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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 noZ!j>f{@l  
    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! YkF52_^_  
     
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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 ?Xj@Sx  
    function z = zernfun(n,m,r,theta,nflag) Kf#iF*  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. <6&Z5mpm$w  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N C8%MKNPd  
    %   and angular frequency M, evaluated at positions (R,THETA) on the w\a6ga!xt"  
    %   unit circle.  N is a vector of positive integers (including 0), and @<koL  
    %   M is a vector with the same number of elements as N.  Each element |3BxNFe`%  
    %   k of M must be a positive integer, with possible values M(k) = -N(k)  0:$pJtx"  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, e4FR)d0x  
    %   and THETA is a vector of angles.  R and THETA must have the same <B!DwMk;.  
    %   length.  The output Z is a matrix with one column for every (N,M) piFZu/~Gq\  
    %   pair, and one row for every (R,THETA) pair. gOr%N!5  
    % qfqL"G  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike |3g'~E?$  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ~Rw][Ys  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral 5 p ,HkV  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, cNMDI  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized erOj(ce  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ccT <UIpq  
    % @U:PXCvh  
    %   The Zernike functions are an orthogonal basis on the unit circle. K/_"ybR7  
    %   They are used in disciplines such as astronomy, optics, and u/ri {neP{  
    %   optometry to describe functions on a circular domain. X|C=Q   
    % %~[@5<p  
    %   The following table lists the first 15 Zernike functions. K{:[0oIHc  
    % Js^(mRv=  
    %       n    m    Zernike function           Normalization %<`sDO6Q?  
    %       -------------------------------------------------- vy-q<6T}:p  
    %       0    0    1                                 1 rdsZ[ii  
    %       1    1    r * cos(theta)                    2 #'n.az=1  
    %       1   -1    r * sin(theta)                    2 <fHN^O0TS  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) D^6Q`o  
    %       2    0    (2*r^2 - 1)                    sqrt(3) yd[4l%G(zS  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) lYmxd8  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) gA gF$H .  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) yb,$UT"]  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) ;rV+eb)I  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) 0"Zxbgu)  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) FiSx"o  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) & Zjs  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) <d O ~;  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) #kE8EhQZ  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) pG22Nx  
    %       -------------------------------------------------- sRZ?Ilua6  
    % 39I|.B"  
    %   Example 1: L`(\ud  
    % 6 X'#F,M  
    %       % Display the Zernike function Z(n=5,m=1) O* lE0~rJ  
    %       x = -1:0.01:1; v]rbm}uU9  
    %       [X,Y] = meshgrid(x,x); M(I%QD  
    %       [theta,r] = cart2pol(X,Y); Dl,sl>{  
    %       idx = r<=1; {$ > .I  
    %       z = nan(size(X)); Y#+Ws0wN  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); V+r&Z<&  
    %       figure nJ$2RN  
    %       pcolor(x,x,z), shading interp .m.Ga|;  
    %       axis square, colorbar Yhjv[9  
    %       title('Zernike function Z_5^1(r,\theta)') pH(X;OC 9S  
    % Z?'?|vM  
    %   Example 2: *j=58d`n  
    % ""Oir!4  
    %       % Display the first 10 Zernike functions q>wO=qWx  
    %       x = -1:0.01:1; VVcli*  
    %       [X,Y] = meshgrid(x,x); 3i\Np =  
    %       [theta,r] = cart2pol(X,Y); 5,-:31(j\  
    %       idx = r<=1; M+%qVwp  
    %       z = nan(size(X)); P4.)kK.3q|  
