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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 bSOxM /N  
    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! 8(yZX4OH>  
     
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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 i D6f/|g  
    function z = zernfun(n,m,r,theta,nflag) (`W_ -PI  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. LtIR)EtB]  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N [&_7w\m  
    %   and angular frequency M, evaluated at positions (R,THETA) on the Rz sgPk  
    %   unit circle.  N is a vector of positive integers (including 0), and [Lck55V+Q  
    %   M is a vector with the same number of elements as N.  Each element #'DrgZ)W  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) UB5CvM28  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, +8<|P&fH  
    %   and THETA is a vector of angles.  R and THETA must have the same X8}m %  
    %   length.  The output Z is a matrix with one column for every (N,M) s ;3k#-w  
    %   pair, and one row for every (R,THETA) pair. lN(|EI  
    % 7aF'E1e'3  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike s3(mkdXv  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), a&^HvXO(>(  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral [b2KBww\  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, .<m${yU{3  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized /M,C%.-  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. -_*ux!  
    % lEZODc+%Y  
    %   The Zernike functions are an orthogonal basis on the unit circle. O/XG}G.x|  
    %   They are used in disciplines such as astronomy, optics, and (vR9vOpJ  
    %   optometry to describe functions on a circular domain. _i3?;Fds  
    % |wxAdPe  
    %   The following table lists the first 15 Zernike functions. H{)DI(,Y^P  
    % c -sc*.&  
    %       n    m    Zernike function           Normalization N8[ &1  
    %       -------------------------------------------------- }WowgY  
    %       0    0    1                                 1 Wg!<V6}  
    %       1    1    r * cos(theta)                    2 J-UqH3({Z,  
    %       1   -1    r * sin(theta)                    2 )r0XQa]@$  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) 1Yk!R9.  
    %       2    0    (2*r^2 - 1)                    sqrt(3) Y>J$OA:  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) < )qJI'u|  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) 0?$jC-@k:  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) e2"<3  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) N9dx^+\  
    %       3    3    r^3 * sin(3*theta)             sqrt(8)  JT,[;  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) qjm6\ii:)  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) \ u*R6z  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) whW% c8  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) #+"1">l  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) +L\Dh.Ir  
    %       -------------------------------------------------- Qi=pP/Y  
    % 5i0vli /L  
    %   Example 1: wbpz,  
    % kEYkd@ {  
    %       % Display the Zernike function Z(n=5,m=1) (v,g=BS,  
    %       x = -1:0.01:1; (y^svXU}a  
    %       [X,Y] = meshgrid(x,x); 1 u~Xk?  
    %       [theta,r] = cart2pol(X,Y); ip+?k<]z  
    %       idx = r<=1;  "d; T1  
    %       z = nan(size(X)); qNuBK6E#4  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); mgd)wZNV  
    %       figure \H4$9lPk  
    %       pcolor(x,x,z), shading interp 3/{,}F$  
    %       axis square, colorbar R:5uZAx  
    %       title('Zernike function Z_5^1(r,\theta)') f-BPT2U+  
    % u2E}DhV  
    %   Example 2: ?mp}_x#=  
    % A4tb>O M  
    %       % Display the first 10 Zernike functions D[ v2#2  
    %       x = -1:0.01:1; Yq-Vwh/  
    %       [X,Y] = meshgrid(x,x); MqAN~<l [  
    %       [theta,r] = cart2pol(X,Y); HkQ rij6  
    %       idx = r<=1; ?:Sqh1-z  
    %       z = nan(size(X)); =c ;.cW  
    %       n = [0  1  1  2  2  2  3  3  3  3]; cK1 Fv6V#  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; |W\U9n  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; M:*)l(  
    %       y = zernfun(n,m,r(idx),theta(idx)); wZg~k\_lF  
    %       figure('Units','normalized') @@z5v bs'{  
    %       for k = 1:10 nIqNhJ+  
    %           z(idx) = y(:,k); p f`vH`r  
    %           subplot(4,7,Nplot(k)) n`X}&(O  
    %           pcolor(x,x,z), shading interp ce<88dL  
    %           set(gca,'XTick',[],'YTick',[]) Zs|m_O G  
    %           axis square B%I<6E[D  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) B'-n ^';  
    %       end SUb:0GUa  
    % E#~J"9k98  
    %   See also ZERNPOL, ZERNFUN2. Ez+8B|0P  
    T0X+\&W  
    %   Paul Fricker 11/13/2006 <xlyk/  
    Y#zHw< <E  
    $EnBigb!  
