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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 {ZSAPq4)L  
    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! 7XT(n v  
     
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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 2Mq@5n  
    function z = zernfun(n,m,r,theta,nflag) z>0$SBQ-  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. =bP<cC=3b  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N rNicg]:\x  
    %   and angular frequency M, evaluated at positions (R,THETA) on the **z^aH?B2  
    %   unit circle.  N is a vector of positive integers (including 0), and ^fsC]9NS  
    %   M is a vector with the same number of elements as N.  Each element 6:8Nz   
    %   k of M must be a positive integer, with possible values M(k) = -N(k) DF-PBVfpu  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, x!W5'DO  
    %   and THETA is a vector of angles.  R and THETA must have the same G9xO>Xp^Al  
    %   length.  The output Z is a matrix with one column for every (N,M) Het>G{  
    %   pair, and one row for every (R,THETA) pair. 6Y6t.j0vN.  
    % y xT}hMa  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike p ^TCr<=  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), J#j3?qrxu  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral 9bRUN<  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, \(=xc2  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized 8[t*VIXI  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. jA2%kX\6//  
    % ge%QbU1J  
    %   The Zernike functions are an orthogonal basis on the unit circle. DzA'MX  
    %   They are used in disciplines such as astronomy, optics, and 8 l= EL7  
    %   optometry to describe functions on a circular domain. T*Ge67  
    % A.7lo  
    %   The following table lists the first 15 Zernike functions. })kx#_o]'d  
    % GV ) "[O  
    %       n    m    Zernike function           Normalization =_3rc\0  
    %       -------------------------------------------------- p/u  
    %       0    0    1                                 1  )h>dD  
    %       1    1    r * cos(theta)                    2 yKK9b  
    %       1   -1    r * sin(theta)                    2 0*kS\R=P  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6)  !a\HdQ  
    %       2    0    (2*r^2 - 1)                    sqrt(3) }X=c|]6i^  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) Voq/0,d  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) H/Ql  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) y=+OC1k\8  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) 0t"Iq71/  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) B]b/(Q+  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) 9mn~57`y  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) f-H"|9  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) =+?OsH v  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) -vc$I=b;  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) &;r'JIp  
    %       -------------------------------------------------- LH @B\ mS  
    % m :~y:.  
    %   Example 1: 7F]Hq  
    % ZdY$NpR,  
    %       % Display the Zernike function Z(n=5,m=1) _\,lv \u  
    %       x = -1:0.01:1; 8KkN "4'  
    %       [X,Y] = meshgrid(x,x); v+trHdSBYE  
    %       [theta,r] = cart2pol(X,Y); `D=d!!1eUi  
    %       idx = r<=1; l= Jw6F+5  
    %       z = nan(size(X)); (Uu5$q(  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); R47y/HG,  
    %       figure lx2%=5+i;  
    %       pcolor(x,x,z), shading interp =oiz@Q@H  
    %       axis square, colorbar T*C F5S  
    %       title('Zernike function Z_5^1(r,\theta)') 5&_")k3$*  
    % r| \""  
    %   Example 2: pXBh^  
    % 0Krh35R_)F  
    %       % Display the first 10 Zernike functions eLgq )  
    %       x = -1:0.01:1; (~5]1S}F  
    %       [X,Y] = meshgrid(x,x); 0Y0`$   
    %       [theta,r] = cart2pol(X,Y); X&rsWk  
    %       idx = r<=1; MF*4E9Ue.  
