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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 @kw=0  
    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! U*cj'`eqC  
     
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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 rVLA"x 9u  
    function z = zernfun(n,m,r,theta,nflag)  m{~r6@  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. QeGU]WU{  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N '?\Hm'8  
    %   and angular frequency M, evaluated at positions (R,THETA) on the b+kb7  
    %   unit circle.  N is a vector of positive integers (including 0), and Y#\e~>K  
    %   M is a vector with the same number of elements as N.  Each element @uc%]V<:k  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) ^VA)vLj@  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, 5QlJX  
    %   and THETA is a vector of angles.  R and THETA must have the same "YivjHa7H  
    %   length.  The output Z is a matrix with one column for every (N,M) }G]]0Oi2  
    %   pair, and one row for every (R,THETA) pair. Mf?4 `LM  
    % Ou/{PK}  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike bcQ$S;U)  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), BJqM=<nQ  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral [.2>=3T  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, !$j'F?2 >  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized xMe[/7)4  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. B|!Re4`0  
    % Xs4`bbap  
    %   The Zernike functions are an orthogonal basis on the unit circle. Ox58L>:0m  
    %   They are used in disciplines such as astronomy, optics, and c Mq|`CM  
    %   optometry to describe functions on a circular domain. *h`zV<j  
    % W)KV"A3C  
    %   The following table lists the first 15 Zernike functions. \hg12],#:@  
    % ur;8uv2o  
    %       n    m    Zernike function           Normalization STO6cNi  
    %       -------------------------------------------------- ~#wq sm  
    %       0    0    1                                 1 IyMKV$"  
    %       1    1    r * cos(theta)                    2 8kk$:8  
    %       1   -1    r * sin(theta)                    2 K1Uur>Pk%  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) d35,[  
    %       2    0    (2*r^2 - 1)                    sqrt(3) S^3I"B  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) }nkX-PG9  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) "/K44(^  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) ondF  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) RK|C*TCnl  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) [-Dx)N  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) ]2?t $"G8  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) hS<+=3 <M  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) B&cC;Hw  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) lV3\5AEW  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) 0C7x1:  
    %       -------------------------------------------------- fxjs"rD5  
    % [8<)^k  
    %   Example 1: #5F\zeo@F?  
    % XSXS;Fh)  
    %       % Display the Zernike function Z(n=5,m=1) DvU(rr\p  
    %       x = -1:0.01:1; d&F8nBIM5  
    %       [X,Y] = meshgrid(x,x); c'[l%4U8[  
    %       [theta,r] = cart2pol(X,Y); >-f`mT  
    %       idx = r<=1; Y7 = *-  
    %       z = nan(size(X)); 3#W>  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); |*Hw6m  
    %       figure f Vw+8[d0  
    %       pcolor(x,x,z), shading interp K^EW*6vB8O  
    %       axis square, colorbar P/4]x@{ih  
    %       title('Zernike function Z_5^1(r,\theta)') 5Osx__6$t  
    % =j6f/8   
    %   Example 2: !M6*A1g5  
    % tAefBFu  
    %       % Display the first 10 Zernike functions I6~.sTl  
    %       x = -1:0.01:1; }5\F<b^@Y  
    %       [X,Y] = meshgrid(x,x); 3V2 "1Ic  
    %       [theta,r] = cart2pol(X,Y); USv: + .  
    %       idx = r<=1; kU0e;r1N  
    %       z = nan(size(X)); I!~5.  
    %       n = [0  1  1  2  2  2  3  3  3  3]; Ab/gY$l  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; |X0h-kX4  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; >2TDYB|;  
    %       y = zernfun(n,m,r(idx),theta(idx)); 2/3,%5j_  
    %       figure('Units','normalized') ng"R[/)In  
    %       for k = 1:10 > T=($:n  
    %           z(idx) = y(:,k); CtfI&rb[  
    %           subplot(4,7,Nplot(k)) %N04k8z  
    %           pcolor(x,x,z), shading interp WL:CBE#  
    %           set(gca,'XTick',[],'YTick',[]) > X<pzD3u  
    %           axis square E)7vuWO O  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) u,I_p[`E  
    %       end B|zJrz0q3  
    % )%I2#Q"Nt-  
    %   See also ZERNPOL, ZERNFUN2. -W<x|ph U  
    q,(U8  
    %   Paul Fricker 11/13/2006 ,3=|a|p  
    %We~k'2f  
    cxn3e,d`  
    % Check and prepare the inputs: D6fry\  
    % ----------------------------- &'Pwz  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) *]:gEO  
        error('zernfun:NMvectors','N and M must be vectors.') 9!&fak _  
    end ux:czZqy  
    wylbs@  
    if length(n)~=length(m) kZ~0fw-  
        error('zernfun:NMlength','N and M must be the same length.') xM"k qRZ  
    end -^yb[b,  
    MEf`&<t  
    n = n(:); )RG@D\t,  
    m = m(:); lV<2+Is  
    if any(mod(n-m,2)) [uqe|< :  
        error('zernfun:NMmultiplesof2', ... Sc#B -4m  
              'All N and M must differ by multiples of 2 (including 0).') }86&? 0j.  
