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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 !^m%O0DT  
    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! eR:b=%T8  
     
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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 +Y}V3(w9X  
    function z = zernfun(n,m,r,theta,nflag) <Mx0\b!  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. s,6`RI%  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N !*aPEf270  
    %   and angular frequency M, evaluated at positions (R,THETA) on the {%C*{,#+8q  
    %   unit circle.  N is a vector of positive integers (including 0), and j%L&jH 6@  
    %   M is a vector with the same number of elements as N.  Each element ]PWDE"  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) 9T7e\<8"vC  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, N|mJg[j@7  
    %   and THETA is a vector of angles.  R and THETA must have the same W3r?7!~  
    %   length.  The output Z is a matrix with one column for every (N,M) OtJ\T/q,  
    %   pair, and one row for every (R,THETA) pair. nOb?-rR  
    % 0fm*`4Q  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike UH? p]4Nz  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), eujK4s  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral lhH`dG D  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ST5V!jz  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized iYJZvN  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. eg/<[ A:  
    % 6KHN&P  
    %   The Zernike functions are an orthogonal basis on the unit circle. IvHh4DU3Z  
    %   They are used in disciplines such as astronomy, optics, and [kV;[c}  
    %   optometry to describe functions on a circular domain. H#i{?RM@l  
    % \3Pv# )  
    %   The following table lists the first 15 Zernike functions. FOwnxYGVf  
    % ~YP Jez  
    %       n    m    Zernike function           Normalization <IJu7t>  
    %       -------------------------------------------------- uR;gVO+QC  
    %       0    0    1                                 1 M;w?[yEZ  
    %       1    1    r * cos(theta)                    2 HOoPrB m  
    %       1   -1    r * sin(theta)                    2 ^/U27B  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) Vw tZLP36  
    %       2    0    (2*r^2 - 1)                    sqrt(3) Bc7V)Y K  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) omSM:f_~  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) s 5WqR 8  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) R=Zn -q  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) r H8@69,B  
    %       3    3    r^3 * sin(3*theta)             sqrt(8)  6e,xDr  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) 0(U#)  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ^n1%OzGK#  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) m)v''`9LU  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 1'.7_EQ4T  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) {@W93=Vq8  
    %       -------------------------------------------------- p;T{i._iL  
    % = ?D(g  
    %   Example 1: B* kcN lW  
    % VhL{'w7f  
    %       % Display the Zernike function Z(n=5,m=1) NLS"eD m  
    %       x = -1:0.01:1; : _e#  
    %       [X,Y] = meshgrid(x,x); %`MQmXgM  
    %       [theta,r] = cart2pol(X,Y); &;yH@@Z  
    %       idx = r<=1; 1CU>L[W)  
    %       z = nan(size(X)); {n#k,b&9B  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); o+w;PP)+=  
    %       figure N`@NiJ(O;  
    %       pcolor(x,x,z), shading interp o?L'Pg  
    %       axis square, colorbar N|N3x7=gs  
    %       title('Zernike function Z_5^1(r,\theta)') {7u[1[L1  
    % S$)*&46g  
    %   Example 2: Hy.AyU|L  
    % d) f@ 5/<  
    %       % Display the first 10 Zernike functions GSclK|#t E  
    %       x = -1:0.01:1; 5=Xy,hmnC  
    %       [X,Y] = meshgrid(x,x); 7SDFz}  
    %       [theta,r] = cart2pol(X,Y); :y*NM,s  
    %       idx = r<=1; 6E(Qx~i L  
    %       z = nan(size(X)); > fnh+M  
    %       n = [0  1  1  2  2  2  3  3  3  3]; CTX9zrY*T  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; 6+r$t#  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; L86n}+ P\  
    %       y = zernfun(n,m,r(idx),theta(idx)); gE#>RM5D  
    %       figure('Units','normalized') ,.eWQK~  
    %       for k = 1:10 <,o>Wx*1C  
    %           z(idx) = y(:,k); 7C#`6:tI  
    %           subplot(4,7,Nplot(k)) ]Chj T}  
    %           pcolor(x,x,z), shading interp C~fjWz' V  
    %           set(gca,'XTick',[],'YTick',[]) r/pH_@  
    %           axis square XL#[ %X9  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) Vk< LJ S  
    %       end KT]Pw\y5  
    % D\IjyZ-O  
    %   See also ZERNPOL, ZERNFUN2. 7Q\|=$2  
    db'/`JeK b  
    %   Paul Fricker 11/13/2006 f#+el y  
    EY*(Bw  
    V5+SWXZ  
    % Check and prepare the inputs: SGb;!T *  
    % ----------------------------- B8E'ddUw  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) n>@oBG)!  
