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

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    离线niuhelen
     
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    只看楼主 倒序阅读 楼主  发表于: 2011-03-12
    小弟不是学光学的,所以想请各位大侠指点啊!谢谢啦 Vsr"W@k_  
    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! KW!+Ws  
     
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    离线phility
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    只看该作者 1楼 发表于: 2011-03-12
    可以用matlab编程,用zernike多项式进行波面拟合,求出zernike多项式的系数,拟合的算法有很多种,最简单的是最小二乘法,你可以查下相关资料,挺简单的
    离线phility
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    只看该作者 2楼 发表于: 2011-03-12
    泽尼克多项式的前9项对应象差的
    离线niuhelen
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    只看该作者 3楼 发表于: 2011-03-12
    回 2楼(phility) 的帖子
    非常感谢啊,我手上也有zernike多项式的拟合的源程序,也不知道对不对,不怎么会有 D8{ ,}@  
    function z = zernfun(n,m,r,theta,nflag) ?L0|$#Iw  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. ksTK'7*  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N [ . }Uzx  
    %   and angular frequency M, evaluated at positions (R,THETA) on the G1\F7A  
    %   unit circle.  N is a vector of positive integers (including 0), and %w?C)$Kn\  
    %   M is a vector with the same number of elements as N.  Each element $f%om)  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) E]}_hZU  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, :5BCW68le  
    %   and THETA is a vector of angles.  R and THETA must have the same  56MY@  
    %   length.  The output Z is a matrix with one column for every (N,M) Zl{9G?abCT  
    %   pair, and one row for every (R,THETA) pair. N.0g%0A.D  
    % !l]_c 5  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike @AM11v\:  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ahQY-%>  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral wWSo+40  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 3@:O1i  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized &er,Wyc(  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 8]oolA:^4s  
    % @biU@[D  
    %   The Zernike functions are an orthogonal basis on the unit circle. 9aNOfs8(  
    %   They are used in disciplines such as astronomy, optics, and Ql%B=vgKL  
    %   optometry to describe functions on a circular domain. Zd88+GS,#  
    % V%z?wDC  
    %   The following table lists the first 15 Zernike functions. )0DgFA6k_  
    % SUv'cld  
    %       n    m    Zernike function           Normalization 3,K\ZUU.,  
    %       -------------------------------------------------- s;..a&C'  
    %       0    0    1                                 1 |28'<BL  
    %       1    1    r * cos(theta)                    2 (> _Lb  
    %       1   -1    r * sin(theta)                    2 #oR`_Dm)P  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) ~)n[Vf  
    %       2    0    (2*r^2 - 1)                    sqrt(3) H^54o$5  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) ca3SE^  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) 8};kNW^2m  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) =<7z :]  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) \yZVn6GVr  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) _/'VD!(MV  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) J@"UFL'^  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) jm@,Ihz=wI  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) FJ4,|x3v[x  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 5 ba e-  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) \#WWJh"W  
    %       -------------------------------------------------- em5~4;&'  
    % wy8Q=X:vP  
    %   Example 1: ;obOr~Jx'5  
    % O!^ >YvOh  
    %       % Display the Zernike function Z(n=5,m=1) J3~%9MCJ  
    %       x = -1:0.01:1; {Z7ixc523  
    %       [X,Y] = meshgrid(x,x); u|T]Ne  
    %       [theta,r] = cart2pol(X,Y); #oFyi @U  
    %       idx = r<=1; ,Q3OQ[Nmh  
    %       z = nan(size(X)); 97$Q?a8S@  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); 8|<</v8i  
    %       figure .@%L8_sMR  
    %       pcolor(x,x,z), shading interp Kh[l};/F  
    %       axis square, colorbar _)~1'tCs}h  
    %       title('Zernike function Z_5^1(r,\theta)') UP$>,05z6  
    % l2:-).7xt  
    %   Example 2: U#]J5'i  
    % #ACT&J  
    %       % Display the first 10 Zernike functions 'RhS%l  
    %       x = -1:0.01:1; 5S2 j5M00  
    %       [X,Y] = meshgrid(x,x); JN4gH4ez)  
    %       [theta,r] = cart2pol(X,Y); !LM`2|3$  
    %       idx = r<=1; HA,8O [jon  
    %       z = nan(size(X)); J*MH`;-  
    %       n = [0  1  1  2  2  2  3  3  3  3]; "]kzt ux  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; M_Q`9  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; ot[ZFF\  
    %       y = zernfun(n,m,r(idx),theta(idx)); [Eccj`\e g  
    %       figure('Units','normalized') Ez"*',(  
    %       for k = 1:10 /]'&cD 1  
    %           z(idx) = y(:,k); > Xh=P%  
    %           subplot(4,7,Nplot(k)) :" Otsb7  
    %           pcolor(x,x,z), shading interp rab$[?]  
