切换到宽版
  • 广告投放
  • 稿件投递
  • 繁體中文
    • 11202阅读
    • 9回复

    [求助]ansys分析后面型数据如何进行zernike多项式拟合? [复制链接]

    上一主题 下一主题
    离线niuhelen
     
    发帖
    19
    光币
    28
    光券
    0
    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 @ 6jKjI  
    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! =w7k@[Bq  
     
    分享到
    离线phility
    发帖
    69
    光币
    11
    光券
    0
    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
    发帖
    69
    光币
    11
    光券
    0
    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
    发帖
    19
    光币
    28
    光券
    0
    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 o2(*5*b!@e  
    function z = zernfun(n,m,r,theta,nflag) rUF= uO(  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. Ps MCs|*  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 'v3> "b  
    %   and angular frequency M, evaluated at positions (R,THETA) on the Oz,/y3_  
    %   unit circle.  N is a vector of positive integers (including 0), and  F_%&,"$  
    %   M is a vector with the same number of elements as N.  Each element U@?Ro enn  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) *$ 7c||J7  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, mm<rdo(`  
    %   and THETA is a vector of angles.  R and THETA must have the same ~4tu*\P  
    %   length.  The output Z is a matrix with one column for every (N,M) RIl+QA  
    %   pair, and one row for every (R,THETA) pair. :-.bXOB(  
    % xBcE>^{1.  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike *X lnEHv  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), wg,w;Gle  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral tm}0kWx  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 3+jqf@fO  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized :u53zX[v  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ) crhF9!4  
    % MY}B)`yx=  
    %   The Zernike functions are an orthogonal basis on the unit circle. o;@T6-VH  
    %   They are used in disciplines such as astronomy, optics, and @ (A[H^E  
    %   optometry to describe functions on a circular domain. `=3:*.T*  
    % m;nT ?kv  
    %   The following table lists the first 15 Zernike functions. A|d(5{:N  
    % ON=6w_  
    %       n    m    Zernike function           Normalization VS\~t  
    %       -------------------------------------------------- !N1DJd  
    %       0    0    1                                 1 7].FdjT.  
    %       1    1    r * cos(theta)                    2 uD''0G\  
    %       1   -1    r * sin(theta)                    2 3 tp'}v  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) 3Ga! )  
    %       2    0    (2*r^2 - 1)                    sqrt(3) H?>R#Ds-  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) qG%'Lt  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) F|XRh6j  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) J_A5,K*r|  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) 0Y9\,y_  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) FHS6Mk26  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) 0)'^vJe  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) /r Hd9^Y  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) Xo>P?^c4?  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Vn6g(:\w  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) *s9 +  
    %       -------------------------------------------------- >I3#ALF  
    % ayJKt03\O\  
    %   Example 1: $!x8XpR8s  
    % L= fz:H  
    %       % Display the Zernike function Z(n=5,m=1) : YU_ \EV  
    %       x = -1:0.01:1; COa"zg  
    %       [X,Y] = meshgrid(x,x);  # xS8  
    %       [theta,r] = cart2pol(X,Y); /bj D*rj  
    %       idx = r<=1; hp]T^  
    %       z = nan(size(X)); g,!6, v@  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); ?j{LE- (  
    %       figure dhW)<  
    %       pcolor(x,x,z), shading interp eFUJASc  
    %       axis square, colorbar <$LVAy"RD  
    %       title('Zernike function Z_5^1(r,\theta)') @O/-~, E68  
    % ! 3O#'CV  
    %   Example 2: R;gN^Yjk:  
    % lUd/^u`  
    %       % Display the first 10 Zernike functions 3PLv;@!#j}  
    %       x = -1:0.01:1; QcGyuS.B  
    %       [X,Y] = meshgrid(x,x); MS-}IHO  
    %       [theta,r] = cart2pol(X,Y); vcnUb$%  
    %       idx = r<=1; w^?uBeqR  
    %       z = nan(size(X)); R g7  O  
    %       n = [0  1  1  2  2  2  3  3  3  3]; W~J@v@..4  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; v_PhJKE  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; {1.t ZCMT  
    %       y = zernfun(n,m,r(idx),theta(idx)); `w1|(Sk$h  
    %       figure('Units','normalized') cTpAU9|(  
    %       for k = 1:10  "MD  
    %           z(idx) = y(:,k); %Uj7 g>  
    %           subplot(4,7,Nplot(k)) ][1 *.7-  
    %           pcolor(x,x,z), shading interp Bkvh]k;F8  
    %           set(gca,'XTick',[],'YTick',[]) Np=IZ npt  
    %           axis square V8w!yc  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) | -l)$i@  
    %       end !fXwX3B  
    % ?r"'JO.w  
    %   See also ZERNPOL, ZERNFUN2. ^fT|Wm<  
    o;+$AU1f  
    %   Paul Fricker 11/13/2006 \*Ro a&<!  
