切换到宽版
  • 广告投放
  • 稿件投递
  • 繁體中文
    • 11132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 bI_MF/r''  
    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! %o{IQ4Lz#  
     
    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 w(t1m]pF[  
    function z = zernfun(n,m,r,theta,nflag) [7[Qw]J  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. }J92TV  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N Gm=&[?}  
    %   and angular frequency M, evaluated at positions (R,THETA) on the 8}4.x3uw  
    %   unit circle.  N is a vector of positive integers (including 0), and ?hR0 MnP  
    %   M is a vector with the same number of elements as N.  Each element pS7y3(_  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) b`^?nD7  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, g}K/ba'  
    %   and THETA is a vector of angles.  R and THETA must have the same Aw4)=-LKO  
    %   length.  The output Z is a matrix with one column for every (N,M) /6yVbo"  
    %   pair, and one row for every (R,THETA) pair. "{H{-`Ni  
    % j \d)#+;  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike },Grg~l  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), KV9~L`=]i  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral >uQjygjj  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, _4#7 ?p  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized B> \q!dX3  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. P 6ka'!z  
    % ]W~M?1 }  
    %   The Zernike functions are an orthogonal basis on the unit circle. Nl%5OBm  
    %   They are used in disciplines such as astronomy, optics, and f4NN?"W)  
    %   optometry to describe functions on a circular domain. \9od*y  
    % .W2w/RayC  
    %   The following table lists the first 15 Zernike functions. d/$e#8  
    % Z.unCf3Q  
    %       n    m    Zernike function           Normalization EQZ/v gho  
    %       -------------------------------------------------- (I>SqM Y  
    %       0    0    1                                 1 $M4_"!  
    %       1    1    r * cos(theta)                    2 hjk]?MC  
    %       1   -1    r * sin(theta)                    2 <8Zs; >YuK  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) ]rg-=Y k  
    %       2    0    (2*r^2 - 1)                    sqrt(3) c*" P+  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) L5 Ai  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) Ju;^^  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) ybv< 1  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) ?@Tsd@s~r  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) HAs/f#zAk6  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) L/vw7XNrX  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) {cdrMP@""  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) >!o!rs  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) HyYJ"54  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) JPgFTr  
    %       -------------------------------------------------- ;4v`FC>  
    % w&]$!g4  
    %   Example 1: ;9J6)zg !n  
    % BI!EmA  
    %       % Display the Zernike function Z(n=5,m=1) wp4  .~E  
    %       x = -1:0.01:1; .O+,1&D5  
    %       [X,Y] = meshgrid(x,x); XZ(<Mo\v  
    %       [theta,r] = cart2pol(X,Y); &Fk|"f+  
    %       idx = r<=1; { W5 _KX  
    %       z = nan(size(X)); WBzPSnS2  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); {sq:vu@NC  
    %       figure evVxzU&  
    %       pcolor(x,x,z), shading interp W3%RB[s-  
    %       axis square, colorbar cN\_1  
    %       title('Zernike function Z_5^1(r,\theta)') M%SNq|Lo  
    % " Z dI~  
    %   Example 2: 2) 2:KX  
    % }ZiJHj'<  
    %       % Display the first 10 Zernike functions V%zo[A  
    %       x = -1:0.01:1; 8N<m V^|}  
    %       [X,Y] = meshgrid(x,x); nmuU*o L  
    %       [theta,r] = cart2pol(X,Y); y4&x`|tv  
    %       idx = r<=1; r,L`@A=v  
    %       z = nan(size(X)); 4 .(5m\s!  
    %       n = [0  1  1  2  2  2  3  3  3  3]; rQTG-& ,  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; `sd H q  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; : ' pK  
    %       y = zernfun(n,m,r(idx),theta(idx)); z38Pi  
    %       figure('Units','normalized') " #U-*Z7  
    %       for k = 1:10 iqsR]mab  
    %           z(idx) = y(:,k); xZQyH  
    %           subplot(4,7,Nplot(k)) Kc>Rd  
    %           pcolor(x,x,z), shading interp lr -+|>M)  
    %           set(gca,'XTick',[],'YTick',[]) 7Q&S [])  
    %           axis square FIQHs"#T  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) b,(<74!#8  
    %       end B!)Tytm9u  
    %  4NIb_E0  
    %   See also ZERNPOL, ZERNFUN2. ^Q!A4 qOQ  
    91k-os(4]  
    %   Paul Fricker 11/13/2006 "!D,9AkZS  
    ,V;HM F.  
