切换到宽版
  • 广告投放
  • 稿件投递
  • 繁體中文
    • 10476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 R]-$]koQO  
    就是我用ansys计算出了镜面的面型的数据,怎样可以得到zernike多项式系数,然后用zemax各阶得到像差!谢谢啦! /eoS$q  
     
    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 @-z#vJ5Qe{  
    function z = zernfun(n,m,r,theta,nflag) XA[G F6W,Y  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. -DO*,Eecv  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 7k<4/|CQ{  
    %   and angular frequency M, evaluated at positions (R,THETA) on the vT<q zN  
    %   unit circle.  N is a vector of positive integers (including 0), and CfMq?.4%E}  
    %   M is a vector with the same number of elements as N.  Each element TtL2}Wdd.%  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) x M1>kbo|  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, \WM*2&  
    %   and THETA is a vector of angles.  R and THETA must have the same :!a9|Fh~  
    %   length.  The output Z is a matrix with one column for every (N,M) {&Kq/sRz  
    %   pair, and one row for every (R,THETA) pair. ~Od4( }/G  
    % wHW";3w2~  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike GHHErXT\a  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), e75 k-  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral U! F~><  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, WUid5e2  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized U*Z P>Vv  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. p[(VhbN  
    % mMqT-jT  
    %   The Zernike functions are an orthogonal basis on the unit circle. \TG!M]D:  
    %   They are used in disciplines such as astronomy, optics, and %Fc, $ =  
    %   optometry to describe functions on a circular domain. I/bED~Z:a  
    % xMsos?5}  
    %   The following table lists the first 15 Zernike functions. ;Ef:mr"Nu  
    % PXGS5,  
    %       n    m    Zernike function           Normalization S;$@?vF  
    %       -------------------------------------------------- 4z-sR/d  
    %       0    0    1                                 1 P'#m1ntxQ  
    %       1    1    r * cos(theta)                    2 s-eC')w~E  
    %       1   -1    r * sin(theta)                    2 Vw*;xek?  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) lrjlkgSN  
    %       2    0    (2*r^2 - 1)                    sqrt(3) G7k0P-r,0  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) tb7Wr1$<  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) <^,w,A  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) ,ZcW+!  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) W[o~AbU  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) BRP9j y  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) 7?K?-Oj  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) wVBY^TE  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) ?;.j)  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ?@9kVB*|  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) b .k J&c  
    %       -------------------------------------------------- KQ3]'2q  
    % c,KT1me  
    %   Example 1: UYH;15s  
    % 4-[L^1%S[  
    %       % Display the Zernike function Z(n=5,m=1) KO(+%>^R  
    %       x = -1:0.01:1; 9+.0ZP?  
    %       [X,Y] = meshgrid(x,x); (LPMEQhI:  
    %       [theta,r] = cart2pol(X,Y); -zg,pK$+  
    %       idx = r<=1; 1)u 3  
    %       z = nan(size(X)); 2O {@W +Mt  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); KyW6[WA9  
    %       figure FG7}MUu  
    %       pcolor(x,x,z), shading interp ?eT^gWX  
    %       axis square, colorbar /-<S FT`  
    %       title('Zernike function Z_5^1(r,\theta)') fGJPZe  
    % #NVtZs!V/  
    %   Example 2: M#on-[  
    % \_FX}1Wc2.  
    %       % Display the first 10 Zernike functions cu|gM[  
    %       x = -1:0.01:1; < pI2}  
    %       [X,Y] = meshgrid(x,x); #M6@{R2_  
    %       [theta,r] = cart2pol(X,Y); ^~(vP:  
    %       idx = r<=1; x^}kG[s  
    %       z = nan(size(X)); (,PO(  
    %       n = [0  1  1  2  2  2  3  3  3  3]; \`o+Le+%  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; ^jb55X}  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; {zBf*x  
    %       y = zernfun(n,m,r(idx),theta(idx)); DW@PPvfs  
    %       figure('Units','normalized') 3 q  
    %       for k = 1:10 jcQ{,9 H`l  
    %           z(idx) = y(:,k); ;rpjXP  
    %           subplot(4,7,Nplot(k)) T%K(opISc(  
    %           pcolor(x,x,z), shading interp VO>A+vx3M  
    %           set(gca,'XTick',[],'YTick',[]) >e*m8gm#  
    %           axis square blph&[`}I  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) @HJ&"72$<  
    %       end ?hvPPEJf  
    % %(Sy XZ  
    %   See also ZERNPOL, ZERNFUN2. }6.R.*Imz  
    `?{QCBVj  
    %   Paul Fricker 11/13/2006 -W wFUm  
    OwV>`BIwns  
    =C 8 t5BZ"  
    % Check and prepare the inputs: *PE 1)bF  
    % ----------------------------- 33|>u+  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) /K2VSj3\  
        error('zernfun:NMvectors','N and M must be vectors.') cu(2BDfiL  
    end 31 4PcSc  
    %5RY Ea  
    if length(n)~=length(m) oAe]/j$  
        error('zernfun:NMlength','N and M must be the same length.') B#AAG*Ai8  
    end U 9TEC)  
    Y8`4K*58%  
    n = n(:); 0G1?  