    % Check and prepare the inputs: C/!7E:  
    % ----------------------------- bMB@${i}  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) uC|bC#;  
        error('zernfun:NMvectors','N and M must be vectors.') f+%s.[;A  
    end #2dH2k\F  
    f~?kx41dq  
    if length(n)~=length(m) xz-?sD/xe  
        error('zernfun:NMlength','N and M must be the same length.') HP,{/ $i:  
    end QT4&Ix,4T1  
    f_z]kA +H  
    n = n(:); }2''}-Nc  
    m = m(:); ";Q}Gs}  
    if any(mod(n-m,2)) }BWT21'-Y  
        error('zernfun:NMmultiplesof2', ... H}cq|hodn  
              'All N and M must differ by multiples of 2 (including 0).') IOY<'t+  
    end PQrc#dfc |  
    k!V@Q!>,  
    if any(m>n) eWr2UXv$  
        error('zernfun:MlessthanN', ... r<[G~n  
              'Each M must be less than or equal to its corresponding N.') BUUc9&f3o  
    end ^g=j`f[T  
    ap<r )<u  
    if any( r>1 | r<0 ) =C- b#4Q  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') '3=@UBs  
    end LaYd7Oyf]  
    $"g'C8  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) pHKc9VC  
        error('zernfun:RTHvector','R and THETA must be vectors.') MxqIB(5k  
    end |`'WEe2  
    #'97mg  
    r = r(:); ZU;nXqjc  
    theta = theta(:); [$@EQ]tt/  
    length_r = length(r); GO3KKuQ=  
    if length_r~=length(theta) $lg{J$ h8  
        error('zernfun:RTHlength', ... qb$M.-\ne  
              'The number of R- and THETA-values must be equal.') h\4enu9[RL  
    end *-&+;|mM  
    CQs,G8 \/  
    % Check normalization: Q[9W{l+  
    % --------------------  = Atyy  
    if nargin==5 && ischar(nflag) eMtQa;Lc9o  
        isnorm = strcmpi(nflag,'norm'); x$z>.4  
        if ~isnorm _adW>-wQ!d  
            error('zernfun:normalization','Unrecognized normalization flag.') 825 QS`  
        end a>&dAo}  
    else 2>g!+p Ox  
        isnorm = false; s=Xg6D  
    end %zN~%mJG  
    Q"K`~QF"  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ;P^}2i[q>[  
    % Compute the Zernike Polynomials k{ulu  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% }" STc&1  
    W$J@|i  
    % Determine the required powers of r: eC@b-q   
    % ----------------------------------- T2TWb  
    m_abs = abs(m); sY* qf=  
    rpowers = []; ,WE2MAjhT  
    for j = 1:length(n) }?*$AVs2q  
        rpowers = [rpowers m_abs(j):2:n(j)]; x,c\q$8yH  
    end ~<"{u-q#K  
    rpowers = unique(rpowers); !?z"d  
    1aezlDc*  
    % Pre-compute the values of r raised to the required powers, U.1&'U*  
    % and compile them in a matrix: P&Wf.qr{:  
    % ----------------------------- 2]Ei4%jo  
    if rpowers(1)==0 |`d-;pk!%  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); xu@+b~C\  
        rpowern = cat(2,rpowern{:}); %?J-0  
        rpowern = [ones(length_r,1) rpowern]; 2+yti,s+/  
    else dB8 e  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); F#z1 sl'  
        rpowern = cat(2,rpowern{:}); n`D-?]*  
    end $\L=RU!c}  
    T3t w.yh  
    % Compute the values of the polynomials: s6!! ty;Y  
    % -------------------------------------- C|RC9b  
    y = zeros(length_r,length(n)); u6 4{w,  
    for j = 1:length(n) EJ(z]M`f  
        s = 0:(n(j)-m_abs(j))/2; #<vzQ\~Y  
        pows = n(j):-2:m_abs(j); IO"q4(&;P4  
        for k = length(s):-1:1 V]/ $ dJ  
            p = (1-2*mod(s(k),2))* ... :M.]-+(  
                       prod(2:(n(j)-s(k)))/              ... @Py?.H   
                       prod(2:s(k))/                     ... G4%dah 5  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... %1 rN6A!%  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); <FwAV=}6p  
            idx = (pows(k)==rpowers); h5lngw  
            y(:,j) = y(:,j) + p*rpowern(:,idx); PQ" v  
        end o`nJJ:Cxq-  
         C\* 0621  
        if isnorm 1~S'' [  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); foe)_  
        end nMOXy\&mI  
    end ;oOv~ YB7H  
    % END: Compute the Zernike Polynomials Mlo:\ST|  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ooj^Z%9P  
    oot kf=  
    % Compute the Zernike functions: 7TA&u'  
    % ------------------------------ *rC%nmJwk!  
