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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 nEfK53i_  
    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! AdmC&!nH  
     
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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 Smh,zCc>s  
    function z = zernfun(n,m,r,theta,nflag) 7^Uv7< pw  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. y} '@R$  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N >lm&iF3y  
    %   and angular frequency M, evaluated at positions (R,THETA) on the zCA2X !7F  
    %   unit circle.  N is a vector of positive integers (including 0), and K:M8h{Ua  
    %   M is a vector with the same number of elements as N.  Each element rOYx b }1  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) xo)P?-  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, ]|@^1we  
    %   and THETA is a vector of angles.  R and THETA must have the same <v2;p}A  
    %   length.  The output Z is a matrix with one column for every (N,M) pCDmXB  
    %   pair, and one row for every (R,THETA) pair. _{>vTBU4F  
    % 3q.q YX  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike K"6vXv4QO  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), Mt$ *a  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral TC('H[ ]  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, Sdo-nt  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized V9vTsmo(  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. $qiya[&G4  
    % Sz~OX6L  
    %   The Zernike functions are an orthogonal basis on the unit circle. U:`Kss`  
    %   They are used in disciplines such as astronomy, optics, and ~u{uZ(~  
    %   optometry to describe functions on a circular domain. }bDm@NU  
    % wkq 66?  
    %   The following table lists the first 15 Zernike functions. 965 jtn  
    % |)&%A%m  
    %       n    m    Zernike function           Normalization 4*L_)z&4;  
    %       -------------------------------------------------- l} /F*  
    %       0    0    1                                 1 .`lCWeHN  
    %       1    1    r * cos(theta)                    2 %>yL1BeA4  
    %       1   -1    r * sin(theta)                    2 Gt1U!dP  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) R-:2HRaA  
    %       2    0    (2*r^2 - 1)                    sqrt(3) s7<AfaJPF  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) /z!%d%"  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) Dv"9qk  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) qM`}{ /i  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) [ 3Gf2_  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) 7v kL1IA  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) 0[`^\Mv4y  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) _#niyW+?~  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) 0@(&eH=  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) |>Vb9:q9Po  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) $ `c:&  
    %       -------------------------------------------------- j.Hf/vi`z  
    % hM{bavd  
    %   Example 1: p?!/+  
    % Z r8*et  
    %       % Display the Zernike function Z(n=5,m=1) f 2.HF@  
    %       x = -1:0.01:1; P?\6@_ Z  
    %       [X,Y] = meshgrid(x,x); M7T5 ~/4  
    %       [theta,r] = cart2pol(X,Y); /(cPfZZ  
    %       idx = r<=1; pkzaNY/q  
    %       z = nan(size(X)); zdYjF|  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); :]KAkhFkbb  
    %       figure |N2#ItBbW  
    %       pcolor(x,x,z), shading interp +R&gqja  
    %       axis square, colorbar uph(V  
    %       title('Zernike function Z_5^1(r,\theta)') #4PN"o@  
    % 6'/ #+,d'  
    %   Example 2: khe}*y  
    % NOva'qk  
    %       % Display the first 10 Zernike functions gJXaPJA{  
    %       x = -1:0.01:1; DI>s-7  
    %       [X,Y] = meshgrid(x,x); 29Ki uP  
    %       [theta,r] = cart2pol(X,Y); 0;k# *#w  
    %       idx = r<=1; cr3^6HB  
    %       z = nan(size(X)); py4 h(04u  
    %       n = [0  1  1  2  2  2  3  3  3  3]; WcAkCH!L  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; b;n[mk  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; xp t:BBo  
    %       y = zernfun(n,m,r(idx),theta(idx)); CrLrw T  
    %       figure('Units','normalized') HtFDlvdy]  
    %       for k = 1:10 [WmM6UEVS  
    %           z(idx) = y(:,k); ;+%rw2Z,B  
    %           subplot(4,7,Nplot(k)) #mF"1QW  
    %           pcolor(x,x,z), shading interp l **X^+=$  
    %           set(gca,'XTick',[],'YTick',[]) C Z;6@{ o  
    %           axis square   ep8  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) CTb%(<r  
    %       end 5O% {{J  
    % aUp g u"  
    %   See also ZERNPOL, ZERNFUN2. A"]YM'.  