    %           set(gca,'XTick',[],'YTick',[]) U#S-x5Gn  
    %           axis square TfT^.p*  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 9jY+0h*uP  
    %       end |aAyWK  S  
    % ?bt;i>O\  
    %   See also ZERNPOL, ZERNFUN2. }e/vKW fT  
    ,zr9*t  
    %   Paul Fricker 11/13/2006 xw_klHL-o  
    ZS4dW_*[  
    {U$XHG  
    % Check and prepare the inputs: pfHjs3A=  
    % ----------------------------- dO%f ;m>#  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) k ,ldi  
        error('zernfun:NMvectors','N and M must be vectors.') y0(.6HI  
    end Dy,MQIM|!  
    i%.k{MY  
    if length(n)~=length(m) E;{CoL  
        error('zernfun:NMlength','N and M must be the same length.') ZD'mwj+K  
    end NK/y,f6  
    LKp;sV  
    n = n(:); #n{4f1TZ  
    m = m(:); KpLaQb  
    if any(mod(n-m,2)) 3@\/5I xn  
        error('zernfun:NMmultiplesof2', ... -,+C*|mu  
              'All N and M must differ by multiples of 2 (including 0).') gC(S(osF  
    end d/j?.\  
    NfPWcK [  
    if any(m>n) u&uFXOc'  
        error('zernfun:MlessthanN', ... ;$zvm`|:  
              'Each M must be less than or equal to its corresponding N.') ;`LG WT-<F  
    end tc[Ld#  
    VBPtM{ g  
    if any( r>1 | r<0 ) ,cS#  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') 9x!kvB6  
    end @ Do.Wgt  
    %LP4RZ  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 6q8}8;STTY  
        error('zernfun:RTHvector','R and THETA must be vectors.') &z40l['4bz  
    end }DM W,+3  
    Z8FgxR  
    r = r(:); Nv.  
    theta = theta(:); P?f${ t+  
    length_r = length(r); :%J;[bS+  
    if length_r~=length(theta) xok T  
        error('zernfun:RTHlength', ... aReJ@  
              'The number of R- and THETA-values must be equal.') He'VqUw_  
    end X"d"a={]  
    RHn3\N  
    % Check normalization: 3{|~'5*  
    % -------------------- 4]tg!ks  
    if nargin==5 && ischar(nflag) M>m!\bb%.  
