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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 Azq#}Oe)u  
    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! 3%vx' 1h[  
     
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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 *]>OCGsr  
    function z = zernfun(n,m,r,theta,nflag) n_xQSVI0F  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. ]Gd]KP@S  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N V)?x*R*T)  
    %   and angular frequency M, evaluated at positions (R,THETA) on the 9TXm Z  
    %   unit circle.  N is a vector of positive integers (including 0), and d'g{K]=tF  
    %   M is a vector with the same number of elements as N.  Each element @=<TA0;LL  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) d~z<,_ r5c  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, Tm~#wL +r  
    %   and THETA is a vector of angles.  R and THETA must have the same {7pE9R5  
    %   length.  The output Z is a matrix with one column for every (N,M) RfKxwo|M<  
    %   pair, and one row for every (R,THETA) pair. k>z-Zg  
    % 2Z IpzH/8  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 1 Z$99  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), EH!EyNNb  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral o7 -h'b-  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, NM.f0{:cj  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized k`4\.m"&  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. B,VSFpPx  
    % $O]E$S${  
    %   The Zernike functions are an orthogonal basis on the unit circle. #35S7G^@`  
    %   They are used in disciplines such as astronomy, optics, and L&gEQDPgq|  
    %   optometry to describe functions on a circular domain. cwW~ *90#  
    % nO.+&kA  
    %   The following table lists the first 15 Zernike functions. Ci#5@Q9#w  
    % \%4+mgiD  
    %       n    m    Zernike function           Normalization C;:1CK  
    %       -------------------------------------------------- ~3-YxCn%  
    %       0    0    1                                 1 H R!>g  
    %       1    1    r * cos(theta)                    2 9:Z~}yX  
    %       1   -1    r * sin(theta)                    2 kV(DnZ#jq  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) , LPFb6o  
    %       2    0    (2*r^2 - 1)                    sqrt(3) RVKaqJ0e<  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) 9q ,Jq B  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) JpHsQ8<  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) r`E1<aCr|  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) W-ND<=:Up  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) 4[EO[x4C  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) hjp?/i%TQ  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Z FrXw+  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) ^CZ|ci6bX  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) -{amzyvLE  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) yNMwd.r[  
    %       -------------------------------------------------- +MoxvW6  
    % AU?YZEAei  
    %   Example 1: R^O)fL0_  
    % }Yl8Q>t  
    %       % Display the Zernike function Z(n=5,m=1) K'rs9v"K|  
    %       x = -1:0.01:1; 7;s0m0<%~  
    %       [X,Y] = meshgrid(x,x); [6gHi.`p'  
    %       [theta,r] = cart2pol(X,Y); ,HO/Q6;N  
    %       idx = r<=1; E#V-F-@2  
    %       z = nan(size(X)); ^l2d?v8  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); Qs[EA_  
    %       figure 68br  
    %       pcolor(x,x,z), shading interp =/'*(\C2  
    %       axis square, colorbar ^d $e^cU  
    %       title('Zernike function Z_5^1(r,\theta)') 8}`8lOE7  
    % Xqva&/-  
    %   Example 2: r_<i*l.  
    % sL`D}_:  
    %       % Display the first 10 Zernike functions C%o/  
    %       x = -1:0.01:1; p`.fYW:p  
    %       [X,Y] = meshgrid(x,x); "N:]d*A\  
    %       [theta,r] = cart2pol(X,Y); j\L$dPZ  
    %       idx = r<=1; Glc4g  
    %       z = nan(size(X)); aTL7"Myp  
    %       n = [0  1  1  2  2  2  3  3  3  3]; <^c0bY1  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; 9 v3Nba  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; MJR\ g3  
    %       y = zernfun(n,m,r(idx),theta(idx)); "&o@%){]  
