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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 S.f5v8  
    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! RVQh2'w  
     
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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 #6HA\dE  
    function z = zernfun(n,m,r,theta,nflag) wG-HF'0L  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. =h5H~G5AT  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N o9dY9o+Z  
    %   and angular frequency M, evaluated at positions (R,THETA) on the N@Uy=?)ZJ  
    %   unit circle.  N is a vector of positive integers (including 0), and lSVp%0jR  
    %   M is a vector with the same number of elements as N.  Each element _v> }_S  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) E vg_q>  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, %2{ %Obp'  
    %   and THETA is a vector of angles.  R and THETA must have the same ud'-;W  
    %   length.  The output Z is a matrix with one column for every (N,M) .Z `av n  
    %   pair, and one row for every (R,THETA) pair. 1oW ED*B  
    % GQUe!G9  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike .7avpOfz  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), EWkLXU6t  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral _8F`cuyW  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, Ssou  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized '9 [vDG~  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. jk[1{I/  
    % &&8IU;J  
    %   The Zernike functions are an orthogonal basis on the unit circle. QLiu2U o  
    %   They are used in disciplines such as astronomy, optics, and 'R'*kxf  
    %   optometry to describe functions on a circular domain. nz=G lO'[  
    % b)qoh^  
    %   The following table lists the first 15 Zernike functions. K1+)4!}%U  
    % "AsKlKz{B  
    %       n    m    Zernike function           Normalization SBfT20z[  
    %       -------------------------------------------------- DN-+osPi  
    %       0    0    1                                 1 qh|_W(`y  
    %       1    1    r * cos(theta)                    2 %4,O 2\0?&  
    %       1   -1    r * sin(theta)                    2 Q/(K$6]j  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) {byBc G  
    %       2    0    (2*r^2 - 1)                    sqrt(3) zck#tht4 n  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) 1AM!8VR2  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) U4C 9<h&  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) q$Zh@  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) 7WkB>cn  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) KyYMfC  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) H Y&DmE  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) g"p%C:NN  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) p93r'&Q  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 6z#acE1)M  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) -w}]fb2Q>  
    %       -------------------------------------------------- 8hOk{xs8  
    % wnEyl[ac  
    %   Example 1: |y!=J$ $_H  
    % ZojI R\F^  
    %       % Display the Zernike function Z(n=5,m=1) =S+wCN  
    %       x = -1:0.01:1; d iL +:H  
    %       [X,Y] = meshgrid(x,x); >~[c|ffyo/  
    %       [theta,r] = cart2pol(X,Y); P2BWuh F  
    %       idx = r<=1; N`5,\TR2f  
    %       z = nan(size(X)); "55skmD.P  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); M"p  
    %       figure Wz49i9e+d  
    %       pcolor(x,x,z), shading interp 7Bzq,2s  
    %       axis square, colorbar {JZZZY!n2  
    %       title('Zernike function Z_5^1(r,\theta)') !;Yg/'vD-  
    % 8dZSi  
    %   Example 2: la0BiLzb]  
    % PV'x+bN5  
    %       % Display the first 10 Zernike functions B}Z63|/N  
    %       x = -1:0.01:1; q<[P6}.  
    %       [X,Y] = meshgrid(x,x); LrM=*R h,O  
    %       [theta,r] = cart2pol(X,Y); ]@j*/IP  
    %       idx = r<=1; 4B =7:r  
    %       z = nan(size(X)); ~:kZgUP_f  
    %       n = [0  1  1  2  2  2  3  3  3  3]; rb5~XnJk  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; QdH\LL^8R4  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; -3t7*  
    %       y = zernfun(n,m,r(idx),theta(idx)); Xx."$l  
    %       figure('Units','normalized') 0%&1\rm+j  
    %       for k = 1:10 R]c+?4J  
    %           z(idx) = y(:,k); 591>rh)  
    %           subplot(4,7,Nplot(k)) DBW[{D E  
    %           pcolor(x,x,z), shading interp :mh_G  
    %           set(gca,'XTick',[],'YTick',[]) S!jTyY7e  
    %           axis square Q('r<v96  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) m[? E  
    %       end L-jJg,eY  
    % qON|4+~u%  
    %   See also ZERNPOL, ZERNFUN2. ]i&6c  
    rdl;M>0@  
    %   Paul Fricker 11/13/2006 V)Z}En["1  
    fxgPhnaC>  
    `18qbot  
    % Check and prepare the inputs: 0bceI  
    % ----------------------------- >BIMi^  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) $UMFNjL  
        error('zernfun:NMvectors','N and M must be vectors.') W98i[Q9A7  
    end ] bM)t<  
    \rx3aJl  
    if length(n)~=length(m) / ;$#d}R  
        error('zernfun:NMlength','N and M must be the same length.') 1tEgl\u\  
    end Fsmycr!R  
    9_# >aOqL  
    n = n(:); dsb`xw  
    m = m(:); 6Z>FTz_  
    if any(mod(n-m,2)) @K\~O__  
        error('zernfun:NMmultiplesof2', ... ^W`<gR  
              'All N and M must differ by multiples of 2 (including 0).') k$R~R-'  
    end N=4G=0 `ke  
    5gb|w\N>  
    if any(m>n) PlU*X8  
        error('zernfun:MlessthanN', ... B-?6M6#  
              'Each M must be less than or equal to its corresponding N.') 4,bv)Im+ `  
    end |'.*K]Yp  
    G"-?&)M#a  
    if any( r>1 | r<0 ) 6LOnU~l,  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') p#01gB  
    end iqC|G/  
    oz,np@f)J  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) <6EeD5{*  
        error('zernfun:RTHvector','R and THETA must be vectors.') PXK7b2fE.  
