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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 \(=xc2  
    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! eHKb`K7C.  
     
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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 Q3$DX, 8?  
    function z = zernfun(n,m,r,theta,nflag) v$JW7CKA  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. z?VjlA(X  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N Z 5P4 H  
    %   and angular frequency M, evaluated at positions (R,THETA) on the P|lDW|}D@  
    %   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 .!lLj1?p  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) XhWo~zh"  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, 1=9GV+`n  
    %   and THETA is a vector of angles.  R and THETA must have the same CK|AXz+EN  
    %   length.  The output Z is a matrix with one column for every (N,M) cH:&S=>h  
    %   pair, and one row for every (R,THETA) pair. -`z%<)!Y  
    % O}2/w2n  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike +R;LHRS%  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), $T66%wX  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral v_v>gPl,  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 8cMX=P  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized pStb j`Eq  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. N'l2$8  
    % 2 w! 0$  
    %   The Zernike functions are an orthogonal basis on the unit circle. vpdPW%B  
    %   They are used in disciplines such as astronomy, optics, and #D?w,<_8,  
    %   optometry to describe functions on a circular domain. QuI!`/N)z  
    % rFm?Bu  
    %   The following table lists the first 15 Zernike functions. hgDFhbHtd6  
    % @8aV*zjB  
    %       n    m    Zernike function           Normalization h -091N  
    %       -------------------------------------------------- S5Pn6'w  
    %       0    0    1                                 1 7zU~ X,  
    %       1    1    r * cos(theta)                    2 vo)W ziHh  
    %       1   -1    r * sin(theta)                    2 {-]K!tWda  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) saQo]6#  
    %       2    0    (2*r^2 - 1)                    sqrt(3) <HS{A$]  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) Vu4LC&q  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) =,qY\@fq  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) EKN<KnU%  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) ZJhI|wRwD  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) e.XD5~Ax  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) /|h+,]< >  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) pX!T; Re;  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) #SI]^T|  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) {,T=Siy  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) {9j0k`A  
    %       -------------------------------------------------- gQu!(7WLI  
    % [ z/G  
    %   Example 1: >Lo'H}[pF  
    % 4@mJEi{  
    %       % Display the Zernike function Z(n=5,m=1) I1dOMu9  
    %       x = -1:0.01:1; -=UvOzw  
    %       [X,Y] = meshgrid(x,x); t%k`)p7O  
    %       [theta,r] = cart2pol(X,Y); yiH;fK+x  
    %       idx = r<=1; rTJqw@]#WH  
    %       z = nan(size(X)); yOXEP  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); j b'M  
    %       figure };Df ><  
    %       pcolor(x,x,z), shading interp Pd `~#!  
    %       axis square, colorbar !mwMSkkq  
    %       title('Zernike function Z_5^1(r,\theta)') 8 K)GH:a  
    % 0A8G8^T  
    %   Example 2: IC$"\7 @  
    % m@L>6;*  
    %       % Display the first 10 Zernike functions )MoHY   
    %       x = -1:0.01:1; /1.Z=@7  
    %       [X,Y] = meshgrid(x,x); Y=<zR9f`  
    %       [theta,r] = cart2pol(X,Y); z 3Z8vq  
    %       idx = r<=1; opzlh@R 3  
    %       z = nan(size(X)); ]z=dRq  
    %       n = [0  1  1  2  2  2  3  3  3  3]; V@gG x  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; f= }!c*l"  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; JLu$UR4  
    %       y = zernfun(n,m,r(idx),theta(idx)); dPV<:uO  
    %       figure('Units','normalized') 0Am\02R.C,  
    %       for k = 1:10 43,*.1;sz  
    %           z(idx) = y(:,k); J5Q.v;  
    %           subplot(4,7,Nplot(k)) qM 3(OvCt  
    %           pcolor(x,x,z), shading interp |A0U 3$S=  
    %           set(gca,'XTick',[],'YTick',[]) <9$Pl%:  
    %           axis square ]S@DVXH  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) wsAb8U C_  
    %       end BPOT!-  