    %       n = [0  1  1  2  2  2  3  3  3  3]; y:_>R=sw  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; u6%\ZK._ \  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; 7U-}Y  
    %       y = zernfun(n,m,r(idx),theta(idx)); p'6XF{  
    %       figure('Units','normalized') =yoR>llbBC  
    %       for k = 1:10 )l/ .<`|  
    %           z(idx) = y(:,k); bdfs'udt9  
    %           subplot(4,7,Nplot(k)) lnK  
    %           pcolor(x,x,z), shading interp Jfo'iNOu  
    %           set(gca,'XTick',[],'YTick',[]) sLFZ 61rT  
    %           axis square mwsdl^c  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ; 6PRi/@  
    %       end u,{R,hTDS  
    % G/#m. =t  
    %   See also ZERNPOL, ZERNFUN2. Lf%=vd  
    Ep:hObWG)  
    %   Paul Fricker 11/13/2006 5Ar gM%  
    i7cUp3  
    78 ]Kv^l^_  
    % Check and prepare the inputs: ,In%r`{i  
    % ----------------------------- FnI}N;"  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 2-jXj9kp`  
        error('zernfun:NMvectors','N and M must be vectors.') o7WAH@g  
    end $M/1pZ  
    +-9-%O.(;  
    if length(n)~=length(m) |=KzQY|u  
        error('zernfun:NMlength','N and M must be the same length.') _l1"X^Aa  
    end =f [/Pv  
    w%..*+P  
    n = n(:); wwQ2\2w>Hm  
    m = m(:); /y|ZAN  
    if any(mod(n-m,2)) FP}I+Ys  
        error('zernfun:NMmultiplesof2', ... Ryh 0r  
              'All N and M must differ by multiples of 2 (including 0).') :U=3*f.{  
    end qL`yaU  
    w w[|| =  
    if any(m>n) fM|s,'Q1x  
        error('zernfun:MlessthanN', ... l9OpaOVfJ  
              'Each M must be less than or equal to its corresponding N.') #I*{_|}=  
    end OU}eTc(FeC  
    1P'A*`!K  
    if any( r>1 | r<0 ) .tppCy  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') wa{!%qu5.R  
    end ngmC~l*,  
    iSR"$H{  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ;\6@s3  
        error('zernfun:RTHvector','R and THETA must be vectors.') G-|c%g!ejf  
    end <SQR";  
    i*$~uuY  
    r = r(:); (6NDY5h~=n  
    theta = theta(:); |)" y  
    length_r = length(r); cruBJZr*  
    if length_r~=length(theta) hdcB*j?4  
        error('zernfun:RTHlength', ... i+_=7(e  
              'The number of R- and THETA-values must be equal.') 6xwjKh:9  
    end ARt{ 2|  
    8x LXXB  
    % Check normalization: "Nb2[R  
    % -------------------- 7R m\#  
    if nargin==5 && ischar(nflag) Ge=^q.  
        isnorm = strcmpi(nflag,'norm'); );_/0:  
        if ~isnorm /5z,G r  
            error('zernfun:normalization','Unrecognized normalization flag.') :T?WN+3  
        end <66%(J>  
    else Lwx J:Kz.  
        isnorm = false; esE!i0%  
    end X }i2qv  
    ,x!r^YO=  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {.p;V  
    % Compute the Zernike Polynomials i2rSP$j  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% TBQ68o  
    6-tIe _5  
    % Determine the required powers of r: Z2`M8xEiH  
    % ----------------------------------- 7l/lY-zO  
    m_abs = abs(m); M!mw6';k  
    rpowers = []; =+Odu  
    for j = 1:length(n) 4c{j9mh  
        rpowers = [rpowers m_abs(j):2:n(j)]; t 4zUj%F  
    end 9-q> W  
    rpowers = unique(rpowers); QV HI}3~  
    2Xk;]-T!  
    % Pre-compute the values of r raised to the required powers, xV`l6QS  
    % and compile them in a matrix: 7&wxnxSk^  
    % ----------------------------- [7~AWZU3  
    if rpowers(1)==0 +9|0\Q  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); G4P*U3&p  
        rpowern = cat(2,rpowern{:}); vu.?@k@  
        rpowern = [ones(length_r,1) rpowern]; U^ , !  
    else dlCiqY: }  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 8#tuB8>  
        rpowern = cat(2,rpowern{:}); ^b`-zFL7  
    end r-L& ee   
    oqysfLJ  
    % Compute the values of the polynomials: _'1 ]CoR  
    % -------------------------------------- oIx|)[  
    y = zeros(length_r,length(n)); E R~RBzp  
    for j = 1:length(n) rC!"<  
        s = 0:(n(j)-m_abs(j))/2; @RszPH1B  
        pows = n(j):-2:m_abs(j); 0A~UuH0.  
        for k = length(s):-1:1 cN?/YkW?]  