    a%a_sR\)  
    =Jd ('r  
    % Check and prepare the inputs: C=& 7V  
    % ----------------------------- bLyU;  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ]5X=u(}  
        error('zernfun:NMvectors','N and M must be vectors.') 1@*qz\ YY  
    end O`_!G`E  
    <<}t&qE%2%  
    if length(n)~=length(m)  :feU  
        error('zernfun:NMlength','N and M must be the same length.') #?/&H;n_8S  
    end SXfuPM  
    V3 j1M?>  
    n = n(:); 42X N*br  
    m = m(:); / 4P+  
    if any(mod(n-m,2)) LWQ.!;HYp  
        error('zernfun:NMmultiplesof2', ... o";5@NH  
              'All N and M must differ by multiples of 2 (including 0).') 'r;C( Gh6  
    end #w\Bc\  
    !;'#f xW[  
    if any(m>n) 7?Vo([8  
        error('zernfun:MlessthanN', ... 4 [2^#t[  
              'Each M must be less than or equal to its corresponding N.') TX7B(JZD  
    end `NIc*B4q.  
    o4I&?d7;"  
    if any( r>1 | r<0 ) >_3+s~  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') (#6E{@eq  
    end  g wM~W  
    2n}nRv/'  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) }1<_  
        error('zernfun:RTHvector','R and THETA must be vectors.') TG ,T>'   
    end E[N5vG<  
    8SmtEV[b3  
    r = r(:); hZ_0lX}  
    theta = theta(:); moO=TGG;F  
    length_r = length(r); +[l52p@a  
    if length_r~=length(theta) <YU+W"jQT  
        error('zernfun:RTHlength', ... .O9 A[s<  
              'The number of R- and THETA-values must be equal.') Yv0;UKd  
    end OE,uw2uaT  
    0fc]RkHs"  
    % Check normalization: &mebpEHUG7  
    % -------------------- `? ayc/TK  
    if nargin==5 && ischar(nflag) C:C9swik"5  
        isnorm = strcmpi(nflag,'norm'); "]BefvE  
        if ~isnorm -OLXRc=  
            error('zernfun:normalization','Unrecognized normalization flag.') z?C;z7eT  
        end I W_:nm6  
    else I"*;fdm  
        isnorm = false; $)OUOv  
    end @&f3zq  
    [.e Y xZ{=  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% .HTX7mA3  
    % Compute the Zernike Polynomials U'i L|JRF  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% yK"OZ2Mv  
    p{:r4!*L  
    % Determine the required powers of r: D7Y5q*F  
    % ----------------------------------- +cAN4  
    m_abs = abs(m);  >m!l5/  
    rpowers = []; FrSeR9b  
    for j = 1:length(n)  YM9oVF-  
        rpowers = [rpowers m_abs(j):2:n(j)]; vfT<%Kl!'  