    idx_pos = m>0; ,<;.'r  
    idx_neg = m<0; ew,g'$drD  
    3A3WD+[L  
    z = y;  @4>?Y=#  
    if any(idx_pos) _ 3{8Zg  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); A s8IjGNs{  
    end 9L>ep&u)^  
    if any(idx_neg) u+ 8wBb5!  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); k"+/DK,:  
    end ^geY Ay  
    US&:UzI.  
    % EOF zernfun
    离线niuhelen
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) ;e Mb$px  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. g8]$BhRIfr  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated rR(\fX!dg  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive {mA#'75a#  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, "S psSQ  
    %   and THETA is a vector of angles.  R and THETA must have the same sX(rJLbD  
    %   length.  The output Z is a matrix with one column for every P-value, I}\`l+  
    %   and one row for every (R,THETA) pair. u4Z Accj  
    % YGZa##i  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike C{YTHN n  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) S>R40T=e  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) (~^KXJ{->  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 xS_tB)C  
    %   for all p. <4*)J9V^s=  
    % sfC/Q"Zs  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 ]u%Y8kBe  
    %   Zernike functions (order N<=7).  In some disciplines it is qO>A 6  
    %   traditional to label the first 36 functions using a single mode oi%IHX(`  
    %   number P instead of separate numbers for the order N and azimuthal (o!i9)  
    %   frequency M. / 16 r_l  
    % Ar`U / %Cu  
    %   Example: n&FRjq9y  
    % E>kgEfzxP  
    %       % Display the first 16 Zernike functions "=UhTE  
    %       x = -1:0.01:1; J!iK W  
    %       [X,Y] = meshgrid(x,x); V.w!]{xm  
    %       [theta,r] = cart2pol(X,Y); 5,du2  
    %       idx = r<=1; lv& y<d;  
    %       p = 0:15; |k)Nf+(}W  
    %       z = nan(size(X)); La si)e=$<  
    %       y = zernfun2(p,r(idx),theta(idx)); W 6CNMI]  
    %       figure('Units','normalized') .7n`]S/  
    %       for k = 1:length(p) { ?]&P  
    %           z(idx) = y(:,k); :BUr8%l  
    %           subplot(4,4,k) ~.Cu,>fV  
    %           pcolor(x,x,z), shading interp NJ(H$tB@  
    %           set(gca,'XTick',[],'YTick',[]) ]Waa7)}DM  
    %           axis square zC!Pb{IaH  
    %           title(['Z_{' num2str(p(k)) '}']) }?Tz=hP  
    %       end zmU>  
    % 0y=lf+xA*  
    %   See also ZERNPOL, ZERNFUN. z?(QM:  
    D__*?frWpW  
    %   Paul Fricker 11/13/2006 Xl/2-'4  
    Ml+O - 3T  
    bYy7Ul6]  
    % Check and prepare the inputs: Pol c.  
    % ----------------------------- h5@JS1cY  
    if min(size(p))~=1 $!:xjb  
        error('zernfun2:Pvector','Input P must be vector.') d0-T\\U  
    end jn>RE   
    rq^VOK|L  
    if any(p)>35 JPkI+0  
        error('zernfun2:P36', ... incUa;  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... CDWchY  
               '(P = 0 to 35).']) jNP%BNd1f  
    end 4"Pf0PD:  
    2$?j'i!  
    % Get the order and frequency corresonding to the function number: ]}BT'fky#  
    % ---------------------------------------------------------------- `MXGEJF  
    p = p(:); =kn-F T  
    n = ceil((-3+sqrt(9+8*p))/2); 7 JVonruaR  
    m = 2*p - n.*(n+2); hi9@U]H#  
    ?., 2EC=+  
    % Pass the inputs to the function ZERNFUN: 3 ~\S]  
    % ---------------------------------------- -\}Ix>  
    switch nargin ]f3R;d  
        case 3 A]OVmw  
            z = zernfun(n,m,r,theta); &y.6Hiy&  
        case 4 @'FOM  
            z = zernfun(n,m,r,theta,nflag); k_7agW  
        otherwise &(rR)cG  
            error('zernfun2:nargin','Incorrect number of inputs.') sQ%gf  
    end _eMY ?  
