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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 0sH~H[ap  
    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! BAi`{?z$<  
     
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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 j"fx|6l)  
    function z = zernfun(n,m,r,theta,nflag) y#8 W1%{x  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle.  <4< y  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N i7cUp3  
    %   and angular frequency M, evaluated at positions (R,THETA) on the 78 ]Kv^l^_  
    %   unit circle.  N is a vector of positive integers (including 0), and ,In%r`{i  
    %   M is a vector with the same number of elements as N.  Each element jatlv/,  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) |MagK$o  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, U$3DIJVI  
    %   and THETA is a vector of angles.  R and THETA must have the same 0-;>O|U3  
    %   length.  The output Z is a matrix with one column for every (N,M) z30 mk  
    %   pair, and one row for every (R,THETA) pair. k+*pg4 '  
    % +@yU `  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike !YI<A\P  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), s:_a.4&Y  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral G e5Yz.Q v  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 9 W|'~r  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized g'{?j~g  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. (y~%6o6  
    % &[ ],rT  
    %   The Zernike functions are an orthogonal basis on the unit circle. <&2<>*/.y  
    %   They are used in disciplines such as astronomy, optics, and >Vg [ A  
    %   optometry to describe functions on a circular domain. VW*?(,#j{  
    % WRwx[[e6z  
    %   The following table lists the first 15 Zernike functions. M d8(P23hS  
    % OU}eTc(FeC  
    %       n    m    Zernike function           Normalization 4_sJ0=z-  
    %       -------------------------------------------------- pLCS\AUTsv  
    %       0    0    1                                 1 <m\<yZ2aa  
    %       1    1    r * cos(theta)                    2 0rz1b6F5,  
    %       1   -1    r * sin(theta)                    2 H1L)9oa  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) AzSu_  
    %       2    0    (2*r^2 - 1)                    sqrt(3) Yl lZ5<}  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) kPiY|EH  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) GAZRQ  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8)  o0>|  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) NZa 7[}H  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) fR~0Fy Gp  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) uv8k ea .(  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ~d1=_p:~T  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) c q[nqjC=  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) aG#d41O  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) mpCu,l+lo  
    %       -------------------------------------------------- 8 hhMuh  
    % J\w4N",  
    %   Example 1: Y .cjEeL@  
    % NZ&ZK@h}.  
    %       % Display the Zernike function Z(n=5,m=1) Rm}5AJ  
    %       x = -1:0.01:1; rx 74v!  
    %       [X,Y] = meshgrid(x,x); _| cSXZ|  
    %       [theta,r] = cart2pol(X,Y); +N7<[hE;  
    %       idx = r<=1; H&%oHyK  
    %       z = nan(size(X)); 6<>1,wbq  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); F?"Gln~;  
    %       figure 0Zp5y@ V8  
    %       pcolor(x,x,z), shading interp nTGZ2C)c<'  
    %       axis square, colorbar 9N{?J"ido  
    %       title('Zernike function Z_5^1(r,\theta)') q }>3NCh  
    % = $^90Q,Z;  
    %   Example 2: (*=>YE'V{  
    % mMOgx   
    %       % Display the first 10 Zernike functions doe3V-if  
    %       x = -1:0.01:1; l2YClK  
    %       [X,Y] = meshgrid(x,x); uDkX{<_Xe  
    %       [theta,r] = cart2pol(X,Y); qyFeq])  
    %       idx = r<=1; q~5zv4NX  
    %       z = nan(size(X)); MffCk!]  
    %       n = [0  1  1  2  2  2  3  3  3  3]; Ok@`<6v  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; 9}a$0H h  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; iAk.pH]a  
    %       y = zernfun(n,m,r(idx),theta(idx)); l0URJRK{*  
    %       figure('Units','normalized') "S6";G^I  
    %       for k = 1:10 )8rF'pxI  
    %           z(idx) = y(:,k); >5Lp;  
    %           subplot(4,7,Nplot(k)) zv0sz])  
    %           pcolor(x,x,z), shading interp zh0T3U0D  
    %           set(gca,'XTick',[],'YTick',[]) .w@B )f*  
    %           axis square !.q99DB  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) `''y,{Fs  
    %       end I= <eCv  
    % 8@(?E[&O>  
    %   See also ZERNPOL, ZERNFUN2. #Y3-P  
    8! !h6dQgI  
    %   Paul Fricker 11/13/2006 f=Pn,.>tIz  
    94dd )/a  
    S ~h*U2  
    % Check and prepare the inputs: =[!(s/+>L  
    % ----------------------------- CueC![pj  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) $N}t)iA  
        error('zernfun:NMvectors','N and M must be vectors.') PN 8#T:E  
    end .K(9=yh  
    H~vrCi~t"  
    if length(n)~=length(m) Sw"h!\c`  
        error('zernfun:NMlength','N and M must be the same length.') .U@u |  
    end Y/I)ECm  
    %xG<hNw/  
    n = n(:); |ka/5o  
    m = m(:); WjK[% ;Z!  