    % Y$|KY/)H)  
    %   See also ZERNPOL, ZERNFUN2.  3(*vZ  
    m|]"e@SF2  
    %   Paul Fricker 11/13/2006 dV*9bDkM/  
    h*Mi/\  
    (58r9WhS  
    % Check and prepare the inputs: 3fYfj  
    % ----------------------------- }h3[QUVf%  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) or7l} X  
        error('zernfun:NMvectors','N and M must be vectors.') Y10  
    end ~0Zy$L/D  
    :Z83*SPc  
    if length(n)~=length(m) !<X/_+G\  
        error('zernfun:NMlength','N and M must be the same length.') tv]9n8v  
    end  7(o:J  
    0/%RrE  
    n = n(:); 9c0  
    m = m(:); &,,:pL[  
    if any(mod(n-m,2)) fX1Ib$v  
        error('zernfun:NMmultiplesof2', ... _tQM<~Y]u\  
              'All N and M must differ by multiples of 2 (including 0).') /7.//klN  
    end y^ st T^  
    Dj0D.}`~  
    if any(m>n) yVpru8+eD  
        error('zernfun:MlessthanN', ... d5=&:cF  
              'Each M must be less than or equal to its corresponding N.') 3?!c<^"e  
    end :#N]s  
    01]W@ \(  
    if any( r>1 | r<0 ) Q5 o0!w  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') "412w^5[T  
    end }`76yH^c  
    *d 4A3|  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) l @E {K|  
        error('zernfun:RTHvector','R and THETA must be vectors.') ^7*zi_Q  
    end oGt2n:  
    F"' (i  
    r = r(:); `C^0YGO%  
    theta = theta(:); 7WNUHLEt  
    length_r = length(r); I(/*pa?m{  
    if length_r~=length(theta) 3A! |M5  
        error('zernfun:RTHlength', ... .rlLt5b%  
              'The number of R- and THETA-values must be equal.') "837b/>/  
    end YYe=E,q  
    8>I4e5Ym  
    % Check normalization: ^i@0P}K<  
    % -------------------- , $cpm=1  
    if nargin==5 && ischar(nflag) D'UIxc8  
        isnorm = strcmpi(nflag,'norm'); _]0<G8|Rv  
        if ~isnorm F$YT4414  
            error('zernfun:normalization','Unrecognized normalization flag.') Ju"c!vu~  
        end sWVapu p?  
    else 1T4#+kW&  
        isnorm = false; ?ihRt+eR~  
    end M~.1:%khM  
    TFXKCl  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% $?;)uoAg  
    % Compute the Zernike Polynomials A5s;<d0  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% a3Xd~Qs  
    7oCY@>(f  
    % Determine the required powers of r: q{L-(!uz7_  
    % ----------------------------------- ;):E 8;B)  
    m_abs = abs(m); /%b nG(4  
    rpowers = []; IGA4"\s  
    for j = 1:length(n) (De>k8  
        rpowers = [rpowers m_abs(j):2:n(j)]; VMu?mqEa  
    end UhU"[^YO  
    rpowers = unique(rpowers); =8Z-ORW51  
    #9HX"<5  
    % Pre-compute the values of r raised to the required powers, g6OPYUPg  
    % and compile them in a matrix: {m_y<  
    % ----------------------------- 7T(&DOGZ  
    if rpowers(1)==0 S>s+ nqcP  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); g$JlpD&  
        rpowern = cat(2,rpowern{:}); A(n3<(O/{Z  
        rpowern = [ones(length_r,1) rpowern]; Ns\};j?TU*  
    else }>b@=5O  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); p?4,YV|#  
        rpowern = cat(2,rpowern{:}); CsjrQ-#9yn  
    end _9<Mo;C  
    ~,x4cOdR#  
    % Compute the values of the polynomials: KppYe9?  
    % -------------------------------------- 5? f!hB|6  
    y = zeros(length_r,length(n)); \GZ|fmYn  
    for j = 1:length(n) nL]eGC  
        s = 0:(n(j)-m_abs(j))/2; R.YUUXT  
        pows = n(j):-2:m_abs(j); O;0VKNn['  
        for k = length(s):-1:1 D&OskM60  
            p = (1-2*mod(s(k),2))* ... y-~_W 6\  
                       prod(2:(n(j)-s(k)))/              ... w,OPM}) il  
                       prod(2:s(k))/                     ... 'oUTY *  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 00yWk_w  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); Qve5qJ  
            idx = (pows(k)==rpowers); ~G.MaSm  
            y(:,j) = y(:,j) + p*rpowern(:,idx); a>,Zp*V(  
        end 27}0  
         x4v&%d=M  
        if isnorm @h/-P'Lc=7  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 7/)0{B4U'  
        end G7r.Jm^q  
    end $Xqc'4YOZ  
    % END: Compute the Zernike Polynomials h\+8eeIl  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% lcVG<*gf-  
    9I''$DVf  
    % Compute the Zernike functions: ~6+>2|wIS  
    % ------------------------------ w zi7pJjXh  
    idx_pos = m>0; q(v|@l|)yO  
    idx_neg = m<0; ST,+]p3L(  
    apnpy\in  
    z = y; ;Nd'GA+1;(  
    if any(idx_pos) 0:c3aq&u  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); _v++NyZXx  
    end |\94a  
    if any(idx_neg) 0IBQE  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); &}\{qFD;  
    end +x<OyjY5?]  
