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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 e2P ds`  
    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! :X66[V&eH  
     
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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  3g#  
    function z = zernfun(n,m,r,theta,nflag) \QZ~w_  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. =MsQ=:ZV  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N XEB1%. p  
    %   and angular frequency M, evaluated at positions (R,THETA) on the x9U(,x6r  
    %   unit circle.  N is a vector of positive integers (including 0), and Cd"cU~HAB  
    %   M is a vector with the same number of elements as N.  Each element &azy1.i~  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) lo!.%PP|  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, BSMM3jXb  
    %   and THETA is a vector of angles.  R and THETA must have the same 5g$]ou  
    %   length.  The output Z is a matrix with one column for every (N,M) _!} L\E~  
    %   pair, and one row for every (R,THETA) pair. *?-,=%,z/  
    % 9Sy|:J0  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike |@+/R .l  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 9c}mAg4  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral 5N_w(B  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, z"vI-~,YU  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized 65>1f  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 8vK$]e36  
    % $$tFP"pZ  
    %   The Zernike functions are an orthogonal basis on the unit circle. X>$s>})Y  
    %   They are used in disciplines such as astronomy, optics, and G%RL8HU  
    %   optometry to describe functions on a circular domain. w`Ss MI  
    % zIeJ[J@  
    %   The following table lists the first 15 Zernike functions. nc.(bb),  
    % q9^6A90  
    %       n    m    Zernike function           Normalization 3rUuRsXn  
    %       -------------------------------------------------- .:nV^+)  
    %       0    0    1                                 1 \D<w:\P  
    %       1    1    r * cos(theta)                    2 /ta5d;@  
    %       1   -1    r * sin(theta)                    2 ,*r}23  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) PE\.JU  
    %       2    0    (2*r^2 - 1)                    sqrt(3) uDWxIP,m  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) 3R=R k  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) ?}tWI7KI  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) W|yF jE&dr  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) ALOS>Bi&  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) 'Wv`^{y <^  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) dP7nR1GS  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) r) SG!;X  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) V(5=-8k  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) b;K]; o-/f  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) dHUcu@,  
    %       -------------------------------------------------- cj5; XK  
    % D J:N  
    %   Example 1: %!vgAH4  
    % JR_s-&GaM  
    %       % Display the Zernike function Z(n=5,m=1) N"M?kk,  
    %       x = -1:0.01:1; iem@ K  
    %       [X,Y] = meshgrid(x,x); nz}} m^-j  
    %       [theta,r] = cart2pol(X,Y); 5x} XiMM  
    %       idx = r<=1; H ({Y  
    %       z = nan(size(X)); O7x'q<PFU  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); ET1>&l:.  
    %       figure nB+UxU@  
    %       pcolor(x,x,z), shading interp M< 1rQW'  
    %       axis square, colorbar ~dm/U7B:  
    %       title('Zernike function Z_5^1(r,\theta)') uHNh|ew21  
    % l"ZfgJ}W  
    %   Example 2: ,o{|W9  
    % ="<S1}.  
    %       % Display the first 10 Zernike functions r@.3.Q  
    %       x = -1:0.01:1; | WN9&  
    %       [X,Y] = meshgrid(x,x); yE6EoC^  
    %       [theta,r] = cart2pol(X,Y); YO3$I!(  
    %       idx = r<=1; {Iu9%uR>@  
    %       z = nan(size(X)); ]JUb;B;Z  
    %       n = [0  1  1  2  2  2  3  3  3  3]; EK JPeeRY  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; f]*_]J/  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; @ a$HJ:  
    %       y = zernfun(n,m,r(idx),theta(idx)); ER)<Twj  
    %       figure('Units','normalized') l|Z<pD  
    %       for k = 1:10 <.N33 7!  
    %           z(idx) = y(:,k); eKT'd#o2R  
    %           subplot(4,7,Nplot(k)) O6Gg?j  
    %           pcolor(x,x,z), shading interp 1I1Z),  
    %           set(gca,'XTick',[],'YTick',[]) 6 pQbh*  
    %           axis square \kQ@G  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) =64%eF  
    %       end |9D;2N(&!  