    .j"@7#tW  
    % Check and prepare the inputs: e_1L J  
    % ----------------------------- xp]9Z]J1l  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) w@,v$4Oi  
        error('zernfun:NMvectors','N and M must be vectors.') i> {0h3Y  
    end &uf|Le4  
    \+C0Rv^^  
    if length(n)~=length(m) F$O$Y[  
        error('zernfun:NMlength','N and M must be the same length.') [X@JH6U r  
    end k<rJm P{  
    q3GkfgY  
    n = n(:); mCQ:< #  
    m = m(:); _gl1Qtv@rf  
    if any(mod(n-m,2)) hF5(1s}e$  
        error('zernfun:NMmultiplesof2', ... Lo=n)cV1,  
              'All N and M must differ by multiples of 2 (including 0).') 1_M}Dc+J  
    end zV"'-iP  
    pK0@H"$8  
    if any(m>n) -Q%Pg<Q-#  
        error('zernfun:MlessthanN', ... RE08\gNIt  
              'Each M must be less than or equal to its corresponding N.') :_o^oi7G  
    end qga?-oz,<6  
    ylQ9Su>o  
    if any( r>1 | r<0 ) A5sz[k  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') lZ) qV!<  
    end :b;`.`@KL_  
    y-"QY[  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) V$dhiP z  
        error('zernfun:RTHvector','R and THETA must be vectors.') !mUO/6Q hq  
    end Au:R]7   
    `X<a(5[vV3  
    r = r(:); q*cEosi'F?  
    theta = theta(:); goJ'z|))  
    length_r = length(r); >G As&\4hs  
    if length_r~=length(theta) <*o V-A  
        error('zernfun:RTHlength', ... "w__AYHV  
              'The number of R- and THETA-values must be equal.') `O0y8  
    end O9AFQ)u   
    be?Bf^O>  
    % Check normalization: M; YJpi  
    % -------------------- WADEDl&,'  
    if nargin==5 && ischar(nflag) + f:!9)C  
        isnorm = strcmpi(nflag,'norm'); n+nZ;GJ5d  
        if ~isnorm %.HLO.A  
            error('zernfun:normalization','Unrecognized normalization flag.') ]ZNFrpq  
        end s-~`Ao' <  
    else RF~G{wz  
        isnorm = false; "F4 3q8P  
    end r`< x@,  
    d1'= \PYr  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% `7[!bCl  
    % Compute the Zernike Polynomials G2-0r.f  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -<M+$hK\  
    *kcc]*6@s  
    % Determine the required powers of r: [70 5[  
    % ----------------------------------- C &y 2I  
    m_abs = abs(m); w\{#nrhYU  
    rpowers = []; yB 'C9wEH  
    for j = 1:length(n) s)KlKh  
        rpowers = [rpowers m_abs(j):2:n(j)]; 34nfL: y  
    end NytodVZ'3  
    rpowers = unique(rpowers); 2A9crL $  
    q?@*  
    % Pre-compute the values of r raised to the required powers, R.vOYzo  
    % and compile them in a matrix: 1}+b4 "7]  
    % ----------------------------- 23 #JmR  
    if rpowers(1)==0 yrl7  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); zQ<;3+*  
        rpowern = cat(2,rpowern{:}); L 4Z+8*  
        rpowern = [ones(length_r,1) rpowern]; OhlK;hvdB*  
    else .w'b%M  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); t1yOAbI  
        rpowern = cat(2,rpowern{:}); RDjw|V  
    end XXA]ukj;r  
    A|YgA66M  
    % Compute the values of the polynomials: >yHtGIHe-  
    % -------------------------------------- %[M0TE=J  
    y = zeros(length_r,length(n)); H)EL0 Kv/  
    for j = 1:length(n) rm$dv%q  
        s = 0:(n(j)-m_abs(j))/2; p<}y'7(  
        pows = n(j):-2:m_abs(j); K<`W>2"  
        for k = length(s):-1:1 iA[o;D#  
            p = (1-2*mod(s(k),2))* ... -Fu,oEj{*  
                       prod(2:(n(j)-s(k)))/              ... 11kyrv  
                       prod(2:s(k))/                     ... *'aouS/?<6  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... )N607 Fa-  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); DT)] [V^w  
            idx = (pows(k)==rpowers); * fj`+J  
            y(:,j) = y(:,j) + p*rpowern(:,idx); ,S(s  
        end 8oXp8CC  
         -3azA7tzz  
        if isnorm ,4 _H{+M  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 1 A0BM  
        end }#%Y eCA?  