    Psf#c:*_)  
    %   Paul Fricker 11/13/2006 @dK Tx#gZ  
    GOPfXtkC  
    vaLSH xi  
    % Check and prepare the inputs: 7dWS  
    % ----------------------------- ]Um/FAW  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 9w"*y#_  
        error('zernfun:NMvectors','N and M must be vectors.') #"!<W0  
    end %EH)&k  
    h{Y",7] !  
    if length(n)~=length(m) ]kSGR  
        error('zernfun:NMlength','N and M must be the same length.') .Mbz3;i0  
    end vP&(-a  
    b}`T Ln  
    n = n(:); 7#XzrT]  
    m = m(:); )`:UP~)H  
    if any(mod(n-m,2))  ?9/G[[(  
        error('zernfun:NMmultiplesof2', ... c{|p.hd  
              'All N and M must differ by multiples of 2 (including 0).') %J(:ADu]  
    end e ,(mR+a8  
    _>+Ld6.T6  
    if any(m>n) T)/eeZ$  
        error('zernfun:MlessthanN', ... -n 1 v3  
              'Each M must be less than or equal to its corresponding N.') V gWRW7Se  
    end @"A4$`Xi3  
    iS^QTuk3%  
    if any( r>1 | r<0 ) C dn J&N{  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') o!Zb0/AP)  
    end )nkY_' BV  
    ^qs $v06  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) SUiOJ[5,  
        error('zernfun:RTHvector','R and THETA must be vectors.') D*jM1w_`  
    end S jqpec8  
    oA 1yIp  
    r = r(:); /uc>@!F  
    theta = theta(:); I7onX,U+  
    length_r = length(r); {: /}NpA$  
    if length_r~=length(theta) X'ag)|5ot  
        error('zernfun:RTHlength', ... $Sq:q0  
              'The number of R- and THETA-values must be equal.') |yCMt:Hk  
    end *4'"2"  
    J.a]K[ci  
    % Check normalization: :WEDAFq0  
    % -------------------- 5pX6t  
    if nargin==5 && ischar(nflag) _BufO7 `.  
        isnorm = strcmpi(nflag,'norm'); 5BIY<B+i  
        if ~isnorm rq{$,/6.  
            error('zernfun:normalization','Unrecognized normalization flag.') 5P2K5,o|n~  
        end 6ujW Nf  
    else X|dlt{Gf   
        isnorm = false; vx =&QavL  
    end 2 ?C)&  
    203 s^K 61  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0GwR~Z}Z  
    % Compute the Zernike Polynomials 8*X4\3:*N  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% KI.unP%  
    0GLM(JmK  
    % Determine the required powers of r: ".%k6W<n  
    % ----------------------------------- WJi]t93  
    m_abs = abs(m); >P(.:_ ^p  
    rpowers = []; HS$r8`S?)  
    for j = 1:length(n) C!gZN9-  
        rpowers = [rpowers m_abs(j):2:n(j)]; i8p6Xht  
    end gXU8hTd8  
    rpowers = unique(rpowers); +`4A$#$+y  
    6Wn1{v0  
    % Pre-compute the values of r raised to the required powers, +@UV?"d  
    % and compile them in a matrix: k6^Z~5 Sy  
    % ----------------------------- Z+SRXKQ  
    if rpowers(1)==0 hH.G#-JO  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); P?<y%c<  
        rpowern = cat(2,rpowern{:}); 'u658Tj  
        rpowern = [ones(length_r,1) rpowern]; [g,}gyeS(  
    else c-w)|-ac.  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); #yen8SskB  
        rpowern = cat(2,rpowern{:}); @EAbF>>  
    end qs6aB0ln  
    f$( e\+ +  
    % Compute the values of the polynomials: ooGM$U  
    % -------------------------------------- xw%0>K[  
    y = zeros(length_r,length(n)); kfNWI#'9  
    for j = 1:length(n) 2oW"'43X  
        s = 0:(n(j)-m_abs(j))/2; d9ihhqq3}  
        pows = n(j):-2:m_abs(j); fA-7VdR`R  
        for k = length(s):-1:1 zs;JJk^  
            p = (1-2*mod(s(k),2))* ... PF2nLb2-  
                       prod(2:(n(j)-s(k)))/              ... *hrd5na  
                       prod(2:s(k))/                     ... 1YA% -~  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... IV-{ve6  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); X&zis1A<  
            idx = (pows(k)==rpowers); g0H[*"hj  
            y(:,j) = y(:,j) + p*rpowern(:,idx); $]1=\ I  
        end G3]4A&h9v~  
         0(I j%Wi,  
        if isnorm 6@o*xK7L  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); oU|c.mYe  
        end b6[j%(   
    end V~bD)?M  
    % END: Compute the Zernike Polynomials e!`i3KYn"  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |{;G2G1[  
    )"LJ hLg  
    % Compute the Zernike functions: g}i61(  
    % ------------------------------ $( )>g>%  
    idx_pos = m>0; v=k$A  
    idx_neg = m<0; ;4a{$Lw~^9  
    mmsPLv6  
    z = y; l2d{ 73h  
    if any(idx_pos) MDN--p08  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); Q\)F;:|  
    end _|p8M!  