            p = (1-2*mod(s(k),2))* ... j<~T:Tk  
                       prod(2:(n(j)-s(k)))/              ... 0gW{6BtPWm  
                       prod(2:s(k))/                     ... $ (xdF  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... &Wb"/Hn2  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); }2e s"  
            idx = (pows(k)==rpowers); iymN|KdpaZ  
            y(:,j) = y(:,j) + p*rpowern(:,idx); Y/I)ECm  
        end u^|cG{i5"  
         1L'Q;?&2H,  
        if isnorm %kop's&?C  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); ABe25Sus  
        end kh=<M{-t  
    end LL (TD&  
    % END: Compute the Zernike Polynomials +[MHl  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ]1>R8  
    [ 't.x=  
    % Compute the Zernike functions: 1:^Xd~X  
    % ------------------------------ #\}FQl6  
    idx_pos = m>0; 7=u Gf$/  
    idx_neg = m<0; V>Z4gZp5sc  
    NyRa.hgZ;  
    z = y; z#PaQp5F  
    if any(idx_pos) PWx%~U.8~j  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); (BxmV1  
    end G'}N?8s1  
    if any(idx_neg) I;E?;i  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); o8<~zeI  
    end 'NCqI  
    siCm)B  
    % EOF zernfun
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) B'(zhjV  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. Udtz zka  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated 'fb\t,  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive T!y 9v5  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, W3>9GY90R  
    %   and THETA is a vector of angles.  R and THETA must have the same < B]qqqP  
    %   length.  The output Z is a matrix with one column for every P-value, 'V:MppQVZ.  
    %   and one row for every (R,THETA) pair. fw,,cu`YA  
    % cf"&22TQ+Z  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike aAGV\o{^  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) yc3/5]E&  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) -`\^_nVC  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 &Lt$~}*&6  
    %   for all p. JZxA:dg l  
    % ?uL-qsU  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 +3-5\t`  
    %   Zernike functions (order N<=7).  In some disciplines it is H9ES|ZJs  
    %   traditional to label the first 36 functions using a single mode bK0(c1*a[e  
    %   number P instead of separate numbers for the order N and azimuthal 3'0vLi  
    %   frequency M. :*]#n  
    % j?|Vx'  
    %   Example: j][&o-Ev  
    % 4Ps;Cor+  
    %       % Display the first 16 Zernike functions Q0jg(=9wP  
    %       x = -1:0.01:1; [@VzpVhXz  
    %       [X,Y] = meshgrid(x,x); |3B<;/v5  
    %       [theta,r] = cart2pol(X,Y); hLZf A rq}  
    %       idx = r<=1; ^1F zs(#.  
    %       p = 0:15; BRY/[QRqZ  
    %       z = nan(size(X)); ><"|>(y  
    %       y = zernfun2(p,r(idx),theta(idx)); yeta)@nH  
    %       figure('Units','normalized') }1k?th  
    %       for k = 1:length(p) \ Xh C  
    %           z(idx) = y(:,k); hO.b?>3NL  
    %           subplot(4,4,k) \rUKP""m  
    %           pcolor(x,x,z), shading interp 965x _ %  
    %           set(gca,'XTick',[],'YTick',[]) q\i&E Rr  
    %           axis square 7"aN7Q+EbI  
    %           title(['Z_{' num2str(p(k)) '}']) g7hI9(8+  
    %       end VUUnB<j  
    % mCg^Y)Q  
    %   See also ZERNPOL, ZERNFUN. z)^.ai,:0  
    ,ps?@lD  
    %   Paul Fricker 11/13/2006 lv!j  
    r`Fs"n#^-4  
    oVHe<zE.  
    % Check and prepare the inputs: l%mp49<  
    % ----------------------------- sj/k';#g  
    if min(size(p))~=1 )ADI[+KW  
        error('zernfun2:Pvector','Input P must be vector.') $X Uck[  
    end qP;1LAX  
    Q~wS2f`)  
    if any(p)>35 s=jH1^  
        error('zernfun2:P36', ... )O+Zbn  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... |gxPuAXa)  
               '(P = 0 to 35).'])  Hk4k  
    end ~PyS;L}  
    tx<^PV2  
    % Get the order and frequency corresonding to the function number: !'# D~   
    % ---------------------------------------------------------------- ' #=n>  
    p = p(:); ZEDvY=@a   
    n = ceil((-3+sqrt(9+8*p))/2); d\3 %5Y  
    m = 2*p - n.*(n+2); aiz ws[C  
    Y{dX[^[  
    % Pass the inputs to the function ZERNFUN: ;J+iwS*Z  
    % ---------------------------------------- &Lzd*}7  
    switch nargin t`hes $E  
        case 3 N\tFK*U^I  
            z = zernfun(n,m,r,theta); ,jz~Np_2  
        case 4 O[U`(A:  
            z = zernfun(n,m,r,theta,nflag); xa]yq%  
        otherwise ;! ?l8R  
            error('zernfun2:nargin','Incorrect number of inputs.') l{^s4  
    end {Z <`@\K3  
    5K?/-0yG  
    % EOF zernfun2
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    function z = zernpol(n,m,r,nflag) ;m.6 ~A  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. zXlerQWUv  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of ,{(XT7hr  
    %   order N and frequency M, evaluated at R.  N is a vector of ~-H3]  
    %   positive integers (including 0), and M is a vector with the uR2|>m  
    %   same number of elements as N.  Each element k of M must be a (4q/LuP^d  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) `mD!z.`U  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is 8"d??3ZXJ  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix se>\5k  
    %   with one column for every (N,M) pair, and one row for every vA6onYjA  
    %   element in R. -Mrt%1g  
    % ] FvGAG.*  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- 6Xo"?f  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is 3 .j/D^  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to  6; )5v  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 gWjr|m<  
    %   for all [n,m]. ;C1#[U1Uy  
    % zHNBX Rx  
    %   The radial Zernike polynomials are the radial portion of the ,1CmB@  
    %   Zernike functions, which are an orthogonal basis on the unit N5K2Hv<"  
    %   circle.  The series representation of the radial Zernike oXk6,b"  
    %   polynomials is =tA;JB  
    % ~9k E.  
