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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 )2t!= ua  
    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! <8Y;9N|94!  
     
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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 %jaB>4.A:  
    function z = zernfun(n,m,r,theta,nflag) ~x<nz/^  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. OU)~ 02|\  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N A)9[.fhx  
    %   and angular frequency M, evaluated at positions (R,THETA) on the gq9D#B  
    %   unit circle.  N is a vector of positive integers (including 0), and 0Y rdu,c  
    %   M is a vector with the same number of elements as N.  Each element 'u@_4wWp  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) @oC# k<  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, lZT9 SDtS  
    %   and THETA is a vector of angles.  R and THETA must have the same Hg8n`a;R  
    %   length.  The output Z is a matrix with one column for every (N,M) Q\(VQ1c  
    %   pair, and one row for every (R,THETA) pair. yn&AMq ]o  
    % X r7pFw  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike Q y(Gy'q~  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), |$[WnYP  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral ]y&w)-0  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, Fua:& 77  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized 3f'dBn5  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. !N5+.E0j  
    % BcJ]bIbKb  
    %   The Zernike functions are an orthogonal basis on the unit circle. en\shc{R]`  
    %   They are used in disciplines such as astronomy, optics, and Fv!zS.)`  
    %   optometry to describe functions on a circular domain. (qn ;MN6<  
    % -QH[gi{%`  
    %   The following table lists the first 15 Zernike functions. M6(oJ*  
    % =n $@  
    %       n    m    Zernike function           Normalization vCC}IDd  
    %       -------------------------------------------------- X8!=Xjl)  
    %       0    0    1                                 1 k+k&}8e  
    %       1    1    r * cos(theta)                    2 :,.g_@wvG  
    %       1   -1    r * sin(theta)                    2 Q _}i8p '  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) =GO/r; 4  
    %       2    0    (2*r^2 - 1)                    sqrt(3) RB]K?  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) u Qy5t:!  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) oicett=5  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) {0(:7IY,  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) a }6Fj&hj  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) L||_Jsu  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) u3{gX{so  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) .H1 kl)~V  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) \Ol3kx|  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) X]'Hz@$N  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) wk { 9  
    %       -------------------------------------------------- M1._{Jw5  
    % _!FM^N}|  
    %   Example 1: w)bLdQ  
    % K`.wj8zGY  
    %       % Display the Zernike function Z(n=5,m=1) x<) %Gs}tb  
    %       x = -1:0.01:1; JyPsRpi\  
    %       [X,Y] = meshgrid(x,x); )k5lA=(Yr+  
    %       [theta,r] = cart2pol(X,Y); u7|{~D&f  
    %       idx = r<=1; ejj|l   
    %       z = nan(size(X)); c"aiZ(aP  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); |xI\)V E^  
    %       figure Ks&~VU  
    %       pcolor(x,x,z), shading interp >V~q`htth  
    %       axis square, colorbar K GlO;Q~7  
    %       title('Zernike function Z_5^1(r,\theta)') i<D}"h|  
    % k*bfq?E a  
    %   Example 2: G9\Bi-'ul  
    % Zl]Zy}p*+  
    %       % Display the first 10 Zernike functions DQg:W |A  
    %       x = -1:0.01:1; \GtZX!0  
    %       [X,Y] = meshgrid(x,x); E-,74B&H  
    %       [theta,r] = cart2pol(X,Y); p},6W,f  
    %       idx = r<=1; T:0X-U  
    %       z = nan(size(X)); @+",f]  
    %       n = [0  1  1  2  2  2  3  3  3  3]; )>LQ{ X.  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; ? WWnt^  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; Kb0OauW  
    %       y = zernfun(n,m,r(idx),theta(idx)); <i'4EnO  
    %       figure('Units','normalized') "Kk3#  
    %       for k = 1:10 /'1UfjW>  
    %           z(idx) = y(:,k); ie$QKoE  
    %           subplot(4,7,Nplot(k)) :oF\?e  
    %           pcolor(x,x,z), shading interp Gy[;yLnX  
    %           set(gca,'XTick',[],'YTick',[]) 5YIi O7@4  
    %           axis square 'l\V{0;mp  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) woZ'T  
    %       end uR"srn;^  
    % `>RJ*_aKEI  
    %   See also ZERNPOL, ZERNFUN2. .<v0y"amJ  
    |0(Z)s,  
    %   Paul Fricker 11/13/2006 F#_7mC   
    lj.z>  
    DLE|ctzj[7  
    % Check and prepare the inputs: )8oI  s  
    % ----------------------------- ~BCSm]j  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 7\^b+*  
        error('zernfun:NMvectors','N and M must be vectors.') c=H(*#  
    end zW%-Z6%D  
    aPB %6c=  
    if length(n)~=length(m) 9>psQ0IRvr  
        error('zernfun:NMlength','N and M must be the same length.') ?n/:1LN,  
    end r&"}zyL  
    `Oys&]vb  
    n = n(:); D_O%[u}  
    m = m(:); oUZwZ_yKW  
    if any(mod(n-m,2)) kgK7 T  
        error('zernfun:NMmultiplesof2', ... hC}A%_S  
              'All N and M must differ by multiples of 2 (including 0).') j._9;HifZ  
    end cl2@p@av  
    J{$C}8V  
    if any(m>n) /woa[7Xe  
        error('zernfun:MlessthanN', ... 3P/T`)V  
              'Each M must be less than or equal to its corresponding N.') }.gDaxj  
    end tjOfekU  
    ksY^w+>(!  