    m = m(:); |E0>-\6  
    if any(mod(n-m,2)) v9INZ1# v  
        error('zernfun:NMmultiplesof2', ... \-N 4G1  
              'All N and M must differ by multiples of 2 (including 0).') {&8-OoH ~  
    end _ 0%sYkUc  
    Jf@M>BT^A  
    if any(m>n) 6+BR5Nr  
        error('zernfun:MlessthanN', ... 'YQ"Lf  
              'Each M must be less than or equal to its corresponding N.') ,i#]&f`c;5  
    end f:\jPkf'  
    Ev%4}GwO4  
    if any( r>1 | r<0 ) 9r@r\-  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') LEvdPG$)  
    end "0 \U>h  
    &Eg>[gAIlp  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) JLm0[1Lzd  
        error('zernfun:RTHvector','R and THETA must be vectors.') H7?C>+ay  
    end 1.!rq,+>1  
    vE7L> 7  
    r = r(:); !OekN,6  
    theta = theta(:); ^RrufwUA  
    length_r = length(r); *DObtS_ 6  
    if length_r~=length(theta) B;Ab`UX#t  
        error('zernfun:RTHlength', ... #>GUfhou)  
              'The number of R- and THETA-values must be equal.') e*jt(p[Ge  
    end |[(4h  
    "AP'' XNi  
    % Check normalization: E.Xf b"]  
    % -------------------- 1uz9zhG><  
    if nargin==5 && ischar(nflag) r<cyxR~  
        isnorm = strcmpi(nflag,'norm'); ZdeRLX  
        if ~isnorm KG)7hja<6g  
            error('zernfun:normalization','Unrecognized normalization flag.') 7lY&/-V  
        end A>(m}P  
    else 7)S`AQ2:)  
        isnorm = false; d$8rzd  
    end \Xc6K!HJM  
    2;r(?ebw  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ?~uTbNR  
    % Compute the Zernike Polynomials }legh:/*?O  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 55MsF}p  
    _%w-y(Sqn  
    % Determine the required powers of r: HE(|x 1C)j  
    % ----------------------------------- Yv }G"-=  
    m_abs = abs(m); frbKi _1  
    rpowers = []; > xkl7D  
    for j = 1:length(n) g* F?  
        rpowers = [rpowers m_abs(j):2:n(j)]; R<{bb'  
    end 9V`/zq?  
    rpowers = unique(rpowers); "{105&c\  
     wX@&Qv  
    % Pre-compute the values of r raised to the required powers, D oX!P|*  
    % and compile them in a matrix: /1ooOq]  
    % ----------------------------- q]YPDdR#  
    if rpowers(1)==0 N~ _GJw@  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); !}|n3wQ  
        rpowern = cat(2,rpowern{:}); `Gzukh  
        rpowern = [ones(length_r,1) rpowern]; F2]v]]F!  