    %          (n-m)/2 "G*$#  
    %            __ WO!OaC?+B,  
    %    m      \       s                                          n-2s {/5aF_0D.  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r T]/>c  
    %    n      s=0 q6sb;?I  
    % G9j f]Ye;  
    %   The following table shows the first 12 polynomials. @}PX:*c  
    % f9y+-GhaD  
    %       n    m    Zernike polynomial    Normalization Dz2Z (EXI~  
    %       --------------------------------------------- Z'5&N5hx  
    %       0    0    1                        sqrt(2) ?+Vi !eS  
    %       1    1    r                           2 Hc|cA(9sh9  
    %       2    0    2*r^2 - 1                sqrt(6) 87S,6Y  
    %       2    2    r^2                      sqrt(6) bV'r9&[_6  
    %       3    1    3*r^3 - 2*r              sqrt(8) D-i, C~W  
    %       3    3    r^3                      sqrt(8) xbcmvJrG  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) aEa+?6;D  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) 726UO#*  
    %       4    4    r^4                      sqrt(10) >6WZSw/Hq  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) H!"TS-s`  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) Ie _{P&J  
    %       5    5    r^5                      sqrt(12) P&[&Dj  
    %       --------------------------------------------- 8sI$  
    % 0/!0W%f[}  
    %   Example: >@?mP$;=  
    % !z2xm3s{]p  
    %       % Display three example Zernike radial polynomials jxhZOLG  
    %       r = 0:0.01:1; HSU?4=Q  
    %       n = [3 2 5]; `YZl2c<w*  
    %       m = [1 2 1]; >mMfZvxl%  
    %       z = zernpol(n,m,r); .qYQ3G'V  
    %       figure `Z{s,!z  
    %       plot(r,z) ~d\V>  
    %       grid on rp1+K4]P  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') <u# 7K\:  
    % #s>'IPc0  
    %   See also ZERNFUN, ZERNFUN2. #k>A,  
    &e*@:5Z:k  
    % A note on the algorithm. 6r@>n_6LY  
    % ------------------------ NN+;I^NqW&  
    % The radial Zernike polynomials are computed using the series %`lJAW[  
    % representation shown in the Help section above. For many special L+@X]O W8  
    % functions, direct evaluation using the series representation can  nsV=  
    % produce poor numerical results (floating point errors), because eFA,xzp  
    % the summation often involves computing small differences between Fe: ~M?]  