    pwV~[+SS_  
    % EOF zernfun
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) ;Eh"]V,e  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. WC b 5  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated *^CN2tm  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive ~yA^6[a=  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, Bj\Us$cZ  
    %   and THETA is a vector of angles.  R and THETA must have the same "~Zdv}^xS  
    %   length.  The output Z is a matrix with one column for every P-value, AoK;6je`K^  
    %   and one row for every (R,THETA) pair. !sYZ1;WAO  
    % </R@)_'  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike *:`fgaIDa  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) L@f&71  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) F*-'8~T  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 - b`  
    %   for all p. Q5_,`r`  
    % lA`-"  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 u`Kc\B Sn  
    %   Zernike functions (order N<=7).  In some disciplines it is ==trl#kQ%%  
    %   traditional to label the first 36 functions using a single mode yh).1Q-D  
    %   number P instead of separate numbers for the order N and azimuthal I*/:rb  
    %   frequency M. Ds{bYK_y  
    % <vu~EY0.  
    %   Example: p4kK" \ln  
    % 3Q2NiYg3  
    %       % Display the first 16 Zernike functions n8D'fvY  
    %       x = -1:0.01:1; i+lq:St  
    %       [X,Y] = meshgrid(x,x); 4ywtE}mp  
    %       [theta,r] = cart2pol(X,Y); \x5>H:\Y  
    %       idx = r<=1; &3)6WD?:U  
    %       p = 0:15; =l6W O*  
    %       z = nan(size(X)); 1`l(H4  
    %       y = zernfun2(p,r(idx),theta(idx)); /q/^B> ]  
    %       figure('Units','normalized') tCGA3t  
    %       for k = 1:length(p) jaMpi^C  
    %           z(idx) = y(:,k); %CgmZTz~<  
    %           subplot(4,4,k) m}2hIhD9  
    %           pcolor(x,x,z), shading interp O"_QDl<ya  
    %           set(gca,'XTick',[],'YTick',[]) Yk*_u}?#  
    %           axis square 6F%6]n  
    %           title(['Z_{' num2str(p(k)) '}']) 4#I=n~8a  
    %       end #G\-ftA&  
    % dkSd Y+Q  
    %   See also ZERNPOL, ZERNFUN. A>(EM}\,  
    "j.Q*Hazg  
    %   Paul Fricker 11/13/2006 auM1k]  
    C[;7i!Dv  
    .'2"83f  
    % Check and prepare the inputs: vq@"y%C4  
    % ----------------------------- FRQkD%k  
    if min(size(p))~=1 D>`{f4Y  
        error('zernfun2:Pvector','Input P must be vector.') %f(4jQ0I  
    end dkg+_V!  
    0XE(vc!  
    if any(p)>35 ;#*.@Or@Ah  
        error('zernfun2:P36', ... t*Z-]P  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... A}3E)Qo=G  
               '(P = 0 to 35).']) SO~pe$c-  
    end 3@<m/%  
    .&K?@T4l  
    % Get the order and frequency corresonding to the function number: _sHeB7K  
    % ---------------------------------------------------------------- c|4_nT 2  
    p = p(:); $A(3-n5=  
    n = ceil((-3+sqrt(9+8*p))/2); l 5f'R  
    m = 2*p - n.*(n+2); ?w)A`G_  
    b}N \h<\G  
    % Pass the inputs to the function ZERNFUN: E>"SC\#7  
    % ---------------------------------------- `"$9L[>  
    switch nargin l8lJ &  
        case 3 9YBlMf`KEf  
            z = zernfun(n,m,r,theta); cL"Ral-qB  
        case 4 paxZlA o  
            z = zernfun(n,m,r,theta,nflag); za8+=?  
