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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 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 b?T  
    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! /?*]lH.  
     
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    离线phility
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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
    离线phility
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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
    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 Av'GB  
    function z = zernfun(n,m,r,theta,nflag) G 2!xPHz  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. jPZaD>!  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N cWyW~Ek  
    %   and angular frequency M, evaluated at positions (R,THETA) on the ^ vilgg~  
    %   unit circle.  N is a vector of positive integers (including 0), and !> }.~[M  
    %   M is a vector with the same number of elements as N.  Each element r.ZF_^y}+  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) 0tg8~H3yy  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, e]=lKxFh&l  
    %   and THETA is a vector of angles.  R and THETA must have the same !V 2/A1?  
    %   length.  The output Z is a matrix with one column for every (N,M) mtz#}qD66  
    %   pair, and one row for every (R,THETA) pair. YH&bD16c3  
    % Xce0~\_ A  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike qt%D'  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), N- H^lqD  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral 29CINC  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 91>fqe  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized fjk\L\1  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ?`zXLY9q7  
    %  Jc&y9]  
    %   The Zernike functions are an orthogonal basis on the unit circle. ';Zi@f"  
    %   They are used in disciplines such as astronomy, optics, and w@JKl5  
    %   optometry to describe functions on a circular domain. ABE@n%|`  
    % ;2'q_Btk4  
    %   The following table lists the first 15 Zernike functions. . 8N.l^0,  
    % om?-WJI  
    %       n    m    Zernike function           Normalization s*U1  
    %       -------------------------------------------------- >{\7&}gz  
    %       0    0    1                                 1  <1%f@}+8  
    %       1    1    r * cos(theta)                    2 e@:sR  
    %       1   -1    r * sin(theta)                    2 ^j-3av=  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) (jU6GJRP  
    %       2    0    (2*r^2 - 1)                    sqrt(3) ;JZS^Wa  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) U!U$x74D5  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) @2'Mt}R>  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) Z R/#V7Pj  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) 4jD2FFG- G  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) 5waKI?4F  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) zg-2C>(6a  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) M%jPH  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) Xd^\@  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ( Jz;W<E  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) y ]?V~%  
    %       -------------------------------------------------- ME'|saP  
    % o sKKt?^?  
    %   Example 1: ;2B{9{  
    % ^%O]P`$  
    %       % Display the Zernike function Z(n=5,m=1) 8\:NMP8W\  
    %       x = -1:0.01:1; sc,Xw:YO  
    %       [X,Y] = meshgrid(x,x); 6k#Jpmmr  
    %       [theta,r] = cart2pol(X,Y); M|:UwqV>  
    %       idx = r<=1; |4'Y/re  
    %       z = nan(size(X)); U8 nH;}i  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); [jmd  
    %       figure q$=#A7H>3)  
    %       pcolor(x,x,z), shading interp 8#vc(04(  
    %       axis square, colorbar C*P7-oE2rh  
    %       title('Zernike function Z_5^1(r,\theta)') ,\NFt`]j  
    % GvBHd%Ot  
    %   Example 2: s6>ZREf#J  
    % u*hSj)vr1  
    %       % Display the first 10 Zernike functions K4kMM*D  
    %       x = -1:0.01:1; 5LOo8xN  
    %       [X,Y] = meshgrid(x,x); IIbYfPiO  
    %       [theta,r] = cart2pol(X,Y); YpqrZWvh  
    %       idx = r<=1; -Z's@'*  
    %       z = nan(size(X)); %n*-VAfE\  
    %       n = [0  1  1  2  2  2  3  3  3  3]; 8YbE`32  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; EY tQw(!Q  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; M3q|l7|9  