        isnorm = strcmpi(nflag,'norm'); 2"Wq=qy\J  
        if ~isnorm (?8i^T?WP=  
            error('zernfun:normalization','Unrecognized normalization flag.') _,60pr3D'  
        end C.:S@{sK  
    else Dt[+HCCY:  
        isnorm = false; N8At N\e  
    end Vf~-v$YI  
    ,)*[Xa_n  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% jQm~F` z  
    % Compute the Zernike Polynomials aV|V C $  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% OYt_i'Q  
    \RR` F .7  
    % Determine the required powers of r: K/Yeh<_&  
    % ----------------------------------- qx1Js3%  
    m_abs = abs(m); 5j.@)XXe  
    rpowers = []; UakVmVN/P  
    for j = 1:length(n) 8CRbo24"s  
        rpowers = [rpowers m_abs(j):2:n(j)]; G \MeJSt*  
    end tjRw bnT"  
    rpowers = unique(rpowers); *j]Bo,AC  
    qGH[kd  
    % Pre-compute the values of r raised to the required powers, $`7Fk%#+e  
    % and compile them in a matrix: [<U=)!Swg  
    % ----------------------------- $8U$.~v  
    if rpowers(1)==0 v5\ALWy+p  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); eL"'-d+]  
        rpowern = cat(2,rpowern{:}); CSoVB[vS  
        rpowern = [ones(length_r,1) rpowern]; C^,b aCX  
    else fi>.X99(G  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); :Ob^b3<t  
        rpowern = cat(2,rpowern{:}); O>h h  
    end &l ]F&-  
    wM N;<  
    % Compute the values of the polynomials: *&IvEu  
    % -------------------------------------- ,.( :b82$  
    y = zeros(length_r,length(n)); E"p _!!1  
    for j = 1:length(n) "}1cQ|0a  
        s = 0:(n(j)-m_abs(j))/2; dF 6od  
        pows = n(j):-2:m_abs(j); -f ~1Id  
        for k = length(s):-1:1 s?m_zJh  
            p = (1-2*mod(s(k),2))* ... BaI-ve  
                       prod(2:(n(j)-s(k)))/              ... ob/<;SrU<  
                       prod(2:s(k))/                     ... 3=oxT6"k  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... bcwb'D\a  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); ]?T^tJ  
            idx = (pows(k)==rpowers); hM!g6\ w  
            y(:,j) = y(:,j) + p*rpowern(:,idx); zL}`7*d:v  
        end r`sKe &  
         ~Azj Y8  
        if isnorm YI*H]V%w  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); @<$m`^H  
        end G)NqIur*Z  
    end >6&Rytcc]  
    % END: Compute the Zernike Polynomials V >eG\  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% hNYO+LrI)  
    p|nPu*R-\  
    % Compute the Zernike functions: VhLfSN>W  
    % ------------------------------ _8y4U  
    idx_pos = m>0; f ]_ki  
    idx_neg = m<0; l x5.50mI  
    XY6Sm{  
    z = y; EX!`Zejf  
    if any(idx_pos) G#`  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 2i #Ekon  
    end $Lbamg->E  
    if any(idx_neg) `?[,1   
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); %wru)  
    end 6 F39'  
    _]ZlGq!L  
    % EOF zernfun
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    function z = zernfun2(p,r,theta,nflag) icIWv  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. vN_ 8qzWk  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated /%jX=S.5h<  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive {0+WVZ4u  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, Q;z!]hjBM  
    %   and THETA is a vector of angles.  R and THETA must have the same pZ*%zt]-a  
    %   length.  The output Z is a matrix with one column for every P-value, M,kO7g  
    %   and one row for every (R,THETA) pair. ^C~R)M:C  
    % KlS#f  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike @^ &p$:  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) O<|pw  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) pJ1\@G  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 =<PEvIn  
    %   for all p. ,UVu.RjXN  
    % =6 zK 1Z  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 KTot40osj  
    %   Zernike functions (order N<=7).  In some disciplines it is F(lJ  
    %   traditional to label the first 36 functions using a single mode .I#_~C'\  
    %   number P instead of separate numbers for the order N and azimuthal +G"YQq'b  