    end l+`f\},  
    o."k7fLB  
    if any(m>n) 1j"_@?H[  
        error('zernfun:MlessthanN', ... 7L)edR [  
              'Each M must be less than or equal to its corresponding N.') ;;;aM:6\  
    end [;~:',vHQf  
    FOz~iS\  
    if any( r>1 | r<0 ) HGM? ?=  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') iYJ:P  
    end S5'ZKk  
    nE;^xMOK!  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) A@M%}h  
        error('zernfun:RTHvector','R and THETA must be vectors.') J'{69<`Dl  
    end :4JqT|nS  
    [&y="6No  
    r = r(:); YD>5zV%!D  
    theta = theta(:); NX.%Rj*  
    length_r = length(r); ;J [ed>v;3  
    if length_r~=length(theta) uzG{jc^  
        error('zernfun:RTHlength', ... /6S% h-#\  
              'The number of R- and THETA-values must be equal.') G4O $gg  
    end CWKN0HB  
    _:"PBN9  
    % Check normalization: !A_<(M<  
    % -------------------- *XN|ZGl/  
    if nargin==5 && ischar(nflag) &ed&2t`Y  
        isnorm = strcmpi(nflag,'norm'); rFn%e  
        if ~isnorm p=13tQS<  
            error('zernfun:normalization','Unrecognized normalization flag.') 0 ]K\G55  
        end o9GtS$ O\  
    else }MU}-6  
        isnorm = false; 8d4:8}  
    end zt,Tda4Y  
    F/8="dM  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% fyHFfPEE  
    % Compute the Zernike Polynomials hv. 33l  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% V/N:Of:\R  
    "!w$7|% T  
    % Determine the required powers of r: uO]^vP]fT  
    % ----------------------------------- 9c pjO  
    m_abs = abs(m); 0 $Ygt0d  
    rpowers = []; TTGk"2 Q'  
    for j = 1:length(n) ui>0?O*G  
        rpowers = [rpowers m_abs(j):2:n(j)]; .C HET]  
    end sWtT"7>x  
    rpowers = unique(rpowers); hKx*V"7/#\  
    x{'3eJ^8  
    % Pre-compute the values of r raised to the required powers, [B0]%!hFw  
    % and compile them in a matrix: BIJlU(aF  
    % ----------------------------- ioJ~k[T  
    if rpowers(1)==0 p-CBsm5P  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); UYw_k\  
        rpowern = cat(2,rpowern{:}); S2APqRg*  
        rpowern = [ones(length_r,1) rpowern]; H]I^?+)9  
    else O\~/J/u <  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); NI<;Lm  
        rpowern = cat(2,rpowern{:}); #saK8; tp  
    end heizO",8.&  
    xbTvv>'U  
    % Compute the values of the polynomials: `S)*(s?T  
    % -------------------------------------- h=a-~= 8  
    y = zeros(length_r,length(n)); mK Ta.  
    for j = 1:length(n) !Py SYY  
        s = 0:(n(j)-m_abs(j))/2; Jm#mC  
        pows = n(j):-2:m_abs(j); ]'"aVGqa.  
        for k = length(s):-1:1 j:Y1  
            p = (1-2*mod(s(k),2))* ... 'nx";[6(  
                       prod(2:(n(j)-s(k)))/              ... n "J+? ~9  
                       prod(2:s(k))/                     ... ^Fop/\E  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... &gv{LJd5b  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); v,eTDgw  
            idx = (pows(k)==rpowers); C<G`wXlP|  
            y(:,j) = y(:,j) + p*rpowern(:,idx); {p90   
        end 178u4$# b  
         \h{M\bSIEa  
        if isnorm U??T>  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); Hyn*O)q!  
        end Le?yzf  
    end p?Rq  
    % END: Compute the Zernike Polynomials U%PII>s'#  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Qnr7Qnb  
    e5z U`R  
    % Compute the Zernike functions: th4yuDPuA  
    % ------------------------------ >}I BPC  
    idx_pos = m>0; d*cAm$  
    idx_neg = m<0; q@+#CUa&n  
    o6b\ w  
    z = y; ^Gt9.  
    if any(idx_pos) +G.F'  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); kv8 /UW  
    end =rL^^MZp  
    if any(idx_neg) ;,&$ob*/  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 'P`L?/_3  
    end v:xfGA nP  
    j34L*?  