    end ,Z _@]D@  
    % END: Compute the Zernike Polynomials jYFmL_{  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% (C"q-0?n  
    o?t H[  
    % Compute the Zernike functions: 5U4V_*V  
    % ------------------------------ R8eBIJ/@_  
    idx_pos = m>0; Y~A I2HS  
    idx_neg = m<0; (xVx|:R[<H  
    (_>Su QK  
    z = y; zwJ&K;"y(  
    if any(idx_pos) VP^Yf_  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); y Rl   
    end ilHf5$  
    if any(idx_neg) 8F`8=L NO  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); %O"Whe  
    end n*na6rV\k  
    -WF((s;<#  
    % 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@&%"nO  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. j7gTVfO  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated {4Kvr4)4  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive `Ng Q>KV!  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, v G\J8s  
    %   and THETA is a vector of angles.  R and THETA must have the same ux(~+<k  
    %   length.  The output Z is a matrix with one column for every P-value, UWV%  y P  
    %   and one row for every (R,THETA) pair. ,pGA|ob  
    % ,-E'059  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike ``VE<:2+  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) s=jYQ5nv  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) *#Ia8^z=p  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 dEKu5GI  
    %   for all p. np6G~0Y`  
    % } f&=}  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 l$9k:#\FD  
    %   Zernike functions (order N<=7).  In some disciplines it is [(#)9/3,  
    %   traditional to label the first 36 functions using a single mode dG5jhkPX  
    %   number P instead of separate numbers for the order N and azimuthal -WqhOZ  
    %   frequency M. ]7W!f 2@  
    % S 4 17.n  
    %   Example: 6#CswSpS  
    %  AkS16A  
    %       % Display the first 16 Zernike functions jwE=  
    %       x = -1:0.01:1; Q`AlK"G,  
    %       [X,Y] = meshgrid(x,x); |w*s:p  
    %       [theta,r] = cart2pol(X,Y); 8o%Vn'^t  
    %       idx = r<=1; w$f_z*/  
    %       p = 0:15; ,Owk;MV@  
    %       z = nan(size(X)); WcKDerc  
    %       y = zernfun2(p,r(idx),theta(idx)); hui #<2{  
    %       figure('Units','normalized') EDgtn)1  
    %       for k = 1:length(p) MjC<N[WO>N  
    %           z(idx) = y(:,k); ^il$t]X5-  
    %           subplot(4,4,k) ff.k1%wr^  
    %           pcolor(x,x,z), shading interp gF)-Ci  
    %           set(gca,'XTick',[],'YTick',[]) Oz-/0;1n  
    %           axis square qF bj~ec  
    %           title(['Z_{' num2str(p(k)) '}']) vkGF_aenk  
    %       end ma*#*4  
    % "M iJM+,  
    %   See also ZERNPOL, ZERNFUN. dE,E,tv  
    8ly)G  
    %   Paul Fricker 11/13/2006 S5>ztK.e  
    Ff/Ap&0+  
    J  Y8Rk=  
    % Check and prepare the inputs: Y2l;NSWU  
    % ----------------------------- nD eVYK  
    if min(size(p))~=1 tb~E.Lm\  
        error('zernfun2:Pvector','Input P must be vector.') L1!~T+%uQ  
    end s;oe Qa}TB  
    TX$dxHSPK  
    if any(p)>35 y$-@|M$GG  
        error('zernfun2:P36', ... c;q=$MO`  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... L{jx'[C  
               '(P = 0 to 35).']) nm<VcCc  
    end 3^ UoK  
    "0PsCr}!  
    % Get the order and frequency corresonding to the function number: hq {{XQ  
    % ---------------------------------------------------------------- UJqh~s  
    p = p(:); d*Mqs}8  
    n = ceil((-3+sqrt(9+8*p))/2); J'ce?_\?PY  
    m = 2*p - n.*(n+2); Ow>u!P!  