    end &lYKi3}x  
    rpowers = unique(rpowers); bTzVmqGY  
    M,[u}Rf^w  
    % Pre-compute the values of r raised to the required powers, S fE^'G\  
    % and compile them in a matrix: SU, t,i  
    % ----------------------------- AR\?bB~`c  
    if rpowers(1)==0 g{8 R+  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); \~xOdqF/  
        rpowern = cat(2,rpowern{:}); 4+mawyM  
        rpowern = [ones(length_r,1) rpowern]; %g(h%V9f  
    else an3HKfv  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 6Rn_@_Nn)f  
        rpowern = cat(2,rpowern{:}); . FT*K[+ih  
    end "r0z( j  
    +P6#7.p`Z  
    % Compute the values of the polynomials: Y_`D5c:  
    % -------------------------------------- Co8b0-Z  
    y = zeros(length_r,length(n)); uPe4Rr  
    for j = 1:length(n) -"5x? \.{m  
        s = 0:(n(j)-m_abs(j))/2; \S;% "0!  
        pows = n(j):-2:m_abs(j); ( 5 d ~0  
        for k = length(s):-1:1 G#'3bxI{f+  
            p = (1-2*mod(s(k),2))* ... K@VXFV  
                       prod(2:(n(j)-s(k)))/              ... 6<H[1PI`,G  
                       prod(2:s(k))/                     ... ,TU!W|($  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 9#%(%s 2 +  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); +7<{yP6wU  
            idx = (pows(k)==rpowers); XzQ=8r>l  
            y(:,j) = y(:,j) + p*rpowern(:,idx); :EyH'v  
        end :5?ti  
         > c7/E  
        if isnorm u*Oz1~  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 'O`jV0aa'  
        end ]^gD@].  
    end mU_?}}aK,  
    % END: Compute the Zernike Polynomials h_]3L/  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 'xb|5_D  
    &+`l $h  
    % Compute the Zernike functions: FStE/2?  
    % ------------------------------ XrC{{K  
    idx_pos = m>0; oKt<s+r  
    idx_neg = m<0; w;QDQ fx0  
    aEdF Z  
    z = y; +,>f-kaV  
    if any(idx_pos) aJMh>  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); >^:g[6Sj  
    end o;Zoj}  
    if any(idx_neg) `#fOY$#XB  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); CpS' 2@6  
    end ~B(]0:  
    LO.4sO  
    % EOF zernfun
    离线niuhelen
    发帖
    19
    光币
    28
    光券
    0
    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) 2Fx<QRz  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. puqLXDjA/  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated +crAkb}i  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive IJ4"X#Q/  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, #0`"gR#+  
    %   and THETA is a vector of angles.  R and THETA must have the same ,xD{A}}V  
    %   length.  The output Z is a matrix with one column for every P-value, FG#j0#|*  
    %   and one row for every (R,THETA) pair. p~8~EQFj  
    % u/.s rK!K  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike R1Pnj  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) hsqUiB tc6  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) ^<E+7  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 rG\m]C3E  
    %   for all p. MT6kJDyLu  
    % &V=54n=O?  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 p,tB  
    %   Zernike functions (order N<=7).  In some disciplines it is =rBFMTllM  