    if any(mod(n-m,2)) ^0cbN[~/ns  
        error('zernfun:NMmultiplesof2', ... {r;_nMfH|[  
              'All N and M must differ by multiples of 2 (including 0).') z80FMulO  
    end Sew*0S(  
    uM_ww6  
    if any(m>n) 3h=kn@I  
        error('zernfun:MlessthanN', ... ik/ X!YTu*  
              'Each M must be less than or equal to its corresponding N.') WwZ3hd  
    end Z'2AsT  
    pg7~%E4  
    if any( r>1 | r<0 ) pU !:  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') ~CV.Ci.dG  
    end 3Og}_  
    3<M yb  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) P*7G?  
        error('zernfun:RTHvector','R and THETA must be vectors.') !vJ$$o6#  
    end I;E?;i  
    Y G8C<g6E7  
    r = r(:); [pm IQ228  
    theta = theta(:); eIF6f& F  
    length_r = length(r); siCm)B  
    if length_r~=length(theta) /Mw;oP{&b  
        error('zernfun:RTHlength', ... :2==7u7v?  
              'The number of R- and THETA-values must be equal.') N*$GP3]  
    end ys`oHS f  
    b/R7 Mk1  
    % Check normalization: DW9MX`!Xc  
    % -------------------- .AO-S)wHR  
    if nargin==5 && ischar(nflag) f sh9-iY8e  
        isnorm = strcmpi(nflag,'norm'); C,eP!_O  
        if ~isnorm RC1bTM  
            error('zernfun:normalization','Unrecognized normalization flag.') N* &T)a  
        end D QxuV1  
    else P/1YN  
        isnorm = false; #;^UW  
    end 3~3tjhw;]9  
    RnRUJNlaG  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% XL aD#J  
    % Compute the Zernike Polynomials EwV$2AK  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ]jVE  
    wn.6l `  
    % Determine the required powers of r: L YB @L06a  
    % ----------------------------------- oNPvksdC;  
    m_abs = abs(m); 5m0lk|`  
    rpowers = []; '5$@ I{z  
    for j = 1:length(n) Q"{Dijc%  
        rpowers = [rpowers m_abs(j):2:n(j)]; O<L=N-  
    end l P=I0A-  
    rpowers = unique(rpowers); 5rck]L'  
    j_}:=3  
    % Pre-compute the values of r raised to the required powers, N1c 0>{  
    % and compile them in a matrix: +3-5\t`  
    % ----------------------------- H9ES|ZJs  
    if rpowers(1)==0 bK0(c1*a[e  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 3'0vLi  
        rpowern = cat(2,rpowern{:}); :*]#n  
        rpowern = [ones(length_r,1) rpowern]; (T pnJq  
    else "xTVu57Z[  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); JmR2skoV,  
        rpowern = cat(2,rpowern{:}); <2 [vR|Q*  
    end #\Y`?  
    J Hm Pa  
    % Compute the values of the polynomials: ey[Z<i1  
    % -------------------------------------- 8r+u!$i!H  
    y = zeros(length_r,length(n)); +8?18@obp  
    for j = 1:length(n) `~=z0I  
        s = 0:(n(j)-m_abs(j))/2; 0vSPeZ  
        pows = n(j):-2:m_abs(j); K*DH_\SPK  
        for k = length(s):-1:1 ;-py h(  
            p = (1-2*mod(s(k),2))* ... 0<@['W}G  
                       prod(2:(n(j)-s(k)))/              ... qQDe'f~  
                       prod(2:s(k))/                     ... t(roj@!x_o  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... )=K8mt0qob  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); 1DAU *^-  
            idx = (pows(k)==rpowers); ETU-6qFtO  
            y(:,j) = y(:,j) + p*rpowern(:,idx); A. tGr(r  
        end c\rP -"C  
         ?K2EK'-q  
        if isnorm ,ps?@lD  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); lv!j  
        end r`Fs"n#^-4  
    end oVHe<zE.  