        error('zernfun:NMvectors','N and M must be vectors.') h(WrL  
    end 2,ECYie^  
    @`\VBW  
    if length(n)~=length(m) *JggU  
        error('zernfun:NMlength','N and M must be the same length.') wFG3KzEq ~  
    end {U&.D [{&  
    rG,5[/l  
    n = n(:); V_plq6z  
    m = m(:); IV\J3N^  
    if any(mod(n-m,2))  >S$Z  
        error('zernfun:NMmultiplesof2', ... gV&z2S~"  
              'All N and M must differ by multiples of 2 (including 0).') .<kqJ|SVi  
    end 'SQG>F Uy  
    h iNEJ_f  
    if any(m>n) l5L.5 $N  
        error('zernfun:MlessthanN', ... !i=nSqW  
              'Each M must be less than or equal to its corresponding N.') 9 \^|6k,  
    end ~]ZpA-*@Ut  
    wAnb Di{W  
    if any( r>1 | r<0 ) =8U&[F  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') Da"j E  
    end cwGbSW$t  
    'X shmZ0&  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) N},n `Yl.  
        error('zernfun:RTHvector','R and THETA must be vectors.') Jx'i2&hGN  
    end 1S@vGq}  
    {Zp\^/  
    r = r(:); )BRKZQN  
    theta = theta(:); ve&zcSeb  
    length_r = length(r); GF3/RT9  
    if length_r~=length(theta) ~'R(2[L!;  
        error('zernfun:RTHlength', ... &=4(l|wcg  
              'The number of R- and THETA-values must be equal.') ~|<m,)!  
    end Bn>8&w/P  
    &+G"k~%  
    % Check normalization: #s!'+|2n  
    % -------------------- aL\nT XakX  
    if nargin==5 && ischar(nflag) 0OGCilOb*  
        isnorm = strcmpi(nflag,'norm'); HF3f)}l$  
        if ~isnorm :O5og[;b  
            error('zernfun:normalization','Unrecognized normalization flag.') {d?$m*YR3`  
        end Qt|c1@J  
    else A&>.74}p  
        isnorm = false; ^iQn'++Q  
    end [Y`,qB<B  
    xLx]_R()  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% j(~ *'&|(  
    % Compute the Zernike Polynomials 4b:s<$TZ  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% NH3cq  
    7 [g/TB  
    % Determine the required powers of r: <8,cuX\  
    % ----------------------------------- OQ9x*TmK  
    m_abs = abs(m); ^{8Gt @  
    rpowers = []; Z}-Vf$O~  
    for j = 1:length(n) iDf,e Kk$'  
        rpowers = [rpowers m_abs(j):2:n(j)]; wY"Q o7  
    end umdG(osR  
    rpowers = unique(rpowers); >2By +/!X  
    t='# |');  
    % Pre-compute the values of r raised to the required powers, cW+t#>' r  
    % and compile them in a matrix: [CAR[ g&  
    % ----------------------------- *3D%<kVl  
    if rpowers(1)==0 , lJ  v  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false);  YBnA+l*  
        rpowern = cat(2,rpowern{:}); `%j~|i)4  
        rpowern = [ones(length_r,1) rpowern]; `)QCn<  
    else frBX{L  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); }W ^: cp  
        rpowern = cat(2,rpowern{:}); Wq^qpN)5Y  
    end yS=oUE$  
    S/ibb&  
    % Compute the values of the polynomials: 9aJ%`i  
    % -------------------------------------- sdS^e`S  
    y = zeros(length_r,length(n)); pAk/Qxl3eo  
    for j = 1:length(n) \cCV6A[  
        s = 0:(n(j)-m_abs(j))/2; mg, j:,  
        pows = n(j):-2:m_abs(j); D$JHs4  
        for k = length(s):-1:1 ZNx$r]4nF  
            p = (1-2*mod(s(k),2))* ... ]~\sA  
                       prod(2:(n(j)-s(k)))/              ... 57 #6yXQ  
                       prod(2:s(k))/                     ... F-*2LMe  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... WQHd[2Z#e  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); Vrvic4  
            idx = (pows(k)==rpowers); vp.ZK[/`  
            y(:,j) = y(:,j) + p*rpowern(:,idx); wM|" I^[  
        end /6_|]ijc  
         2W$cFC  
        if isnorm HEGKX]  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); )Jv[xY~  
        end f0T ,ul,  