    %       y = zernfun(n,m,r(idx),theta(idx)); z*-2.}&U<  
    %       figure('Units','normalized') b9!FC$^J  
    %       for k = 1:10 L*:jXmUM_~  
    %           z(idx) = y(:,k); rW=Z>1  
    %           subplot(4,7,Nplot(k)) 0=?<y'=  
    %           pcolor(x,x,z), shading interp j:VbrR  
    %           set(gca,'XTick',[],'YTick',[]) !jTcsN%  
    %           axis square :8OZ#D_Hl  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) Oeok ;:  
    %       end :5{wf Am  
    % %\:[ o  
    %   See also ZERNPOL, ZERNFUN2. ,k;^G>< =  
    ;5)P6S.D  
    %   Paul Fricker 11/13/2006 Om5Y|v"*  
    K57&yVX  
    3U0`,c\ao*  
    % Check and prepare the inputs: (=om,g}  
    % ----------------------------- p9x(D/YP0  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) \pVXimam  
        error('zernfun:NMvectors','N and M must be vectors.') <_-hRbS  
    end NGbG4-w-  
    | AozR ~  
    if length(n)~=length(m) rogT~G}q  
        error('zernfun:NMlength','N and M must be the same length.') %4gg@Z9  
    end 2I,^YWR  
    |E JD3 &  
    n = n(:); H["`Mn7j2  
    m = m(:); =Lf,?"S  
    if any(mod(n-m,2)) ^y<<>Y'I  
        error('zernfun:NMmultiplesof2', ... '2 PF  
              'All N and M must differ by multiples of 2 (including 0).') H<PtAYFS  
    end %yv<y+yP~  
    3v1iy / /  
    if any(m>n) AHXSt  
        error('zernfun:MlessthanN', ... T9Nb`sbV]  
              'Each M must be less than or equal to its corresponding N.') & tg&5_  
    end kH G"XTL  
    mj W8 Q\D  
    if any( r>1 | r<0 ) {?:X8&Sf  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') e4>_v('  
    end =4FXBPoQK  
    0.8  2kl  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) NTYg[VTr  
        error('zernfun:RTHvector','R and THETA must be vectors.') JzQ)jdvp  
    end YoBDvV":@  
    AP'*Nh@Ik(  
    r = r(:); R#%(5-Zu#R  
    theta = theta(:); 7/I,HxXp!  
    length_r = length(r); i OW#>66d  
    if length_r~=length(theta) 5kCUaPu  
        error('zernfun:RTHlength', ... E87Ww,z8  
              'The number of R- and THETA-values must be equal.') e4? >-  
    end Vf] "L .G  
    W*Zkc:{eB  
    % Check normalization: =T HpdtL  
    % -------------------- :bwjJ}F  
    if nargin==5 && ischar(nflag) Vl& ?U  
        isnorm = strcmpi(nflag,'norm'); ,$s8GAmq  
        if ~isnorm ChGYTn`X   
            error('zernfun:normalization','Unrecognized normalization flag.') _`&m\Qe>  
        end I ?gSG*m  
    else l]Ax:Z  
        isnorm = false; (k5We!4[1  
    end L^@'q6*}  
    ~A'!2  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% F\KjEl0  
    % Compute the Zernike Polynomials 4T|b Cs?e  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% c;Pe/d  
    M2OIBH4!  
    % Determine the required powers of r: a_f~N1kq  
    % ----------------------------------- PgtJ3oq [}  
    m_abs = abs(m); ON=@ O  
    rpowers = []; JTSlWq4  
    for j = 1:length(n) zzTfYf)  
        rpowers = [rpowers m_abs(j):2:n(j)]; 6e9,PS  
    end  D~S<U  
    rpowers = unique(rpowers); )dbB =OZ  
    m% -g~q  
    % Pre-compute the values of r raised to the required powers, e7Xeo+/  
    % and compile them in a matrix: [ 9 {*94M  
    % ----------------------------- dJJP3} M/  
    if rpowers(1)==0 7}f}$1   
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); e!N:,`R 5  
        rpowern = cat(2,rpowern{:}); JTO~9>$ B  
        rpowern = [ones(length_r,1) rpowern]; _aGOb;h  
    else $PTP/^  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); l{I6&^!KS  
        rpowern = cat(2,rpowern{:}); ^1iSn)&  
    end $HHs^tW  
    DFZkh^PFd  
    % Compute the values of the polynomials: {.?ZHy\Rk  
    % -------------------------------------- {C=NUK%?  
    y = zeros(length_r,length(n)); 4>F'oqFF  
    for j = 1:length(n) xST8|H  
        s = 0:(n(j)-m_abs(j))/2; 6& e3Nt  
        pows = n(j):-2:m_abs(j); \KMToN&2  
        for k = length(s):-1:1 adCU61t  
            p = (1-2*mod(s(k),2))* ... `q}I"iS  
                       prod(2:(n(j)-s(k)))/              ... _<k\FU r  
                       prod(2:s(k))/                     ... F, W~,y  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... v- T$:cL  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); z>58dA@f  
            idx = (pows(k)==rpowers); ML w7}[  
            y(:,j) = y(:,j) + p*rpowern(:,idx); `eE&5.   