    %   traditional to label the first 36 functions using a single mode %+I(S`}  
    %   number P instead of separate numbers for the order N and azimuthal raqLXO!j  
    %   frequency M. d:pp,N~2o  
    % ]#BXaBVMY  
    %   Example: Mxd fuFss  
    % \B}W(^\wg;  
    %       % Display the first 16 Zernike functions (Wx)YI  
    %       x = -1:0.01:1; JG( <  
    %       [X,Y] = meshgrid(x,x); 8fe"#^"sR  
    %       [theta,r] = cart2pol(X,Y); ^1~/FU  
    %       idx = r<=1; 4|2$b:t  
    %       p = 0:15; Q}=RG//0*  
    %       z = nan(size(X)); Ro<!n>H  
    %       y = zernfun2(p,r(idx),theta(idx)); \s+MHa&  
    %       figure('Units','normalized') E`?BaCrG~  
    %       for k = 1:length(p) .R gfP'M  
    %           z(idx) = y(:,k); <rC#1wR4  
    %           subplot(4,4,k) %'=oMbi>i4  
    %           pcolor(x,x,z), shading interp  Nl_;l  
    %           set(gca,'XTick',[],'YTick',[]) XS!ZTb>[  
    %           axis square @wFm])}0  
    %           title(['Z_{' num2str(p(k)) '}'])  *$cp"  
    %       end gx6&'${=#  
    % I2W2B3D` c  
    %   See also ZERNPOL, ZERNFUN. 'I<j`)4`d  
    ReE3742@  
    %   Paul Fricker 11/13/2006 OeAPBhTmFj  
    ky~x4_y5  
    * QR7t:([  
    % Check and prepare the inputs: e`~q ;?:  
    % ----------------------------- Z~c7r n  
    if min(size(p))~=1 {30<Vc=  
        error('zernfun2:Pvector','Input P must be vector.') xg^fM@#m  
    end -$0S#/)Z  
    ZGbZu  
    if any(p)>35 B_&PK7vA  
        error('zernfun2:P36', ... Za f)  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... As\5Ze9|  
               '(P = 0 to 35).']) 3LxJ}>]TO  
    end W" !amMQ  
    ^KaMi_--  
    % Get the order and frequency corresonding to the function number: )KuvG:+9W  
    % ---------------------------------------------------------------- d+;wDu   
    p = p(:); MMAC,4  
    n = ceil((-3+sqrt(9+8*p))/2); *Vc=]Z2G^  
    m = 2*p - n.*(n+2); +|H'I j$  
    FO5SXwx  
    % Pass the inputs to the function ZERNFUN: 4bBxZY  
    % ---------------------------------------- "n e'iJf_(  
    switch nargin 2G-? P"4l@  
        case 3 E+)Go-rS(  
            z = zernfun(n,m,r,theta); y2U:( H:l!  
        case 4 6,)y{/ENC  
            z = zernfun(n,m,r,theta,nflag); bpx=&74,6m  
        otherwise 1v4kN -  
            error('zernfun2:nargin','Incorrect number of inputs.') mTPj@F>  
    end D1n2Z :9  
    :kVV.a#g  
    % EOF zernfun2
    离线niuhelen
    发帖
    19
    光币
    28
    光券
    0
    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) l(5-Cr  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. r+\z0_' w6  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of [86'/:L\2  
    %   order N and frequency M, evaluated at R.  N is a vector of nB5[]x'  
    %   positive integers (including 0), and M is a vector with the tTt3D]h(  
    %   same number of elements as N.  Each element k of M must be a 3+-(;>>\  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) _}&]`,s>  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is is(!_Iv  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix 4I-p/&Q  
    %   with one column for every (N,M) pair, and one row for every hwzUCh 5!  