    end +DW~BS3  
    }\z.)B4,  
    r = r(:); 8;d:-Cp  
    theta = theta(:); 8ZM?)# `@{  
    length_r = length(r); `n#H5Oyn  
    if length_r~=length(theta) ;\a YlV-  
        error('zernfun:RTHlength', ... 5QW=&zI`=  
              'The number of R- and THETA-values must be equal.') mPOGidxix  
    end ]9YJ,d@J  
    $Z!`Hb  
    % Check normalization: wF IegC(  
    % -------------------- -|J"s$yO4  
    if nargin==5 && ischar(nflag) D8inB+/-  
        isnorm = strcmpi(nflag,'norm'); ujDd1Bxf?  
        if ~isnorm 9i'jj N  
            error('zernfun:normalization','Unrecognized normalization flag.') v/Py"hQ  
        end VvvRRP^q  
    else *i\Qo  
        isnorm = false; ?+_Gs;DGVE  
    end zO~8?jDN4|  
    gD,1 06%  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |*oZ _gI  
    % Compute the Zernike Polynomials un)4eo!7  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% M}`B{]lLz  
    G^~k)6v=m  
    % Determine the required powers of r: $:cE ^8K  
    % ----------------------------------- qOe+ZAJ{%N  
    m_abs = abs(m); E.r>7`E  
    rpowers = []; 1_o],? Q  
    for j = 1:length(n) oo,uO;0G  
        rpowers = [rpowers m_abs(j):2:n(j)]; pf%=h |  
    end nc~F_i=  
    rpowers = unique(rpowers); I CZ4 A{I  
    f*!j[U/r_  
    % Pre-compute the values of r raised to the required powers,  W,4QzcQR  
    % and compile them in a matrix: yL%K4$z  
    % ----------------------------- QP@%(]fG  
    if rpowers(1)==0 jq-p;-i  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 8 BY j  
        rpowern = cat(2,rpowern{:}); o]+z)5zC  
        rpowern = [ones(length_r,1) rpowern]; E%+Dl=  
    else :H7D~ n  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); L; T8?+x  
        rpowern = cat(2,rpowern{:}); u6M.'  