        end 2|3)S`WZl  
         ~ELNyI11  
        if isnorm _ky,;9G]  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); LJd5;so-  
        end -I*^-+>H  
    end .AR#&mL9  
    % END: Compute the Zernike Polynomials K&POyOvT  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% .a O,8M  
    Rp.Sj{<2  
    % Compute the Zernike functions: 7mI:| G  
    % ------------------------------ LPZF)@|`  
    idx_pos = m>0; EN$2,qf  
    idx_neg = m<0; M2PAy! J  
    F"&~*m^+  
    z = y; q$I;dOCJ,  
    if any(idx_pos) QQ%D8$k"  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); .>= (' -  
    end H5DC[bZMb%  
    if any(idx_neg) >.Chl$)<  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 9>l*lCA  
    end rSZd!OQ  
    0H6(EzN  
    % EOF zernfun
    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) iX0i2ek  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. W#^2#sjO  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated 9{RB{<Se!  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive  3L< wQ(  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, HBy[FYa4  
    %   and THETA is a vector of angles.  R and THETA must have the same / : L?~  
    %   length.  The output Z is a matrix with one column for every P-value, lpQSup  
    %   and one row for every (R,THETA) pair. i*|\KM?P  
    % SG6kud\b  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike P^ A!.}d  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) j}%ja_9S  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) LgKaPg$  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 iL7DRQ1  
    %   for all p. n!NS(. o  
    % K?[q% W]%  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 /I1h2 E  
    %   Zernike functions (order N<=7).  In some disciplines it is JW{rA6?   
    %   traditional to label the first 36 functions using a single mode E0Y-7&Fv  
    %   number P instead of separate numbers for the order N and azimuthal mkYqpD7  
    %   frequency M. y+X2Pl  
    % (pY 7J  
    %   Example: x}_]A$nV  
    % % W=b? :  
    %       % Display the first 16 Zernike functions ZjgsR|i  
    %       x = -1:0.01:1; xAK6pDp  
    %       [X,Y] = meshgrid(x,x); qlb- jL  
    %       [theta,r] = cart2pol(X,Y); 9{(.Il J>  
    %       idx = r<=1; ySx>L uY#3  
    %       p = 0:15; /q<__N  
    %       z = nan(size(X)); eFaO7mz5V%  
    %       y = zernfun2(p,r(idx),theta(idx)); F<L EQ7T  
    %       figure('Units','normalized') (a[y1{DLy  
    %       for k = 1:length(p) G f,`  
    %           z(idx) = y(:,k); IAw{P08+  
    %           subplot(4,4,k) 7Nk!1s :  
    %           pcolor(x,x,z), shading interp zfc'=ODX  
    %           set(gca,'XTick',[],'YTick',[]) uEktQ_u[  
    %           axis square ,5|&A  
    %           title(['Z_{' num2str(p(k)) '}']) Yn@lr6s  
    %       end n2]/v{E;/  
    % o<[#0T^K   
    %   See also ZERNPOL, ZERNFUN. S#MZV@nGF  
    xCg52zkH#  
    %   Paul Fricker 11/13/2006 qT%FmX  
    { vKLAxc  
    4(|cG7>9-  
    % Check and prepare the inputs: &X|#R1\  
    % ----------------------------- M[mF8Zf  
    if min(size(p))~=1 I,0q4  
        error('zernfun2:Pvector','Input P must be vector.') P* w9 ,  
    end yUZb #%n  
    F ~^Jmp7Y  
    if any(p)>35 LW[9  
        error('zernfun2:P36', ... I(V!Mv8j  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... O_ChxX0KP  
               '(P = 0 to 35).']) |.F$G<  
    end *pSQU=dmS  
    jzDuE{  
    % Get the order and frequency corresonding to the function number: ?\t#1"d  
    % ---------------------------------------------------------------- pimtiQqC  
    p = p(:); yKa{08X:  
    n = ceil((-3+sqrt(9+8*p))/2); Fx;QU)1l3  
    m = 2*p - n.*(n+2); P>s[tM  
    #:[t^}  
    % Pass the inputs to the function ZERNFUN: q=%RDG+  
    % ---------------------------------------- 4x  
    switch nargin {[+mpKq  
        case 3 9f hsIe  
            z = zernfun(n,m,r,theta); PmKeF}  
        case 4 ~io szX  
            z = zernfun(n,m,r,theta,nflag); @)|C/oA  
        otherwise ,cB\  
            error('zernfun2:nargin','Incorrect number of inputs.') P{wF"vf  
    end TygW0b 1  
    K POa|$  
    % EOF zernfun2
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    function z = zernpol(n,m,r,nflag) v2B0q4*BS?  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. d-k%{eBV  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of $npT[~U5  
    %   order N and frequency M, evaluated at R.  N is a vector of y%%}k  
    %   positive integers (including 0), and M is a vector with the qU#1i:(F*  
