%th  
    if length(n)~=length(m) 8vchLl#  
        error('zernfun:NMlength','N and M must be the same length.') ,@GI3bl  
    end /y NU0/  
    %"{SGp  
    n = n(:); ! 5]/2  
    m = m(:); E*k=8$Y  
    if any(mod(n-m,2)) M|e@N  
        error('zernfun:NMmultiplesof2', ... T} U`?s`)  
              'All N and M must differ by multiples of 2 (including 0).') 539[,jH  
    end rw58bkh6  
    :5p`H  
    if any(m>n) bY]aADv\  
        error('zernfun:MlessthanN', ... KZ&8aulP  
              'Each M must be less than or equal to its corresponding N.') ^F_c'  
    end %m{h1UQQ +  
    ^Z;5e@S  
    if any( r>1 | r<0 ) 2}hEBw68  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') f`vB$r>  
    end , @(lYeD"  
    -R| v&h%T  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) *\-6p0~A  
        error('zernfun:RTHvector','R and THETA must be vectors.') @#;~_?$?C  
    end Y+lZT4w  
    0riTav8  
    r = r(:); n{=vP`V_  
    theta = theta(:); 2gukK8R$  
    length_r = length(r); o5A@U0c_  
    if length_r~=length(theta) ,uK }$l  
        error('zernfun:RTHlength', ... %n T!u!#  
              'The number of R- and THETA-values must be equal.')  ig jr=e  
    end ?3"lI,!0  
    +>Y2luR1  
    % Check normalization: }eSaF@.  
    % -------------------- #sN]6  
    if nargin==5 && ischar(nflag) _-^a8F>/19  
        isnorm = strcmpi(nflag,'norm'); -=@d2LY  
        if ~isnorm eUQrn>`  
            error('zernfun:normalization','Unrecognized normalization flag.') lfK sqe"  
        end `l'z#\  
    else z'j4^Xz?%$  
        isnorm = false; N-y[2]J90  
    end !CY: XQm  
    <V>]-bl/  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% _-$(=`8|<{  
    % Compute the Zernike Polynomials <D%.'=%pZ  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 4ba[*R2  
    Y2W|b5  
    % Determine the required powers of r: MA6(VII  
    % ----------------------------------- 3c}@_Yn  
    m_abs = abs(m); o7;lR?  
    rpowers = []; gwm!Pw j  
    for j = 1:length(n) WB K6Ug  
        rpowers = [rpowers m_abs(j):2:n(j)]; Z8%?ej`8  
    end X@RS /  
    rpowers = unique(rpowers); _ VKBzOH  
    Uc^eIa@  
    % Pre-compute the values of r raised to the required powers, A+de;&  
    % and compile them in a matrix: g]vo."}5E  
    % ----------------------------- Je5}Z.3m  
    if rpowers(1)==0 ose(#n40  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); qILb>#  
        rpowern = cat(2,rpowern{:}); T\?$7$/V  
        rpowern = [ones(length_r,1) rpowern]; z{`K_s%5  
    else w;W# 'pE  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); kOdXbw9v  
        rpowern = cat(2,rpowern{:}); %<8`(Uu5  
    end iO+,U}&  
    \2)D  
    % Compute the values of the polynomials: Swa0TiT(  
    % -------------------------------------- jVi> 9[rz  
    y = zeros(length_r,length(n)); h! =h0  
    for j = 1:length(n) @ <(4J   
        s = 0:(n(j)-m_abs(j))/2; Pm&hv*D  
        pows = n(j):-2:m_abs(j); =HMa<"-8  
        for k = length(s):-1:1 n&OM~Vs  
            p = (1-2*mod(s(k),2))* ... }C4wED.  
                       prod(2:(n(j)-s(k)))/              ... U}@xMt8@l  
                       prod(2:s(k))/                     ... ;`Nh@*_  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ckGmwYP9  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); HxSq &j*F  
            idx = (pows(k)==rpowers); O,6Wdw3+-3  
            y(:,j) = y(:,j) + p*rpowern(:,idx); 3{$vN).  