    %   frequency M. +`1~zcu  
    % tEo-Mj5:  
    %   Example: ]2|fc5G'  
    % &\cS{35  
    %       % Display the first 16 Zernike functions A*/8j\{n  
    %       x = -1:0.01:1; za 7+xF  
    %       [X,Y] = meshgrid(x,x); .:Sk=r4u\  
    %       [theta,r] = cart2pol(X,Y); R)SY#*Y  
    %       idx = r<=1; b]xoXC6@t  
    %       p = 0:15; k#\j\t-  
    %       z = nan(size(X)); ,=G]tnsv^  
    %       y = zernfun2(p,r(idx),theta(idx)); #+U1QOsz  
    %       figure('Units','normalized') `s UY$Q  
    %       for k = 1:length(p) y4IQa.F  
    %           z(idx) = y(:,k); .Oc j|A6  
    %           subplot(4,4,k) PXtF#,roP  
    %           pcolor(x,x,z), shading interp UA~ 4O Q]  
    %           set(gca,'XTick',[],'YTick',[]) xz.M'az\  
    %           axis square 6L\]Ee  
    %           title(['Z_{' num2str(p(k)) '}']) ~a_X 7  
    %       end @b.,pwZF  
    % bnanTH9-  
    %   See also ZERNPOL, ZERNFUN. gzat!>*  
    O(=9&PRi  
    %   Paul Fricker 11/13/2006 !||Gfia  
    3}mg7KV&  
    Rmn{Vui9\  
    % Check and prepare the inputs: H7Z`aQC  
    % -----------------------------  qbS6#7D  
    if min(size(p))~=1 Rcw[`q3/  
        error('zernfun2:Pvector','Input P must be vector.') 4<E <sD  
    end 2.MUQ;OX  
    -}!mi V  
    if any(p)>35  #$2/<  
        error('zernfun2:P36', ... <,/7:n  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... cjg~?R  
               '(P = 0 to 35).']) 4J(-~  
    end BV7P_!vt  
    5=#d#dDc  
    % Get the order and frequency corresonding to the function number:  s7:H  
    % ---------------------------------------------------------------- "sDs[Lcq  
    p = p(:); ~Sm6{L  
    n = ceil((-3+sqrt(9+8*p))/2); G'w!Aw s  
    m = 2*p - n.*(n+2); ~$[fG}C.K  
    qAbmQ{|w  
    % Pass the inputs to the function ZERNFUN: aL90:,V  
    % ---------------------------------------- Tl[*(| /C  
    switch nargin 8{i}^.p  
        case 3 4~FRE)8  
            z = zernfun(n,m,r,theta); 0pEM0M  
        case 4 RU6c 8>"  
            z = zernfun(n,m,r,theta,nflag); 4L{]!dox  
        otherwise .{`C>/"}  
            error('zernfun2:nargin','Incorrect number of inputs.') LJ@r+|>  
    end X;)/<:mX  
    A4#F AFy  
    % EOF zernfun2
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    function z = zernpol(n,m,r,nflag) Qb/:E}h]$  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. `#`jU"T|  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of .7b%7dQ<\  
    %   order N and frequency M, evaluated at R.  N is a vector of h\3-8m  
    %   positive integers (including 0), and M is a vector with the VR&dy|5BO  
    %   same number of elements as N.  Each element k of M must be a CyYr5 Dz  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) il !B={  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is ,&M#[>\(3  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix rQ]JM  
    %   with one column for every (N,M) pair, and one row for every vGh>1U:  
    %   element in R. MO7R3PP  
    % vBF9!6X.  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- a*.#Zgy:lK  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is ?H@<8Ra=3  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to j_<!y(W  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 zixG}'  
    %   for all [n,m]. rEj[XK  
    % ?(d1;/0v>  
    %   The radial Zernike polynomials are the radial portion of the 1kL8EPT%o  
    %   Zernike functions, which are an orthogonal basis on the unit FE,mUpHIR  
    %   circle.  The series representation of the radial Zernike b$#b+G{y  
    %   polynomials is 7f ub^'_  
    % FibZT1-k  
    %          (n-m)/2 _[Imwu}  
    %            __ HSRO gBNI:  
    %    m      \       s                                          n-2s pl1CPxSdO  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r Bh cp=#  
    %    n      s=0 W~D_+[P|_  
    % YkB@fTTS  
    %   The following table shows the first 12 polynomials. _\tv ${  
    % w@cW`PlF  
    %       n    m    Zernike polynomial    Normalization BPt? 3tC  