    %   element in R. pPZ/O 6  
    % j''Iai_  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- i .N1Cvp&  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is 32iI :u  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to f}X8|GlBo  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 ymZ/(:3_  
    %   for all [n,m].  J31M:<  
    % ]M(mq`K  
    %   The radial Zernike polynomials are the radial portion of the (UF!Zb]{  
    %   Zernike functions, which are an orthogonal basis on the unit XijLS7Aw|  
    %   circle.  The series representation of the radial Zernike ~vGtNMQg  
    %   polynomials is RfBb{?PP)  
    % L"iyjL<M  
    %          (n-m)/2 fmJWd|  
    %            __ X~he36-+<  
    %    m      \       s                                          n-2s ik *)j  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r ~5NGDT#L*  
    %    n      s=0 ~ n^G<iXLp  
    % <83gn :$  
    %   The following table shows the first 12 polynomials. nV3 7` I  
    % ;p 5v3<PC  
    %       n    m    Zernike polynomial    Normalization 66<\i ltUQ  
    %       --------------------------------------------- o#P3lz  
    %       0    0    1                        sqrt(2) \Byk`} 9  
    %       1    1    r                           2 WcwW@cY7\  
    %       2    0    2*r^2 - 1                sqrt(6) EQJ_$6  
    %       2    2    r^2                      sqrt(6) V;-.38py  
    %       3    1    3*r^3 - 2*r              sqrt(8) n u8j_grW  
    %       3    3    r^3                      sqrt(8) g&3#22z  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) `Kw"XGT  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) 2A}uqaF  
    %       4    4    r^4                      sqrt(10) 1MI7l)D?  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) TucAs 0-bF  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) B21AcE  
    %       5    5    r^5                      sqrt(12) @JEmybu  
    %       --------------------------------------------- p=+*g.,O  
    % (oitCIV  
    %   Example: =)- Q?1q  
    % |3:=qpT-  
    %       % Display three example Zernike radial polynomials iW@Vw{|i I  
    %       r = 0:0.01:1; {!1n5a3" 1  
    %       n = [3 2 5]; < }wAP_y  
    %       m = [1 2 1]; O*03PF^  
    %       z = zernpol(n,m,r); Q f-k&d  
    %       figure a\69,%!:  
    %       plot(r,z) :"P hkR  
    %       grid on //M4Sq(  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') GHJ=-9{YL  
    % !""!sFx)R  
    %   See also ZERNFUN, ZERNFUN2. ; G59}d p~  
    s`iNbW="  
    % A note on the algorithm. Sq|1f?_gU  
    % ------------------------ {vT55i<mk  
    % The radial Zernike polynomials are computed using the series j]B $(pt  
    % representation shown in the Help section above. For many special >?9 WeXG  
    % functions, direct evaluation using the series representation can e#6&uFce  
    % produce poor numerical results (floating point errors), because o`K^Wy~+k#  
    % the summation often involves computing small differences between UW/3{2  
    % large successive terms in the series. (In such cases, the functions R;E"Qdt  
    % are often evaluated using alternative methods such as recurrence :gD0EqV  
    % relations: see the Legendre functions, for example). For the Zernike @yn^6cE  
    % polynomials, however, this problem does not arise, because the |/\U^AHm"h  
    % polynomials are evaluated over the finite domain r = (0,1), and SZ2q}[o`R  
    % because the coefficients for a given polynomial are generally all a^XTW7]r  
    % of similar magnitude. ;WS7.  
    % \ ~LU 'j  
    % ZERNPOL has been written using a vectorized implementation: multiple 5'kTe=  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] @b!R2Yq  
    % values can be passed as inputs) for a vector of points R.  To achieve 3 2 1={\X  
    % this vectorization most efficiently, the algorithm in ZERNPOL I4<{R  
    % involves pre-determining all the powers p of R that are required to {|5$1v   
    % compute the outputs, and then compiling the {R^p} into a single '!fFI1s  
    % matrix.  This avoids any redundant computation of the R^p, and 9O=05CQ  
    % minimizes the sizes of certain intermediate variables. UJz4>JF  
    % `F2*o47|t  