    %       z = nan(size(X)); d ( ru5*p  
    %       n = [0  1  1  2  2  2  3  3  3  3]; 9H:J&'Xi7  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; "H@I~X=  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; 0yMHU[):~  
    %       y = zernfun(n,m,r(idx),theta(idx)); i-p,x0th  
    %       figure('Units','normalized') ZWjje6  
    %       for k = 1:10 Bf+~&I#E  
    %           z(idx) = y(:,k); M$>Nd6,@N  
    %           subplot(4,7,Nplot(k)) '^7UcgugB  
    %           pcolor(x,x,z), shading interp X_bB6A6  
    %           set(gca,'XTick',[],'YTick',[]) KyP@ hhj  
    %           axis square vo)W ziHh  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) Lc]hwMGR*  
    %       end ;p <BiC$b  
    % oOubqx  
    %   See also ZERNPOL, ZERNFUN2. JX&%5sn(  
    ePaC8sd0  
    %   Paul Fricker 11/13/2006 <pKOFN%m  
    1;{nU.If  
    G-]<+-Q$4  
    % Check and prepare the inputs: Nr)DU.f  
    % ----------------------------- %Q.M& U  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) u$ci{<  
        error('zernfun:NMvectors','N and M must be vectors.') P%Q'w  
    end 1~2+w]-kU  
    2,Z@<  
    if length(n)~=length(m) 5 /oW/2"  
        error('zernfun:NMlength','N and M must be the same length.') `qCL&(`%  
    end RX^8`}N  
    Q.V@Sawe5  
    n = n(:); k!$$ *a*  
    m = m(:); E(1G!uu<  
    if any(mod(n-m,2)) =eDC{/K  
        error('zernfun:NMmultiplesof2', ... 0HbCT3g.  
              'All N and M must differ by multiples of 2 (including 0).') 'iwTvkf{  
    end Ytqx 0  
    a%6=sqxE  
    if any(m>n) n<b}6L}  
        error('zernfun:MlessthanN', ... {3K ]Q=  
              'Each M must be less than or equal to its corresponding N.') 3G^A^]h  
    end ma) + G!  
    _Vt9ckaA  
    if any( r>1 | r<0 ) f8f3[O!x  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') }"%mP 4]&  
    end gF2 93Ez  
    d#ab"&$bv  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) E6 T=lwOZ  
        error('zernfun:RTHvector','R and THETA must be vectors.') ^Mhh2v  
    end L/GV Qjb  
    P-yVc2YH  
    r = r(:); !Zc#E,  
    theta = theta(:); -sDl[  
    length_r = length(r); GH3RRzp r  
    if length_r~=length(theta) ka(3ONbG  
        error('zernfun:RTHlength', ... Y(T$k9%}+  
              'The number of R- and THETA-values must be equal.') ?LvU7  
    end 5s4x%L (~}  
    M A%g-}  
    % Check normalization: H xc>?  
    % -------------------- q8GCO\(  
    if nargin==5 && ischar(nflag) &=T>($3r94  
        isnorm = strcmpi(nflag,'norm'); @cx#'  
        if ~isnorm W!=ur,F+  
            error('zernfun:normalization','Unrecognized normalization flag.') fti0Tz'  
        end K 4{[s z  
    else OP_\V8=  
        isnorm = false; o(D_ /]'8  
    end Pe11a zJ  
    {D,- Whi  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% W~l.feW$i  
    % Compute the Zernike Polynomials Go]y{9+(7  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% l6MBnvi   
    .~^A!t  
    % Determine the required powers of r: 1NrNTBI@  
    % ----------------------------------- u,`V%J?vW  
    m_abs = abs(m); 4Y G\<Zf  
    rpowers = []; 6aWnj*dF  
    for j = 1:length(n) bpDlFa  
        rpowers = [rpowers m_abs(j):2:n(j)]; \"5p )(  
    end lm+s5}*%o  
    rpowers = unique(rpowers); M3JV^{O/DV  
    ,d^HAg^j  
    % Pre-compute the values of r raised to the required powers, )hVn/*mH  
    % and compile them in a matrix: onv0gb/J  
    % ----------------------------- 9%MgAik(  
    if rpowers(1)==0 DoICf1  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); QV#HN"F/K  
        rpowern = cat(2,rpowern{:}); $HRl:KDdP~  
        rpowern = [ones(length_r,1) rpowern]; T=g2gmo9  
    else 5pff}Ru`  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 8ddBQfCY  
        rpowern = cat(2,rpowern{:}); Y%zWaH  
    end Y|KT3  
     Tx'anP  
    % Compute the values of the polynomials: &$~irI  
    % -------------------------------------- ^7*zi_Q  
    y = zeros(length_r,length(n)); Tj6Czq=*%T  
    for j = 1:length(n) {817Svp@  