    9E]7Etfw  
    % 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) y7Y g$)sL  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. 5`[B:<E4  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of Zcxj.F(,  
    %   order N and frequency M, evaluated at R.  N is a vector of C\; $RH  
    %   positive integers (including 0), and M is a vector with the nAW`G'V#  
    %   same number of elements as N.  Each element k of M must be a |iB svI:  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) 'Mm=<Bh  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is ;n=A245W\  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix f)!7/+9>  
    %   with one column for every (N,M) pair, and one row for every W1p5F\ wt  
    %   element in R. {Aq:Kh`&  
    % ajALca4  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- ^ l9NF  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is \o{rw0w0  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to 6T{SRN{  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 UvM_~qo  
    %   for all [n,m]. (TFo]c  
    % nV&v@g4Tt  
    %   The radial Zernike polynomials are the radial portion of the ~F=,)GE  
    %   Zernike functions, which are an orthogonal basis on the unit # dxS QmG  
    %   circle.  The series representation of the radial Zernike \t@4)+s/)  
    %   polynomials is hZNA I  
    % lF.yQ  
    %          (n-m)/2 :_"%o=  
    %            __ "N*i!h  
    %    m      \       s                                          n-2s c %.vI  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r ?tFsSU  
    %    n      s=0 "4e{Cq  
    % 2@#`x"0  
    %   The following table shows the first 12 polynomials. 088"7 s  
    % ##clReS  
    %       n    m    Zernike polynomial    Normalization 1rQKHC:|  
    %       --------------------------------------------- zV"oB9\9O  
    %       0    0    1                        sqrt(2) It3.  
    %       1    1    r                           2 caC( KK#<  
    %       2    0    2*r^2 - 1                sqrt(6) F_C_K"[s  
    %       2    2    r^2                      sqrt(6) (GI]Uyn  
    %       3    1    3*r^3 - 2*r              sqrt(8) f!aE/e\  
    %       3    3    r^3                      sqrt(8) !E|k#c9  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) Seb J}P1x  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) I`8jJpGA  
    %       4    4    r^4                      sqrt(10) 26<Wg7/,  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) <tp#KZE  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) qmM%MPv  
    %       5    5    r^5                      sqrt(12) J}J7A5P  
    %       --------------------------------------------- dw]wQ\4B  
    % *QT|J6ng  
    %   Example: ,3E9H&@j  
    % J=C63YB  
    %       % Display three example Zernike radial polynomials [.`%]Z(  
    %       r = 0:0.01:1; sCE2 F_xjL  
    %       n = [3 2 5]; J,=: ] t  
    %       m = [1 2 1]; &b7i> ()  
    %       z = zernpol(n,m,r); %:WM]dc  
    %       figure ;_hL  
    %       plot(r,z) I~.d/!>Z  
    %       grid on nlkQ'XGAI  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') 0x/3Xz  
    % T^~9'KDd  
    %   See also ZERNFUN, ZERNFUN2. ^HasT4M+x  
    Zc9j_.?*  
    % A note on the algorithm. }./_fFN@  
    % ------------------------ )mbRG9P  
    % The radial Zernike polynomials are computed using the series |ZnRr  
    % representation shown in the Help section above. For many special Lc:DJA  
    % functions, direct evaluation using the series representation can ST[+k  
    % produce poor numerical results (floating point errors), because +)gXU Vwd  
    % the summation often involves computing small differences between mv+K!T6  
    % large successive terms in the series. (In such cases, the functions r06M.r   
    % are often evaluated using alternative methods such as recurrence }lzN)e  
    % relations: see the Legendre functions, for example). For the Zernike p&#*  
    % polynomials, however, this problem does not arise, because the WgK|r~  
    % polynomials are evaluated over the finite domain r = (0,1), and bn$}U.m$-  
    % because the coefficients for a given polynomial are generally all >7^+ag~&  
    % of similar magnitude. Vt!<.8&`  
    % t|Ipxk.)  