    if any( r>1 | r<0 ) {AIP\  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') ` e~/  
    end U*/  
    =,-80WNsX  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) [?W3XUJ,Y  
        error('zernfun:RTHvector','R and THETA must be vectors.') m&,d8Gss^  
    end I Jq$GR  
    [x!T<jJ  
    r = r(:); U_!"&O5lr  
    theta = theta(:); O<f_-n@G|  
    length_r = length(r); qfppJ8L  
    if length_r~=length(theta) gq[}/E0e  
        error('zernfun:RTHlength', ... '. atbl  
              'The number of R- and THETA-values must be equal.') mMrvr9%  
    end @Sub.z&T{  
    i1vBg}WHN  
    % Check normalization: OjMDxG w  
    % -------------------- GfQMdLy\Z  
    if nargin==5 && ischar(nflag) "rc}mq  
        isnorm = strcmpi(nflag,'norm'); S~YrXQ{_>-  
        if ~isnorm <3HW!7Ad1  
            error('zernfun:normalization','Unrecognized normalization flag.') lZ^XZjwoM  
        end &[I#5 bGk  
    else oX3Q9)  
        isnorm = false; nUmA  
    end lhQ*;dMj%"  
    LLgN%!&  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ,Q(n(m'  
    % Compute the Zernike Polynomials D8`,PXtV  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ,a$LT   
    G7%Nwe~Y  
    % Determine the required powers of r: 9]|[z{v'>l  
    % -----------------------------------  +aP %H  
    m_abs = abs(m); jc;&g)Rv  
    rpowers = []; l:Ci'=  
    for j = 1:length(n) PhKJ#D Rbr  
        rpowers = [rpowers m_abs(j):2:n(j)]; u9mMkzgSkP  
    end 3dadeu^{A  
    rpowers = unique(rpowers); z(1h^.  
    QHMXQyr(  
    % Pre-compute the values of r raised to the required powers, plfz)x3  
    % and compile them in a matrix: , X$S4>  
    % ----------------------------- _sZ/tU@_-K  
    if rpowers(1)==0 BT d$n!'$n  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); LfOGq%&  
        rpowern = cat(2,rpowern{:}); FD_0FMZ9,  
        rpowern = [ones(length_r,1) rpowern]; gADt%K2 #Z  
    else $C#~c1w  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); U@f3V8CPy  
        rpowern = cat(2,rpowern{:}); .&r] ?O  
    end seAkOIc  
    '-w G  
    % Compute the values of the polynomials: An]*J|nFIY  
    % -------------------------------------- XFK$p^qu  
    y = zeros(length_r,length(n)); \FVR'A1  
    for j = 1:length(n) 9Od Kh\F (  
        s = 0:(n(j)-m_abs(j))/2; v~uwQ&AH  
        pows = n(j):-2:m_abs(j); Ku,Efr  
        for k = length(s):-1:1 ?_<ZCH  
            p = (1-2*mod(s(k),2))* ... D ?,P\cp  
                       prod(2:(n(j)-s(k)))/              ... +/Y )s5@<  
                       prod(2:s(k))/                     ... zKfb  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... *WMcE$w/D  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); 4 Iy\   
            idx = (pows(k)==rpowers); e5`{*g$i).  