        s = 0:(n(j)-m_abs(j))/2; B_3N:K Y 9  
        pows = n(j):-2:m_abs(j); ]x'd0GH"]  
        for k = length(s):-1:1 DTdqwe6pi  
            p = (1-2*mod(s(k),2))* ... <e@4;Z(h04  
                       prod(2:(n(j)-s(k)))/              ... I%z,s{9p  
                       prod(2:s(k))/                     ... Z:,`hW*A6  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... (7??5gjh  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); R|*Eg,1g -  
            idx = (pows(k)==rpowers); =&: |a$C  
            y(:,j) = y(:,j) + p*rpowern(:,idx); B,ao%3t  
        end %w/vKB"nO  
         v++&%  
        if isnorm 5n e&6  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); nHLMF7\  
        end Q>G% *?  
    end JE eXoGKd  
    % END: Compute the Zernike Polynomials vI"BNC*Q1  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% z6Nz)$!_i  
    mWMtz]M}  
    % Compute the Zernike functions: "|E'E"_1  
    % ------------------------------ +'[/eW  
    idx_pos = m>0; iBY16_q  
    idx_neg = m<0; hN\Q&F!  
    VLbbn  
    z = y; .k,,PuP  
    if any(idx_pos) [z'jL'\4  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); B@8lD\  
    end ~bw=;xF{3  
    if any(idx_neg) /.t1Ow  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Y/L*0 M.<  
    end EO/41O  
    {s:"mkR  
    % 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) Q5Epq sKyC  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. *rYPjk6g[  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated EZZE(dq@gf  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive \0FwxsL  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, ]VS:5kOj`  
    %   and THETA is a vector of angles.  R and THETA must have the same rrG}; A  
    %   length.  The output Z is a matrix with one column for every P-value, `4ti?^BNm  
    %   and one row for every (R,THETA) pair. ({cWb:+r  
    % Us%g&MWdpb  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike PlwM3lrj  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) i*T -9IP  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) 60WlC0Y~u  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 Fv:x>qZr@  
    %   for all p. NIp]n[ =.q  
    % tY#Zl 54~{  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 yqP=6   
    %   Zernike functions (order N<=7).  In some disciplines it is G\~?.s|^  
    %   traditional to label the first 36 functions using a single mode 6lUC$B Y  
    %   number P instead of separate numbers for the order N and azimuthal ~m[Gp;pL  
    %   frequency M. |fo#pwX  
    % ``>z8t[ks  
    %   Example: C(M?$s`  
    % 3jHE,5m  
    %       % Display the first 16 Zernike functions 7R,;/3wWjG  
    %       x = -1:0.01:1; #oN}DP  
    %       [X,Y] = meshgrid(x,x); qI<c47d;q  
    %       [theta,r] = cart2pol(X,Y); bEmzigN[  
    %       idx = r<=1; .0MY$0s  
    %       p = 0:15; #8y"1I=i&  
    %       z = nan(size(X)); JkKbw&65  
    %       y = zernfun2(p,r(idx),theta(idx)); E H|L1g  
    %       figure('Units','normalized') ^ ~dC&!D  
    %       for k = 1:length(p) VH vL:z  
    %           z(idx) = y(:,k); xE!b)@>S  
    %           subplot(4,4,k) -C* 6>$A  
    %           pcolor(x,x,z), shading interp L^K,YlNBR  
    %           set(gca,'XTick',[],'YTick',[]) D Q c pIV  
    %           axis square ant2];0p  
    %           title(['Z_{' num2str(p(k)) '}']) 5f2=`C0_  
    %       end "Jdi>{o8  
    % K>n@8<7  
    %   See also ZERNPOL, ZERNFUN. ^AERGB\36  
    ^oNcZK>  
    %   Paul Fricker 11/13/2006 +Eel|)Z*Q  
    Y' 5X4Ks|  
    RMdU1@  
    % Check and prepare the inputs: &-m}w:j=  
    % ----------------------------- ,bP8"|e  
    if min(size(p))~=1 *e:2iM)8~  
        error('zernfun2:Pvector','Input P must be vector.') ?8;WP&  
    end ?yu@eo  
    fUPYCw6F  
    if any(p)>35 Dn#UcMO>W  
        error('zernfun2:P36', ... -#R63f&  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... ;vn0b"Fi3  
               '(P = 0 to 35).']) 12: Q`   
    end  `YO&  
    @q{.  