    end K).n.:vYZ  
    % END: Compute the Zernike Polynomials (xq%  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% r0m*5rd1  
    z'`y,8Y1l  
    % Compute the Zernike functions: 4WB-Ec  
    % ------------------------------ TB;o~>9U  
    idx_pos = m>0; ^OErq&`u  
    idx_neg = m<0; w/L `  
    5#QXR+ T  
    z = y; FW.$5*f='  
    if any(idx_pos) `N5|Ho*C  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); BAO|)~1Pd  
    end c_" ~n|  
    if any(idx_neg) P<K){V  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ti &J  
    end CX m+)a-L  
    CpQN,-4  
    % EOF zernfun
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) _Ob@`  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. dZ _zg<  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated 1 n%?l[o  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive wR{'y)$  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, -)~SM&  
    %   and THETA is a vector of angles.  R and THETA must have the same X<R?uI?L  
    %   length.  The output Z is a matrix with one column for every P-value, jg&E94}+  
    %   and one row for every (R,THETA) pair. !a~`Bs$'jr  
    % Nl' )l"  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike s#uJ ;G  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) _{|D  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) a5G/[[cwTV  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 "YB** Y  
    %   for all p. Pm$q]A~  
    % G7=p Bf  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 *SO{\bu  
    %   Zernike functions (order N<=7).  In some disciplines it is X=C1/4wU  
    %   traditional to label the first 36 functions using a single mode zB? V_aT  
    %   number P instead of separate numbers for the order N and azimuthal sN("+ sZ.n  
    %   frequency M. qp}Ma8+  
    % 7r{83_B  
    %   Example: CB&iI'  
    % srV.)Ur  
    %       % Display the first 16 Zernike functions mYc.x  
    %       x = -1:0.01:1; w}{5#   
    %       [X,Y] = meshgrid(x,x); x 2QIPUlf  
    %       [theta,r] = cart2pol(X,Y); cJgBI(S5  
    %       idx = r<=1; f7I{WfZ\P  
    %       p = 0:15; RaTH\ >n  
    %       z = nan(size(X)); xF8}:z0  
    %       y = zernfun2(p,r(idx),theta(idx)); IG3,XW  
    %       figure('Units','normalized') ?EX"k+G  
    %       for k = 1:length(p) X w.p  
    %           z(idx) = y(:,k); ``A 0WN  
    %           subplot(4,4,k) Z#.f&K )xX  
    %           pcolor(x,x,z), shading interp Jdy=_88MD  
    %           set(gca,'XTick',[],'YTick',[]) +w8R!jdA  
    %           axis square CU7F5@+  
    %           title(['Z_{' num2str(p(k)) '}']) _ %%Z6x(  
    %       end -TZ p FT"  
    % CDi<< ,  
    %   See also ZERNPOL, ZERNFUN. {NJfNu  
     8L*GE  
    %   Paul Fricker 11/13/2006 =:SN1#G3n  
    3Oa*%kP+  
    1&:@  
    % Check and prepare the inputs: *z'yk*  
    % ----------------------------- wDwH.~3!  
    if min(size(p))~=1 O@iu aeEW  
        error('zernfun2:Pvector','Input P must be vector.') >\w&6 i~  
    end G4 G5PXi  
    @t1V o}c  
    if any(p)>35 `Bn=?9  
        error('zernfun2:P36', ... )f dE6  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... k-*Mzm]kb  
               '(P = 0 to 35).']) RmI1`  
    end _73h<|0  
    5( _6+'0  
    % Get the order and frequency corresonding to the function number: 8^B;1`#  
    % ---------------------------------------------------------------- MCh#="L2  
    p = p(:); .qob_dRA  
    n = ceil((-3+sqrt(9+8*p))/2); !6}O.Nu  
    m = 2*p - n.*(n+2); u =gt<1U  
    =e?$M  
    % Pass the inputs to the function ZERNFUN: TEsnNi 1  
    % ---------------------------------------- dC}`IR  
    switch nargin !AJ]j|@VBd  
        case 3 3/JyUh?  