    % zq?Iwyo  
    %   See also ZERNPOL, ZERNFUN2. :AzP3~BI  
    ?#cX_  
    %   Paul Fricker 11/13/2006 uINm>$G,5  
    .AzGPcJY  
    $:aKb#l)  
    % Check and prepare the inputs: >d{O1by=d9  
    % ----------------------------- #G/ _FRo`  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) j+E[ [  
        error('zernfun:NMvectors','N and M must be vectors.') !:7aXT*D$  
    end J6s55 v  
    -H;%1y$A-  
    if length(n)~=length(m) _ 1? PN8  
        error('zernfun:NMlength','N and M must be the same length.') x,3oa_'E  
    end S[_Hc$7U  
    JuD$CHg;#  
    n = n(:); ^&|$&7  
    m = m(:); R8ui LZd  
    if any(mod(n-m,2)) +EnJyli  
        error('zernfun:NMmultiplesof2', ... Q.dHg7+D  
              'All N and M must differ by multiples of 2 (including 0).') QvF UFawN  
    end fV` R7m.  
    k/|j e~$  
    if any(m>n) CL U[')H0  
        error('zernfun:MlessthanN', ... ua'dm6",:  
              'Each M must be less than or equal to its corresponding N.') gkN|3^  
    end dF- d  
    qZ:--,9+  
    if any( r>1 | r<0 ) :<`hsKy&  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') =}G `i**  
    end -i}@o1o\  
    #$qhxYyd  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) /^ d!$v  
        error('zernfun:RTHvector','R and THETA must be vectors.') e?&4;  
    end ),K!| 7#h  
    LAoX'^6  
    r = r(:); - /s2'  
    theta = theta(:); (S@H'G"  
    length_r = length(r); Dyx3N5?C  
    if length_r~=length(theta) CDz-IQi  
        error('zernfun:RTHlength', ... ^<@9ph  
              'The number of R- and THETA-values must be equal.') wN])"bmB  
    end X5@rPGc  
    <.d0GD`^  
    % Check normalization: oXR%A7  
    % -------------------- ,a I0Aw  
    if nargin==5 && ischar(nflag) *d,u)l :S  
        isnorm = strcmpi(nflag,'norm'); xY/ S;dE  
        if ~isnorm *(~=L%s  
            error('zernfun:normalization','Unrecognized normalization flag.') RyGce' q  
        end Mb I';Mq  
    else %rz.>4i)(  
        isnorm = false; YdI|xu>0A^  
    end k^pf)*p  
    ypuW}H%`  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% !5~{?sr>  
    % Compute the Zernike Polynomials 0!n6tz lT  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% !J}Bv  
    _Z:WgO].  
    % Determine the required powers of r: CbJ ]}Z  
    % ----------------------------------- 0<+=Ew5Z  
    m_abs = abs(m); B=:7N;BT  
    rpowers = []; \h%/Cp+p  
    for j = 1:length(n) ~CQYF,[Th  
        rpowers = [rpowers m_abs(j):2:n(j)]; H1,;Xrm  
    end :VPZGzK4  
    rpowers = unique(rpowers); o0>z6Ya<  
    3N) bJ  
    % Pre-compute the values of r raised to the required powers, 0ih=<@1K  
    % and compile them in a matrix: [Hn4&PET  
    % ----------------------------- xQ `>\f  
    if rpowers(1)==0 zkdyfl5  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); N U*6MT4  
        rpowern = cat(2,rpowern{:}); D]9I-|  
        rpowern = [ones(length_r,1) rpowern]; R}Ih~zw  
    else p;dH[NW  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); RZ!-,|"cwL  
        rpowern = cat(2,rpowern{:}); ( Ck|RojC  
    end pJ6Z/3]  
    HR55|`]  
    % Compute the values of the polynomials: .`84Y  
    % -------------------------------------- +'$=\d^  
    y = zeros(length_r,length(n)); 'H<0:bQ=I  
    for j = 1:length(n) .kSx>3  
        s = 0:(n(j)-m_abs(j))/2; igp[cFN  
        pows = n(j):-2:m_abs(j); ocCC63J  
        for k = length(s):-1:1 P1b5=/}:V  
            p = (1-2*mod(s(k),2))* ... **V^8'W<  
                       prod(2:(n(j)-s(k)))/              ... [q/=%8qLUA  
                       prod(2:s(k))/                     ... 3 T$gT  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... dnVl;L8L3  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); 0/d+26lR  
            idx = (pows(k)==rpowers); LL+ROX^M  
            y(:,j) = y(:,j) + p*rpowern(:,idx); )miY>7K  
        end GZ# 6}/;b  
         gG0P &9xz  
        if isnorm Lrgv:n  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi);  T|NNd1>  
        end =dD<[Iz6  
    end vgSs]g  
    % END: Compute the Zernike Polynomials )6#dxb9  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% TC1#2nE&T  
    [ Y_6PR  
    % Compute the Zernike functions: |:SBkM,  
    % ------------------------------ W$7db%qFx  
    idx_pos = m>0; OPR+K ?  