            y(:,j) = y(:,j) + p*rpowern(:,idx); ynP^|Ou  
        end Qt>yRt  
         f+<-Jc  
        if isnorm y0(k7D|\  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 8{?Oi'-|0  
        end %HYC-TF#  
    end 8(Z*Vz uu  
    % END: Compute the Zernike Polynomials P7u5Ykc*  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% /jj}.X7yH  
    LgUaX  
    % Compute the Zernike functions: +hXph  
    % ------------------------------ [FyE{NfiJ%  
    idx_pos = m>0; [gv2fqpP  
    idx_neg = m<0; OkzfQ hC}  
    |:H[Y"$1;  
    z = y; |&RdOjw$u  
    if any(idx_pos) 7!MW`L/`  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); $: Qi9N   
    end FpW{=4yk  
    if any(idx_neg) !(SaE'  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); GVEjB;  
    end uE5kL{Fv  
    :kFWUs=  
    % 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) 7J>n;8{%?  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. kTC6fNj[  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated CiPD+I  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive >mt<`s  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, v5<Ext rV  
    %   and THETA is a vector of angles.  R and THETA must have the same -} Z  
    %   length.  The output Z is a matrix with one column for every P-value, {SROg;vA  
    %   and one row for every (R,THETA) pair. i 3?zYaT  
    % H%])>  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike <[a9"G 7  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) >Y{.)QS  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) iY*Xm,#  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 -{L[Wt{1  
    %   for all p. $f C=v  
    % *AxKV5[H  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 },[j+wx  
    %   Zernike functions (order N<=7).  In some disciplines it is XM8C{I1  
    %   traditional to label the first 36 functions using a single mode y4shW|>5_  
    %   number P instead of separate numbers for the order N and azimuthal pV>/ "K  
    %   frequency M. }A2@1TTPX  
    % &:{| nDT_2  
    %   Example: Iqo4INGIi  
    % t3bDi/m  
    %       % Display the first 16 Zernike functions ^(&:=r.PC  
    %       x = -1:0.01:1; QiwZk<rb  
    %       [X,Y] = meshgrid(x,x); V?5_J%  
    %       [theta,r] = cart2pol(X,Y); vEfX'gyk  
    %       idx = r<=1; (dO4ww@O  
    %       p = 0:15; C(?lp  
    %       z = nan(size(X)); b5H[~8mf  
    %       y = zernfun2(p,r(idx),theta(idx)); B>~E6j7[Mp  
    %       figure('Units','normalized') A?6b)B/e?  
    %       for k = 1:length(p) d~[ >%&  
    %           z(idx) = y(:,k); Q7#Q6-Q  
    %           subplot(4,4,k) p]pFZ";70  
    %           pcolor(x,x,z), shading interp D;:lw]  
    %           set(gca,'XTick',[],'YTick',[]) ,P9B8oIq  
    %           axis square ^. Pn)J  
    %           title(['Z_{' num2str(p(k)) '}'])  '5[L []A  
    %       end [ ra [~  
    % ~8|$KD4I  
    %   See also ZERNPOL, ZERNFUN. }25{"R}K  
    fh,Y#.V`  
    %   Paul Fricker 11/13/2006 %7V?7BE  
    $RF"m"  
    /nC"'d(#  
    % Check and prepare the inputs: Z/ThY bk  
    % ----------------------------- TJO|{Lxm  
    if min(size(p))~=1 Bpqq-_@  
        error('zernfun2:Pvector','Input P must be vector.') =@ON>SmPs  
    end I#9K/[  
    8K1+ttjm  
    if any(p)>35 0e/~H^,SQ  
        error('zernfun2:P36', ... ,cbP yg  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... \3Xt\1qN4  
               '(P = 0 to 35).']) lz# inC|  
    end [vIO  
    -98bX]8  
    % Get the order and frequency corresonding to the function number: ~[,E i k  
    % ---------------------------------------------------------------- 9Uk9TG5  
    p = p(:); 8(S'g+p  
    n = ceil((-3+sqrt(9+8*p))/2); g[Yok` e[  
    m = 2*p - n.*(n+2); oTfEX4 t {  
    n){\KIU/O  
    % Pass the inputs to the function ZERNFUN: ,@>B#%Nz  
    % ---------------------------------------- \w`Il"}V  
    switch nargin L- =^GNh  
        case 3 * 9*I:Uh57  
            z = zernfun(n,m,r,theta); c{&sf y  
        case 4 iF`E> %#  
            z = zernfun(n,m,r,theta,nflag); LWIU7dw  
        otherwise EcP"GO5  
            error('zernfun2:nargin','Incorrect number of inputs.') JbG+ysn  
    end 8BWLi5R[  
    ?{^T&<18t  
    % EOF zernfun2