    % END: Compute the Zernike Polynomials ZLKbF9lo  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% IZ>l  
    VV$#<D<)  
    % Compute the Zernike functions: ue7D' UZL>  
    % ------------------------------ hV,T889'  
    idx_pos = m>0; "DvZCf[}  
    idx_neg = m<0; O-p`9(_m  
    ]C"?xy  
    z = y; G?,3Zn0  
    if any(idx_pos) tF/Ni*\^rV  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); |H^v8^%>zm  
    end #U%HG TE0  
    if any(idx_neg) PD S( /x&  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); x(EwHg>;  
    end nPI$<yW7F  
    (fl$$$  
    % 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) s1[.L~;J  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. |dHtv6I  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated q!U$\Q&  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive g^|R;s{  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, 0w TOdCvmb  
    %   and THETA is a vector of angles.  R and THETA must have the same R%2.N!8v  
    %   length.  The output Z is a matrix with one column for every P-value, qk^/ &j  
    %   and one row for every (R,THETA) pair. =IX-n$d`>  
    % NM:$Q<n  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike SfY 5Xgp  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) *wJz0ex7R/  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) C]JK'K<7-  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 abS~'r14  
    %   for all p. & uwOyb  
    % 7>r[.g  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 cQd?,B3#F  
    %   Zernike functions (order N<=7).  In some disciplines it is #M!{D  
    %   traditional to label the first 36 functions using a single mode jbZTlG  
    %   number P instead of separate numbers for the order N and azimuthal {*8G<&  
    %   frequency M. ?771e:>S-  
    % ^uw]/H3?L  
    %   Example: \@h$|nb  
    % jzpDKc%  
    %       % Display the first 16 Zernike functions jp4-w(  
    %       x = -1:0.01:1; /L(}VJg-  
    %       [X,Y] = meshgrid(x,x); 2)$-L'YS  
    %       [theta,r] = cart2pol(X,Y); R64f0N K.  
    %       idx = r<=1; 9o?\*{'KT  
    %       p = 0:15; 4IY|<  
    %       z = nan(size(X)); ppLLX1S  
    %       y = zernfun2(p,r(idx),theta(idx)); JP( tf+  
    %       figure('Units','normalized') +zDRed_]=_  
    %       for k = 1:length(p) lqrI*@>Tz  
    %           z(idx) = y(:,k); Jo;&~/ V   
    %           subplot(4,4,k) !^NZp%Yd  
    %           pcolor(x,x,z), shading interp hCgk78O?  
    %           set(gca,'XTick',[],'YTick',[]) t(6i4c>  
    %           axis square QH7 GEj]  
    %           title(['Z_{' num2str(p(k)) '}']) `h :!^"G  
    %       end yXEI%2~)  
    % >Sw?F&  
    %   See also ZERNPOL, ZERNFUN. E&t8nlTx  
    w9"~NK8xzM  
    %   Paul Fricker 11/13/2006 WQ:Y NmQ1p  
    Zi\ex\ )5  
    ()t~X Q  
    % Check and prepare the inputs: Dz2Z (EXI~  
    % ----------------------------- Z'5&N5hx  
    if min(size(p))~=1 ?+Vi !eS  
        error('zernfun2:Pvector','Input P must be vector.') Hc|cA(9sh9  
    end 87S,6Y  
    bV'r9&[_6  
    if any(p)>35 LN$T.r+  
        error('zernfun2:P36', ... ?5};ONjN  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... x)$0Nr62D  
               '(P = 0 to 35).']) %`]!atH  
    end NZ8X@|N  
    T?Z^2.Pvc  
    % Get the order and frequency corresonding to the function number: PX23M|$!  