    idx_neg = m<0; jk2h"):B>  
    @.f@N;z  
    z = y; wt4uzg8  
    if any(idx_pos) P g.PD,&U  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)');  H  
    end .7TQae%  
    if any(idx_neg) |ahleu  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 6R V]9  
    end 0x!XE|7I  
    0[@ 9f1Nk4  
    % EOF zernfun
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) S 8mqz.  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. SovK|b &  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated x g~q'>  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive 1J<Wth{  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, r+fR^hv  
    %   and THETA is a vector of angles.  R and THETA must have the same rMIr&T  
    %   length.  The output Z is a matrix with one column for every P-value, bj4cW\b(  
    %   and one row for every (R,THETA) pair. ^& ZlV  
    % uj|{TV>v9  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike 1UX"iO x(  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) y#8| @?  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) 09<O b[%h  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 |LA./%U  
    %   for all p. kD:O$8[J8  
    % XYIZ^_My  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 hko0 ?z  
    %   Zernike functions (order N<=7).  In some disciplines it is Uj&2'>MJ$  
    %   traditional to label the first 36 functions using a single mode Z-;<R$  
    %   number P instead of separate numbers for the order N and azimuthal hD!W&Er  
    %   frequency M. O1IR+"0  
    % PV vNu5k  
    %   Example: 8qL*Nf  
    % UA$ XjP  
    %       % Display the first 16 Zernike functions preKg $U  
    %       x = -1:0.01:1; "M5P-l$p}  
    %       [X,Y] = meshgrid(x,x); Ub=g<MYHV  
    %       [theta,r] = cart2pol(X,Y); b[,J-/;JNL  
    %       idx = r<=1; :RR<-N5+  
    %       p = 0:15; _#xS1sD  
    %       z = nan(size(X)); v'0A$`w`  
    %       y = zernfun2(p,r(idx),theta(idx)); xlG/$`Ab  
    %       figure('Units','normalized') l<qK' P4  
    %       for k = 1:length(p) ^Ts|/+}'i  
    %           z(idx) = y(:,k); T8E=}!68w}  
    %           subplot(4,4,k) kx8\]'  
    %           pcolor(x,x,z), shading interp x'_I{$C &  
    %           set(gca,'XTick',[],'YTick',[]) &1Dq3%$c  
    %           axis square "']I.  
    %           title(['Z_{' num2str(p(k)) '}']) %cMX]U  
    %       end $}&Y$w>S  
    % p x1y#Q  
    %   See also ZERNPOL, ZERNFUN. IMLk{y%6  
     {h/[!I `  
    %   Paul Fricker 11/13/2006 {pMbkA Q@  
     f.acH]p  
    (&R /ns~  
    % Check and prepare the inputs: e9/:q"*)/  
    % ----------------------------- Pn|A>.)z  
    if min(size(p))~=1 4eTfb  
        error('zernfun2:Pvector','Input P must be vector.') eAN]*: ]g  
    end yi*)g0M  
    )*@n G$i99  
    if any(p)>35 !Kr|04Qp#x  
        error('zernfun2:P36', ... 8px@sXI*`  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... dC8 $Ql^<  
               '(P = 0 to 35).']) @qk$ 6X  
    end jY'svD~  
    D@ut -J(.  
