切换到宽版
  • 广告投放
  • 稿件投递
  • 繁體中文
    • 11070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 3@KX|-  
    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! /=lrdp!a  
     
    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 AcwLs%'sx  
    function z = zernfun(n,m,r,theta,nflag) _Qt  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. VA&_dU]*  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ==RYf*d  
    %   and angular frequency M, evaluated at positions (R,THETA) on the ;/XWX$G@  
    %   unit circle.  N is a vector of positive integers (including 0), and ||;V5iR:  
    %   M is a vector with the same number of elements as N.  Each element D. fP Hq  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) _s[ohMlh  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, t3}>5cAxy  
    %   and THETA is a vector of angles.  R and THETA must have the same E].hoq7WiB  
    %   length.  The output Z is a matrix with one column for every (N,M) _K<H*R  
    %   pair, and one row for every (R,THETA) pair. ,6=j'j1#a  
    % ve49m%NQ  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike W4%I%&j  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), C< 3` ]l  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral <U%4$83$  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, M+j V`J!  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized !nQ_<  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. fd! bs*\X  
    % ++w7jVi9  
    %   The Zernike functions are an orthogonal basis on the unit circle. 97l<9^$  
    %   They are used in disciplines such as astronomy, optics, and :[xFp}w{  
    %   optometry to describe functions on a circular domain. ^SM>bJ1Z_  
    % NOM6},rp  
    %   The following table lists the first 15 Zernike functions. a> qB k})  
    % `yJ3"{uO  
    %       n    m    Zernike function           Normalization O$z XDxn  
    %       -------------------------------------------------- >!sxX = <  
    %       0    0    1                                 1 Nk?eVJ)  
    %       1    1    r * cos(theta)                    2 dD YD6  
    %       1   -1    r * sin(theta)                    2 V1di#i:  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) q>|&u  
    %       2    0    (2*r^2 - 1)                    sqrt(3) 3MX&%_wUhB  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) eFKF9m  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) [GQn1ZLc  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) RK )1@Tz7!  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) !aQb Kp  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) {z#!3a  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) tVQq,_9C  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Y % 9$!  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) EDAtC  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) W {A4*{  
    %       4    4    r^4 * sin(4*theta)             sqrt(10)  )OHGg  
    %       -------------------------------------------------- H46N!{<;@  
    % s!<RWy+  
    %   Example 1: B/O0 ~y!n  
    % or,:5Z  
    %       % Display the Zernike function Z(n=5,m=1) [[$dPa9  
    %       x = -1:0.01:1; JAx0(MZO  
    %       [X,Y] = meshgrid(x,x); w)N~u%  
    %       [theta,r] = cart2pol(X,Y); "?%2`*\  
    %       idx = r<=1; ^XX_ qC'1  
    %       z = nan(size(X)); :W^\ } UX4  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); +Tt.5>N  
    %       figure %@9c'6  
    %       pcolor(x,x,z), shading interp ?wP/l  
    %       axis square, colorbar EDT9O  
    %       title('Zernike function Z_5^1(r,\theta)') _?> x{![  
    %  .0YcB  
    %   Example 2: fUMjLA|*I<  
    % WEYZ(a|  
    %       % Display the first 10 Zernike functions 4#qZ`H,Ur)  
    %       x = -1:0.01:1; 3xk_ZK82  
    %       [X,Y] = meshgrid(x,x); 8WE@ X)e  
    %       [theta,r] = cart2pol(X,Y); > ^=n|%  