    end o 4`hY/<t  
    ,oN8HpGs  
    % Compute the values of the polynomials: ?5U2D%t  
    % -------------------------------------- Da&vb D-Bg  
    y = zeros(length_r,length(n)); IC#>X5  
    for j = 1:length(n) |M>eEE*F<  
        s = 0:(n(j)-m_abs(j))/2; FqkDKTS\&  
        pows = n(j):-2:m_abs(j); H9KKed47d/  
        for k = length(s):-1:1 hhSy0  
            p = (1-2*mod(s(k),2))* ... dA-2%uJ  
                       prod(2:(n(j)-s(k)))/              ... kQ4dwF~  
                       prod(2:s(k))/                     ... BHd&yIyI  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... _9faBrzd  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); R?v>Q` Qi  
            idx = (pows(k)==rpowers); ]Oh@,V8  
            y(:,j) = y(:,j) + p*rpowern(:,idx); ln$&``L  
        end \X<bH&x:z  
         ~hZ"2$(0  
        if isnorm .clP#r{U  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); tna .52*/  
        end x1Lb*3Fe  
    end ` BDLW%aL  
    % END: Compute the Zernike Polynomials kv8Fko  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 4A@NxihH  
    So{x]x:f  
    % Compute the Zernike functions: j;']cWe  
    % ------------------------------ .EpV;xq}  
    idx_pos = m>0; P.6nA^hXB  
    idx_neg = m<0; _6O\W%it  
    P6E3-?4j  
    z = y; N<f"]  
    if any(idx_pos) CJ(NgYC h  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); B,M(@5wz  
    end uJOJ-5}yt  
    if any(idx_neg) $>*3/H  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); (>F%UY  
    end f _[<L  
    I{ HN67O  
    % 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) &f!z1d-qg?  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. Dpvk\t  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated 0.dgoq 3u  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive LAVAFlK5  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, W)9K`hM6  
    %   and THETA is a vector of angles.  R and THETA must have the same VGtC)mG8)  
    %   length.  The output Z is a matrix with one column for every P-value, }tsYJlh5  
    %   and one row for every (R,THETA) pair. aD=a,  
    % ElS9?Q+  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike Is]aj-#r  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) !vX D  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) 5V5%/FU m  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 *_R]*o!W'  
    %   for all p. [`p=(/I&L  
    % m0LTx\w!  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 $}lbT15a  
    %   Zernike functions (order N<=7).  In some disciplines it is N5*u]j  
    %   traditional to label the first 36 functions using a single mode hZh9uI7.  
    %   number P instead of separate numbers for the order N and azimuthal mu?Eco`~  
    %   frequency M. x 8Retuv  
    % b|cyjDMAA  
    %   Example: uIcn{RZ_z  
    % qP{/[uj[K  
    %       % Display the first 16 Zernike functions .{ 44a$)  
    %       x = -1:0.01:1; Of{/t1o?  
    %       [X,Y] = meshgrid(x,x); 1c<=A!"{  
    %       [theta,r] = cart2pol(X,Y); + ` s@  
    %       idx = r<=1; m_=$0m J$  
    %       p = 0:15; ^\\Tx*#i  
    %       z = nan(size(X)); ~\=1'D^6CK  
    %       y = zernfun2(p,r(idx),theta(idx)); t=_J9|  
    %       figure('Units','normalized') 7h6,c/<  
    %       for k = 1:length(p)  yyv8gH  
    %           z(idx) = y(:,k); M7+nW ; e%  
    %           subplot(4,4,k) `VKf3&|<A  
    %           pcolor(x,x,z), shading interp bA\<.d  
    %           set(gca,'XTick',[],'YTick',[]) ?"zY" *>4  
    %           axis square '3TW [!m  
    %           title(['Z_{' num2str(p(k)) '}']) %6L^2 X  
    %       end ~.A)bp  
    % &krwf ]|  
    %   See also ZERNPOL, ZERNFUN. /rq VB|M  
    ox:[f9.5  
    %   Paul Fricker 11/13/2006 Y|8:;u'  
    >tO`r.5u9  
    5QPM t^  
    % Check and prepare the inputs: <@}I0  
    % ----------------------------- } @K FB  
    if min(size(p))~=1 A3B56K  
        error('zernfun2:Pvector','Input P must be vector.') Mu{;vf|j  
    end ?_"+^R z  
    bx]N>k J  
    if any(p)>35 Q-MQ9'  
        error('zernfun2:P36', ... 2P/K K  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... bHg,1y)UC  
               '(P = 0 to 35).']) sXi=70o  
    end )Psb>'X  
    F;gx%[$GX  
    % Get the order and frequency corresonding to the function number: 0{dz5gUde  
    % ---------------------------------------------------------------- )K,F]fc+O  
    p = p(:); UNPezHaz  
    n = ceil((-3+sqrt(9+8*p))/2); w" SoeU  
    m = 2*p - n.*(n+2); ogL EtqT  
    ms!ref4`+  
    % Pass the inputs to the function ZERNFUN: d+X}cq=  
    % ---------------------------------------- UilMv~0  
    switch nargin kGd<5vCs  
        case 3 jeGj<m  
            z = zernfun(n,m,r,theta); A,%C,*)Cg  
        case 4 0PU8 #2pR  
            z = zernfun(n,m,r,theta,nflag); AtF3%Z v2  
        otherwise ,z;ky5Ct  
            error('zernfun2:nargin','Incorrect number of inputs.') uL3Eq>~x  
    end ;]gP@h/  
    ~4s'0 w^  
    % EOF zernfun2
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    function z = zernpol(n,m,r,nflag) c)A{p  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. FBpH21|/y  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of &g.@u~SI1  
    %   order N and frequency M, evaluated at R.  N is a vector of >nw++[K_  
    %   positive integers (including 0), and M is a vector with the "y_#7K  
    %   same number of elements as N.  Each element k of M must be a '=1KVE^Fk  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) (tCUlX2  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is HcedE3Rg  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix Kx=4~  
    %   with one column for every (N,M) pair, and one row for every `)T~psT  
    %   element in R. I!>\#K  
    % mcn 2Wt  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- W -  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is `ORECg)  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to $2M#qkik-  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 -s|}Rh?Y  
    %   for all [n,m]. 3/b;7\M  
    % zoDH` h_  
    %   The radial Zernike polynomials are the radial portion of the esHQoIhd  
    %   Zernike functions, which are an orthogonal basis on the unit ?gPKcjgoH!  