    %   Paul Fricker 11/13/2006 N^Hn9n  
    YoF\ MT]W  
    <Id1:  
    % Check and prepare the inputs: g2aT`=&Z  
    % ----------------------------- fa4951_  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) LN.Bd,  
        error('zernpol:NMvectors','N and M must be vectors.') }+dM1O  
    end y/@;c)1b9  
    b>bgUDq  
    if length(n)~=length(m) lpfwlB'~9  
        error('zernpol:NMlength','N and M must be the same length.') :^H9W^2  
    end n~,]KdU]  
    k,;lyE  
    n = n(:); t*Vao  
    m = m(:); qP;{3FSkAF  
    length_n = length(n); Ho)t=qn  
    [>$\s=` h  
    if any(mod(n-m,2)) V`g\ja*Y  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') d^Zo35X  
    end Iuu<2#gb8"  
    q'%[[<  
    if any(m<0) G>=9gSLM  
        error('zernpol:Mpositive','All M must be positive.') PG^j}  
    end '<?v:pb9  
    #JL&]Z+X6  
    if any(m>n) n&0mz1rw  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') 3@PUg(M  
    end 3R<ME c  
    jU]]:S4xD/  
    if any( r>1 | r<0 ) #.E\,N'  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') DnNt@e2|  
    end d,Cz-.'sOf  
    KFTf~!|  
    if ~any(size(r)==1) < v|%K.yd  
        error('zernpol:Rvector','R must be a vector.') }[>RxHd  
    end J ?ztn  
    9{)Z5%Kz  
    r = r(:); i"%JFj_G  
    length_r = length(r); V17>j0Ev$W  
    Vqa5RVnI  
    if nargin==4 :LZ-da"QR  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); ( oQ'4,F  
        if ~isnorm RNv{n mf  
            error('zernpol:normalization','Unrecognized normalization flag.') bGZ hUEq  
        end !dfS|BA]  
    else *F\T}k7  
        isnorm = false; b$d J?%W  
    end b4Ricm  
    Ci]'G>F@"  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% =7 Jy  
    % Compute the Zernike Polynomials BW'L.*2  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% H!unIy|  
    tt+>8rxF:;  
    % Determine the required powers of r: " Sc5qG  
    % ----------------------------------- 0M$#95n  
    rpowers = []; |7svA<<[  
    for j = 1:length(n) SX1X< 9  
        rpowers = [rpowers m(j):2:n(j)]; t2=a(N-/,  
    end Dh hG$  
    rpowers = unique(rpowers); SQh+5  
    N7e`6d!  
    % Pre-compute the values of r raised to the required powers, jMN)?6$=  
    % and compile them in a matrix: &?SX4c~?u  
    % ----------------------------- KKLR'w,A>  
    if rpowers(1)==0 c:=7lI  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); P7's8KOoS  
        rpowern = cat(2,rpowern{:}); &}vR(y*#c  
        rpowern = [ones(length_r,1) rpowern]; \:]DFZ=!  
    else yOX&cZ[  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ~c9>Nr9|`  
        rpowern = cat(2,rpowern{:}); L/Hv4={  
    end "Z;~Y=hC13  
    ;c1relR2  
    % Compute the values of the polynomials: i~9?:plS  
    % -------------------------------------- HwB {8S?sm  
    z = zeros(length_r,length_n); t(}/g  
    for j = 1:length_n A/!<kp{S  
        s = 0:(n(j)-m(j))/2; y&1%1 #8F  
        pows = n(j):-2:m(j); ID)^vwn  
        for k = length(s):-1:1 2$M,*Dnr  
            p = (1-2*mod(s(k),2))* ... K'x4l,rq  
                       prod(2:(n(j)-s(k)))/          ... St6aYK  
                       prod(2:s(k))/                 ... Iha[G u  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... *~c qr  
                       prod(2:((n(j)+m(j))/2-s(k))); 5gSylts8  
            idx = (pows(k)==rpowers); Ffnk1/ Zy  
            z(:,j) = z(:,j) + p*rpowern(:,idx); 0Y7$d`  
        end gdl| ^*tc  
         S"zk!2@C  
        if isnorm %Ys$@dB  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); H#+\nT2m  
        end (/&;jV2DD[  
    end Hk@r5<{  
    uG;?vvg>  
    % EOF zernpol
    离线niuhelen
    发帖
    19
    光币
    28
    光券
    0
    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
    发帖
    59
    光币
    0
    光券
    0
    只看该作者 7楼 发表于: 2012-09-28
    我也正在找啊
    离线guapiqlh
    发帖
    856
    光币
    846
    光券
    0
    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
    发帖
    4352
    光币
    5478
    光券
    1
    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)  F9flSeN  
    z|N3G E(.@  
    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。 fex,z%}p  
    yI3kvh  
    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)