    %       idx = r<=1; 7Kf  
    %       z = nan(size(X)); b(oe^jeGz  
    %       n = [0  1  1  2  2  2  3  3  3  3];  5@DCo  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; $K.DLqDt  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; 9a[1s|>w-  
    %       y = zernfun(n,m,r(idx),theta(idx)); )DmydyQ'  
    %       figure('Units','normalized') #+QJ5VI :  
    %       for k = 1:10 -AD@wn!wCJ  
    %           z(idx) = y(:,k); IsmZEVuC  
    %           subplot(4,7,Nplot(k)) :zX^H9'E<(  
    %           pcolor(x,x,z), shading interp eL>wKu:r  
    %           set(gca,'XTick',[],'YTick',[]) A_l\ij$Y  
    %           axis square Ni8%K6]z  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ^KdT,^6T  
    %       end _CPj] m{  
    % B `.aQ  
    %   See also ZERNPOL, ZERNFUN2. +m]-)  
    aGBd~y@e  
    %   Paul Fricker 11/13/2006 A@Q6}ESD  
    >|, <9z`D  
    e/cHH3 4  
    % Check and prepare the inputs: >;XtJJS  
    % ----------------------------- CuK>1_Dq  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) L@z[b^  
        error('zernfun:NMvectors','N and M must be vectors.') J90:c@O"w  
    end yH=<KYk  
    A +=#  
    if length(n)~=length(m) L*dGo,oN  
        error('zernfun:NMlength','N and M must be the same length.') T*mR9 8i  
    end ,}\LC;31,  
    DLP@?]BBOA  
    n = n(:); ? A;RTM  
    m = m(:); A9N8Hav  
    if any(mod(n-m,2)) 6\u. [2lE^  
        error('zernfun:NMmultiplesof2', ... P5h*RV>oS  
              'All N and M must differ by multiples of 2 (including 0).') O'B3sy  
    end :-#7j} R&  
    EZ{{p+e ^  
    if any(m>n) ovOV&Zt  
        error('zernfun:MlessthanN', ... Xp| 4WM  
              'Each M must be less than or equal to its corresponding N.') WY QVe_<z:  
    end n|?sNM<J3  
    AA)pV-  
    if any( r>1 | r<0 ) ;hODzfNkS  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') @{{L1[~:0  
    end Du +_dr^4  
    K|\0jd)N  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 0[fBP\H"Wr  
        error('zernfun:RTHvector','R and THETA must be vectors.') !~RK2d  
    end O-ENFA~E;v  
    KPDJ$,:  
    r = r(:); d&L  
    theta = theta(:); AX&Emz-  
    length_r = length(r); !]}C!dXd  
    if length_r~=length(theta) '5*&  
        error('zernfun:RTHlength', ... 0}`.Z03fy  
              'The number of R- and THETA-values must be equal.') sr[[xzL  
    end A@?-"=h}  
    -K$ugDi  
    % Check normalization: 9=6BQ`u  
    % -------------------- J!RRG~  
    if nargin==5 && ischar(nflag) J E5qR2VA  
        isnorm = strcmpi(nflag,'norm'); %-$ :/ N  
        if ~isnorm jj;TS%  
            error('zernfun:normalization','Unrecognized normalization flag.') etX(~"gG_  
        end P.Cn[64a+@  
    else ~ArRD-_t  
        isnorm = false; !mWm@ }Ujg  
    end 4_CL1g  
    5+Tx01 )  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {|OXiRm'  
    % Compute the Zernike Polynomials pRxVsOb  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% htrtiJ1  
    @`nG &U  
    % Determine the required powers of r: !B#lZjW#  
    % ----------------------------------- SYQP7oG9oQ  
    m_abs = abs(m); \+/ciPzA-  
    rpowers = []; ^?\|2H  
    for j = 1:length(n) Uc ,..  
        rpowers = [rpowers m_abs(j):2:n(j)]; )MTf  
    end [g: cG  
    rpowers = unique(rpowers); 7I]?:%8 h  
    b KIL@AI  
    % Pre-compute the values of r raised to the required powers, W?!rqo2SP  
    % and compile them in a matrix: ,JbP~2M~%  
    % ----------------------------- 2?:OsA}  
    if rpowers(1)==0 Q3$DX, 8?  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); c05-1  
        rpowern = cat(2,rpowern{:}); Pk(%=P ,  
        rpowern = [ones(length_r,1) rpowern]; Z-_Xt^N  
    else XhWo~zh"  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); )a'`  
        rpowern = cat(2,rpowern{:}); car|&b  
    end +eKLwM  
    x;} 25A|  
    % Compute the values of the polynomials: =b1 y*?  