    %   same number of elements as N.  Each element k of M must be a 1JztFix  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) 7UdM  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is y#U+c*LB  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix ] lrWgm  
    %   with one column for every (N,M) pair, and one row for every H@%GSE  
    %   element in R. 0:9.;x9_  
    % cc~O&?)i  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- n)^i/ nXb'  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is 5@+,Xh,H|t  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to I'uSp-Sfy  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 orWbU UC  
    %   for all [n,m]. "#{4d),r  
    % KX'{[7}m'  
    %   The radial Zernike polynomials are the radial portion of the 6)Y.7XR  
    %   Zernike functions, which are an orthogonal basis on the unit n:yTeZ=-s4  
    %   circle.  The series representation of the radial Zernike &6ZD136  
    %   polynomials is @~YYD#'vNY  
    % FaDjLo2'o  
    %          (n-m)/2 Rm255z p  
    %            __ ^(f"v e#7v  
    %    m      \       s                                          n-2s rA%usaW  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r xI($Uu}S  
    %    n      s=0 VJickXA  
    % _j< K=){  
    %   The following table shows the first 12 polynomials. tjdaaN#,V  
    %  UA48Ug  
    %       n    m    Zernike polynomial    Normalization 19E 8'@  
    %       --------------------------------------------- \=:~ki=@B  
    %       0    0    1                        sqrt(2) Y@N,qHtz  
    %       1    1    r                           2 $}>+kHoT{  
    %       2    0    2*r^2 - 1                sqrt(6) +trC,D  
    %       2    2    r^2                      sqrt(6) ;"dV"W  
    %       3    1    3*r^3 - 2*r              sqrt(8) /v- 6WSN  
    %       3    3    r^3                      sqrt(8) l5Gq|!2yxD  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) \s=t|Wpu2  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) :,'wVS8"]  
    %       4    4    r^4                      sqrt(10) '>cKH$nVC}  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) l49*<nkmq  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) <<+\X:,  
    %       5    5    r^5                      sqrt(12) 3%E }JU?MM  
    %       --------------------------------------------- $\]&rZVi  
    % ;7?kl>5]  
    %   Example: _AAaC_q  
    % 8FKXSqhVM  
    %       % Display three example Zernike radial polynomials [RLN;(0n  
    %       r = 0:0.01:1; nD`w/0hT<  
    %       n = [3 2 5]; K 1 a\b"  
    %       m = [1 2 1]; 9x>d[-#y:J  
    %       z = zernpol(n,m,r); g;qx">xJ`o  
    %       figure 6p?,(  
    %       plot(r,z) y9q8i(E0  
    %       grid on  >qS9PX  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') YwDbPX  
    % N G "C&v  
    %   See also ZERNFUN, ZERNFUN2. rH_\ d?b  
    \;qW 3~  
    % A note on the algorithm. kYG/@7f/  
    % ------------------------ + +M$#Er&  
    % The radial Zernike polynomials are computed using the series YG@t5j#b  
    % representation shown in the Help section above. For many special 5*lT.  
    % functions, direct evaluation using the series representation can 3Z5D)zuc  
    % produce poor numerical results (floating point errors), because i V'k}rXC  
    % the summation often involves computing small differences between VH9dleZ  
    % large successive terms in the series. (In such cases, the functions xTj|dza  
    % are often evaluated using alternative methods such as recurrence i~I%D%;  
    % relations: see the Legendre functions, for example). For the Zernike $ M`hh{ -  
    % polynomials, however, this problem does not arise, because the [@J/eWB  
    % polynomials are evaluated over the finite domain r = (0,1), and A mNW0.}  
    % because the coefficients for a given polynomial are generally all ,l !Ta "  
    % of similar magnitude. [fAV5U  
    % wQ^EYKD  
    % ZERNPOL has been written using a vectorized implementation: multiple tnH2sHby  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] "P 7nNa  
    % values can be passed as inputs) for a vector of points R.  To achieve L^}_~PO N5  
    % this vectorization most efficiently, the algorithm in ZERNPOL ad*m%9Y1Q  
    % involves pre-determining all the powers p of R that are required to _I@9HC 4  
    % compute the outputs, and then compiling the {R^p} into a single SxOC1+Oy  
    % matrix.  This avoids any redundant computation of the R^p, and ,K)_OVB  
    % minimizes the sizes of certain intermediate variables. h"X;3b^ m  
    % qh9Z50E9  
    %   Paul Fricker 11/13/2006 pT=JP> nd^  
    /t+f{VX$  
    B"h#C!E  
    % Check and prepare the inputs: NQBpX  
    % ----------------------------- D{GfL ib"U  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ^2Fei.?T.  