    % EOF zernfun
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) "'9[c"Iz  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. H}f} Y8J{  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated 9|K3xH  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive +#wh`9[wBt  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, vi8)U]6  
    %   and THETA is a vector of angles.  R and THETA must have the same 9~lC/I')t  
    %   length.  The output Z is a matrix with one column for every P-value, &}+^*X  
    %   and one row for every (R,THETA) pair. &}."sGK  
    % 1 Vq)& N  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike + m+v1(@  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) 5;4bZ3e,0  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) 3mCf>qj73  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 <5fb, @YN  
    %   for all p. = VFPZ  
    % ,T<q"d7-#  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 mAZfo53  
    %   Zernike functions (order N<=7).  In some disciplines it is V!XT=Ou?6  
    %   traditional to label the first 36 functions using a single mode 65g"$:0  
    %   number P instead of separate numbers for the order N and azimuthal .kT}E5  
    %   frequency M. d:"#_  
    % ,+mH1#-3  
    %   Example: L^b /+R#  
    % ~IhLjE  
    %       % Display the first 16 Zernike functions  tOEY|  
    %       x = -1:0.01:1; %fH&UFby  
    %       [X,Y] = meshgrid(x,x); %+F%C=GqI  
    %       [theta,r] = cart2pol(X,Y); %c`P`~sp  
    %       idx = r<=1; m&&Y=2  
    %       p = 0:15; =IC cN|  
    %       z = nan(size(X)); W5c?f,  
    %       y = zernfun2(p,r(idx),theta(idx)); $sa5aUg }  
    %       figure('Units','normalized') a|5^4 J \%  
    %       for k = 1:length(p) %jc"s\  
    %           z(idx) = y(:,k); "IE*MmsEz  
    %           subplot(4,4,k) L';b908r2  
    %           pcolor(x,x,z), shading interp R8(Bt73  
    %           set(gca,'XTick',[],'YTick',[]) uNoP8U%*  
    %           axis square J4VyP["m  
    %           title(['Z_{' num2str(p(k)) '}']) <Z:Fnp  
    %       end %X^K5Io  
    % D$&LCW#x  
    %   See also ZERNPOL, ZERNFUN. ~bsL W:.'  
    vXUq[,8yf  
    %   Paul Fricker 11/13/2006 (t%+Z"j  
    b>_eD-  
    E,EpzB$_dj  
    % Check and prepare the inputs: I4RUXi 5  
    % ----------------------------- NBaXfWh  
    if min(size(p))~=1 `= FDNOwp  
        error('zernfun2:Pvector','Input P must be vector.') VQe@H8>3  
    end ;"T,3JQPn6  
    R|OY5@  
    if any(p)>35 48p< ~#<W\  
        error('zernfun2:P36', ... 8V f]K}d  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... 0[QVU,]<  
               '(P = 0 to 35).']) _ W +  
    end G~$[(Fhk  
    L32[IL|  
    % Get the order and frequency corresonding to the function number: 1nAAs;`'  
    % ---------------------------------------------------------------- \7elqX`.yY  
    p = p(:); [/'=M h  
    n = ceil((-3+sqrt(9+8*p))/2); Y [ p  
    m = 2*p - n.*(n+2); ~IIlCmMl,  
    K2gg"#ft?  