    % large successive terms in the series. (In such cases, the functions 'q}f3u>  
    % are often evaluated using alternative methods such as recurrence "j8=%J{  
    % relations: see the Legendre functions, for example). For the Zernike (w(k*b/  
    % polynomials, however, this problem does not arise, because the cmCD}Skk  
    % polynomials are evaluated over the finite domain r = (0,1), and Y8lZ]IB  
    % because the coefficients for a given polynomial are generally all 9Nv?j=*$  
    % of similar magnitude. kv?DE4=;  
    % $XkO\6kh  
    % ZERNPOL has been written using a vectorized implementation: multiple Gp)J[8j  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] tW-[.Y -M,  
    % values can be passed as inputs) for a vector of points R.  To achieve x`IWo:j  
    % this vectorization most efficiently, the algorithm in ZERNPOL "VoufXM:  
    % involves pre-determining all the powers p of R that are required to >O~V#1 H  
    % compute the outputs, and then compiling the {R^p} into a single CS-jDok  
    % matrix.  This avoids any redundant computation of the R^p, and _]D 6m2R  
    % minimizes the sizes of certain intermediate variables. .O#7X  
    % OqF8KJnO;  
    %   Paul Fricker 11/13/2006 Sx0{]1J  
    N@!PhP  
    uKD }5M?{  
    % Check and prepare the inputs: BYa#<jXtAT  
    % ----------------------------- GM)q\Hx{  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) /PR 4ILed  
        error('zernpol:NMvectors','N and M must be vectors.') UH\{:@GjNO  
    end WT1y7+_g(d  
    yxf #@Je"  
    if length(n)~=length(m) EymSrZw  
        error('zernpol:NMlength','N and M must be the same length.') {7%HK2='  
    end f:~$x  
    Y}Y~?kE>M|  
    n = n(:); CW/L(RQ  
    m = m(:); tc\ZYCFr  
    length_n = length(n); 7kT X  
    V+l7W  
    if any(mod(n-m,2)) OTJMS_IT  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') !)W#|sys&  
    end Q0x?OL]A  
    UVgSO|Tg  
    if any(m<0) m^TN6/])  
        error('zernpol:Mpositive','All M must be positive.') M_r[wYt!  
    end & 2q<#b  
    J}#2Wy^{  
    if any(m>n) Iij$ce`nx  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') @qx$b~%  
    end XAtRA1.  
    w8J8III\~  
    if any( r>1 | r<0 ) A2A_F|f  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') 'Yc^9;C(  
    end zM<L_l&  
    5tLb o  
    if ~any(size(r)==1) \$ss  
        error('zernpol:Rvector','R must be a vector.') oK4xRv8Hd  
    end CY[3%7 fv  
    +Kg }R5+  
    r = r(:); X6qgApyE  
    length_r = length(r); pFwJ:  
    BS?$eai@:9  
    if nargin==4 Rr[Wka9[  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); tXZMr   
        if ~isnorm 3MKu!  
            error('zernpol:normalization','Unrecognized normalization flag.') wrXn|aV  
        end PCV#O63[  
    else *W>, 98  
        isnorm = false; :2C <;o  
    end Tm7LaM  
    Y> }\'$\b  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% S'NLj(  
    % Compute the Zernike Polynomials S{f,EBE  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 8Djki]  
    D<<q5gG  
    % Determine the required powers of r: G#6Z@|kVw  
    % ----------------------------------- -!li,&,A1  
    rpowers = []; IXR'JZ?fH  
    for j = 1:length(n) Em5,Zr_  
        rpowers = [rpowers m(j):2:n(j)]; ]+B.=mO_  
    end 2efdJ&eIV  
    rpowers = unique(rpowers); yx:+Xy*N  
    "zCT S  
    % Pre-compute the values of r raised to the required powers, P4eH:0=#  
    % and compile them in a matrix: d"Wuu1tEY  
    % ----------------------------- (uhE'IQ{(  
    if rpowers(1)==0 [/VpvQ'  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 1\Pjz Lj  
        rpowern = cat(2,rpowern{:}); 6%>'n?  
        rpowern = [ones(length_r,1) rpowern]; o90g;Vog  
    else |JHNFs  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); NJ-Ji> w  
        rpowern = cat(2,rpowern{:}); B'`25u_e<  
    end W3\E; C-g0  
    j|N<6GSke  
    % Compute the values of the polynomials: BGwD{6`U  
    % -------------------------------------- re}PpXRC  
    z = zeros(length_r,length_n); bqp^\yu-E  
    for j = 1:length_n {&B_b|g*fW  
        s = 0:(n(j)-m(j))/2; ~/z%yg  
        pows = n(j):-2:m(j); 3]9Rmx  
        for k = length(s):-1:1 H{S+^'5Y.  
            p = (1-2*mod(s(k),2))* ... ^~7Mv^A  
                       prod(2:(n(j)-s(k)))/          ... 8e,F{>N  
                       prod(2:s(k))/                 ... mU?~s7  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... S_OtY]gF  
                       prod(2:((n(j)+m(j))/2-s(k))); j6XHH&ZEb  
            idx = (pows(k)==rpowers); ?_A[E]/H  
            z(:,j) = z(:,j) + p*rpowern(:,idx); /93z3o7D>  
        end ]j6K3  
         ;Oqf{em];  
        if isnorm P3Lsfi.  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); x a<KF  
        end 1TR+p? "  
    end M"/Jn[  
    |O oczYf  
    % 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)  #'%ii,;w Q  
    3-:^mRPJ  
    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。 FO_}9<s  
    We}9'X}  
    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)