        end (qglD  
         ' _d4[Olu  
        if isnorm Yw] 7@  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); >3ax `8  
        end Xii>?sA5Z"  
    end "i#aII+T  
    % END: Compute the Zernike Polynomials 0civXZgj  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \?SvO  
    <qg4Rz\c]  
    % Compute the Zernike functions: TZ *>MySiF  
    % ------------------------------ vd?Bk_d9k,  
    idx_pos = m>0; ?4A/?Z]ub  
    idx_neg = m<0; w 5 yOSz  
    %UAF~2]g  
    z = y; 2ah%,o  
    if any(idx_pos) U0gZf5;*  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); a`L:E'|B9  
    end Ve2{;`t  
    if any(idx_neg) KL9k9|!p  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); }}"pQ!Z  
    end 84vd~Cf 9  
    e2f+Fv 9  
    % 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) Wl |5EY  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. 2 /FQ;<L  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated jMgXIK\  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive Hs*["zFc  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, ,Cb3R|L8  
    %   and THETA is a vector of angles.  R and THETA must have the same #8|LPfA  
    %   length.  The output Z is a matrix with one column for every P-value, L8 L1_  
    %   and one row for every (R,THETA) pair. ,Klv[_x7  
    % MC* Hl`C  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike <%HRs>4  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) ,;_+o]  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) 0?<#!  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 < cvh1~>(  
    %   for all p. h:nybLw?  
    % 7~ PL8  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 OvtE)u l@  
    %   Zernike functions (order N<=7).  In some disciplines it is sU"%,Q5  
    %   traditional to label the first 36 functions using a single mode DcW?L^Mst  
    %   number P instead of separate numbers for the order N and azimuthal G 5;6q  
    %   frequency M. >> zd  
    % VG);om7`PD  
    %   Example: O\6U2b~  
    % >#w;67he2  
    %       % Display the first 16 Zernike functions !R=@Nr>  
    %       x = -1:0.01:1; $@>0;i ::  
    %       [X,Y] = meshgrid(x,x); #;$]M4  
    %       [theta,r] = cart2pol(X,Y); j{@6y  
    %       idx = r<=1; TxX=(7V  
    %       p = 0:15; ){*+s RBW  
    %       z = nan(size(X)); u= NLR\  
    %       y = zernfun2(p,r(idx),theta(idx)); O$<>v\NC?  
    %       figure('Units','normalized') $"r9U|6kk  
    %       for k = 1:length(p) m1l6QcT1  
    %           z(idx) = y(:,k); 7;s#QqG`I  
    %           subplot(4,4,k) uh )S;3|  
    %           pcolor(x,x,z), shading interp !y= R)k  
    %           set(gca,'XTick',[],'YTick',[]) 8R,<S-+v  
    %           axis square BmG(+;;&  
    %           title(['Z_{' num2str(p(k)) '}']) zxbf h/=  
    %       end %2?+:R5.  
    % U ? +_\  
    %   See also ZERNPOL, ZERNFUN. DN*5q9.  
    WMSJU/-P  
    %   Paul Fricker 11/13/2006 l4OrlS/5  
    aQCu3T  
    DxJ;C09xNa  
    % Check and prepare the inputs: tAdE<).!  
    % ----------------------------- UEU/505  
    if min(size(p))~=1 CL|/I:%0  
        error('zernfun2:Pvector','Input P must be vector.') *MP.YI:h  
    end +$h  
    l/&.HF  
    if any(p)>35 9a}9cMJ^"  
        error('zernfun2:P36', ... e$# *t  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... 4:`D3  
               '(P = 0 to 35).']) 5 4gr'qvr  
    end fw%`[( hK  
    Fx9-A8oIR  
    % Get the order and frequency corresonding to the function number: 8xAV[i  
    % ---------------------------------------------------------------- UB/> Ro  
    p = p(:); WsI`!ez;D  
    n = ceil((-3+sqrt(9+8*p))/2); Cn{Hk)6  
    m = 2*p - n.*(n+2); lW+mH=  
    CMa6':~  
    % Pass the inputs to the function ZERNFUN: 2 !s&|lI  
    % ---------------------------------------- |$RNY``J  
    switch nargin lQn" 6o1  
        case 3 b 7UJ  
            z = zernfun(n,m,r,theta); IH]9%d)  
        case 4 *'%V}R[>  
            z = zernfun(n,m,r,theta,nflag); %FO{:@CH  
        otherwise (l{vlFWd  
            error('zernfun2:nargin','Incorrect number of inputs.') TNX9Z)=>g  
    end b)LT[>f  
    !"rPSGK*  
    % 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) z/@_?01T=  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. Ei;tfB  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of y.rN(  
    %   order N and frequency M, evaluated at R.  N is a vector of IGlR,tw_/  
    %   positive integers (including 0), and M is a vector with the )!T~l(g  
    %   same number of elements as N.  Each element k of M must be a O'y8q[2KE  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) 2]>O ZhS  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is v}B%:1P4  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix S;|:ci<[=  
    %   with one column for every (N,M) pair, and one row for every vQAFgG  
    %   element in R. ^h(wi`i  