    % Get the order and frequency corresonding to the function number: Qh* }v!3Jo  
    % ---------------------------------------------------------------- 5xU}}[|~-  
    p = p(:); >ul&x!?@  
    n = ceil((-3+sqrt(9+8*p))/2); J/PK #<  
    m = 2*p - n.*(n+2); XinKG< 3!  
    vFeR)Ox's  
    % Pass the inputs to the function ZERNFUN: 9E|QPT  
    % ---------------------------------------- L=P8;Gj)  
    switch nargin ^==Tv+T9U  
        case 3 %Z8pPH~T  
            z = zernfun(n,m,r,theta); ?v'CuWS  
        case 4 C IRMAX  
            z = zernfun(n,m,r,theta,nflag); IoV"t,  
        otherwise 5glEV`.je  
            error('zernfun2:nargin','Incorrect number of inputs.') e)3Mg^  
    end ;ZkY[5  
    4w]<1V  
    % 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) XjYMp3  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. !~ j9Oc^  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of 0BAZWm  
    %   order N and frequency M, evaluated at R.  N is a vector of [FBc&HN  
    %   positive integers (including 0), and M is a vector with the Wu l8ej:  
    %   same number of elements as N.  Each element k of M must be a ucbtPTFYvr  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) |C,]-mJG  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is %:dd#';g  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix T>&dPVmG,  
    %   with one column for every (N,M) pair, and one row for every A.YK=_J  
    %   element in R. )ub!tm  
    % vi[~Qt  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- j-qg{oIJ  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is {yi!vw  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to >z,Y%A  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 Upm#:i|"  
    %   for all [n,m]. !L_xcov!Y  
    % #}8VUbJ  
    %   The radial Zernike polynomials are the radial portion of the YYvX@f  
    %   Zernike functions, which are an orthogonal basis on the unit |@?='E?h  
    %   circle.  The series representation of the radial Zernike "'>fTk_  
    %   polynomials is :73T9/  
    % dLf ;g}W  
    %          (n-m)/2 r 2{7h>  
    %            __ `G>|g^6%i  
    %    m      \       s                                          n-2s ")fgQ3XZ  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r kjSzu qB  
    %    n      s=0 NCm=l  
    % 4&6cDig7*2  
    %   The following table shows the first 12 polynomials. `2xt%kC  
    % >as+#rz1p  
    %       n    m    Zernike polynomial    Normalization 5Iv"  
    %       --------------------------------------------- ADMeOdgca  
    %       0    0    1                        sqrt(2) %H}M[_f  
    %       1    1    r                           2 w}29#F\]R  
    %       2    0    2*r^2 - 1                sqrt(6) bf\ Uq<&IJ  
    %       2    2    r^2                      sqrt(6) 9g$fFO  
    %       3    1    3*r^3 - 2*r              sqrt(8) U:$`M,762Z  
    %       3    3    r^3                      sqrt(8) ~{6}SXp4U  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) h@s i)5"  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) 9,}Z1 f\%  
    %       4    4    r^4                      sqrt(10) ^q<EnsY  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) #EH\Q%  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) aecvz0}@R  
    %       5    5    r^5                      sqrt(12) mrVN&.  