    ?'f  
    % Pass the inputs to the function ZERNFUN: *8,W$pe3  
    % ---------------------------------------- N5zWeFq@6  
    switch nargin )s:kQ~+  
        case 3 E@k'uyIu  
            z = zernfun(n,m,r,theta); IYq#|^)5+  
        case 4 QpQ2hNf  
            z = zernfun(n,m,r,theta,nflag); sdO8;v>  
        otherwise xva e^gr  
            error('zernfun2:nargin','Incorrect number of inputs.') lbt8S.fx  
    end K!ogpd&X&  
    3t+{~{Dj  
    % 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) X4:84  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. |*J;X<Vm  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of [-l>f P0  
    %   order N and frequency M, evaluated at R.  N is a vector of cu5}(  
    %   positive integers (including 0), and M is a vector with the ) C~#W  
    %   same number of elements as N.  Each element k of M must be a Q`J U[nY  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) 9HN&M*}  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is UK^w;w2F  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix n7ZJ< ~wl  
    %   with one column for every (N,M) pair, and one row for every o92BGqA>&  
    %   element in R. * wqR.n?  
    % ;9)nG,P3  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- K($+ILZ  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is t4;gY298  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to 0[ "CP:u  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 h){#dU+&  
    %   for all [n,m]. dmkGIg}  
    % }3Ke  
    %   The radial Zernike polynomials are the radial portion of the gJwX  
    %   Zernike functions, which are an orthogonal basis on the unit {be|G^.c  
    %   circle.  The series representation of the radial Zernike w(+ L&IBC  
    %   polynomials is ?^-fivzS>  
    % +  rN#  
    %          (n-m)/2 j1Sjw6}GCH  
    %            __ UZiL NKc  
    %    m      \       s                                          n-2s [}Rs  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r ~Odclrs  
    %    n      s=0 x5rm 2C  
    % &kWT<*;J)  
    %   The following table shows the first 12 polynomials. g(jn /Cx  
    % A #pH$s  
    %       n    m    Zernike polynomial    Normalization 9ygNJX'~  
    %       --------------------------------------------- F DGzh/  
    %       0    0    1                        sqrt(2) {}Afah  
    %       1    1    r                           2 ?g K|R  
    %       2    0    2*r^2 - 1                sqrt(6) n ]l3 )u  
    %       2    2    r^2                      sqrt(6) }DvT6  
    %       3    1    3*r^3 - 2*r              sqrt(8) \dB z-H'@  
    %       3    3    r^3                      sqrt(8) p4uObK,  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) gz:US 77  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) h^H)p`[Gme  
    %       4    4    r^4                      sqrt(10) $G /p[JG6-  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) n? ^oQX}.\  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) JsX}PVuL  
    %       5    5    r^5                      sqrt(12) se_Oi$VZ{  
    %       --------------------------------------------- (r.y   
    % ?|hYtV  
    %   Example: zP!j {y4w  
    % >\lBbq a#  
    %       % Display three example Zernike radial polynomials p{0rHu[  
    %       r = 0:0.01:1; @C~gU@F  
    %       n = [3 2 5]; 0}>p)k3&A  
    %       m = [1 2 1]; eN  TKX  
    %       z = zernpol(n,m,r); l n09_Lr  
    %       figure SmP&wNHQf  
    %       plot(r,z) @7?L+.r$9  
    %       grid on Q,o"[ &Gp  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') ~#rmw6y  
    % Q\le3KB  
    %   See also ZERNFUN, ZERNFUN2. R{Me~L?  
    w[;5]z  
    % A note on the algorithm. ,V+,3TT  
    % ------------------------ {k<mN Y  
    % The radial Zernike polynomials are computed using the series ~5~Cpu2v7  
    % representation shown in the Help section above. For many special _$gP-J  
    % functions, direct evaluation using the series representation can 6oR5q 4  
    % produce poor numerical results (floating point errors), because j@jUuYuDgl  
    % the summation often involves computing small differences between @aWd0e]  
    % large successive terms in the series. (In such cases, the functions KN\tRE  
    % are often evaluated using alternative methods such as recurrence >yt8gw0J  
    % relations: see the Legendre functions, for example). For the Zernike N MH'4R  
    % polynomials, however, this problem does not arise, because the |#O>DdKHT  
    % polynomials are evaluated over the finite domain r = (0,1), and ^r 9  