    % ZERNPOL has been written using a vectorized implementation: multiple w>W`8P_b@  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] ax)j$  
    % values can be passed as inputs) for a vector of points R.  To achieve }_{QsPx9  
    % this vectorization most efficiently, the algorithm in ZERNPOL sUP !'Av  
    % involves pre-determining all the powers p of R that are required to \O7Vo<B&D  
    % compute the outputs, and then compiling the {R^p} into a single r\-25F<e5  
    % matrix.  This avoids any redundant computation of the R^p, and * (4TasQu  
    % minimizes the sizes of certain intermediate variables. 6Q6l?!|W4  
    % KtFxG6a  
    %   Paul Fricker 11/13/2006 9IRvbE~2  
    Dxwv\+7]  
    Hw"UJP  
    % Check and prepare the inputs: gxku3<S  
    % ----------------------------- *KXg;777  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) k9^Vw+$m  
        error('zernpol:NMvectors','N and M must be vectors.') M5Twulz/w  
    end 6!3Jr  
    MK<VjpP0(  
    if length(n)~=length(m) .u_k?.8|  
        error('zernpol:NMlength','N and M must be the same length.') >Lo!8Hen  
    end G{cTQH|  
    ~BVK6  
    n = n(:); Sl{nS1q  
    m = m(:); <9s=K\-  
    length_n = length(n); L#`9# Q  
    BU="BB/[  
    if any(mod(n-m,2)) .2b) rKo~  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') $pT%7jV}  
    end _uO#0 )l  
    /I' n]  
    if any(m<0) F)0I7+lP  
        error('zernpol:Mpositive','All M must be positive.') ikGH:{  
    end Y"uFlHN&i  
    QS~;C&1Hl  
    if any(m>n) Ur626}  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') e<> Lr  
    end >t"]gQHtx  
    p.2>- L  
    if any( r>1 | r<0 ) LaE;{jY  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') axY-Vj  
    end {tE/Jv $  
    {1GW,T!#  
    if ~any(size(r)==1) s'^zudx  
        error('zernpol:Rvector','R must be a vector.') T$]2U>=<J  
    end ~PNO|]8j  
    X*~NE\  
    r = r(:); I_yIVw;  
    length_r = length(r); ]kmOX  
    v`hv5wQ  
    if nargin==4 n;"4`6L~  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); L!L/QG|wdf  
        if ~isnorm UZdE ^Q[  
            error('zernpol:normalization','Unrecognized normalization flag.') 0<L@f=i  
        end xQ7n$.?y@  
    else s9A'{F  
        isnorm = false; ISr~JQr  
    end 13Z6dhZu  
    J1<fE(X  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 1V`]sfRK  
    % Compute the Zernike Polynomials <LW|m7  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 4(4JQ(5  
    bIm$7a`T  
    % Determine the required powers of r: Xh]\q)  
    % ----------------------------------- L\og`L)5\  
    rpowers = []; yj$S?B Ee  
    for j = 1:length(n) .Qh8I+Q%  
        rpowers = [rpowers m(j):2:n(j)]; Hcd>\0  
    end 7o z(hO~  
    rpowers = unique(rpowers); x#0C+cU  
    DuvP3(K  
    % Pre-compute the values of r raised to the required powers, ^@L[0Z`  
    % and compile them in a matrix: <nsl`C~6g0  
    % ----------------------------- 5?kA)!|UB  
    if rpowers(1)==0 gE=~.P[ZX  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); )C2d)(baEJ  
        rpowern = cat(2,rpowern{:}); `Ik}Xw  
        rpowern = [ones(length_r,1) rpowern]; savz>E &  
    else 7IJb$af:;  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); &SN$D5U'  
        rpowern = cat(2,rpowern{:}); /&j4IlT  
    end %,,h )9  
    b+p!{  
    % Compute the values of the polynomials: 8 (^2  
    % -------------------------------------- ;D8Nya>%  
    z = zeros(length_r,length_n); Vd<= y  
    for j = 1:length_n Dlj=$25  
        s = 0:(n(j)-m(j))/2; W,{`)NWg  
        pows = n(j):-2:m(j); H5RHA^p|  
        for k = length(s):-1:1 'v|2} T*  
            p = (1-2*mod(s(k),2))* ... 6 qKIz{;  
                       prod(2:(n(j)-s(k)))/          ... 0v7;Z xD  
                       prod(2:s(k))/                 ... 4_.k Q"'DH  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... 3 n:<oOV  
                       prod(2:((n(j)+m(j))/2-s(k))); el|t6ZT*  
            idx = (pows(k)==rpowers); j|G-9E  
            z(:,j) = z(:,j) + p*rpowern(:,idx); P)ZSxU  
        end ex!^&7Q(  
         "(efd~.]  
        if isnorm NuO>zAu  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); MZjiJZaO:L  
        end }BogE$tc  
    end 6vaxp|D  
    c?R.SBr,'  
    % 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)  0lCd,a 2:  
    ennz/'  
    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。 @izi2ND  
    t4/eB<fP  
    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)