    if any(idx_neg) *I'yH8Fcn  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); h![#;>(  
    end +"(jjxJm  
    uEY tE7  
    % 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) R3! t$5HG  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. 1cGmg1U;  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated ~Z+%d9ode  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive $N\Ja*g  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, 9cgU T@a  
    %   and THETA is a vector of angles.  R and THETA must have the same 2%> FR4a  
    %   length.  The output Z is a matrix with one column for every P-value, C7vxw-o|&p  
    %   and one row for every (R,THETA) pair. Tr|JYLwF  
    % P$sxr  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike X|[`P<'N<  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) 8_tQa^.n\  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) ]~%6JJN7  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 2(nlJ7R  
    %   for all p. I|J/F}@p  
    % OH"XrCX7n  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 {U1m.30n  
    %   Zernike functions (order N<=7).  In some disciplines it is w:l"\Tm  
    %   traditional to label the first 36 functions using a single mode s7EinI{^  
    %   number P instead of separate numbers for the order N and azimuthal TKjFp%  
    %   frequency M. @H<q"-J  
    % <X5 fUU"+U  
    %   Example: <1 pEwI~  
    % KF/-wZ"1s  
    %       % Display the first 16 Zernike functions ?}7p"3j'z  
    %       x = -1:0.01:1; KU;9}!#  
    %       [X,Y] = meshgrid(x,x); �{x7,  
    %       [theta,r] = cart2pol(X,Y); gJhiGYx  
    %       idx = r<=1; 875od  
    %       p = 0:15; 1sCR4L:+  
    %       z = nan(size(X)); y?0nI<}}HK  
    %       y = zernfun2(p,r(idx),theta(idx)); b[7 ]F  
    %       figure('Units','normalized') 8X0z~ &  
    %       for k = 1:length(p) 'n|5ZhXPB  
    %           z(idx) = y(:,k); ^t"'rD-I  
    %           subplot(4,4,k) uGt-l4  
    %           pcolor(x,x,z), shading interp Sc   
    %           set(gca,'XTick',[],'YTick',[]) Tf)*4O4@'  
    %           axis square o Rzi>rr  
    %           title(['Z_{' num2str(p(k)) '}']) oE~Bq/p  
    %       end 5-G@L?~Vw  
    % pNIf=lA  
    %   See also ZERNPOL, ZERNFUN. yEoV[K8k  
    2"5v[,$1H  
    %   Paul Fricker 11/13/2006 ty`DJO=Omj  
    g1o8._f.  
    NCx%L-GPi  
    % Check and prepare the inputs: H.2QKws^F  
    % ----------------------------- 0RK!/:'  
    if min(size(p))~=1 Z)\@i=m  
        error('zernfun2:Pvector','Input P must be vector.') T^v}mWCZ  
    end  *,m;  
    ERt{H3eCcJ  
    if any(p)>35 E!#WnSpnK  
        error('zernfun2:P36', ... ]tDDq=+v  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... k'"%.7$U!  