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    function z = zernpol(n,m,r,nflag) }.e*=/"MB  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. {L.0jAwB  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of ^8We}bs-c  
    %   order N and frequency M, evaluated at R.  N is a vector of b/<n:*$   
    %   positive integers (including 0), and M is a vector with the o<%Sr*  
    %   same number of elements as N.  Each element k of M must be a m#8mU,7  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) V_Y SYG9f  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is gKBcD\F  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix VE{t]>*-u  
    %   with one column for every (N,M) pair, and one row for every Vwl`A3Y  
    %   element in R. 6h;$^3x$  
    % Tw|=;m  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- $L;7SY?  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is ;2& (]1X  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to 'fZHtnmc0  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 6B|IbQ^  
    %   for all [n,m]. fq\E$'o$  
    % 9n44 *sZ  
    %   The radial Zernike polynomials are the radial portion of the x+^iEj`gk  
    %   Zernike functions, which are an orthogonal basis on the unit @'~v~3 $S  
    %   circle.  The series representation of the radial Zernike V =1Y&y  
    %   polynomials is O(wt[AEA  
    % ?1?m4i  
    %          (n-m)/2 l$.C40v  
    %            __ _`ot||J  
    %    m      \       s                                          n-2s p?5`+Z  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r /z~;.jRg  
    %    n      s=0 \W #M]Q  
    % p6I@o7f  
    %   The following table shows the first 12 polynomials. T2^ @x9  
    % 'rU 5VrK  
    %       n    m    Zernike polynomial    Normalization WM*7p;t@)  
    %       --------------------------------------------- / 1E6U6  
    %       0    0    1                        sqrt(2) o^5xCK:Oi2  
    %       1    1    r                           2 e|xRK?aVBu  
    %       2    0    2*r^2 - 1                sqrt(6) kg9ZSkJr  
    %       2    2    r^2                      sqrt(6) aq[kKS`  
    %       3    1    3*r^3 - 2*r              sqrt(8) @K2q*d  
    %       3    3    r^3                      sqrt(8) F&I^bkvh  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) APy&~`  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) u9sffX5x[J  
    %       4    4    r^4                      sqrt(10) /R@eOl}D  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) D0tI  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) Q+Jzab  
    %       5    5    r^5                      sqrt(12) zjB8~ku#  
    %       --------------------------------------------- 1j3mTP  
    % :\gdQG  
    %   Example: Tig`4d-%  
    % l.Qj?G  
    %       % Display three example Zernike radial polynomials -=2tKH`Q  
    %       r = 0:0.01:1; ,in`JM<o  
    %       n = [3 2 5]; $)z(4Ev  
    %       m = [1 2 1]; jSpmE  
    %       z = zernpol(n,m,r); ~B|K]&/]  
    %       figure ,Q2`N{f  
    %       plot(r,z) dk-Y!RfNx  
    %       grid on D+#QQH  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') kf.w:X"i  
    % ]KLj Qpd  
    %   See also ZERNFUN, ZERNFUN2. [y64%|m  
    7s1FJm=Y/  
    % A note on the algorithm. y kwS-e  
    % ------------------------ eBBqF!WDb  
    % The radial Zernike polynomials are computed using the series ( *UMpdj  
    % representation shown in the Help section above. For many special -05#/-Z=  
    % functions, direct evaluation using the series representation can EL5gMs  
    % produce poor numerical results (floating point errors), because fPa FL}&  
    % the summation often involves computing small differences between W=%}~ 7*  
    % large successive terms in the series. (In such cases, the functions /vBOf;L  
    % are often evaluated using alternative methods such as recurrence 34&n { xv  
    % relations: see the Legendre functions, for example). For the Zernike L+(5`Y  
    % polynomials, however, this problem does not arise, because the M7BJ$fA0E  
    % polynomials are evaluated over the finite domain r = (0,1), and DF P0WXbOE  
    % because the coefficients for a given polynomial are generally all k7'B5zVd  
    % of similar magnitude. ggXg4~WL  
    % (Lp<T!"  