    %   circle.  The series representation of the radial Zernike Yr w$  
    %   polynomials is >[ Ye  
    % 63.wL0~  
    %          (n-m)/2 )r[&RGz6  
    %            __ ?Q-h n:F)  
    %    m      \       s                                          n-2s Hew d4k  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r e]T`ot#/  
    %    n      s=0 ZxlAk+<]  
    % vTaJqEE  
    %   The following table shows the first 12 polynomials. 7C$ 5  
    % G NS`.fS  
    %       n    m    Zernike polynomial    Normalization ?[& 2o|  
    %       --------------------------------------------- = <j"M85.  
    %       0    0    1                        sqrt(2) ;U<rc'qE  
    %       1    1    r                           2 \~ BDm  
    %       2    0    2*r^2 - 1                sqrt(6) &z]K\-xp  
    %       2    2    r^2                      sqrt(6) =7m}yDs6$  
    %       3    1    3*r^3 - 2*r              sqrt(8) quvanx V-L  
    %       3    3    r^3                      sqrt(8) @ JvPx0  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) ;L|uIg;.s  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) ?7 M.o  
    %       4    4    r^4                      sqrt(10) 0<8XI>.3D  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) 70lfb`  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) "Tm[t?FMbe  
    %       5    5    r^5                      sqrt(12) ]$p{I)d&  
    %       --------------------------------------------- `Pw*_2  
    % F  q!fWl  
    %   Example: M:P0m6ie  
    % }lK3-2Pk  
    %       % Display three example Zernike radial polynomials k ^ YO%_  
    %       r = 0:0.01:1; dJv!Dts')C  
    %       n = [3 2 5]; 4GR!y)  
    %       m = [1 2 1]; 8/t$d#xHI  
    %       z = zernpol(n,m,r); +rIL|c}J  
    %       figure 1Nu1BLPm  
    %       plot(r,z) 5OO'v07b  
    %       grid on T \CCF  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') *=fr8  
    % %?aS#4jI  
    %   See also ZERNFUN, ZERNFUN2. G[8in   
    >6oOZbUY0  
    % A note on the algorithm. p-%|P ]&  
    % ------------------------ ~>0qZ{3J_  
    % The radial Zernike polynomials are computed using the series TRZRYm"  
    % representation shown in the Help section above. For many special Ne $"g[uFU  
    % functions, direct evaluation using the series representation can %L [&,a  
    % produce poor numerical results (floating point errors), because VyRsPg[(  
    % the summation often involves computing small differences between q %0Cg=  
    % large successive terms in the series. (In such cases, the functions G60R9y47c  
    % are often evaluated using alternative methods such as recurrence l<Q>N|1#k%  
    % relations: see the Legendre functions, for example). For the Zernike |7B!^ K  
    % polynomials, however, this problem does not arise, because the t8+_/BXv  
    % polynomials are evaluated over the finite domain r = (0,1), and H'MJ{r0,  
    % because the coefficients for a given polynomial are generally all X[2[!)Rk  
    % of similar magnitude. 7~ztwL  
    % Z_gC&7+  
    % ZERNPOL has been written using a vectorized implementation: multiple k'$!(*]\b  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] m=Q[\.Ra  
    % values can be passed as inputs) for a vector of points R.  To achieve N)S!7%ne  
    % this vectorization most efficiently, the algorithm in ZERNPOL p'sc0@}_O  
    % involves pre-determining all the powers p of R that are required to }pa9%BQI  
    % compute the outputs, and then compiling the {R^p} into a single -dv %H{  
    % matrix.  This avoids any redundant computation of the R^p, and w'X]M#Q><  
    % minimizes the sizes of certain intermediate variables. ;. wX@  
    % UZEI:k,dv  
    %   Paul Fricker 11/13/2006 (_r EAEo  
    <`!PCuR  
    mR8W]'gl.L  