               '(P = 0 to 35).']) %#}Zy   
    end _l]fkk[T  
    qv*^fiT  
    % Get the order and frequency corresonding to the function number: mQ=#nk$~g  
    % ---------------------------------------------------------------- * H9 8Du  
    p = p(:); `p7=t)5k  
    n = ceil((-3+sqrt(9+8*p))/2); 39|MX21k  
    m = 2*p - n.*(n+2); )Beiu*  
    kxRV )G  
    % Pass the inputs to the function ZERNFUN: &w~d_</  
    % ---------------------------------------- ukY"+&  
    switch nargin +U.I( 83F  
        case 3 "Yca%:  
            z = zernfun(n,m,r,theta); l\?c}7k  
        case 4 OC:T O|S:4  
            z = zernfun(n,m,r,theta,nflag); |&[EZ+[  
        otherwise 3{h_&Gbo'D  
            error('zernfun2:nargin','Incorrect number of inputs.') ,u g@f-T  
    end 2>H24F  
    : \}(& >  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) 4 N7^?  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M.  _\HQvH  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of :Xd<74Nu  
    %   order N and frequency M, evaluated at R.  N is a vector of t!\tF[9e  
    %   positive integers (including 0), and M is a vector with the IyPnp&_  
    %   same number of elements as N.  Each element k of M must be a >6pf$0  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) K!]/(V(}  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is jMDY(mwt  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix ,-e{(L  
    %   with one column for every (N,M) pair, and one row for every |id <=Xf  
    %   element in R. {$Gd2g O  
    % 9 5RBO4w%w  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- O s.4)  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is ]}(H0?OQR  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to E\2%E@0#  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 @k/NY *+  
    %   for all [n,m]. $"&{aa  
    % 7 ^mL_SMj  
    %   The radial Zernike polynomials are the radial portion of the [\b 0Lem  
    %   Zernike functions, which are an orthogonal basis on the unit `I5wV/%ib  
    %   circle.  The series representation of the radial Zernike [=^3n#WW  
    %   polynomials is oF GhNk  
    % 6qd\)q6T&x  
    %          (n-m)/2 fe#\TNeQJ[  
    %            __ rI-%be==  
    %    m      \       s                                          n-2s mcX/GO}  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r om-omo&,X=  
    %    n      s=0 m<qJcZk  
    % {3{"8-18  
    %   The following table shows the first 12 polynomials. xLZG:^(I  
    % 1\rz%E  
    %       n    m    Zernike polynomial    Normalization 7;(UF=4  
    %       --------------------------------------------- JO"<{ngsQ  
    %       0    0    1                        sqrt(2) {LQ#y/H?  
    %       1    1    r                           2 v+=BCyT  
    %       2    0    2*r^2 - 1                sqrt(6) Uwx E<=z  
    %       2    2    r^2                      sqrt(6) 'D"C4;X  
    %       3    1    3*r^3 - 2*r              sqrt(8) \K]0JH  
    %       3    3    r^3                      sqrt(8) [o5Hl^  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) ~B(4qK1G  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) %O;bAC_M  
    %       4    4    r^4                      sqrt(10) df#$ 9 -  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) -701j'q{  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) o"BoZsMk  
    %       5    5    r^5                      sqrt(12) {9aE5kR  
    %       --------------------------------------------- Y6L ~K?  
    % <)-Sj,  
    %   Example: (%W&4a1di  
    % 8rS:5:Hi  
    %       % Display three example Zernike radial polynomials e?ly H  
    %       r = 0:0.01:1; ?r2` Q  
    %       n = [3 2 5]; *6F[t.Or  
    %       m = [1 2 1]; Eq\M;aDq  
    %       z = zernpol(n,m,r); T+K):u g  
    %       figure E5lBdM>2  
    %       plot(r,z) !*. -`$x  
    %       grid on t#pS{.I  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') dg"3rs /?A  
    % Zt.|oYH$  
    %   See also ZERNFUN, ZERNFUN2. FfPar:PHj  
    6N S201o  
    % A note on the algorithm. -f>%+<k=  
    % ------------------------ >R! jB]5  
    % The radial Zernike polynomials are computed using the series //<nr\oP  
    % representation shown in the Help section above. For many special ,.1Psz^U  
    % functions, direct evaluation using the series representation can QR0Q{}wbqU  
    % produce poor numerical results (floating point errors), because iBgx  
    % the summation often involves computing small differences between CxG#"{&  
    % large successive terms in the series. (In such cases, the functions % pd,%pg  
    % are often evaluated using alternative methods such as recurrence f-n1I^|  
    % relations: see the Legendre functions, for example). For the Zernike  K;z7/[%  