    % Get the order and frequency corresonding to the function number: \U $'3M  
    % ---------------------------------------------------------------- ;Z|X` <6g  
    p = p(:); I$!rNfrs  
    n = ceil((-3+sqrt(9+8*p))/2); +-YMW;5  
    m = 2*p - n.*(n+2); ek!x:G$'  
    y<(q<V#0!S  
    % Pass the inputs to the function ZERNFUN: _7r<RZ  
    % ---------------------------------------- Ik2y If5d  
    switch nargin qYFOHu  
        case 3 6lw)L  
            z = zernfun(n,m,r,theta); .lnyn|MVb  
        case 4 u;p.:{'  
            z = zernfun(n,m,r,theta,nflag); FJo  ?~  
        otherwise 7(cRm$)L  
            error('zernfun2:nargin','Incorrect number of inputs.') X!Ag7^E  
    end +{U0PI82  
    F N6 GV  
    % EOF zernfun2
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    function z = zernpol(n,m,r,nflag) Er/h:=  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. 7\x7ySM  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of buV {O[  
    %   order N and frequency M, evaluated at R.  N is a vector of u#(VR]u\7  
    %   positive integers (including 0), and M is a vector with the M)4-eo  
    %   same number of elements as N.  Each element k of M must be a  `{w.OK  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) 2;h4$^`dt  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is q?} /q  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix /)RyRS8c  
    %   with one column for every (N,M) pair, and one row for every vbEAd)*S  
    %   element in R. }j<:hD QP  
    % 8~*<s5H  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- u5D@,wSNz  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is dH:z _$Mg  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to ptJ58U$Bb  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 S -$ L2N  
    %   for all [n,m]. o/0cd  
    % r7B.@+QK  
    %   The radial Zernike polynomials are the radial portion of the a#3+PB #  
    %   Zernike functions, which are an orthogonal basis on the unit &b :u~puM  
    %   circle.  The series representation of the radial Zernike {"{kWbXZ  
    %   polynomials is "&h{+DHS  
    % '-9B`O,&  
    %          (n-m)/2 [+ ,%T;d;  
    %            __ ]q!,onJ  
    %    m      \       s                                          n-2s xvo""R/g8  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r \S #Mc  
    %    n      s=0 581Jp'cje  
    % %L;z~C  
    %   The following table shows the first 12 polynomials. h8 Wv t's  
    % z~Zu >Q1u[  
    %       n    m    Zernike polynomial    Normalization ])`+ 78  
    %       --------------------------------------------- _0HCtx ;  
    %       0    0    1                        sqrt(2) FPAy.cljJ  
    %       1    1    r                           2 rl:6N*kK  
    %       2    0    2*r^2 - 1                sqrt(6) %3p~5jhm1  
    %       2    2    r^2                      sqrt(6) >O0z+tj  
    %       3    1    3*r^3 - 2*r              sqrt(8) @4sEHk 3  
    %       3    3    r^3                      sqrt(8) HJ5 Ktt  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) ;MSdTHN"  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) ~|y$^qy?U  
    %       4    4    r^4                      sqrt(10) ,_SE!iL  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) 8 ckcTNPu  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) vP k\b 3E  
    %       5    5    r^5                      sqrt(12) YoC{ t&rY  
    %       --------------------------------------------- 3K)12x$.K  
    % D1xIRyc/  
    %   Example: :vsBobiJ  
    % 1=U(ZX+u  
    %       % Display three example Zernike radial polynomials 6Bv!t2  
    %       r = 0:0.01:1; pTzwyj!SD  
    %       n = [3 2 5]; s,z$Vt"h*K  
    %       m = [1 2 1]; KImBQ2^Tu  
    %       z = zernpol(n,m,r); w_gFN%8  
    %       figure WA'4y\N  
    %       plot(r,z) !V%h0OE\  
    %       grid on a"ct"g=  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') b\{34z,  
    % QmHj=s:x\  
    %   See also ZERNFUN, ZERNFUN2. $!?tJ@{  
    B7'rbc'  
    % A note on the algorithm. p?@R0]  
    % ------------------------ Ai*R%#  
    % The radial Zernike polynomials are computed using the series HACY  
    % representation shown in the Help section above. For many special s)dL^lj;  
    % functions, direct evaluation using the series representation can jRn5)u  
    % produce poor numerical results (floating point errors), because blVt:XS{,m  
    % the summation often involves computing small differences between ;FQ<4PR$  
    % large successive terms in the series. (In such cases, the functions k U75  
    % are often evaluated using alternative methods such as recurrence Q4;%[7LU  
    % relations: see the Legendre functions, for example). For the Zernike 9` a1xnL  