    %       --------------------------------------------- bl^Ihza  
    %       0    0    1                        sqrt(2) ^w~B]*A :"  
    %       1    1    r                           2 [yQ%g;m  
    %       2    0    2*r^2 - 1                sqrt(6) L98T!5)  
    %       2    2    r^2                      sqrt(6) r=Lgh#9S  
    %       3    1    3*r^3 - 2*r              sqrt(8) `{Q'iydU  
    %       3    3    r^3                      sqrt(8) +8N6tw/&  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) &5b 3k[K"  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) GVnDN~[  
    %       4    4    r^4                      sqrt(10) N=2T~M 1  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) /R=MX>JA;  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) o&%v"#H2  
    %       5    5    r^5                      sqrt(12) %ZWt 45A  
    %       --------------------------------------------- X]@"ZV[  
    % (iir,Ks2C  
    %   Example: U6@c)_* <  
    % QtKcv7:4  
    %       % Display three example Zernike radial polynomials ,7)hrA$(  
    %       r = 0:0.01:1; r\ft{Z<P  
    %       n = [3 2 5]; xLoQ0rt 6  
    %       m = [1 2 1]; 0,ryy,2  
    %       z = zernpol(n,m,r); ,jis@]:  
    %       figure Y 2ANt w@  
    %       plot(r,z) |JYb4J4Ni  
    %       grid on 9N9&y^SmD  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') #d\&6'O  
    % C){Q;`M-<  
    %   See also ZERNFUN, ZERNFUN2. ( uD^_N]3  
    -]zb3P  
    % A note on the algorithm. &Z]}rn  
    % ------------------------ P%e7c,  
    % The radial Zernike polynomials are computed using the series '_.qhsS  
    % representation shown in the Help section above. For many special Bgc]t  
    % functions, direct evaluation using the series representation can mXyP;k  
    % produce poor numerical results (floating point errors), because bGeIb-|(  
    % the summation often involves computing small differences between ;HXk'xN  
    % large successive terms in the series. (In such cases, the functions Ei@  
    % are often evaluated using alternative methods such as recurrence ye$_=KARP  
    % relations: see the Legendre functions, for example). For the Zernike f- 9t  
    % polynomials, however, this problem does not arise, because the tjXg  
    % polynomials are evaluated over the finite domain r = (0,1), and \~%+)a%%  
    % because the coefficients for a given polynomial are generally all NLL"~  
    % of similar magnitude. e3;D1@  
    % 96V8R<   
    % ZERNPOL has been written using a vectorized implementation: multiple <>\|hno}  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] { %X2K  
    % values can be passed as inputs) for a vector of points R.  To achieve 2\1+M)  
    % this vectorization most efficiently, the algorithm in ZERNPOL J Ah!#S(  
    % involves pre-determining all the powers p of R that are required to zT,@PIC(  
    % compute the outputs, and then compiling the {R^p} into a single cHFW"g78  
    % matrix.  This avoids any redundant computation of the R^p, and d0I s|Gs  
    % minimizes the sizes of certain intermediate variables. tf6m .  
    % hp'oiR;~w  
    %   Paul Fricker 11/13/2006 C 6 \  
    "73y}'  
    %[*-aA  
    % Check and prepare the inputs: )9yQ C  
    % ----------------------------- +-|""`I1I  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) LQPQ !):;  
        error('zernpol:NMvectors','N and M must be vectors.') "{D/a7]lC  
    end -@w,tbc$  
    ?;W"=I*3  
    if length(n)~=length(m) F7JO/U^oU  
        error('zernpol:NMlength','N and M must be the same length.') ]ouoRlb/  
    end }?Y -I> w  
    c%doNY9Q  
    n = n(:); X.4WVI  
    m = m(:); .2JZ7  
    length_n = length(n); Ljz)%y[s  
    Y0,{fw<  
    if any(mod(n-m,2)) .euA N8L  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') K _VIk'RB  
    end #0<pRDXj  
    Z. gb'  
    if any(m<0) L1RD`qXu.  
        error('zernpol:Mpositive','All M must be positive.') | 9S8sfw  
    end |(%<FY$  
    {"Sv~L|J;  
    if any(m>n) h 8UhrD<:  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') #h=V@Dh  
    end U!|)M  
    uZn_*_J!  