    % polynomials, however, this problem does not arise, because the 364`IC( a  
    % polynomials are evaluated over the finite domain r = (0,1), and os={PQRD  
    % because the coefficients for a given polynomial are generally all iv;Is[<o  
    % of similar magnitude. }n2M G  
    % m~d]a$KQ5-  
    % ZERNPOL has been written using a vectorized implementation: multiple EbE-}>7OO  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] B1C-J/J  
    % values can be passed as inputs) for a vector of points R.  To achieve usCt#eZK  
    % this vectorization most efficiently, the algorithm in ZERNPOL s<eb;Z2D  
    % involves pre-determining all the powers p of R that are required to {U m)15K  
    % compute the outputs, and then compiling the {R^p} into a single 4 f'V8|QM{  
    % matrix.  This avoids any redundant computation of the R^p, and lqZ5?BD1  
    % minimizes the sizes of certain intermediate variables. 5}]"OXQ  
    %  jQ  
    %   Paul Fricker 11/13/2006 7Vo$(kj  
    ?D*/*Gk{  
    ~%=MpQ3  
    % Check and prepare the inputs: &NoS=(s,  
    % ----------------------------- >kp?vK;'B  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) (ECnM ti+  
        error('zernpol:NMvectors','N and M must be vectors.') ;n=.>s*XL'  
    end C3],n   
    J| bd)0  
    if length(n)~=length(m) $#S&QHyEe  
        error('zernpol:NMlength','N and M must be the same length.') Xudg2t)+K  
    end _FVcx7l!u  
    &6YIn|}  
    n = n(:); TQ*1L:X7M&  
    m = m(:); uPG4V2  
    length_n = length(n); A?%H=>v$  
    lWc:$qnR-K  
    if any(mod(n-m,2)) E}p&2P+MR  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') s<_)$}  
    end W 7\f1}]H  
    +hT:2TXn  
    if any(m<0) M#VE]J  
        error('zernpol:Mpositive','All M must be positive.') @EpIh&  
    end Q/_f zg  
    6%Pdy$ P  
    if any(m>n) n3Z 5t  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') - 'W++tH=  
    end s4SG[w!d  
    R0vIbFwj  
    if any( r>1 | r<0 ) `[)YEg s  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') #Xb+`'  
    end \r,Q1n?7  
    S=nzw-(I  
    if ~any(size(r)==1) hKjt'N:~ZY  
        error('zernpol:Rvector','R must be a vector.')  Q&g^c2  
    end MLWM&cFG  
    #=f?0UTA  
    r = r(:); 5sJJGv#6  
    length_r = length(r); &twf,8  
    xp72>*_9&  
    if nargin==4 `gb5 "`EZ  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); }J1tdko#  
        if ~isnorm jvFTR'R)=  
            error('zernpol:normalization','Unrecognized normalization flag.') NchXt6$i9  
        end (+3Wgl+]/  
    else A"D,Kg S  
        isnorm = false; .!,z:l$Kh  
    end :Q_<Z@2Y{  
    #KXa&C  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% >W`4aA  
    % Compute the Zernike Polynomials *"n vX2iz  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% "7V2lu  
    ;Tc`}2  
    % Determine the required powers of r: [P7N{l=I  
    % ----------------------------------- <-S%kA8  
    rpowers = []; cwWodPNm  
    for j = 1:length(n) p2udm!)J  
        rpowers = [rpowers m(j):2:n(j)]; !PJ6%"  
    end 5qoSEI-m  
    rpowers = unique(rpowers); Zx  bq  
    WRDjh7~Efn  
    % Pre-compute the values of r raised to the required powers, 88h3|'*  
    % and compile them in a matrix: Qx47l  
    % ----------------------------- LLXVNO@e+  
    if rpowers(1)==0 ehG/zVgn  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); q]N:Tpm9  
        rpowern = cat(2,rpowern{:}); C[Dav&=^F  
        rpowern = [ones(length_r,1) rpowern]; x,S P'fcP  
    else ) ^3avRsC  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); hQHnwr  
        rpowern = cat(2,rpowern{:}); _b.qkTWUB  
    end <_Q:'cx'  
    A\#P*+k0  
    % Compute the values of the polynomials: ]U7KLUY>:  
    % -------------------------------------- /3:q#2'v  
    z = zeros(length_r,length_n); mJ`A_0  
    for j = 1:length_n 'hv k  
        s = 0:(n(j)-m(j))/2; ~Oq +IA~9  
        pows = n(j):-2:m(j); *`Yv.=cd  
        for k = length(s):-1:1 mL`5u f  
            p = (1-2*mod(s(k),2))* ... `zt_7MD  
                       prod(2:(n(j)-s(k)))/          ... z,:a8LB#[  
                       prod(2:s(k))/                 ... Y.U[wL>  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... 0`A~HH}  
                       prod(2:((n(j)+m(j))/2-s(k))); ZwerDkd  
            idx = (pows(k)==rpowers); UaViI/ks  
            z(:,j) = z(:,j) + p*rpowern(:,idx); $aPfGZ<i  
        end _#}n~}d  
         F. =Bnw/-  
        if isnorm 9Xo[(h)5d  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); *[R eb %  
        end V{&rQ@{W  
    end Css l{B  
    dVo.Czyd  
    % 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)  >AI<60/<  
    ad`_>lA4Lp  
    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"+dTq8W  
    u=qPzmywt  
    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)