    %       figure('Units','normalized') x3F L/^S  
    %       for k = 1:10 jP6G.aiO  
    %           z(idx) = y(:,k); 0$h$7'a  
    %           subplot(4,7,Nplot(k)) Y~ ?YA/.x  
    %           pcolor(x,x,z), shading interp hfa_M[#Q-  
    %           set(gca,'XTick',[],'YTick',[]) jN{xpd  
    %           axis square X10TZ  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) w)SxwlW}  
    %       end -ns a3P  
    % {"AYOc>2|  
    %   See also ZERNPOL, ZERNFUN2. Pw{{+PBu R  
    t4W0~7   
    %   Paul Fricker 11/13/2006 |2` $g  
    YZu# 0)  
    x(6.W"-S  
    % Check and prepare the inputs: _BaS\U%1(  
    % ----------------------------- !b8|{#qh.  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) j|8{Vyqd  
        error('zernfun:NMvectors','N and M must be vectors.') X"59`Yh  
    end @!HMd{r  
    ptL}F~  
    if length(n)~=length(m) (&x\,19U$  
        error('zernfun:NMlength','N and M must be the same length.') zq>"a&Y,  
    end |L-juT X9  
    j'b4Sb s-f  
    n = n(:); j 0NPd^  
    m = m(:); A^7Zy79  
    if any(mod(n-m,2)) rxA)&  
        error('zernfun:NMmultiplesof2', ... ^Iq.0E9_  
              'All N and M must differ by multiples of 2 (including 0).') aV#;o9H{  
    end pODo[Rkq  
    v333z<<S  
    if any(m>n) S$:S*6M@"  
        error('zernfun:MlessthanN', ... O@&I.d$  
              'Each M must be less than or equal to its corresponding N.') Rzj!~`&N  
    end v^E2!X  
    KywT Oq  
    if any( r>1 | r<0 ) !t{!.  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') \K=PIcH  
    end U^S:2  
    c=E.-  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) QCnVZ" !(  
        error('zernfun:RTHvector','R and THETA must be vectors.') ds[~Cp   
    end 9Dkgu ^`  
    0kEq|k9  
    r = r(:); O/@[VPf  
    theta = theta(:); @3D%i#2o&[  
    length_r = length(r); .v8=zi:7Y  
    if length_r~=length(theta) v65r@)\`  
        error('zernfun:RTHlength', ... CBHWMetJ*  
              'The number of R- and THETA-values must be equal.') Kwau:_B  
    end :fUmMta  
    6-}9m7#Y  
    % Check normalization: t')I c6.?i  
    % -------------------- B}T72!a  
    if nargin==5 && ischar(nflag) mJqP#Unik  
        isnorm = strcmpi(nflag,'norm'); Z)4P>{  
        if ~isnorm `/iN%ZKum  
            error('zernfun:normalization','Unrecognized normalization flag.') p 1fnuN |,  
        end -OAH6U9^  
    else $o^}<)DW  
        isnorm = false; Etk<`GRfA  
    end I<#kw)W!  
    6P $q7G  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %QkvBg*  
    % Compute the Zernike Polynomials 69L&H!<i:  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 1Pc'wfj  
    }DwXs`M7  
    % Determine the required powers of r: vsR&1hs  
    % ----------------------------------- Vngi8%YWp  
    m_abs = abs(m); IRY2H#:$  
    rpowers = []; 9`b3=&i\  
    for j = 1:length(n) Kep?=9r4+  
        rpowers = [rpowers m_abs(j):2:n(j)]; s=+G%B'  
    end T[J_/DE@  
    rpowers = unique(rpowers); XoOe=V?I )  
    0U~JSmj:2K  
    % Pre-compute the values of r raised to the required powers, B5S1F4  
    % and compile them in a matrix: uEY5&wX`  
    % ----------------------------- ^a r9$$~/!  
    if rpowers(1)==0 u[@*}|uXM  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ~*Wb MA  
        rpowern = cat(2,rpowern{:}); S([De"y  
        rpowern = [ones(length_r,1) rpowern]; z@}~2K  
    else 2Ev,dWV  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); P'';F}NwfX  
        rpowern = cat(2,rpowern{:}); 6ZJQ '9f  
    end b1"wQM9  
    ,.B8hr@H6-  
    % Compute the values of the polynomials: s,= ^V/c  
    % -------------------------------------- 6w#v,RDEu  
    y = zeros(length_r,length(n)); OYkd?LN  
    for j = 1:length(n) ~<3yTl>  
        s = 0:(n(j)-m_abs(j))/2; ~Fh(4'  
        pows = n(j):-2:m_abs(j); hR2.w/2j  
        for k = length(s):-1:1 _L ].n)b  
            p = (1-2*mod(s(k),2))* ... *{bqHMd4L  
                       prod(2:(n(j)-s(k)))/              ... $6[]c)(  
                       prod(2:s(k))/                     ... G<I5%Yo6G  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 'tj4;+xf^  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); r}w 9?s^rB  
            idx = (pows(k)==rpowers); SQ[}]Tm;n  
            y(:,j) = y(:,j) + p*rpowern(:,idx); &-9D.'WzP  
        end xYq8\9Qb  