    % -------------------------------------- ci:|x =  
    y = zeros(length_r,length(n)); <}c7E3Uc  
    for j = 1:length(n) 9` VY)"rJ  
        s = 0:(n(j)-m_abs(j))/2; ;.=0""-IF  
        pows = n(j):-2:m_abs(j); FjiIB1 T  
        for k = length(s):-1:1 3fZoF`<a  
            p = (1-2*mod(s(k),2))* ... )"{}L.gC6  
                       prod(2:(n(j)-s(k)))/              ... xb9^WvV  
                       prod(2:s(k))/                     ... + !nf?5;  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... Vj8-[ww!  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); )$a6l8  
            idx = (pows(k)==rpowers); ]-a/)8  
            y(:,j) = y(:,j) + p*rpowern(:,idx); gVJh@]8)  
        end %Q.M& U  
         'IVC!uL,%  
        if isnorm {9j0k`A  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); X>o*eN  
        end /!6 VP |  
    end (6[/7e)  
    % END: Compute the Zernike Polynomials |DVFi2   
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% --c)!Vxzx  
    k{lXK\zN  
    % Compute the Zernike functions: jJ2{g> P0P  
    % ------------------------------ b`DPlQHj  
    idx_pos = m>0; 8-kR {9r  
    idx_neg = m<0; }&s |~  
    JP ;SO  
    z = y; E6 T=lwOZ  
    if any(idx_pos) V}Q`dEk2r  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); W4(  
    end JLu$UR4  
    if any(idx_neg) E\9HZ;}G  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); +~ Y.m8  
    end M A%g-}  
    XGYsTquSe  
    % EOF zernfun
    离线niuhelen
    发帖
    19
    光币
    28
    光券
    0
    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) Z`%^?My  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. 7kMO);pO  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated c2Y\bKeN  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive iUqD>OV  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, ~eiD(04^r*  
    %   and THETA is a vector of angles.  R and THETA must have the same RH$YM `cZ  
    %   length.  The output Z is a matrix with one column for every P-value, YCdtf7P=q  
    %   and one row for every (R,THETA) pair.  Tx'anP  
    % >Wd_?NaI  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike oGt2n:  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) :j32 :/u  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) QUz4 Kt  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 I%z,s{9p  
    %   for all p. }+)q/]%  
    % |#(y?! A^  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 ow'CwOj$  
    %   Zernike functions (order N<=7).  In some disciplines it is _]0<G8|Rv  
    %   traditional to label the first 36 functions using a single mode O`9vEovjs  
    %   number P instead of separate numbers for the order N and azimuthal @$~;vS  
    %   frequency M. vI"BNC*Q1  
    % ;2gO(  
    %   Example: Dh68=F0  
    % iBY16_q  
    %       % Display the first 16 Zernike functions q{L-(!uz7_  
    %       x = -1:0.01:1; "z*?#&?,  
    %       [X,Y] = meshgrid(x,x); 8/"C0I (G  
    %       [theta,r] = cart2pol(X,Y); }Am5b@g"$Y  
    %       idx = r<=1; YQR[0Y&e=  
    %       p = 0:15; @oD2_D2  
    %       z = nan(size(X)); j=u) z7J  
    %       y = zernfun2(p,r(idx),theta(idx)); 79+i4(H  
    %       figure('Units','normalized') ^SIA%S3  
    %       for k = 1:length(p) tLP Er@  
    %           z(idx) = y(:,k); l,UOP[j  
    %           subplot(4,4,k) -'^:+FU  
    %           pcolor(x,x,z), shading interp 5? f!hB|6  
    %           set(gca,'XTick',[],'YTick',[]) xiqeKoAD  
    %           axis square {f;DhB-jj  
    %           title(['Z_{' num2str(p(k)) '}']) @qB>qD~WsD  
    %       end 5+q dn|9%T  
    % /RWD\u<l  
    %   See also ZERNPOL, ZERNFUN. "1 UpoF'w  
    #S[Y}-]T  
    %   Paul Fricker 11/13/2006 G\~?.s|^  
    S.pXo'}  
    i\x@s>@x}  
    % Check and prepare the inputs: & s:\t L  
    % ----------------------------- 3jHE,5m  
    if min(size(p))~=1 *] !r T&E  
        error('zernfun2:Pvector','Input P must be vector.') ST,+]p3L(  
    end #8y"1I=i&  
    sj6LrE=1  
    if any(p)>35 $"MGu^0;1  
        error('zernfun2:P36', ... &}\{qFD;  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... uG-S$n"7K  
               '(P = 0 to 35).']) ~g;)8X;;+  
    end K#VGG,h7Y  
    cg9*+]rc  
    % Get the order and frequency corresonding to the function number: q#\B}'I{  
    % ---------------------------------------------------------------- t3=K>Y@w  
    p = p(:); !/X>k{  
    n = ceil((-3+sqrt(9+8*p))/2); ubc k{\.  
    m = 2*p - n.*(n+2); 5Fbb5`(  
    b;NVvc(  
    % Pass the inputs to the function ZERNFUN: p&D7&Sb[  
    % ---------------------------------------- TP)o0U  
    switch nargin :z6?  