    if any( r>1 | r<0 ) @QmN= X5  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') lF(v<drkB  
    end qA7,txQ:  
    C%yH}T\s  
    if ~any(size(r)==1) TzaeE  
        error('zernpol:Rvector','R must be a vector.') &qXobJRM  
    end AJi+JO-  
    ?Sh]kJ O  
    r = r(:); -r cEG!  
    length_r = length(r); fU$Jh/#":  
    8wkhbD|;  
    if nargin==4 xFp$JN  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); 2K>1,[C'Z  
        if ~isnorm 'Jf LTG.  
            error('zernpol:normalization','Unrecognized normalization flag.') $;Fx Zkp  
        end y60aJ)rAX  
    else J8Wits]A]$  
        isnorm = false; a+LK~mC*  
    end O"~[njwkE  
    dM^EYW  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% yGtTD9j  
    % Compute the Zernike Polynomials |E6_TZ#=  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% }TMO>eB'  
    Fj<*!J$,  
    % Determine the required powers of r: >|%3j,<U  
    % ----------------------------------- 0a:oC(Ak  
    rpowers = []; 'M6+(`x  
    for j = 1:length(n) kB@gy}  
        rpowers = [rpowers m(j):2:n(j)]; r*b+kSh  
    end |Yw k  
    rpowers = unique(rpowers); ddN(L`nd  
    )=GPhC/sw  
    % Pre-compute the values of r raised to the required powers, b(N\R_IQ~  
    % and compile them in a matrix: 7 w,D2T  
    % ----------------------------- 26aDPTP$<  
    if rpowers(1)==0 _(J#RH  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); MUl7o@{'  
        rpowern = cat(2,rpowern{:}); >U* p[FGW  
        rpowern = [ones(length_r,1) rpowern]; | I:@:  
    else _s@PL59,  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); npzp/mcIe)  
        rpowern = cat(2,rpowern{:}); 1#3|PA#>  
    end ')q4d0B`"  
    \ejHM}w3,  
    % Compute the values of the polynomials: 3\}u#/Vb  
    % -------------------------------------- A^).i_&#  
    z = zeros(length_r,length_n); d%k7n+ICQ4  
    for j = 1:length_n >Du=(pB  
        s = 0:(n(j)-m(j))/2; yH" i5L9  
        pows = n(j):-2:m(j); Q SF0?Puf  
        for k = length(s):-1:1 (]cL5o9  
            p = (1-2*mod(s(k),2))* ... KJyCfMH&:@  
                       prod(2:(n(j)-s(k)))/          ... RYCiO,+  
                       prod(2:s(k))/                 ... f$^wu~  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... QBGm)h?=  
                       prod(2:((n(j)+m(j))/2-s(k))); Z4Q]By:/L  
            idx = (pows(k)==rpowers); R /" f  
            z(:,j) = z(:,j) + p*rpowern(:,idx); AHn!>w,  
        end ,*W~M&n"m  
         a6 vej  
        if isnorm G?@W;o)  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); AR( gI]1  
        end F<{,W-my `  
    end :y7K3:d3  
    [c=P)t7 V  
    % EOF zernpol
    离线niuhelen
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    只看该作者 6楼 发表于: 2011-03-12
    这三个文件,我不知道该怎样把我的面型节点的坐标及轴向位移用起来,还烦请指点一下啊,谢谢啦!
    离线li_xin_feng
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    只看该作者 7楼 发表于: 2012-09-28
    我也正在找啊
    离线guapiqlh
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    只看该作者 8楼 发表于: 2014-03-04
    我也一直想了解这个多项式的应用,还没用过呢
    离线phoenixzqy
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    只看该作者 9楼 发表于: 2014-04-22
    回 guapiqlh 的帖子
    guapiqlh:我也一直想了解这个多项式的应用,还没用过呢 (2014-03-04 11:35)  PC@H Nto{  
    pHVDug3  
    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。 H h](n<Bs  
    5[zr(FuE  
    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)