         ;DOz92X94  
        if isnorm VrG|/2  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 'lF|F+8   
        end PC5FfX  
    end mCo5 Gdt  
    % END: Compute the Zernike Polynomials +( d2hSIF  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% !~#31kL&  
    l%O-c}X  
    % Compute the Zernike functions: {_JLmyaerZ  
    % ------------------------------ &DV'%h>i=  
    idx_pos = m>0; 4KKNw9L)  
    idx_neg = m<0; 6r`g+Js/  
    ~*qGH  
    z = y; Vl%k:  
    if any(idx_pos) C%&7,F7  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); J&?kezs  
    end iT5%X   
    if any(idx_neg) 0qv)'[O  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 5y)kQ<x"  
    end Us<lWEX;k  
    uE2Y n`Ha  
    % 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) c:bB4ch}  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. Ldu!uihx  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated al2v1.Y}  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive $t]DxMd  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, rtI4W  
    %   and THETA is a vector of angles.  R and THETA must have the same ]Vubz54  
    %   length.  The output Z is a matrix with one column for every P-value, cIX59y#7  
    %   and one row for every (R,THETA) pair. 5]{YERa'  
    % ;rFa I^  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike zAH+{4lC+  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) 2a G<^3  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) q&+GpR  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 wNPZ[V:  
    %   for all p. Og["X0j  
    % V3-LVgM%  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 j6\{j#q  
    %   Zernike functions (order N<=7).  In some disciplines it is =\3*;59\  
    %   traditional to label the first 36 functions using a single mode 1ayxE(vMcX  
    %   number P instead of separate numbers for the order N and azimuthal 6 3HxQH  
    %   frequency M. XDn$=`2  
    % =($qiL'h  
    %   Example: .sLx6J%  
    % 2`z+_DA  
    %       % Display the first 16 Zernike functions 1F=x~FMvY  
    %       x = -1:0.01:1; r"n)I$  
    %       [X,Y] = meshgrid(x,x); \3KCZ  
    %       [theta,r] = cart2pol(X,Y); T57S!CJ^$5  
    %       idx = r<=1; W&"FejD  
    %       p = 0:15; rnW i<Se  
    %       z = nan(size(X)); d&fENnt?h  
    %       y = zernfun2(p,r(idx),theta(idx)); Pvtf_Qo^  
    %       figure('Units','normalized') fhC=MJ @  
    %       for k = 1:length(p) f_ ::?  
    %           z(idx) = y(:,k); FnCHbPlb  
    %           subplot(4,4,k) *33Zt+  
    %           pcolor(x,x,z), shading interp 6 2LZ}yn_"  
    %           set(gca,'XTick',[],'YTick',[]) CV`  I.  
    %           axis square XW19hG  
    %           title(['Z_{' num2str(p(k)) '}']) q3;HfZ  
    %       end $FAl9  
    % ie_wJ=s  
    %   See also ZERNPOL, ZERNFUN. wk3yz6V2  
    ''V:+@Toh  
    %   Paul Fricker 11/13/2006 55#H A?cR  
    X<1# )xC  
    FNUue  
    % Check and prepare the inputs: O9qEKW)a  
    % ----------------------------- LOQEU? z  
    if min(size(p))~=1 Tx!mW-Lt  
        error('zernfun2:Pvector','Input P must be vector.') >FY`xl\m}<  
    end 8U-}%D<a  
    NZ Xmrc{S  
    if any(p)>35 $,R|$0B7  
        error('zernfun2:P36', ... )37|rB E  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... Q4UaqiL  
               '(P = 0 to 35).']) U?kJXM2  
    end j/9'L^]  
    |=SaI%%Be  
    % Get the order and frequency corresonding to the function number: J3K!@m_\  
    % ---------------------------------------------------------------- wc ^z9y  
    p = p(:); *t~( _j  
    n = ceil((-3+sqrt(9+8*p))/2); %=GnGgu  
    m = 2*p - n.*(n+2); :imp~~L;  
    7VR+EV  
    % Pass the inputs to the function ZERNFUN: +4)Kc9S#  
    % ---------------------------------------- VG)kPKoi  
    switch nargin &'R\yX<J)  
        case 3 dfk=%lZYd9  
            z = zernfun(n,m,r,theta); ]A'E61t<n  
        case 4 Ix}:!L  
            z = zernfun(n,m,r,theta,nflag); JD,/oL.KA  
        otherwise Iz VtiX  
            error('zernfun2:nargin','Incorrect number of inputs.') =n9|r.\&uJ  
    end p0[ %+n%  
    5*~G7/hT  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) 81`-xVd  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. { "=d7i  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of ?9.SwIxU&  