    % ---------------------------------------------------------------- K(lVAKiP]  
    p = p(:); CsT&}-C  
    n = ceil((-3+sqrt(9+8*p))/2); ;0 +Dx~  
    m = 2*p - n.*(n+2); CHO_3QIz  
    +mR^I$9  
    % Pass the inputs to the function ZERNFUN: p9\*n5{  
    % ---------------------------------------- ([rSYKpi  
    switch nargin : #n>Q1}x  
        case 3 `@ ,Vbn^_  
            z = zernfun(n,m,r,theta); OfA+|xT&  
        case 4 !:esdJH  
            z = zernfun(n,m,r,theta,nflag); z_KCG2=5  
        otherwise 1BEc"  
            error('zernfun2:nargin','Incorrect number of inputs.') cZoj|=3a  
    end =;I+: K  
    't*]6^  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) `e3$jy@  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. H*f2fyC1\  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of +`V<& Y-5l  
    %   order N and frequency M, evaluated at R.  N is a vector of X+,0;% p  
    %   positive integers (including 0), and M is a vector with the =_@) KWeX$  
    %   same number of elements as N.  Each element k of M must be a cuy9QBB :  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) 8)"lCIf  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is 8uW%jG3/  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix tgl 4pAc  
    %   with one column for every (N,M) pair, and one row for every O kT@ _U  
    %   element in R.  <|82)hO  
    % SlT>S1`rnG  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- -rgdKA@)(  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is O%F*i2I:+k  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to ~MYE8xrId  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 aiE\r/k8s  
    %   for all [n,m]. Q uw|KL  
    % =i;T?*@  
    %   The radial Zernike polynomials are the radial portion of the gnxD'1_  
    %   Zernike functions, which are an orthogonal basis on the unit \>n[x; $  
    %   circle.  The series representation of the radial Zernike 4"!kCUB  
    %   polynomials is IQ-l%x[fue  
    % )z4eRs F|  
    %          (n-m)/2 w5/6+@}  
    %            __ >@4AxV\  
    %    m      \       s                                          n-2s cF9oo%3  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r lHTr7uF(  
    %    n      s=0 }ALli0n`V)  
    % FDGG$z?>m  
    %   The following table shows the first 12 polynomials. BTG_c_ ?]e  
    % m9&%A0  
    %       n    m    Zernike polynomial    Normalization jWh)bsqI!  
    %       --------------------------------------------- YH^@8   
    %       0    0    1                        sqrt(2) KQw>6)  
    %       1    1    r                           2 |68/FJZ,5  
    %       2    0    2*r^2 - 1                sqrt(6) 0,.|-OZ  
    %       2    2    r^2                      sqrt(6) ?gvu E1  
    %       3    1    3*r^3 - 2*r              sqrt(8) _RZ"WA^[  
    %       3    3    r^3                      sqrt(8) |m@>AbR5dk  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) kDM?`(r  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) aU[!*n 4Ux  
    %       4    4    r^4                      sqrt(10) D+~*nc~ g  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) T*8K.yw2  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) ^~@3X[No  
    %       5    5    r^5                      sqrt(12) 1cRF0MI  
    %       --------------------------------------------- 7-u'x[=m  
    % fy|I3  
    %   Example: R?- zJ ;  
    % FS!)KxC/-  
    %       % Display three example Zernike radial polynomials 5a)$:oO!  
    %       r = 0:0.01:1; BD86t[${W  
    %       n = [3 2 5]; ({!H ()  
    %       m = [1 2 1]; 0]=Bqyg  
    %       z = zernpol(n,m,r); du#f_|xG  
    %       figure Vx> Q  
    %       plot(r,z) [fo#){3K  
    %       grid on Yw5-:w0f  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') N#$]W"U  
    % CQrP%}`r  
    %   See also ZERNFUN, ZERNFUN2. ozl!vf# kv  
    y c 8 h}`  
    % A note on the algorithm. X(3| (1;sV  
    % ------------------------ d\JB jT1g  
    % The radial Zernike polynomials are computed using the series ,IE.8h)H  
    % representation shown in the Help section above. For many special X')l04P@%  
    % functions, direct evaluation using the series representation can EVw{G<  
    % produce poor numerical results (floating point errors), because ux| QGT2LY  
    % the summation often involves computing small differences between 83{P7PBQ;]  
    % large successive terms in the series. (In such cases, the functions V7p hD3Y  
    % are often evaluated using alternative methods such as recurrence l+hOD{F4pS  
    % relations: see the Legendre functions, for example). For the Zernike .jtv Hr}U  
    % polynomials, however, this problem does not arise, because the ;c DMcKKIA  
    % polynomials are evaluated over the finite domain r = (0,1), and ;'08-Et  
    % because the coefficients for a given polynomial are generally all 6 v~nEw  
    % of similar magnitude. 6PS[OB{3  
    % oayu*a.  
    % ZERNPOL has been written using a vectorized implementation: multiple ki/Cpfq40*  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] 8c_X`0jy  
    % values can be passed as inputs) for a vector of points R.  To achieve X-,oL.:c  
    % this vectorization most efficiently, the algorithm in ZERNPOL *,*O.#<6  
    % involves pre-determining all the powers p of R that are required to @EyB^T/  
    % compute the outputs, and then compiling the {R^p} into a single "Jg.)1Jw  
    % matrix.  This avoids any redundant computation of the R^p, and V`W']  
    % minimizes the sizes of certain intermediate variables. &1&OXm$  
    % J]{<Z?%  
    %   Paul Fricker 11/13/2006 nKnrh]hX  
    o76!7  
    =NI?Jk*iAq  
    % Check and prepare the inputs: e#mqerpJ  
    % ----------------------------- S?C.:  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) >w,jaQ  
        error('zernpol:NMvectors','N and M must be vectors.') 3]9Rmx  
    end H{S+^'5Y.  