        otherwise CKDg3p';  
            error('zernfun2:nargin','Incorrect number of inputs.') va.Ve# N  
    end qtP*O#1q  
    4@-Wp]  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) 'RR,b*Ql  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. >lugHF$G  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of Fk?KR  
    %   order N and frequency M, evaluated at R.  N is a vector of Ft>,  
    %   positive integers (including 0), and M is a vector with the n$"B F\eM  
    %   same number of elements as N.  Each element k of M must be a D,s[{RW+q  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) u 0 K1n_  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is 1mx;b)4t  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix "3?:,$*  
    %   with one column for every (N,M) pair, and one row for every I>>X-}  
    %   element in R. w1= f\  
    % hud'@O"R+  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- f:9qId ;/M  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is "l6Ob  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to z'EphL7r   
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 Aac7k m  
    %   for all [n,m]. _6 yrd.H  
    % ~hU^5R-%  
    %   The radial Zernike polynomials are the radial portion of the kwFo*1 {  
    %   Zernike functions, which are an orthogonal basis on the unit * @&V=l  
    %   circle.  The series representation of the radial Zernike  /?6  
    %   polynomials is v/3Vsd  
    % [g: KFbEY  
    %          (n-m)/2 $tebNi P  
    %            __ (DTkK5/%  
    %    m      \       s                                          n-2s t%Y}JKLR  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r Uql7s:!,U  
    %    n      s=0 hQDl&A  
    % e\]CZ5hs3  
    %   The following table shows the first 12 polynomials. "3NE%1T  
    % mmEe@-lE  
    %       n    m    Zernike polynomial    Normalization bw[K^/  
    %       --------------------------------------------- diF2:80o  
    %       0    0    1                        sqrt(2) @z(s\T  
    %       1    1    r                           2 $ 2k9gO  
    %       2    0    2*r^2 - 1                sqrt(6) y Xi$w.gr  
    %       2    2    r^2                      sqrt(6) Fi#b0S  
    %       3    1    3*r^3 - 2*r              sqrt(8) 5U/C 0{6  
    %       3    3    r^3                      sqrt(8) <t.  w(?  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) W; ?'  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) IN!IjInaT@  
    %       4    4    r^4                      sqrt(10) w;T?m,"  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) +/8KN  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) a]R1Fi0n  
    %       5    5    r^5                      sqrt(12) 1?T^jcny:M  
    %       --------------------------------------------- X!0m,  
    % EA!I& mBq  
    %   Example: }Ym~[S*x  
    % 4m/L5W:K  
    %       % Display three example Zernike radial polynomials <(2,@_~@r  
    %       r = 0:0.01:1; +~M`rR*  
    %       n = [3 2 5]; 7vK}aOs0  
    %       m = [1 2 1]; +?bOGUik  
    %       z = zernpol(n,m,r); |",/  
    %       figure 62J -)~_  
    %       plot(r,z) 4031~A8  
    %       grid on 1;y?!;FD  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') \-)augq([  
    % sVT\e*4m}  
    %   See also ZERNFUN, ZERNFUN2. GAcU8  MD  
    8E\6RjM  
    % A note on the algorithm. lnRbvulH  
    % ------------------------ ik|iAWy  
    % The radial Zernike polynomials are computed using the series 8w4cqr4m  
    % representation shown in the Help section above. For many special 4krK CD>|G  
    % functions, direct evaluation using the series representation can R9V v*F]m@  
    % produce poor numerical results (floating point errors), because VtC1TZ3-7  
    % the summation often involves computing small differences between s4Z5t$0|  
    % large successive terms in the series. (In such cases, the functions i^u5j\pfY*  
    % are often evaluated using alternative methods such as recurrence %$I@7Es>  
    % relations: see the Legendre functions, for example). For the Zernike @K.[;-;g  
    % polynomials, however, this problem does not arise, because the 6/eh~ME=  
    % polynomials are evaluated over the finite domain r = (0,1), and j`kw2(  
    % because the coefficients for a given polynomial are generally all 1t7S:IZ  
    % of similar magnitude. Ym "Nj  
    % A!j6JY.w  