    % !l:GrT8J  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- D /eH~  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is +#O+%!  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to ><V*`{bD9)  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 Dl,QCZeM  
    %   for all [n,m]. %y1!'R:ZW  
    % d*(aue=  
    %   The radial Zernike polynomials are the radial portion of the K,b M9>}  
    %   Zernike functions, which are an orthogonal basis on the unit YeH!v, >  
    %   circle.  The series representation of the radial Zernike @u~S!(7.Wi  
    %   polynomials is &Y@i:O  
    % 8|u4xf<  
    %          (n-m)/2 HU3:6R&  
    %            __ m'4f'tbN  
    %    m      \       s                                          n-2s PwY/VGT  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r 9}573M  
    %    n      s=0 &w@]\7L,:  
    % +-9vrEB  
    %   The following table shows the first 12 polynomials. D=tZ}_'{t  
    % 0I}e>]:I  
    %       n    m    Zernike polynomial    Normalization @"@a70WHk  
    %       --------------------------------------------- D6 B-#u!M  
    %       0    0    1                        sqrt(2) ;KeU f(tH  
    %       1    1    r                           2 u9lZHh#V-  
    %       2    0    2*r^2 - 1                sqrt(6) b 2gng}  
    %       2    2    r^2                      sqrt(6) ."Ms7=  
    %       3    1    3*r^3 - 2*r              sqrt(8) iD^,O)b  
    %       3    3    r^3                      sqrt(8) nl@an!z  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) RObnu*  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) .@1+}0  
    %       4    4    r^4                      sqrt(10) \kADh?phV  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) TpjiKM  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) Z6!Up1  
    %       5    5    r^5                      sqrt(12) Z!p\=M,%  
    %       --------------------------------------------- RLF&-[mr3  
    % N&9o  1_}  
    %   Example: k,h602(  
    % v.0qE}' |  
    %       % Display three example Zernike radial polynomials o%d TcoCN  
    %       r = 0:0.01:1; @]\fO)\f  
    %       n = [3 2 5]; Fs+ tcr/\[  
    %       m = [1 2 1]; QX,$JM3  
    %       z = zernpol(n,m,r); G0FzXtu)q  
    %       figure BK$y>= `  
    %       plot(r,z) j3-YZKpg  
    %       grid on n1[c\1   
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') &kb`)F3nU  
    % P_bB{~$4  
    %   See also ZERNFUN, ZERNFUN2. xR7ZqTcw  
    ^r}c&@  
    % A note on the algorithm. STKL  
    % ------------------------ Zxk~X}K\P  
    % The radial Zernike polynomials are computed using the series FO{=^I5YA  
    % representation shown in the Help section above. For many special C.j+Zb1Z(  
    % functions, direct evaluation using the series representation can U(&c@u%  
    % produce poor numerical results (floating point errors), because qwTz7r  
    % the summation often involves computing small differences between @"w4R6l+*  
    % large successive terms in the series. (In such cases, the functions JWVV?~1  
    % are often evaluated using alternative methods such as recurrence HC`0Ni1  
    % relations: see the Legendre functions, for example). For the Zernike X>(1fra4  
    % polynomials, however, this problem does not arise, because the _]:b@gXUw  
    % polynomials are evaluated over the finite domain r = (0,1), and q'3{M]Tk  
    % because the coefficients for a given polynomial are generally all WM bkKC.{J  
    % of similar magnitude. _&KqmQ8$7  
    % ) u?f| D  
    % ZERNPOL has been written using a vectorized implementation: multiple pEyZH!W  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] z]7 WC  
    % values can be passed as inputs) for a vector of points R.  To achieve zzmC[,u}  
    % this vectorization most efficiently, the algorithm in ZERNPOL {v={q1  
    % involves pre-determining all the powers p of R that are required to ULx:2jz  
    % compute the outputs, and then compiling the {R^p} into a single 'nmGHorp  
    % matrix.  This avoids any redundant computation of the R^p, and 0uy'Py@2<  
    % minimizes the sizes of certain intermediate variables. !$I~3_c  
    % ];bRRBEU  
    %   Paul Fricker 11/13/2006 _~FfG!H ^X  
    DP_b9o \5  
    r6<;bO(  
    % Check and prepare the inputs: Bk8}K=%w  
    % ----------------------------- nz 10/nw  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) G?Et$r7:R  
        error('zernpol:NMvectors','N and M must be vectors.') bpu`'Vx  
    end *)^6'4=  
    %-hSa~20  
    if length(n)~=length(m) {X,%GI  
        error('zernpol:NMlength','N and M must be the same length.') 8t+eu O  
    end /<[0o]  
    ixTjXl2g  
    n = n(:); UB~K/r`.|  
    m = m(:); eqtZU\GI>  
    length_n = length(n); )@]%:m!ER  
    iSfRJ:_&6  
    if any(mod(n-m,2)) (Tx_`rO4VY  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') O]:9va  
    end ammi4k/  
    M1jT+  
    if any(m<0) :s)cTq|3  
        error('zernpol:Mpositive','All M must be positive.') KGt:  
    end &X4anH>O  
    2H%9l@}u  
    if any(m>n) Ir;JYY!0?  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') ^}  {r@F  
    end IIk_!VzT  
    s.M39W?  