    %N`_g' r!  
    if length(n)~=length(m) 3/yt  
        error('zernpol:NMlength','N and M must be the same length.') b09xf"D  
    end lz36;Fp  
    Rt&5s)O'  
    n = n(:); m.1-[2{8~  
    m = m(:); d!Gy#<H  
    length_n = length(n); gH\>", [  
    o^* :  
    if any(mod(n-m,2)) ud BIEW,`  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') yg* #~,  
    end bj6Yz,g F  
    jWi~Q o+  
    if any(m<0) [[6" qq  
        error('zernpol:Mpositive','All M must be positive.') Yg,b ;H  
    end qBh@^GxY),  
    bDJ!Fc/  
    if any(m>n) ``$$yS~d};  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') WG,1%=M@  
    end vR2);ywX  
    <*dcl2xS  
    if any( r>1 | r<0 ) cg17e  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') y%61xA`#  
    end D M+MBK  
    e!gNd>b {  
    if ~any(size(r)==1) r^<,f[yH  
        error('zernpol:Rvector','R must be a vector.') V\{@c%xW  
    end F8{gJaP x  
    &ZkJ,-  
    r = r(:); [MwL=9;!H  
    length_r = length(r); a=A12<  
    &B[*L+-E  
    if nargin==4 b$fmU"%&|  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); GIcq|Pe  
        if ~isnorm L8f+uI   
            error('zernpol:normalization','Unrecognized normalization flag.') p5vQ.Ni*\-  
        end #0uu19+}  
    else 1hgIR^;[b  
        isnorm = false; Ax;?~v4Z  
    end Zy;jp*Q  
    mI4GBp  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% vN],9 q  
    % Compute the Zernike Polynomials |9]-_a  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% b+J|yM<`  
    <3@nv%  
    % Determine the required powers of r: $hcv}<$/  
    % ----------------------------------- vfv?QjR  
    rpowers = []; 7=}tJ  
    for j = 1:length(n)  ] ?D$n  
        rpowers = [rpowers m(j):2:n(j)]; ecz-jZ! `  
    end /}>8|#U3y  
    rpowers = unique(rpowers); %%?}db1n  
    Ao`e{  
    % Pre-compute the values of r raised to the required powers, YtSYe%  
    % and compile them in a matrix: /*Q3=Dse]  
    % ----------------------------- *<:X3|3E  
    if rpowers(1)==0 5 0-7L,  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); #t(/wa4  
        rpowern = cat(2,rpowern{:}); *~U.36  
        rpowern = [ones(length_r,1) rpowern]; W$SV+q(rT  
    else lqL5V"2Y  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); {S9gOg  
        rpowern = cat(2,rpowern{:}); 9=]HOUn  
    end [#Gu?L_W  
    %po;ih$jr*  
    % Compute the values of the polynomials: `Kf@<=  
    % -------------------------------------- 6:B,ir _  
    z = zeros(length_r,length_n); hDW_a y4  
    for j = 1:length_n .$x}~Sw  
        s = 0:(n(j)-m(j))/2; !]g[u3O  
        pows = n(j):-2:m(j); /]Y#*r8jRi  
        for k = length(s):-1:1 oYf+I  
            p = (1-2*mod(s(k),2))* ... E Zf|>^N  
                       prod(2:(n(j)-s(k)))/          ... 8\' tfHL  
                       prod(2:s(k))/                 ... vvxD}p=y  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... *kK +Nvt8s  
                       prod(2:((n(j)+m(j))/2-s(k))); /N*<Fq7w~  
            idx = (pows(k)==rpowers); L3wj vq^  
            z(:,j) = z(:,j) + p*rpowern(:,idx); ';Nc;9  
        end HP[B%  
          wk8fa  
        if isnorm R"O%##Ws  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); VpHwc!APq  
        end 4C_1wk('  
    end SWI\;:k  
    ,pTZ/#vP#  
    % 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)  ^e]O >CJ  
    w-km qh  
    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。 YT[=o}jS  
    tJtp1$h  
    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)