    else =:n>yZ3T  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); `4__X;  
        rpowern = cat(2,rpowern{:}); f1(V~{N,+  
    end 7tMV*{+Z  
    ` `j..v,  
    % Compute the values of the polynomials: *T$o" *}  
    % -------------------------------------- U:m[* }+<  
    y = zeros(length_r,length(n)); T^g2N`w2  
    for j = 1:length(n) j9u/R01d  
        s = 0:(n(j)-m_abs(j))/2; ^5j|   
        pows = n(j):-2:m_abs(j); IlG)=?8XZ  
        for k = length(s):-1:1 -;&aU;k  
            p = (1-2*mod(s(k),2))* ... Pj>r(Cv  
                       prod(2:(n(j)-s(k)))/              ... Ls)y.u  
                       prod(2:s(k))/                     ... Q( .d!CQ>  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 7YsBwo  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); *gfx'$  
            idx = (pows(k)==rpowers); <DP_`[+C  
            y(:,j) = y(:,j) + p*rpowern(:,idx); #Mw|h^ Wm  
        end ~Z!!wDHS  
         |E-/b6G  
        if isnorm +gqtW8 6  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); >;kCcfS3ct  
        end YMOy 6C  
    end -jnx0{/  
    % END: Compute the Zernike Polynomials azR<Y_tw  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% md:$O C3  
    ac"Pn? q  
    % Compute the Zernike functions: Og[NRd+  
    % ------------------------------ { 2G9>'  
    idx_pos = m>0; sE@t$'=  
    idx_neg = m<0; tgK$}#.*  
    h~haA8i?{  
    z = y; ^IGutZov  
    if any(idx_pos) &S}%)g%Iv9  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); gQ4Q h;  
    end 5!u.w  
    if any(idx_neg) 5_Yl!=  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Wc [@,  
    end BV,P;T0"D  
    \PU3{_G]  
    % EOF zernfun
    离线niuhelen
    发帖
    19
    光币
    28
    光券
    0
    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) &}wr N(?w  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. =S/$h}Vi  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated `l,=iy$  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive 0Ci\(  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, g.OBh_j-v  
    %   and THETA is a vector of angles.  R and THETA must have the same #R$d6N[H  
    %   length.  The output Z is a matrix with one column for every P-value, 08AC 9  
    %   and one row for every (R,THETA) pair. "]J4BZD  
    % le*mr0a  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike W$LaXytmak  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) Cc*"cQe  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) CqX2R:#  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 IPmSkK  
    %   for all p. EeGP E  
    % hNBv|&D#  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 4GWt.+{J$  
    %   Zernike functions (order N<=7).  In some disciplines it is 'W>Bz,M6yo  
    %   traditional to label the first 36 functions using a single mode Nf8."EDUW  
    %   number P instead of separate numbers for the order N and azimuthal T P'  
    %   frequency M. :q=u+h_  
    % (\m4o   
    %   Example: [.4R ,[U  
    % Qj: D=j8  
    %       % Display the first 16 Zernike functions JwWW w1  
    %       x = -1:0.01:1; ,9<}V;(  
    %       [X,Y] = meshgrid(x,x); kG_ K&,;@  
    %       [theta,r] = cart2pol(X,Y); Ug>yTc_(7  
    %       idx = r<=1; ^2E\{$J  
    %       p = 0:15; ry9%Y3  
    %       z = nan(size(X)); 3a PCi>i!_  
    %       y = zernfun2(p,r(idx),theta(idx)); Jj+|>(P  
    %       figure('Units','normalized') usEd p  
    %       for k = 1:length(p) dr0<K[S_  
    %           z(idx) = y(:,k); $z,lq#zzl  
    %           subplot(4,4,k) "j BrPCB 8  
    %           pcolor(x,x,z), shading interp 4cJ7W_ >i6  
    %           set(gca,'XTick',[],'YTick',[]) H59}d oKH  
    %           axis square +c4]}9f!  
    %           title(['Z_{' num2str(p(k)) '}']) *y[i~{7:  
    %       end hZ NS$  
    % vQB;a?)o  
    %   See also ZERNPOL, ZERNFUN. O="# yE)  
    &!> )EHGV  
    %   Paul Fricker 11/13/2006 X`bN/sI  
    f)w>V3~w,  
    >qh8em  
    % Check and prepare the inputs: SA_5..  
    % ----------------------------- -w nlJi1f  
    if min(size(p))~=1 S^nshQI  
        error('zernfun2:Pvector','Input P must be vector.') A41*4!L=  
    end OZ 4uk.)  