            z = zernfun(n,m,r,theta); Iak0 [6Ey  
        case 4 gK|R =J  
            z = zernfun(n,m,r,theta,nflag); 2l5KJlfj>k  
        otherwise htP|3B  
            error('zernfun2:nargin','Incorrect number of inputs.') D5?phyC[Z  
    end [Vf}NF  
    (qwdQMj`  
    % 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) u&QKwD Uh  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. 6aOp[-Le  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of N]5m(@h  
    %   order N and frequency M, evaluated at R.  N is a vector of oojiJ~  
    %   positive integers (including 0), and M is a vector with the FbACTeB  
    %   same number of elements as N.  Each element k of M must be a jw6Tj;c  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) (P6vOo  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is v[<Bjs\q5  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix 0=v{RQ;W4  
    %   with one column for every (N,M) pair, and one row for every TJ6#P<M  
    %   element in R. oN(-rWdhZ  
    % ED} 31L  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- u~Tg&0V30  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is [;O^[Iybf:  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to ZEbLL4n  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 `0#H]=$2h  
    %   for all [n,m]. Ul Mi.;/^  
    % |)_-Bi;MW`  
    %   The radial Zernike polynomials are the radial portion of the 6-,m}Ce\  
    %   Zernike functions, which are an orthogonal basis on the unit Ua*&_~7kJ  
    %   circle.  The series representation of the radial Zernike _>bRv+RVR  
    %   polynomials is N~,_`=yRx  
    % rVA L|0;3  
    %          (n-m)/2 qX>Q+_^  
    %            __ MmU%%2QG  
    %    m      \       s                                          n-2s 8 |h9sn;P  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r `{FwTZ=6{  
    %    n      s=0 KO]N%]:&~  
    % ecH/Wz1  
    %   The following table shows the first 12 polynomials. nb dGt  
    % fAj2LAK  
    %       n    m    Zernike polynomial    Normalization SlT*C6f  
    %       --------------------------------------------- 1(`M~vFDK  
    %       0    0    1                        sqrt(2) [EHrIn  
    %       1    1    r                           2  ?)tK!'  
    %       2    0    2*r^2 - 1                sqrt(6) e-D4'lu  
    %       2    2    r^2                      sqrt(6) rcbP$t vz  
    %       3    1    3*r^3 - 2*r              sqrt(8) nJ3vi}`  
    %       3    3    r^3                      sqrt(8) J+<p+(^*v  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) @Hr+/52B  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) |LYKc.xo  
    %       4    4    r^4                      sqrt(10) wFlV=!>,  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) P0\eB S  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) DacJ,in_I{  
    %       5    5    r^5                      sqrt(12) xNdIDj@  
    %       --------------------------------------------- xDrV5bg  
    % u39FN?<^  
    %   Example: 6GOcI#C9C  
    % ~-Oa8ww  
    %       % Display three example Zernike radial polynomials <Dw]yGK@  
    %       r = 0:0.01:1; oP$kRfXS!<  
    %       n = [3 2 5]; (M% ;~y\  
    %       m = [1 2 1]; &`:rp!Lc  
    %       z = zernpol(n,m,r); `B#Z;R  
    %       figure 4'At.<]jL  
    %       plot(r,z) A'eAu  
    %       grid on <]#o*_aFP  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') x% XT2+  
    % kP,7Li\  
    %   See also ZERNFUN, ZERNFUN2. lpEDPvD_Vm  
     Q&+c.S  
    % A note on the algorithm. |6B6?'  
    % ------------------------ %RFYm  
    % The radial Zernike polynomials are computed using the series Kd{#r/HZ  
    % representation shown in the Help section above. For many special ZzL@[g  
    % functions, direct evaluation using the series representation can -Z?Ck!00  
    % produce poor numerical results (floating point errors), because ^L O]Z  
    % the summation often involves computing small differences between `Wf5  
    % large successive terms in the series. (In such cases, the functions 2lz {_9  
    % are often evaluated using alternative methods such as recurrence o`U|`4,  
    % relations: see the Legendre functions, for example). For the Zernike M]ap:  
    % polynomials, however, this problem does not arise, because the =WRO\lgv.  
    % polynomials are evaluated over the finite domain r = (0,1), and vw r RZ"2  
    % because the coefficients for a given polynomial are generally all k&DGJ5m$.  