    %   order N and frequency M, evaluated at R.  N is a vector of 1^$ vmULj  
    %   positive integers (including 0), and M is a vector with the E{|j  
    %   same number of elements as N.  Each element k of M must be a p"3_u;cN  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) /dj r_T  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is u 6;SgPw  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix WF`y j%0  
    %   with one column for every (N,M) pair, and one row for every ?X~Keb  
    %   element in R. ``DS?pUY  
    % % ,1bh  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- Fn1|Wt*  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is }GRZCX>  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to )BmK'H+l  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 1UT&kD!si  
    %   for all [n,m]. .3M=|rE   
    % #[ipJ %  
    %   The radial Zernike polynomials are the radial portion of the Z2!O)8  
    %   Zernike functions, which are an orthogonal basis on the unit cba ~  
    %   circle.  The series representation of the radial Zernike IXc"gO  
    %   polynomials is z^'3f!:3  
    % |Q[[WHqj2f  
    %          (n-m)/2 f+d[Q1  
    %            __ ha&2V=  
    %    m      \       s                                          n-2s Pm*FA8a7  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r .0}]/%al  
    %    n      s=0 Z]jm.'@z@  
    % 2guWWFS  
    %   The following table shows the first 12 polynomials. _vr> -:G  
    % C5"=%v[gQv  
    %       n    m    Zernike polynomial    Normalization $t}t'uJ  
    %       --------------------------------------------- %a$ l%8j&  
    %       0    0    1                        sqrt(2) )! +~q!A  
    %       1    1    r                           2 ?H3Ls~R  
    %       2    0    2*r^2 - 1                sqrt(6) s"gNHp.oF  
    %       2    2    r^2                      sqrt(6) 1 CXO=Q  
    %       3    1    3*r^3 - 2*r              sqrt(8) [+j }:u  
    %       3    3    r^3                      sqrt(8) G |033(j  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) `\Z7It?aDs  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) V $Y=JK@  
    %       4    4    r^4                      sqrt(10) .ww~'5b0  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) #2{H!jr  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) }04 EM  
    %       5    5    r^5                      sqrt(12) tX)l_ ?jVH  
    %       --------------------------------------------- Okxuhzn>"  
    % X"lPXoCN  
    %   Example: U|yXJ.Z3  
    % ~?E.U,R  
    %       % Display three example Zernike radial polynomials 9 M>.9~  
    %       r = 0:0.01:1; dPvRbwH<  
    %       n = [3 2 5]; s aY;[bz}  
    %       m = [1 2 1]; _/xA5/V  
    %       z = zernpol(n,m,r); ~FCkr&Ky3  
    %       figure 3}hJ`xQ  
    %       plot(r,z) hc#Sy:T>  
    %       grid on 9+S$,|9  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') ; D'6sd"  
    % cCa+UTxaJ  
    %   See also ZERNFUN, ZERNFUN2. EIdEXAC(  
    'ip2|UG  
    % A note on the algorithm. rlMahY"C  
    % ------------------------ VO u/9]a  
    % The radial Zernike polynomials are computed using the series ?/3'j(Gk  
    % representation shown in the Help section above. For many special d0U-:S-  
    % functions, direct evaluation using the series representation can m!#'4  
    % produce poor numerical results (floating point errors), because ykMdH:  
    % the summation often involves computing small differences between X?f\j"v  
    % large successive terms in the series. (In such cases, the functions C6` Tck!  
    % are often evaluated using alternative methods such as recurrence  VB&` S+-  
    % relations: see the Legendre functions, for example). For the Zernike h[*:\P`  
    % polynomials, however, this problem does not arise, because the e2F{}N  
    % polynomials are evaluated over the finite domain r = (0,1), and )wqG^yv  
    % because the coefficients for a given polynomial are generally all >8;EeRvI  
    % of similar magnitude. j;TXZ`|(  
    % "WF@T  
    % ZERNPOL has been written using a vectorized implementation: multiple fmgXh)=  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] ?q{HS&k  
    % values can be passed as inputs) for a vector of points R.  To achieve /Pv dP#!  