    % Pass the inputs to the function ZERNFUN: z pV+W-j]  
    % ---------------------------------------- c!20(( 2|I  
    switch nargin xmp^`^v*  
        case 3 oy< q;'  
            z = zernfun(n,m,r,theta); ^\Gukkmh}  
        case 4 n+qa/<  
            z = zernfun(n,m,r,theta,nflag); 9 %MHIY5  
        otherwise hOH DXc"  
            error('zernfun2:nargin','Incorrect number of inputs.') R.rxpJ+kU  
    end @b2JR^  
    ^`< %Pk  
    % 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) 2C@s-`b   
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. =U)e_q  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of {L9WeosQ  
    %   order N and frequency M, evaluated at R.  N is a vector of v5Qp[O_  
    %   positive integers (including 0), and M is a vector with the l:?w{'i$  
    %   same number of elements as N.  Each element k of M must be a W:V:Ej7 h  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) ,MRAEa2  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is Q xg)Wb#  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix NL7CeHs5  
    %   with one column for every (N,M) pair, and one row for every U5!T-o;3}  
    %   element in R. mYRW/8+g  
    % IJz=SV  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- f 3t&Bcw$  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is N-cLp}D}WB  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to 0g&#hW};[6  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 g[ dI%  
    %   for all [n,m]. B!X;T9^d  
    % ehe;<A  
    %   The radial Zernike polynomials are the radial portion of the +`D,7"{Eu  
    %   Zernike functions, which are an orthogonal basis on the unit R-\a3q  
    %   circle.  The series representation of the radial Zernike [ _N w5_  
    %   polynomials is ^_C]?D?  
    % $lYy`OuC  
    %          (n-m)/2 I&lb5'6D  
    %            __ N~] 4,~  
    %    m      \       s                                          n-2s l*(Ml= O{  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r m(6SiV=D9  
    %    n      s=0 ~[H+,+XLY+  
    % h:qt?$]J  
    %   The following table shows the first 12 polynomials. {@tqeu%IM  
    % brXLx +H8  
    %       n    m    Zernike polynomial    Normalization DE659=Tq  
    %       --------------------------------------------- 52H'aHO1  
    %       0    0    1                        sqrt(2) /yhGc}h  
    %       1    1    r                           2 g(`m#&P>G  
    %       2    0    2*r^2 - 1                sqrt(6) $22_>OsA  
    %       2    2    r^2                      sqrt(6) @_ Q  
    %       3    1    3*r^3 - 2*r              sqrt(8) ~0 5p+F)  
    %       3    3    r^3                      sqrt(8) aUVJ\ ;V  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) zUNWcv!& "  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) \4qw LM?E^  
    %       4    4    r^4                      sqrt(10) 5&QDZnsl  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) oMNgyAp^  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) dd{pF\a  
    %       5    5    r^5                      sqrt(12) _/(7:  
    %       --------------------------------------------- t<%S_J\  
    % |y T-N3H@  
    %   Example: md.#n  
    % EqB3f_  
    %       % Display three example Zernike radial polynomials H,I k&{@j  
    %       r = 0:0.01:1; g82_KUkB  
    %       n = [3 2 5]; 1B#Z<p  
    %       m = [1 2 1]; }/cMG/%  
    %       z = zernpol(n,m,r); W:z?w2{VI(  
    %       figure NrTK+6 z  
    %       plot(r,z) wE*jN~  
    %       grid on 14LOeo5O  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') 6%nKrK  
    % )08mG_&atL  
    %   See also ZERNFUN, ZERNFUN2. A3jT;D9Y%  
    <XAW-m9SC  
    % A note on the algorithm. @2HNYW)  
    % ------------------------ b{WEux{)  
    % The radial Zernike polynomials are computed using the series h9J%NH  
    % representation shown in the Help section above. For many special ?oKY"C8/  
    % functions, direct evaluation using the series representation can [ S_8;j  
    % produce poor numerical results (floating point errors), because p l.D h  
    % the summation often involves computing small differences between D%k`udz<  
    % large successive terms in the series. (In such cases, the functions E,fG<X{  
    % are often evaluated using alternative methods such as recurrence L4wKG&  
    % relations: see the Legendre functions, for example). For the Zernike ~ R:=zGDV  
    % polynomials, however, this problem does not arise, because the 4Z"JC9As  
    % polynomials are evaluated over the finite domain r = (0,1), and 3$E\B=7/U  
    % because the coefficients for a given polynomial are generally all XX@@tzN  
    % of similar magnitude. p~h)@  
    % afJ`1l  
    % ZERNPOL has been written using a vectorized implementation: multiple X?Yp=%%  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] Ry+Ax4#+(y  
    % values can be passed as inputs) for a vector of points R.  To achieve NE'4atQ|  
    % this vectorization most efficiently, the algorithm in ZERNPOL e,0y+~  
    % involves pre-determining all the powers p of R that are required to equTKM  
    % compute the outputs, and then compiling the {R^p} into a single n,.ZLuBEX  
    % matrix.  This avoids any redundant computation of the R^p, and F_ Cp,  
    % minimizes the sizes of certain intermediate variables. 2G4OK7x  
    % "N|gU;~W  
    %   Paul Fricker 11/13/2006 7j <:hF~  
    /6$8djw  
    R"0fZENTG  
    % Check and prepare the inputs: N8s2v W  
    % ----------------------------- T9,T'y>BD  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Sjogv  
        error('zernpol:NMvectors','N and M must be vectors.') ^71!.b%  
    end |KM<\v(A{  
    @\_l%/z{  
    if length(n)~=length(m) n qR8uL>  
        error('zernpol:NMlength','N and M must be the same length.') y&$mN  
    end v;r!rZX  
    D02_ Jrg  
    n = n(:); mROXwzL  
    m = m(:); $G_,$U !  