        case 3 'ITZz n*  
            z = zernfun(n,m,r,theta); c5pK%I}O  
        case 4 Yu9VtC1  
            z = zernfun(n,m,r,theta,nflag); w{N8Y ~O  
        otherwise z)Yb9y>2  
            error('zernfun2:nargin','Incorrect number of inputs.') I*/:rb  
    end C:f^&4 3  
    %4BQY>O)@  
    % 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) {96NtR0Z  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. j J54<.D  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of ?q0a^c?A^  
    %   order N and frequency M, evaluated at R.  N is a vector of =c]We:I  
    %   positive integers (including 0), and M is a vector with the C[+?gQJ[9  
    %   same number of elements as N.  Each element k of M must be a WMFn#.aY5  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) {yi!vw  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is R1.Yx?  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix $lJ!f  
    %   with one column for every (N,M) pair, and one row for every 715J1~aRNr  
    %   element in R. "'>fTk_  
    % +RK/u  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- s,D GFK  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is P#;pQC  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to NCm=l  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 IEfm>N-]  
    %   for all [n,m]. P+3 ]g{2w  
    % [ .3Gb}B  
    %   The radial Zernike polynomials are the radial portion of the Exat_ L'?  
    %   Zernike functions, which are an orthogonal basis on the unit jank<Q&w  
    %   circle.  The series representation of the radial Zernike ~{6}SXp4U  
    %   polynomials is U/7jK40  
    % ZV07;`I  
    %          (n-m)/2 M@0S*[O{"  
    %            __ -3XnUGK  
    %    m      \       s                                          n-2s !Z;Nv  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r lB|.TCbW  
    %    n      s=0  ~"h V-3U  
    % J!'IkC$>  
    %   The following table shows the first 12 polynomials. ;!m_RQPFF  
    % 5%DHF-W)  
    %       n    m    Zernike polynomial    Normalization rE\&FVx  
    %       --------------------------------------------- <*p  
    %       0    0    1                        sqrt(2) fl*49-d  
    %       1    1    r                           2 B^x}=Z4  
    %       2    0    2*r^2 - 1                sqrt(6) Ft>,  
    %       2    2    r^2                      sqrt(6) !,*Uvs@b  
    %       3    1    3*r^3 - 2*r              sqrt(8) g;y*F;0@  
    %       3    3    r^3                      sqrt(8) ZXWm?9uw  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) *(VwD)*  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) j*_#{niy:  
    %       4    4    r^4                      sqrt(10) L!2Ef4,wAz  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) ab<7jfFIa  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) mS]soYTQ  
    %       5    5    r^5                      sqrt(12) #YLI"/Kn  
    %       --------------------------------------------- c5{3  
    % 'Ub\8<HfJU  
    %   Example: Zor Q2>  
    % hHsO?([99  
    %       % Display three example Zernike radial polynomials w{Y:p[}  
    %       r = 0:0.01:1; DZ5h<1  
    %       n = [3 2 5]; 1.j;Xo/+:V  
    %       m = [1 2 1]; QWK\6  
    %       z = zernpol(n,m,r); ~"vRH  
    %       figure U6_GEBz~y  
    %       plot(r,z) ,j\UZ  
    %       grid on })ic@ Mmd$  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') 0I>[rxal  
    % z)p p{  
    %   See also ZERNFUN, ZERNFUN2. IZ+ZIR@}ci  
    X FvPc  
    % A note on the algorithm. ZN(@M@}  
    % ------------------------ %|By ?i  
    % The radial Zernike polynomials are computed using the series Dad*6;+N  
    % representation shown in the Help section above. For many special iYzm<3n?  