    if any( r>1 | r<0 ) +!).'  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') A}fm).Wp@  
    end SQMl5d1d:  
    py6<QoGV  
    if ~any(size(r)==1) 0 kJ8H!~u  
        error('zernpol:Rvector','R must be a vector.') .zb  
    end Xj, %t}  
    M)13'B.  
    r = r(:); 2EgvS!"  
    length_r = length(r); `IN!#b+Eo  
    i)l0[FNI}  
    if nargin==4 Y9BQLu4F  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); Zo UeLU  
        if ~isnorm n7> |$2Y  
            error('zernpol:normalization','Unrecognized normalization flag.') 8Qi)E 1n  
        end "{<X! ^u>  
    else lxd{T3LU  
        isnorm = false; r8"2C#  
    end bvD}N<>3N  
    ` wa;@p+j8  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% t?hfP2&6  
    % Compute the Zernike Polynomials coCT]<  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% _2KIe(,;  
    RvG=GJJ9  
    % Determine the required powers of r: [aSuEu?mC  
    % ----------------------------------- 9]Jv >_W*  
    rpowers = []; ?}`- ?JB1  
    for j = 1:length(n) ^%!{qAp}Z  
        rpowers = [rpowers m(j):2:n(j)]; 8K4^05*S   
    end l8~(bq1  
    rpowers = unique(rpowers); >/ _#+,  
    (iKJ~bJ  
    % Pre-compute the values of r raised to the required powers, xLed];2G  
    % and compile them in a matrix: S(@kdL  
    % ----------------------------- |GMo"[  
    if rpowers(1)==0 iM!Ya!  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ")KqPD6k  
        rpowern = cat(2,rpowern{:}); V u")%(ix  
        rpowern = [ones(length_r,1) rpowern]; s.4+5rE  
    else A=kOSq 4Q  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ge`GQ>  
        rpowern = cat(2,rpowern{:}); )4rt-_t<  
    end /<_!Gz.@uG  
    +-tFgXG  
    % Compute the values of the polynomials: J4+WF#xI2  
    % -------------------------------------- x[mz`0  
    z = zeros(length_r,length_n); ;PaU"z+Je~  
    for j = 1:length_n -sJ1q^;f@  
        s = 0:(n(j)-m(j))/2; =]%,&Se  
        pows = n(j):-2:m(j); et5lfj  
        for k = length(s):-1:1 7R:j^"I@  
            p = (1-2*mod(s(k),2))* ... A~xw:[zy$a  
                       prod(2:(n(j)-s(k)))/          ... he(K   
                       prod(2:s(k))/                 ... S ,F[74K  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... z5gVP8*z5  
                       prod(2:((n(j)+m(j))/2-s(k))); wa<k%_# M  
            idx = (pows(k)==rpowers); +TbAtkEF*  
            z(:,j) = z(:,j) + p*rpowern(:,idx); xHt7/8wF  
        end Jqb~RP~  
         XaCvBQ  
        if isnorm U!uPf:p2  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); Xz@#,F:@  
        end c:7V..   
    end Hc\C0V<  
    #b/L~Bw[  
    % 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)  ={;pg(  
    86)2\uan  
    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。 I9 zs  
    h,R Isq;`  
    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)