    length_n = length(n); 8LH"j(H  
    ~S='~ g)  
    if any(mod(n-m,2)) z 5T_  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') X9d~r_2&m<  
    end I`O)I&KH  
    +Je(]b @  
    if any(m<0) &$!'Cw`,  
        error('zernpol:Mpositive','All M must be positive.') -X)KY_Xn@/  
    end U6R"eQUTV  
    bkZ~O=uv$-  
    if any(m>n) FK8G BkQ!  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') b.<>CG'  
    end JMnk~8O  
    v@6TC1M,  
    if any( r>1 | r<0 ) .Y?/J,Ch  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') [ NSsT>C  
    end 7xmyjy%c  
    d_0r  
    if ~any(size(r)==1) w# t[sI"IT  
        error('zernpol:Rvector','R must be a vector.') 7:Jyu/*]  
    end h7EKb-@  
    c]y"5;V8  
    r = r(:); YRkp(}*!\  
    length_r = length(r); #\Q{?F!4  
    d]v4`nc  
    if nargin==4 S;582H9D  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); UP@a ?w  
        if ~isnorm q66+x)  
            error('zernpol:normalization','Unrecognized normalization flag.') 1>doa1  
        end f-V8/  
    else ?Q~6\xA  
        isnorm = false; 1lxsj{>U  
    end a!;]9}u7  
    y5XHJUTu  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% IQZBH2R  
    % Compute the Zernike Polynomials s[V$f vW  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% J| orvnkK  
    o4)^U t+  
    % Determine the required powers of r: =ZsM[wd  
    % ----------------------------------- yMQuM :d  
    rpowers = []; N}Ol`@@#h  
    for j = 1:length(n) x,s Ma*vd  
        rpowers = [rpowers m(j):2:n(j)]; H$:Z`CQt<  
    end 8-N8v *0  
    rpowers = unique(rpowers); nt/+?Sj  
    >bf29tr  
    % Pre-compute the values of r raised to the required powers, YH VJg?H3  
    % and compile them in a matrix: 6AZJ,Q\E@  
    % ----------------------------- v;E7UL .w  
    if rpowers(1)==0 d(>7BV  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); .b4_O CGg  
        rpowern = cat(2,rpowern{:}); `ym@ U(;N  
        rpowern = [ones(length_r,1) rpowern]; y\'t{>U/  
    else qsB,yckml  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); d9zI A6y  
        rpowern = cat(2,rpowern{:}); w1J&c'-  
    end |g5B==KI  
    Q0K4_iN)&  
    % Compute the values of the polynomials: U/ncD F%C  
    % -------------------------------------- 6]i"lqb  
    z = zeros(length_r,length_n); _}X_^taTZS  
    for j = 1:length_n 't$(Ruw  
        s = 0:(n(j)-m(j))/2; f\Bd lOJ>  
        pows = n(j):-2:m(j); 73$^y)AvY  
        for k = length(s):-1:1 H61 ,pr>  
            p = (1-2*mod(s(k),2))* ... m6a q_u{W  
                       prod(2:(n(j)-s(k)))/          ... _P!J0  
                       prod(2:s(k))/                 ... f-9& n4=H  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... D6iHkDTg  
                       prod(2:((n(j)+m(j))/2-s(k))); S~\i"A)4  
            idx = (pows(k)==rpowers); /znW$yh o  
            z(:,j) = z(:,j) + p*rpowern(:,idx); (+<SR5,/3  
        end / r#.BXP  
         DnA}!s  
        if isnorm 1 l'Wb2g>A  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); })Og sBk  
        end 3K2`1+kBVG  
    end pYo]lO  
    VGoD2,(b^  
    % 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)  1vw [{.wC  
    aQzDOeTi  
    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。 V0 70oZ  
    ZBYFQTEE  
    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)