    % ZERNPOL has been written using a vectorized implementation: multiple rp{q.fy'U  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] K;k&w; j  
    % values can be passed as inputs) for a vector of points R.  To achieve _cQTQ  
    % this vectorization most efficiently, the algorithm in ZERNPOL Pur~Rz\ \  
    % involves pre-determining all the powers p of R that are required to e4j:IK>  
    % compute the outputs, and then compiling the {R^p} into a single G?#f@N0.5p  
    % matrix.  This avoids any redundant computation of the R^p, and `og 3P:y  
    % minimizes the sizes of certain intermediate variables. q`{crY30  
    % ,n-M!y  
    %   Paul Fricker 11/13/2006 -1DQO|q#  
    z~# .Ey  
    vB hpD  
    % Check and prepare the inputs: 3#!}W#xv  
    % ----------------------------- &k+ jVymH  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) DwMq  
        error('zernpol:NMvectors','N and M must be vectors.') -M/DOTc  
    end Oc}4`?oy<O  
    ,73J#  
    if length(n)~=length(m) ^M0e0  
        error('zernpol:NMlength','N and M must be the same length.') o cotO  
    end G\@ uj>Z  
    G0Eqo$W)S  
    n = n(:); %scSp&X  
    m = m(:); {U= Mfo?AH  
    length_n = length(n); 2"cUBFc1I  
    rF'_YYpr>  
    if any(mod(n-m,2)) ds}:t.3}6  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') 'avzESe~'  
    end iO4Yfj#?  
    ]+@@{?0  
    if any(m<0) Oe:+%p  
        error('zernpol:Mpositive','All M must be positive.') Y@)/iwq  
    end UFr ]$m&  
    $@QF<?i~  
    if any(m>n) h mC. 5mY  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') OuWG.Za  
    end \qj4v^\  
    "#Qqwsw7  
    if any( r>1 | r<0 ) X,DG2HT  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') j> Ce06G  
    end #fk#RNt  
    l6] :Zcd0  
    if ~any(size(r)==1) Cak `}J 2  
        error('zernpol:Rvector','R must be a vector.') {]-AuC2E/0  
    end *DfwTbg|  
    "ld4v+o8l  
    r = r(:); 0&XdCoIe  
    length_r = length(r); 6b`3AAGU"  
    #29m <f_n  
    if nargin==4 rWsUWA T*  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); 1- Jd Qs6  
        if ~isnorm @#m@ .   
            error('zernpol:normalization','Unrecognized normalization flag.') di $\\ Ah  
        end }rK9M$2]u  
    else T rK-XTev  
        isnorm = false; i286 J.  
    end as%@dUK?  
    `#J0@ -  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {HoeK>rd  
    % Compute the Zernike Polynomials CCx_|>  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% b{4@ ~>i  
    G)5R iRcs  
    % Determine the required powers of r: 'y_<O|-  
    % ----------------------------------- -|_#6-9  
    rpowers = []; &gGh%:`B  
    for j = 1:length(n) 9vX~gh{]~  
        rpowers = [rpowers m(j):2:n(j)]; A><w1-X&=o  
    end JO3"$s|t  
    rpowers = unique(rpowers); p SMF1Oy  
    *DBm"{q%&k  
    % Pre-compute the values of r raised to the required powers, TQ hu$z<  
    % and compile them in a matrix: )w/ #T  
    % ----------------------------- B L^?1x  
    if rpowers(1)==0 1V/?p<A  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); $>if@}u  
        rpowern = cat(2,rpowern{:}); SrVo0$5)  
        rpowern = [ones(length_r,1) rpowern]; ' 5tk0A  
    else :[P)t %  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); }(MI}o}  
        rpowern = cat(2,rpowern{:}); dbVMG-z8  
    end '\g-z  
    4zo^ b0v  
    % Compute the values of the polynomials: Pk{eGG<F$  
    % -------------------------------------- kz30! L  
    z = zeros(length_r,length_n); ,76xa%k(U|  
    for j = 1:length_n >G#SfE$0  
        s = 0:(n(j)-m(j))/2; 9szUN;:ZZ  
        pows = n(j):-2:m(j); ;zF3e&e(  
        for k = length(s):-1:1 o*5iHa(Qm  
            p = (1-2*mod(s(k),2))* ... E!Ljq3iT`  
                       prod(2:(n(j)-s(k)))/          ... A  [c1E[  
                       prod(2:s(k))/                 ... B:X,vE  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... z.vE RP56  
                       prod(2:((n(j)+m(j))/2-s(k))); Fi!BXngbd  
            idx = (pows(k)==rpowers); |^:qJ;dOP  
            z(:,j) = z(:,j) + p*rpowern(:,idx); qz_'v{uAj  
        end Sj(5xa[  
         .Tm m  
        if isnorm ^osXM`  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); WWBm*?U  
        end Acix`-<  
    end 70Z#Ej  
    I,8f{T!O@"  
    % 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)  $1y8X K7r  
    q8!X^1F7  
    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。 tx7B?/5D  
    2jF}n*[OW  
    07年就写过这方面的计算程序了。
    让光学不再神秘,让光学变得容易,快速实现客户关于光学的设想与愿望。