    %       --------------------------------------------- 2J` LZS  
    % frWY8&W^H  
    %   Example: %Wc$S]>i  
    % >waA\C}  
    %       % Display three example Zernike radial polynomials |J ^I8gx+  
    %       r = 0:0.01:1; J/[PA[Rf  
    %       n = [3 2 5];  uHTm  
    %       m = [1 2 1]; r&XxF >  
    %       z = zernpol(n,m,r); >Q)S-4iR  
    %       figure ;!m_RQPFF  
    %       plot(r,z) TQ5kT?/{  
    %       grid on e"2x!(&n(  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') nYK!'x$  
    % vLCm,Bb2L  
    %   See also ZERNFUN, ZERNFUN2. E7]a#  
    ^9|&w.:@Q  
    % A note on the algorithm. ;ZZmX]kz,M  
    % ------------------------ EFs\zWF  
    % The radial Zernike polynomials are computed using the series QmsS,Zljo  
    % representation shown in the Help section above. For many special 'gk^NAG2^E  
    % functions, direct evaluation using the series representation can 9O:-q[K**  
    % produce poor numerical results (floating point errors), because f:9qId ;/M  
    % the summation often involves computing small differences between "l6Ob  
    % large successive terms in the series. (In such cases, the functions z'EphL7r   
    % are often evaluated using alternative methods such as recurrence Aac7k m  
    % relations: see the Legendre functions, for example). For the Zernike c*)PS`]t  
    % polynomials, however, this problem does not arise, because the (HeIO  
    % polynomials are evaluated over the finite domain r = (0,1), and @h7 i;Ok  
    % because the coefficients for a given polynomial are generally all ]T>YYz  
    % of similar magnitude. ?:RWHe.P  
    % l{_1`rC'  
    % ZERNPOL has been written using a vectorized implementation: multiple OEHw%  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] 8noo^QO  
    % values can be passed as inputs) for a vector of points R.  To achieve TI3@/SB>  
    % this vectorization most efficiently, the algorithm in ZERNPOL t%Y}JKLR  
    % involves pre-determining all the powers p of R that are required to &F +hh{  
    % compute the outputs, and then compiling the {R^p} into a single (m=F  
    % matrix.  This avoids any redundant computation of the R^p, and hx@E,  
    % minimizes the sizes of certain intermediate variables. p2 m`pT  
    % <*$IZl6I  
    %   Paul Fricker 11/13/2006 4n.EA,:g:(  
    wpm $?X  
    $yMNdBI[  
    % Check and prepare the inputs: $60]RCu  
    % ----------------------------- d^XRkB:h  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) |JCn=v@  
        error('zernpol:NMvectors','N and M must be vectors.') U9q6m3#$  
    end p%CcD]o  
    R6G%_,p$7  
    if length(n)~=length(m) y1Yrf,E m=  
        error('zernpol:NMlength','N and M must be the same length.') .A <n2-  
    end b#_u.vP  
    K_BF=C.k  
    n = n(:); m?<5-"hz  
    m = m(:); ]N1gzHaS  
    length_n = length(n); `~ R%}ID  
    1$ {Cwb/F  
    if any(mod(n-m,2)) c(!{_+q"  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') B,ZLX/c9  
    end u_ym=N57`  
    `z`"0;,7S  
    if any(m<0) <ApzcyC  
        error('zernpol:Mpositive','All M must be positive.') )Ft>X9$  
    end >J@hqW  
     Z_?r5M;  
    if any(m>n) ^2!l/(?  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') Vu1X@@z  
    end A(qy>x-BI  
    &V7{J9  