    % Check and prepare the inputs: uY< H#k  
    % ----------------------------- O)kg B rB  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) s4%(>Q  
        error('zernpol:NMvectors','N and M must be vectors.') \([WH!7  
    end b dJ+@r  
    \<vNVz7.D  
    if length(n)~=length(m) v(l eide  
        error('zernpol:NMlength','N and M must be the same length.') -[OXSaf6  
    end ]NhS=3*i+  
    VD4C::J  
    n = n(:); O+=vEp(  
    m = m(:); H0a/(4/xg  
    length_n = length(n); i)Lp7m z  
    j5 Un1  
    if any(mod(n-m,2)) 7([h4bg{  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') 55t\Bms{  
    end d7cg&9+  
    _{jP;W  
    if any(m<0) 'SLE;_TD  
        error('zernpol:Mpositive','All M must be positive.') M}0eu(_|  
    end uhV0J97  
    nK3 k]gLc{  
    if any(m>n) :)jJge&^p  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') $jI>[%  
    end _,6f#t  
    Ufo>|A6;$  
    if any( r>1 | r<0 ) BpO9As 1um  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') kC$&:\Rh  
    end w:o-klKXY  
    # x>ga  
    if ~any(size(r)==1) }a&mY^  
        error('zernpol:Rvector','R must be a vector.') 9umGIQHnil  
    end `ya;:$(6  
     Voh hQ  
    r = r(:); oUx[+Gnv  
    length_r = length(r);  .Qt4&B  
    O`cu_  
    if nargin==4 @\(vX]  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); I~'*$l  
        if ~isnorm '?o9VrO  
            error('zernpol:normalization','Unrecognized normalization flag.') 92dF`sv  
        end SW(q$i  
    else ,`td@Y  
        isnorm = false; K:yr-#(P/  
    end ~9D~7UR  
    |!d"*.Q@F  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% YEYY}/YX  
    % Compute the Zernike Polynomials A%Z)wz{  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% h5|.Et  
    -%IcYzyA  
    % Determine the required powers of r: yy2Ie  
    % ----------------------------------- <XQ.A3SG!  
    rpowers = []; < /p 8r  
    for j = 1:length(n) TUp%FJXA|  
        rpowers = [rpowers m(j):2:n(j)]; Em13dem  
    end t~K%.|'0  
    rpowers = unique(rpowers); K.>wQA&  
    ;n#%G^!H  
    % Pre-compute the values of r raised to the required powers, m4ApHM2  
    % and compile them in a matrix: oB c@]T5>  
    % ----------------------------- h. hjz?  
    if rpowers(1)==0 ]Ql 0v"` F  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 4cCF \&yU  
        rpowern = cat(2,rpowern{:}); ^9"KTZc-*  
        rpowern = [ones(length_r,1) rpowern]; fW0$s`  
    else ^E>CGGS4  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); d-!<C7O}  
        rpowern = cat(2,rpowern{:}); xc'vS>&  
    end ;Fl<v@9  
    X=p"5hhfn  
    % Compute the values of the polynomials: &hZwZgV +3  
    % -------------------------------------- )JgC$ <  
    z = zeros(length_r,length_n); yOHXY&  
    for j = 1:length_n &m{'nRU}c  
        s = 0:(n(j)-m(j))/2; z YDK $  
        pows = n(j):-2:m(j); 4\ $3  
        for k = length(s):-1:1 B0mLI%B  
            p = (1-2*mod(s(k),2))* ... OOy}]uYF`  
                       prod(2:(n(j)-s(k)))/          ... k{Lv37H  
                       prod(2:s(k))/                 ... s_wUM)!  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... @YL}km&Fw  
                       prod(2:((n(j)+m(j))/2-s(k))); z[!x:# q8`  
            idx = (pows(k)==rpowers); )3E,D~1e%  
            z(:,j) = z(:,j) + p*rpowern(:,idx); /NBTvTI  
        end X:q_c=X  
         #n})X,ip2  
        if isnorm gT1P*N;v  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); 6* rcR]  
        end Px4/O~bLk  
    end ,jh~;, w2  
    f{Qp  
    % 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)  soxfk+ 9  
    N+hedF@ZU  
    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。 Mb~~A5  
    'DeW<Sa~  
    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)