    % because the coefficients for a given polynomial are generally all qZh}gu*>  
    % of similar magnitude. AbOF/ g)C  
    % 89%#;C  
    % ZERNPOL has been written using a vectorized implementation: multiple z7]GZF  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] 2fMKS  
    % values can be passed as inputs) for a vector of points R.  To achieve 6F3FcUL  
    % this vectorization most efficiently, the algorithm in ZERNPOL 43BqNQ0  
    % involves pre-determining all the powers p of R that are required to m}sh (W5\  
    % compute the outputs, and then compiling the {R^p} into a single ![aa@nOSa  
    % matrix.  This avoids any redundant computation of the R^p, and e6I7N?j  
    % minimizes the sizes of certain intermediate variables. [|APMMYK1  
    % '0jn|9l58  
    %   Paul Fricker 11/13/2006 Kr@6m80E5  
    OCbwV7q:  
    l !:kwF  
    % Check and prepare the inputs: )cBO_  
    % ----------------------------- Fz@9 @  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) AkVgFQg" n  
        error('zernpol:NMvectors','N and M must be vectors.') nm]m!.$d  
    end OH6-\U'.Z  
    5T]dQ3[v4  
    if length(n)~=length(m) $'93:9tg  
        error('zernpol:NMlength','N and M must be the same length.') (A\\s$fE/1  
    end I"2*}v|  
    $9?<mP2-*  
    n = n(:); ..UA*#%1  
    m = m(:); CK(`]-q>,  
    length_n = length(n); s-JS[  
    ;]^% 6B n  
    if any(mod(n-m,2)) cng166}1A  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') X \qG WpN%  
    end #X qnH  
    <,,X\>B  
    if any(m<0) F $1f8U8  
        error('zernpol:Mpositive','All M must be positive.') u Fn?U)  
    end 1;eWnb(  
    8\Hz FB  
    if any(m>n) >Xw0i\G  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') O9_SVXWVw  
    end CI^s~M >  
    w/csLi.O  
    if any( r>1 | r<0 ) vF1Fcp.@  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') ]D[\l$(  
    end d:=Z<Y?d/  
    8yk4#CZ  
    if ~any(size(r)==1) 6o4Y]C2W{1  
        error('zernpol:Rvector','R must be a vector.') kO4'|<  
    end l"/E,X  
    2@Oz_?O=  
    r = r(:); GI6]Ecc  
    length_r = length(r); fNz(z\  
    ^G4@cR.An  
    if nargin==4 h 27f0x9  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); d .%2QkL  
        if ~isnorm l2ie\4dK@  
            error('zernpol:normalization','Unrecognized normalization flag.') nf1O8FwRb  
        end tXcZl!3x  
    else 8.Ufw. 5  
        isnorm = false; C-XJe~  
    end 5+yy:#J]  
    \>x1#Vr>#V  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% d:"7Tw2v+  
    % Compute the Zernike Polynomials HuR774f[  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Vh>|F}%E  
    Gn<0Fy2  
    % Determine the required powers of r: 3aU5rbi|B  
    % ----------------------------------- sf/m@425  
    rpowers = []; ESUO I  
    for j = 1:length(n) QAAuFZs  
        rpowers = [rpowers m(j):2:n(j)]; X *EseC  
    end iiF`2  
    rpowers = unique(rpowers); 4ETHaIiWp  
    P^!g0K  
    % Pre-compute the values of r raised to the required powers, -xbs'[  
    % and compile them in a matrix: <v/aquLN  
    % ----------------------------- -}PE(c1%?q  
    if rpowers(1)==0 o")"^@Zh i  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); oUO3,2bn  
        rpowern = cat(2,rpowern{:}); UBJYs{zz  
        rpowern = [ones(length_r,1) rpowern]; P&=YLL<W  
    else hnH<m7  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 8:)[.  
        rpowern = cat(2,rpowern{:}); 7r^Cs#b+I  
    end ("F$r$9S  
    Z=Cw7E  
    % Compute the values of the polynomials: v9FR  
    % -------------------------------------- Wp+lI1t  
    z = zeros(length_r,length_n); ]uF7HX7F  
    for j = 1:length_n *C0a,G4  
        s = 0:(n(j)-m(j))/2; IZm6.F  
        pows = n(j):-2:m(j); <Np Mv!g  
        for k = length(s):-1:1 ,X1M!'  
            p = (1-2*mod(s(k),2))* ... |vm-(HY!  
                       prod(2:(n(j)-s(k)))/          ... VcXr!4 M  
                       prod(2:s(k))/                 ... . AOc$Nt  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... G[]%1 _QCO  
                       prod(2:((n(j)+m(j))/2-s(k))); $]Ix(7@W  
            idx = (pows(k)==rpowers); J%|;  
            z(:,j) = z(:,j) + p*rpowern(:,idx); D "5|\  
        end vea{o 35!  
         5'+g[eNyBV  
        if isnorm ^|6#Vx  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); G7|d$!%  
        end w'A*EWO  
    end N\&VJc  
    t#_6GL  
    % 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)  K<tkNWasQ  
    E)l@uPA'1  
    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。 ;;- I<TL  
    -n$fh::^  
    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)