    ?U'c;*O-  
    if any(p)>35 4i0~t~vDpr  
        error('zernfun2:P36', ... >zcR ?PPs  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... iu!j#VO  
               '(P = 0 to 35).']) Xv ;} !z  
    end <)n8lIK  
    *Pj[r  
    % Get the order and frequency corresonding to the function number: t.wB\Kmt\  
    % ---------------------------------------------------------------- sLiKcR8^  
    p = p(:); >7%Gd-;l  
    n = ceil((-3+sqrt(9+8*p))/2); r.1/ * i  
    m = 2*p - n.*(n+2); uk%C:4T  
    d*3;6ZLy  
    % Pass the inputs to the function ZERNFUN: N8a+X|3]0  
    % ---------------------------------------- Y1 RiuJtL  
    switch nargin } :U'aa  
        case 3 0<f\bY02  
            z = zernfun(n,m,r,theta); <Stfqa6FJ  
        case 4 lx9tUTaus/  
            z = zernfun(n,m,r,theta,nflag); uNnx i  
        otherwise 4+&4  
            error('zernfun2:nargin','Incorrect number of inputs.') +~~FfIzf#  
    end xb/L AlJ  
    iW.4'9   
    % 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) ,G2]3 3Z  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. xWkCP2$?P  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of :4 9ttJl  
    %   order N and frequency M, evaluated at R.  N is a vector of EC[2rROn\  
    %   positive integers (including 0), and M is a vector with the 8@`"ZzM  
    %   same number of elements as N.  Each element k of M must be a H/!_D f  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) {}F?eI  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is Q'?{_  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix s$g"6;_\  
    %   with one column for every (N,M) pair, and one row for every + 3+^J?N  
    %   element in R. r\6 "mU  
    % E]G#"EV!Y  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- Q-<,+[/  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is #H]cb#  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to A2Je*Gz  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 Y@)iPK@z  
    %   for all [n,m]. osKM3}Sb  
    % [S{KGe:g  
    %   The radial Zernike polynomials are the radial portion of the MUo}Qi0K  
    %   Zernike functions, which are an orthogonal basis on the unit e[txJ*SuO  
    %   circle.  The series representation of the radial Zernike W i a%rm  
    %   polynomials is h\+U+ ?u  
    % x13t@b  
    %          (n-m)/2 R@lmX%Z1  
    %            __ =`VA_xVu  
    %    m      \       s                                          n-2s G$X+g{  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r hh\\api  
    %    n      s=0 H>8B$fi)$  
    % =,Y i" E  
    %   The following table shows the first 12 polynomials. +T}:GBwD7  
    % wps`2`z  
    %       n    m    Zernike polynomial    Normalization h-0sDt pR  
    %       --------------------------------------------- #BA=?7  
    %       0    0    1                        sqrt(2) 2-G he3  
    %       1    1    r                           2 INca  
    %       2    0    2*r^2 - 1                sqrt(6) |\g=ua+h  
    %       2    2    r^2                      sqrt(6) t*<@>]k  
    %       3    1    3*r^3 - 2*r              sqrt(8) ,TrrqCw>  
    %       3    3    r^3                      sqrt(8) o *5<Cxg  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) /cn=8%!N  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) c^^[~YW j  
    %       4    4    r^4                      sqrt(10) yKJKQ9  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) j$%KKl8j  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) sGx"j a +  
    %       5    5    r^5                      sqrt(12) Uj!3H]d  
    %       --------------------------------------------- ?L=@Zs  
    % 2qEm,x'S  
    %   Example: o(~QuHOp8>  
    % sflH{!;p  
    %       % Display three example Zernike radial polynomials Wj2s+L7,  
    %       r = 0:0.01:1; #X&`gDW  
    %       n = [3 2 5]; Ap}^6_YXd  
    %       m = [1 2 1]; yya"*]*S  
    %       z = zernpol(n,m,r); gg ;&a(  
    %       figure y%.^| G  
    %       plot(r,z) U=_O*n?N-d  
    %       grid on T7f>u}T  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') (_^pX  
    % 20[_eu)  
    %   See also ZERNFUN, ZERNFUN2. l7G&[\~  
    7CfHL;+m<4  
    % A note on the algorithm. %T:~N<8)  
    % ------------------------ _YVp$aKDR  
    % The radial Zernike polynomials are computed using the series Gc=#  
    % representation shown in the Help section above. For many special c"X`OB  
    % functions, direct evaluation using the series representation can N?`-$C ]  
    % produce poor numerical results (floating point errors), because [a~|{~?8  
    % the summation often involves computing small differences between ;B[*f?y-  
    % large successive terms in the series. (In such cases, the functions WLr\ l29  
    % are often evaluated using alternative methods such as recurrence xA|72!zk0P  
    % relations: see the Legendre functions, for example). For the Zernike +bwSu)k  