    % polynomials, however, this problem does not arise, because the N}|<P[LW  
    % polynomials are evaluated over the finite domain r = (0,1), and /JcfAY  
    % because the coefficients for a given polynomial are generally all A'zXbp:%  
    % of similar magnitude. pxGDzU  
    % -(oFO'Lbg  
    % ZERNPOL has been written using a vectorized implementation: multiple t[r<&1[&  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] " Y1]6 Zu  
    % values can be passed as inputs) for a vector of points R.  To achieve k vu SE  
    % this vectorization most efficiently, the algorithm in ZERNPOL ^i"~6QYE  
    % involves pre-determining all the powers p of R that are required to bmid;X|  
    % compute the outputs, and then compiling the {R^p} into a single !^Ly#$-X  
    % matrix.  This avoids any redundant computation of the R^p, and <2.87:  
    % minimizes the sizes of certain intermediate variables. ~10>mg  
    % `] fud{  
    %   Paul Fricker 11/13/2006 Wx8oTN  
    4uX|2nJ2!;  
    B2kKEMdGg  
    % Check and prepare the inputs: w'r?)WW$  
    % ----------------------------- R(^2+mV?  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) HL`=zB%  
        error('zernpol:NMvectors','N and M must be vectors.') H{d;, KfX  
    end Hxr)`i46  
    )%zOq:{\5  
    if length(n)~=length(m) 7u=R5  
        error('zernpol:NMlength','N and M must be the same length.') |T; ]%<O3E  
    end 78MQoG<  
    mVs<XnA47  
    n = n(:); ,N1I\f  
    m = m(:); ! ^ DQX=1  
    length_n = length(n); xHpB/P~  
    ahUc ;S:v#  
    if any(mod(n-m,2)) <i$ud&D  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') 1;E^3j$  
    end 2zPO3xL,  
    fxOa(mt  
    if any(m<0) C$x r)_  
        error('zernpol:Mpositive','All M must be positive.') AE@N:a  
    end +7V4mF!u  
    /QW-#K|S&  
    if any(m>n) \i.Yhl:O  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') ?= R C?K  
    end nYb{?{_ca8  
    q(XO_1W0V  
    if any( r>1 | r<0 ) X+%5q =N  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') /@O$jlX5I  
    end n# 4e1n+I  
    ]n 'FD|  
    if ~any(size(r)==1) xgQ&'&7l  
        error('zernpol:Rvector','R must be a vector.') \2^_v' >K  
    end v?L`aj1ox  
    \s@7pM=(  
    r = r(:); ?.~hex#M@  
    length_r = length(r); y?-zQs0  
    3*C|"|lJ  
    if nargin==4 [B1h0IR  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); Q~-MB]'  
        if ~isnorm w7e+~8|  
            error('zernpol:normalization','Unrecognized normalization flag.') ]w5j?h"b  
        end sKniqWi  
    else KM9)  
        isnorm = false; "V3f"J?  
    end ]m=2 $mK  
    2_C&p6VGj  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @\?QZX(H  
    % Compute the Zernike Polynomials M@a=|N~  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% OaF[t*]D3  
    :YUQKy  
    % Determine the required powers of r: Z[%vO?,  
    % ----------------------------------- !WgVk7aP`  
    rpowers = []; XdV(=PS!a@  
    for j = 1:length(n) 5tUN'KEbN  
        rpowers = [rpowers m(j):2:n(j)]; 2 e9lk$  
    end u d$*/ )/  
    rpowers = unique(rpowers); ~ \3j{pr  
    "bmWr)  
    % Pre-compute the values of r raised to the required powers, S7kZpD $  
    % and compile them in a matrix: %<rV~9:  
    % ----------------------------- "yG*Kh7ur  
    if rpowers(1)==0 ~fz9AhU8  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); NA{?DSP  
        rpowern = cat(2,rpowern{:}); .[:2M9Rx  
        rpowern = [ones(length_r,1) rpowern]; U6Xi-@XP  
    else S&z8-D=8k  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); BW{&A&j  
        rpowern = cat(2,rpowern{:}); h/xV;oj  
    end BWev(SF{Ny  
    k3VRa|Y")  
    % Compute the values of the polynomials: z$b'y;k  
    % -------------------------------------- +et)!2N  
    z = zeros(length_r,length_n); Xd@_:ds  
    for j = 1:length_n 9^2l<4^Z  
        s = 0:(n(j)-m(j))/2; `CqF&b  
        pows = n(j):-2:m(j); v?<Tkw ^F  
        for k = length(s):-1:1 $(N+E,XB  
            p = (1-2*mod(s(k),2))* ... `S:LuU8e  
                       prod(2:(n(j)-s(k)))/          ...  .H7xG'$  
                       prod(2:s(k))/                 ... D]*<J"/]d  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... f@aFs]xV  
                       prod(2:((n(j)+m(j))/2-s(k))); %WO4uOi:@  
            idx = (pows(k)==rpowers); DEN (pA\  
            z(:,j) = z(:,j) + p*rpowern(:,idx); ~(Xzm  
        end Wo, "$Z6B  
         K<~J*k<v  
        if isnorm OqlP_^Zz7p  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); Ek,$XH  
        end Xb\de_8!  
    end uK5 C-  
    x2.YEuSMC  
    % 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)  hd~0qK  
    CjZIBMGc  
    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。 R&FO-{S  
    lOu&4Kq{g  
    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)