    % ZERNPOL has been written using a vectorized implementation: multiple QBfsdu<@^  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] O U3KB  
    % values can be passed as inputs) for a vector of points R.  To achieve a).bk!G  
    % this vectorization most efficiently, the algorithm in ZERNPOL ~T<o?98  
    % involves pre-determining all the powers p of R that are required to )mMHwLDwH  
    % compute the outputs, and then compiling the {R^p} into a single RA~%Cw4t  
    % matrix.  This avoids any redundant computation of the R^p, and $^4URH  
    % minimizes the sizes of certain intermediate variables. U.HeIJ#  
    %  7ehs+GI  
    %   Paul Fricker 11/13/2006 :TzHI    
    l~V^  
    s'|^6/  
    % Check and prepare the inputs: U[UjL)U  
    % ----------------------------- 2,O;<9au<  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) S+EC!;@Xg  
        error('zernpol:NMvectors','N and M must be vectors.') J 4EG  
    end RwC1C(ZP  
    o {bwWk7v6  
    if length(n)~=length(m) U`fxe`nVa  
        error('zernpol:NMlength','N and M must be the same length.') 4^mpQ.]lO  
    end ?`%)3gx|  
    > %U  
    n = n(:); C!8XFf8e  
    m = m(:); m# ]VdO'f  
    length_n = length(n); J9 iQW  
    FF"`F8-w>Z  
    if any(mod(n-m,2)) ovn)lIs  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') vpGeG  
    end 7Bhi72&6  
    K2glkGK  
    if any(m<0) F(i@Gm=J]  
        error('zernpol:Mpositive','All M must be positive.') GX+oA]  
    end L-lDvc?5c  
    P]4C/UDS-~  
    if any(m>n) {b^JH2,  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') v<z%\`y  
    end  OBCRZ   
    v~N8H+! d  
    if any( r>1 | r<0 ) M#Vl{ b  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') C1@6 r%YD  
    end E}V8+f54S  
    @,RrAL }|  
    if ~any(size(r)==1) 'K=n}}&:  
        error('zernpol:Rvector','R must be a vector.') [D=3:B&f  
    end  d!%:Ok  
    #lM :BO  
    r = r(:); U[b $VZ}  
    length_r = length(r); 4Y[uqn[  
    h<50jnH!  
    if nargin==4 p}j$p'D.RI  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); 8%s_~Yc  
        if ~isnorm 8pfQAzl  
            error('zernpol:normalization','Unrecognized normalization flag.') 9:!<=rk  
        end 4|*H0}HOm  
    else E5P?(5Nv  
        isnorm = false; |7V:~MTkk&  
    end $4\,a^  
    _-^Lr /`G!  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% TM8WaH   
    % Compute the Zernike Polynomials =8?gx$r2  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% xe;1D'(   
    ei>8{v&g  
    % Determine the required powers of r: xG05OqKpE  
    % ----------------------------------- gu[3L  
    rpowers = []; &>I4-D[  
    for j = 1:length(n) k{3:$, b  
        rpowers = [rpowers m(j):2:n(j)]; jj,Y:  
    end U[1Ir92:  
    rpowers = unique(rpowers); R<+K&_  
    C ebl"3Q  
    % Pre-compute the values of r raised to the required powers, ?3wEO>u  
    % and compile them in a matrix: Z?H#=|U  
    % ----------------------------- YPraf$  
    if rpowers(1)==0 * _puW x  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); _ 13M  
        rpowern = cat(2,rpowern{:}); E4^zW_|xE  
        rpowern = [ones(length_r,1) rpowern]; 0^VA,QkQ\  
    else jrGVC2*rD  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); A`|OPi)  
        rpowern = cat(2,rpowern{:}); Tz H*?bpP  
    end ho#]i$b}f2  
    5Uc!;Gd?b  
    % Compute the values of the polynomials: 2YD;Gb[8  
    % -------------------------------------- 2 w2JFdm  
    z = zeros(length_r,length_n); Yl[GO}M  
    for j = 1:length_n 8-Ik .,}  
        s = 0:(n(j)-m(j))/2; DW ^E46k)A  
        pows = n(j):-2:m(j); '"{ IV  
        for k = length(s):-1:1 &[*F!=%8  
            p = (1-2*mod(s(k),2))* ... FpjpsD~ Qu  
                       prod(2:(n(j)-s(k)))/          ... A+Nf]([  
                       prod(2:s(k))/                 ... wXv\[z L`  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... ln#Jb&u  
                       prod(2:((n(j)+m(j))/2-s(k))); _@[M0t}g_  
            idx = (pows(k)==rpowers); tH0=ysf  
            z(:,j) = z(:,j) + p*rpowern(:,idx); 3+gp_7L  
        end lLy^@s  
         c!Gnd*!?-  
        if isnorm 5`oVyxJ<  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); J>(I"K%  
        end R=jIVw'  
    end >r] bfN,  
    6$TE-l  
    % 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)  u Ey>7I  
    &sJZSrk|  
    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。 6{/HNEI*1  
    rap`[O|l=  
    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)