    % of similar magnitude. 0(TvQ{  
    % Iu6W=A  
    % ZERNPOL has been written using a vectorized implementation: multiple w4OW4J#  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] 91;HiILgT  
    % values can be passed as inputs) for a vector of points R.  To achieve :J%'=_I&H  
    % this vectorization most efficiently, the algorithm in ZERNPOL LkaG[^tfN  
    % involves pre-determining all the powers p of R that are required to ^uBwj }6  
    % compute the outputs, and then compiling the {R^p} into a single .;%q/hP  
    % matrix.  This avoids any redundant computation of the R^p, and 8?Wgawx  
    % minimizes the sizes of certain intermediate variables. 9}n,@@  
    % o4'v> b  
    %   Paul Fricker 11/13/2006 -h9#G{2W[  
    Y2vj}9jK  
    {h^c  
    % Check and prepare the inputs: BQU/QoDY  
    % ----------------------------- j1Fw U  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) xvO 3BU~2  
        error('zernpol:NMvectors','N and M must be vectors.') {*__B} ,N  
    end T/7vM6u  
    3jg'1^c  
    if length(n)~=length(m) zn[QvY  
        error('zernpol:NMlength','N and M must be the same length.') kC|Tubs(  
    end n'3u] ~7^  
    KZ4zF  
    n = n(:); gD5P!}s[u0  
    m = m(:); a],h<wGEx  
    length_n = length(n); . z].:$J&  
    X4 Y  
    if any(mod(n-m,2)) V@Kn24''  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') r+TK5|ke  
    end e7's)C>/'  
    _y-B";Vmm  
    if any(m<0) ~%KM3Vap  
        error('zernpol:Mpositive','All M must be positive.') EJ8I[(  
    end rV U:VL`2  
    \L %q[  
    if any(m>n) n`QO(pZ6+  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') 1(# RN9   
    end CnQg*+  
    U%n,XOJ  
    if any( r>1 | r<0 ) p~FQcW'a~  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') 9[,s4sxH  
    end 9IOGc}  
    qTS @D  
    if ~any(size(r)==1) 5Fr;  
        error('zernpol:Rvector','R must be a vector.') Y@ObwKcG  
    end m6eFXP1U  
    "kU>~~y,  
    r = r(:); -3\7vpcdN  
    length_r = length(r); k~R{Y~W!!  
    EM=xd~H  
    if nargin==4 44Q9* ."  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); )]tvwEo  
        if ~isnorm p}&Md-$1  
            error('zernpol:normalization','Unrecognized normalization flag.') yz8-&4YRNd  
        end )ib7K1GJ  
    else O%prD}x  
        isnorm = false; OH0S2?,{>  
    end ;Na^]32  
    Nr%(2[$ =  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 5^b i 7J  
    % Compute the Zernike Polynomials x'zihDOI  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% CJm.K  
    (Hl8U  
    % Determine the required powers of r: 8H7O/n  
    % ----------------------------------- /MhS=gVxM  
    rpowers = []; Rnzqw,q  
    for j = 1:length(n) %N>\:8 5?  
        rpowers = [rpowers m(j):2:n(j)]; XzW7eO ,A  
    end -.^3;-[  
    rpowers = unique(rpowers); eQ$e*|}"m  
    Oy:QkV9  
    % Pre-compute the values of r raised to the required powers, fEB195#@9  
    % and compile them in a matrix: l 4!kxXf-<  
    % ----------------------------- W"dU1]  
    if rpowers(1)==0 n}0za#G  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); |PI)A`  
        rpowern = cat(2,rpowern{:}); {=s:P|ah  
        rpowern = [ones(length_r,1) rpowern]; O@nqHZ  
    else `bzr_fJ  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); {>wI8  
        rpowern = cat(2,rpowern{:}); 5dqQws-,?1  
    end ;i.I&*t  
    xrfPZBLy  
    % Compute the values of the polynomials: sZ]'DH&_(  
    % -------------------------------------- ^p$1D  
    z = zeros(length_r,length_n); U@#?T  
    for j = 1:length_n xLe =d|6  
        s = 0:(n(j)-m(j))/2; |3S'8Oe CI  
        pows = n(j):-2:m(j); ZH_FA  
        for k = length(s):-1:1 "\4]X"3<+  
            p = (1-2*mod(s(k),2))* ... &S<? 07Z  
                       prod(2:(n(j)-s(k)))/          ... qC\]"Z`m  
                       prod(2:s(k))/                 ... 2H[=l Y  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... +mivqR~{{  
                       prod(2:((n(j)+m(j))/2-s(k))); GhtbQM1[H  
            idx = (pows(k)==rpowers); I<c@uXXV;!  
            z(:,j) = z(:,j) + p*rpowern(:,idx); /X@7ju;   
        end ('T4Db  
         l8er$8S}  
        if isnorm (L`l+t1  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); MJ1W*'9</W  
        end 5LO4P>fq  
    end cT_uJbP+  
    $< A8gTJ  
    % 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)  37ri b  
    skmDsZzw  
    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。 z- {"pI  
    ?j^?@%f0  
    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)