    % functions, direct evaluation using the series representation can Vu1X@@z  
    % produce poor numerical results (floating point errors), because 9vz"rHV  
    % the summation often involves computing small differences between _cXLQ)-  
    % large successive terms in the series. (In such cases, the functions :jljM(\  
    % are often evaluated using alternative methods such as recurrence Fu#mMn0c  
    % relations: see the Legendre functions, for example). For the Zernike RU GhhK  
    % polynomials, however, this problem does not arise, because the swT/ tesj  
    % polynomials are evaluated over the finite domain r = (0,1), and a$'= a09  
    % because the coefficients for a given polynomial are generally all "]D2}E>U;  
    % of similar magnitude. j`kw2(  
    % Dz>v;%$S-  
    % ZERNPOL has been written using a vectorized implementation: multiple 7\1bq&a<  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] X3;|h93.a  
    % values can be passed as inputs) for a vector of points R.  To achieve c_^-`7g  
    % this vectorization most efficiently, the algorithm in ZERNPOL _,;|,  
    % involves pre-determining all the powers p of R that are required to eZ~ZWb,%  
    % compute the outputs, and then compiling the {R^p} into a single 5n'C6q "  
    % matrix.  This avoids any redundant computation of the R^p, and d*xKq"+ &E  
    % minimizes the sizes of certain intermediate variables. 4RV5:&ALLS  
    % !mLY W  
    %   Paul Fricker 11/13/2006 }2eP~3  
    +iYy^oXxw  
    {qHf%y&[  
    % Check and prepare the inputs: 2_]"9d4  
    % ----------------------------- p%v+\T2r  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 0*KU"JcXd  
        error('zernpol:NMvectors','N and M must be vectors.') k6vY/)-S  
    end r!vSYgee  
    3tlA! e  
    if length(n)~=length(m) T>ds<MaLP  
        error('zernpol:NMlength','N and M must be the same length.') `|i[*+WC  
    end wv8WqYV  
    b4$-?f?V  
    n = n(:); H1FSN6'  
    m = m(:); Dog Tj  
    length_n = length(n); "3"9sIZ(  
    +) 4_1i4"x  
    if any(mod(n-m,2)) gL+8fX2G6  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') N| dwuBW  
    end lxpi   
    +8 avA:o  
    if any(m<0) NyTv~8A`)  
        error('zernpol:Mpositive','All M must be positive.') ?-P]m&nh|  
    end #lM :BO  
    U[b $VZ}  
    if any(m>n) 4Y[uqn[  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') h<50jnH!  
    end 09{B6l6P  
    i-'rS/R  
    if any( r>1 | r<0 ) R&BbXSIDX  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') 85<zl|ZD  
    end IG1+_-H:  
    %z&=A%'a  
    if ~any(size(r)==1) 4  |E`  
        error('zernpol:Rvector','R must be a vector.') 4%TY` II  
    end  'mz _JM  
    TixXA:Mf  
    r = r(:); -o\r]24  
    length_r = length(r); 0^Vc,\P?  
    fT-yY`  
    if nargin==4 LB|FVNW/S  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); X#$mBRK7  
        if ~isnorm %G& Zm$u=  
            error('zernpol:normalization','Unrecognized normalization flag.') $:R"IqDG  
        end dHnR)[?e  
    else gOpGwpYZ,  
        isnorm = false; OQ>r;)/  
    end G!J{$0.  
    ,[rh7 _  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ,ufB*[~  
    % Compute the Zernike Polynomials +SGM3tY  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 72qbxPY13h  
    URbu=U  
    % Determine the required powers of r: Z_oBZs  
    % ----------------------------------- $0C1';=^}  
    rpowers = []; <!#6c :(Q  
    for j = 1:length(n) ^[{\ZX  
        rpowers = [rpowers m(j):2:n(j)]; Y4Hi<JWo  
    end ;Jex#+H(:D  
    rpowers = unique(rpowers); kDM\IyM<\  
    _q >>]{5  
    % Pre-compute the values of r raised to the required powers, ~ l~ai>/  
    % and compile them in a matrix: /F;b<kIy8  
    % ----------------------------- v Dgf}  
    if rpowers(1)==0 LEoL6ga  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); __\Tv>Y  
        rpowern = cat(2,rpowern{:}); 0p\cDrB ?  
        rpowern = [ones(length_r,1) rpowern]; 6mr5`5~w  
    else 1=x4m=wV  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); /xmUu0H$R  
        rpowern = cat(2,rpowern{:}); I4kN4*d!N,  
    end t&+f:)n  
    /79_3;^  
    % Compute the values of the polynomials: {O-,JCq/  
    % -------------------------------------- #!d@;= [\  
    z = zeros(length_r,length_n); 5`oVyxJ<  
    for j = 1:length_n J>(I"K%  
        s = 0:(n(j)-m(j))/2; 1s4+a^ &  
        pows = n(j):-2:m(j); |cwGc\ES  
        for k = length(s):-1:1 B[:-SWd  
            p = (1-2*mod(s(k),2))* ... d(RSn|[0  
                       prod(2:(n(j)-s(k)))/          ... ZzA4iT=KO  
                       prod(2:s(k))/                 ... 9/[3xhB4  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... l$$N~FN  
                       prod(2:((n(j)+m(j))/2-s(k))); x35(i  
            idx = (pows(k)==rpowers); |A".Mo_5  
            z(:,j) = z(:,j) + p*rpowern(:,idx); ^&iUC&8W  
        end 1{B^RR.  
         <^?64  
        if isnorm L4I1nl  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); lYkm1  
        end (J(JB}[X,  
    end VkCv`E  
    +$Q33@F5l  
    % 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)  WVPnyVDc  
    G:TM k4  
    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。 'xO5Le(=M  
    KrwG><+j  
    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)