    if any( r>1 | r<0 ) J"]P" `/  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') *'ex>4^  
    end 4 O~zkg  
    '~kAsn*/  
    if ~any(size(r)==1) =Ev* Q[  
        error('zernpol:Rvector','R must be a vector.') ADN  
    end pL)o@-k#%  
    (]p,Z <f  
    r = r(:); !l1ycQM  
    length_r = length(r); i^u5j\pfY*  
    %$I@7Es>  
    if nargin==4 \^rAH@  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); CZ_ (IT7  
        if ~isnorm NhA_dskvo  
            error('zernpol:normalization','Unrecognized normalization flag.') 0=Z_5.T>  
        end 3 ~^}R  
    else E;^~}  
        isnorm = false; *%xmCP J  
    end kkE1CHY  
    m\xE8D(,  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 9hIcnPu  
    % Compute the Zernike Polynomials #97w6,P+  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% y9L:2f\  
    t9B]V  
    % Determine the required powers of r: :If1zB)  
    % ----------------------------------- X"qC&oZmf  
    rpowers = []; .I&]G  
    for j = 1:length(n) RtVG6'Y  
        rpowers = [rpowers m(j):2:n(j)]; i@}/KT  
    end rwUKg[ 1N  
    rpowers = unique(rpowers); ?1u2P$d  
    I`e |[k2  
    % Pre-compute the values of r raised to the required powers, Dk XB  
    % and compile them in a matrix: wx|eO[14  
    % ----------------------------- i-4pdK u  
    if rpowers(1)==0 %XieKL  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); @4N@cM0   
        rpowern = cat(2,rpowern{:}); lIj2w;$v  
        rpowern = [ones(length_r,1) rpowern]; [EETx-  
    else 1]uHaI(  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); T};fy+iq  
        rpowern = cat(2,rpowern{:}); t K+K lz  
    end Zn`vL52_  
    7TjK;w7xS.  
    % Compute the values of the polynomials: LL1HDG >l  
    % -------------------------------------- ]"< ` ^  
    z = zeros(length_r,length_n); _pv<_ Sm  
    for j = 1:length_n [9E<z2H  
        s = 0:(n(j)-m(j))/2; <ZV !fn  
        pows = n(j):-2:m(j); P]4C/UDS-~  
        for k = length(s):-1:1 P+Ta|-  
            p = (1-2*mod(s(k),2))* ... nRmZu\(Ow|  
                       prod(2:(n(j)-s(k)))/          ... gUoTOA,  
                       prod(2:s(k))/                 ... ]#N~r&hmQ  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... Jn_;  cN  
                       prod(2:((n(j)+m(j))/2-s(k))); Nn],sEs  
            idx = (pows(k)==rpowers); "&ElKy 7j  
            z(:,j) = z(:,j) + p*rpowern(:,idx); a0oM KGW:  
        end `4'['x  
         i#M$i*H*A  
        if isnorm H *gF>1  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); csV.AN'obq  
        end *LbRLwt  
    end gZ*8F|sg  
    C"I:^&sL  
    % EOF zernpol
    离线niuhelen
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    只看该作者 6楼 发表于: 2011-03-12
    这三个文件,我不知道该怎样把我的面型节点的坐标及轴向位移用起来,还烦请指点一下啊,谢谢啦!
    离线li_xin_feng
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    只看该作者 7楼 发表于: 2012-09-28
    我也正在找啊
    离线guapiqlh
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    只看该作者 8楼 发表于: 2014-03-04
    我也一直想了解这个多项式的应用,还没用过呢
    离线phoenixzqy
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    只看该作者 9楼 发表于: 2014-04-22
    回 guapiqlh 的帖子
    guapiqlh:我也一直想了解这个多项式的应用,还没用过呢 (2014-03-04 11:35)  E"7 iU  
    vy?Zz<c;  
    数值分析方法看一下就行了。其实就是正交多项式的应用。zernike也只不过是正交多项式的一种。 <$)F_R~T3  
    >]?!c5=  
    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)