    % this vectorization most efficiently, the algorithm in ZERNPOL X^o0t^  
    % involves pre-determining all the powers p of R that are required to 2pQ29  
    % compute the outputs, and then compiling the {R^p} into a single KATu7)e&~^  
    % matrix.  This avoids any redundant computation of the R^p, and 'LX]/ D  
    % minimizes the sizes of certain intermediate variables. aWS_z6[t#6  
    % ,::f? Gc7j  
    %   Paul Fricker 11/13/2006 z ?L]5m` H  
    K6Z/  
    fug F k  
    % Check and prepare the inputs: 8.WZC1N  
    % ----------------------------- _<^mi!Y  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Wd>gOE  
        error('zernpol:NMvectors','N and M must be vectors.') X+7@8)1(  
    end >S}^0vNZX  
    IoKN.#;^  
    if length(n)~=length(m) 3Z_\.Z1R@  
        error('zernpol:NMlength','N and M must be the same length.') a1dkB"Zp.p  
    end *e,GXU@  
    O_ 4 j"0  
    n = n(:); /0 2-0mNv  
    m = m(:); .dPy<6E  
    length_n = length(n); Q ym=L(X  
    T|^KG<uPV!  
    if any(mod(n-m,2)) a 8}!9kL  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') 1|XC$0  
    end 2A&Y})D  
    p N+1/m,  
    if any(m<0) wX+KW0|>  
        error('zernpol:Mpositive','All M must be positive.') PblO?@~O  
    end i87+9X  
    + 'V ,z  
    if any(m>n) FR^(1+lx&  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') H^fErl  
    end |%$mN{  
    :{=2ih-}  
    if any( r>1 | r<0 ) _1ax6MwX  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') -izZ D  
    end -'Oq.$Qq  
    .azA1@V|  
    if ~any(size(r)==1) I><sK-3  
        error('zernpol:Rvector','R must be a vector.') hZtJ LY  
    end U2CCjAgRs  
    W.m2`] &  
    r = r(:); B&Iy_;  
    length_r = length(r); w,VUWja  
    5pz(6gA  
    if nargin==4 `nv82v  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); DAVgP7h'  
        if ~isnorm J_7&nIH7  
            error('zernpol:normalization','Unrecognized normalization flag.') 94'0X  
        end _ lE d8Cb  
    else tdi^e;:?  
        isnorm = false; k:DAko}  
    end RxUzJ  
    ZIp"X  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ->*'Y;t4  
    % Compute the Zernike Polynomials &BE'~G  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% js F96X{  
    ^"{txd?6  
    % Determine the required powers of r: ZU K'z  
    % ----------------------------------- VQ/Jz5^  
    rpowers = []; |m>{< :  
    for j = 1:length(n) l~'NqmXe  
        rpowers = [rpowers m(j):2:n(j)]; ~9JLqN"  
    end [`U9  
    rpowers = unique(rpowers); &uLC{Ik}  
    56!/E5qgW  
    % Pre-compute the values of r raised to the required powers, IgNL1KRD  
    % and compile them in a matrix: ";*Iwd*V  
    % ----------------------------- k*k 9hv?  
    if rpowers(1)==0 ^k}%k#)  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); =x-@-\m  
        rpowern = cat(2,rpowern{:}); $[M5V v  
        rpowern = [ones(length_r,1) rpowern]; 57rH`UFXH  
    else tish%Qnpd  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); o}6d[G>  
        rpowern = cat(2,rpowern{:}); /2fQM_ ,P  
    end TW!>~|U)y  
    $94l('B6H  
    % Compute the values of the polynomials: .?LP$O=  
    % -------------------------------------- hM~zO1XW  
    z = zeros(length_r,length_n); 3fhlMOm  
    for j = 1:length_n yQdoy^d/4  
        s = 0:(n(j)-m(j))/2; 0})mCVBY  
        pows = n(j):-2:m(j); #9 u2LK  
        for k = length(s):-1:1 KpSho<  
            p = (1-2*mod(s(k),2))* ... Uv%?z0F<C  
                       prod(2:(n(j)-s(k)))/          ... xy Pz_9  
                       prod(2:s(k))/                 ...  HV\l86}  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... 65AG# O5R  
                       prod(2:((n(j)+m(j))/2-s(k))); &'A8R;b}-?  
            idx = (pows(k)==rpowers); N3?@CM^hHw  
            z(:,j) = z(:,j) + p*rpowern(:,idx); +5oK91o[y  
        end oa:30@HSb  
         Qv/Kbw N{  
        if isnorm \zv?r :1t  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); @ !m+s~~]h  
        end Hl b%/&  
    end r0fEW9wL  
    zqs|~W]c  
    % 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)  znxP.=GB   
    {BS}9jZx  
    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。 rQ*Fc~^L  
    Z|k>)pv@  
    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)