    % polynomials, however, this problem does not arise, because the Hm=!;xAFX  
    % polynomials are evaluated over the finite domain r = (0,1), and 0pP;[7k\  
    % because the coefficients for a given polynomial are generally all 83,ATQg  
    % of similar magnitude. `TNW LD@Z  
    % HorFQ?8  
    % ZERNPOL has been written using a vectorized implementation: multiple =,B44:`r  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] NFT&\6!o  
    % values can be passed as inputs) for a vector of points R.  To achieve 'bN\8t\S  
    % this vectorization most efficiently, the algorithm in ZERNPOL s53 Pw>f  
    % involves pre-determining all the powers p of R that are required to #K1VPezN  
    % compute the outputs, and then compiling the {R^p} into a single I$@0FSl  
    % matrix.  This avoids any redundant computation of the R^p, and <ptZY.8N  
    % minimizes the sizes of certain intermediate variables. w4YuijhW  
    % <uUHr,#  
    %   Paul Fricker 11/13/2006 W7 9wz\a  
    ]qZs^kQ  
    __Kn 1H{  
    % Check and prepare the inputs: BM+v,hGY  
    % ----------------------------- O)g\/uRy  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) tuzw% =Ey  
        error('zernpol:NMvectors','N and M must be vectors.') ,k}-I65M*t  
    end Ew>E]Ys  
    V4]t=3>  
    if length(n)~=length(m) 9&R. <I  
        error('zernpol:NMlength','N and M must be the same length.') gkDyWZG B  
    end Q/EHvb]  
    #'}?.m  
    n = n(:); @1_M's;  
    m = m(:); V gLnpPOQ  
    length_n = length(n); C*A!`Q?1Y  
    FsI51@V72Q  
    if any(mod(n-m,2)) d{?X:*F  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') wO6 D\#  
    end 37Z@a!#  
    |E/r64T  
    if any(m<0)  /="~Jo  
        error('zernpol:Mpositive','All M must be positive.') 4\qnCf3  
    end Ke0j8|  
    |tl4I2AV  
    if any(m>n) WEgJ_dB  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') xVOoYr>O  
    end }GV5':W@WG  
    ,,_$r7H`  
    if any( r>1 | r<0 ) R-Y07A  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') S>AM?  
    end EqW/Wxv7b  
    '!8'Xo@Go3  
    if ~any(size(r)==1) (* WO<V  
        error('zernpol:Rvector','R must be a vector.') Z_iVOctP  
    end < {1'cx  
    #~(J J  
    r = r(:); 6_KO6O7g  
    length_r = length(r); n#F:(MSOp  
    T0=8 U; =  
    if nargin==4 `_`\jd@  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); mUFg(;ya  
        if ~isnorm MM_c{gFF  
            error('zernpol:normalization','Unrecognized normalization flag.') HV_5 +  
        end 8UY[$lc  
    else Aj9<4N  
        isnorm = false; AUZ^XiK  
    end K"lZwU\:On  
    b#ih= qE  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% TGUlJLT  
    % Compute the Zernike Polynomials pP JhF8Dt  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 6x?3%0Km  
    :xd)]Ns  
    % Determine the required powers of r: yHrYSEM  
    % ----------------------------------- ei TG  
    rpowers = []; ;3~+M:{2  
    for j = 1:length(n) /r Q4JoR>  
        rpowers = [rpowers m(j):2:n(j)]; J['pBlEb\  
    end U]!~C 1cmw  
    rpowers = unique(rpowers); Q]n a_'_  
    Jat|n97$  
    % Pre-compute the values of r raised to the required powers, 'JA<q-Gn  
    % and compile them in a matrix: {3s=U"\  
    % ----------------------------- t@hE}R  
    if rpowers(1)==0 >M`ryM2=D  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); NT3Ti ?J,  
        rpowern = cat(2,rpowern{:}); X:3W9`s )*  
        rpowern = [ones(length_r,1) rpowern]; >ZX&2 {  
    else  nIWZo ~  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); |2ttdc.  
        rpowern = cat(2,rpowern{:}); 0+H4sz%.  
    end > St]MS  
    <G+IbUG:  
    % Compute the values of the polynomials: MlbQLtw  
    % -------------------------------------- Zt3Y<3o  
    z = zeros(length_r,length_n); 8v|?g8e3  
    for j = 1:length_n k;)t}7(  
        s = 0:(n(j)-m(j))/2; iL vzoQ  
        pows = n(j):-2:m(j); x'OYJ>l|  
        for k = length(s):-1:1 VB(S]N)F^  
            p = (1-2*mod(s(k),2))* ... y9/x:n&]  
                       prod(2:(n(j)-s(k)))/          ... oQsls9t  
                       prod(2:s(k))/                 ... hXF#KVqx  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... `Z)]mH\X  
                       prod(2:((n(j)+m(j))/2-s(k))); GxFmw:  
            idx = (pows(k)==rpowers); A9:dHOmT^U  
            z(:,j) = z(:,j) + p*rpowern(:,idx); f`^\v  
        end ?G|*=-8  
         c)5d-3"  
        if isnorm oZ CvEVUk  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); 295U<  
        end dE ,NG)MH  
    end b~5Q|3P9  
    0vi)m y;!  
    % 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
    850
    光币
    833
    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
    5476
    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)  ^/H9`z;  
    |IL/F]I  
    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。 e{*-_j "I  
    AxbQN.E  
    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)