        error('zernpol:NMvectors','N and M must be vectors.') oI }VV6vO  
    end #|L8tuWW  
    yv t.  
    if length(n)~=length(m) %j.0G`x9 +  
        error('zernpol:NMlength','N and M must be the same length.') ULs\+U  
    end */sS`/Lx  
    b$N 2z  
    n = n(:); X{5vXT\/y  
    m = m(:); eD,.~Y#?=  
    length_n = length(n); GeyvId03H  
    [xSF6  
    if any(mod(n-m,2)) ) i;1*jK  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') u+Y\6~=+  
    end Cn,d?H  
    r)y=lAyF>  
    if any(m<0) nV"~-On  
        error('zernpol:Mpositive','All M must be positive.') ((H^2KJn  
    end |Luqoa  
    zd2)M@  
    if any(m>n) arIf'CG6  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') MqW7cjg  
    end |:nn>E}ZA/  
    smlpD3?va  
    if any( r>1 | r<0 ) gH12[Us'`  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') q ?|,O;?  
    end <c2E'U)X  
    j'Gt&\4  
    if ~any(size(r)==1) 00(on28b  
        error('zernpol:Rvector','R must be a vector.') <^&ehy:7y  
    end z>LUH  
    Si_ _8D  
    r = r(:); ni.cTOSx  
    length_r = length(r); (up~[  
    9B{k , 1  
    if nargin==4 \nXtH}9ZF  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); ?4+9fE<Q  
        if ~isnorm um jt]Gu[  
            error('zernpol:normalization','Unrecognized normalization flag.') 2GP=&K/A  
        end gqZ'$7So  
    else v:IpMU-+\  
        isnorm = false; N4v~;;@(  
    end ( l\1n;s*B  
    ASKf '\,dV  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ,vr? 2k  
    % Compute the Zernike Polynomials Njxv4cc  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% /Gd=n  
    Q A< Rhv,  
    % Determine the required powers of r: (7vF/7BZ|_  
    % ----------------------------------- IbT=8l,Li  
    rpowers = []; FtpK)9/4  
    for j = 1:length(n) h?AS{`.1  
        rpowers = [rpowers m(j):2:n(j)]; @3) (BpFe  
    end X$HIVxyq2  
    rpowers = unique(rpowers); M\o9I  
    o2nv+fy W  
    % Pre-compute the values of r raised to the required powers, Q 8T]\6)m  
    % and compile them in a matrix: qB~rQPa  
    % ----------------------------- +NeOSQSj  
    if rpowers(1)==0 /$i.0$L  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); v_BcTzQ0S  
        rpowern = cat(2,rpowern{:}); q8FTi^=Kb  
        rpowern = [ones(length_r,1) rpowern]; rV2WnAb[H&  
    else L9r8BK;  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); G/k2Pe{SL  
        rpowern = cat(2,rpowern{:}); <lgYcdJ   
    end !<j'Ea  
    \9cbI3rGz  
    % Compute the values of the polynomials: :G [|CPm-  
    % -------------------------------------- /$ w%Q-p  
    z = zeros(length_r,length_n); ,`|3KE9  
    for j = 1:length_n 69PE9zz  
        s = 0:(n(j)-m(j))/2; dz:E?  
        pows = n(j):-2:m(j); &TnS4O  
        for k = length(s):-1:1 \RNNg  
            p = (1-2*mod(s(k),2))* ... p ?*Q- f  
                       prod(2:(n(j)-s(k)))/          ... - \ 5v^l  
                       prod(2:s(k))/                 ... rxe >}ZO  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... Cl5uS%g  
                       prod(2:((n(j)+m(j))/2-s(k))); 2L:_rR#w  
            idx = (pows(k)==rpowers); GOj-)i/_  
            z(:,j) = z(:,j) + p*rpowern(:,idx); DH[p\Wy'  
        end v]'ztFA  
         RU'=ERYC  
        if isnorm Z 6t56"u  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); $3W;=Id=+  
        end VEH&&@d  
    end BHIRH mM<Y  
    %oF}HF.  
    % EOF zernpol
    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)  z+ s6)Ad  
    W egtyO  
    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。 "b?v?V0%C  
    0?sRDYaX;c  
    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)