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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 [oD u3Qn  
    就是我用ansys计算出了镜面的面型的数据,怎样可以得到zernike多项式系数,然后用zemax各阶得到像差!谢谢啦! ' o(7@   
     
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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 b^R_8x  
    function z = zernfun(n,m,r,theta,nflag) l5FKw;=K}:  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. )QW p[bV  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N {y`n _  
    %   and angular frequency M, evaluated at positions (R,THETA) on the guk{3<d:Jy  
    %   unit circle.  N is a vector of positive integers (including 0), and gt\*9P   
    %   M is a vector with the same number of elements as N.  Each element cCv@f ks  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) W.nr&yiQ  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, mWTV)z57  
    %   and THETA is a vector of angles.  R and THETA must have the same UO4z~  
    %   length.  The output Z is a matrix with one column for every (N,M) #k|f%!-Vo  
    %   pair, and one row for every (R,THETA) pair.  ?)2;W  
    % 5%]O'h  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike En{< OMg  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), go?}M]c%7  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral ;4k/h/o1#  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, hxkwT  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized #L+ZHs~  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 85vyt/.,k  
    % ?X@uR5?{  
    %   The Zernike functions are an orthogonal basis on the unit circle. mbXW$E-&R2  
    %   They are used in disciplines such as astronomy, optics, and !@[@&.  
    %   optometry to describe functions on a circular domain. `{H!V~42  
    % nG~^-c+  
    %   The following table lists the first 15 Zernike functions. t/J|<Ooj?  
    % d@ef+-  
    %       n    m    Zernike function           Normalization K>_~|ZN1C8  
    %       -------------------------------------------------- |Ge!;v  
    %       0    0    1                                 1 FJ2~SKWT  
    %       1    1    r * cos(theta)                    2 r#B{j$Rw   
    %       1   -1    r * sin(theta)                    2 #{5h6IC  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) gg@Ew4L&  
    %       2    0    (2*r^2 - 1)                    sqrt(3) (l}nwyh5  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) FL[w\&fp  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) Dop,_94G  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) og`g]Z<I  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) grE'ySX0  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) ^C~t)U  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) x,Z:12H0  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ,'byJlw_pv  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) %Mf3OtPiJW  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) V(M7d>N5G  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) 22R ,  
    %       -------------------------------------------------- wDKA1i%G  
    % $fwj8S7$  
    %   Example 1: naM=oSB(  
    % emG1Wyl  
    %       % Display the Zernike function Z(n=5,m=1) e#^ vA$d  
    %       x = -1:0.01:1; m6o o-muAr  
    %       [X,Y] = meshgrid(x,x); B3Ws)nF"  
    %       [theta,r] = cart2pol(X,Y); o"g<Vz  
    %       idx = r<=1; OySn[4`(i  
    %       z = nan(size(X)); qv8B$}FU  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); gM&4Ur  
    %       figure lh-zE5;  
    %       pcolor(x,x,z), shading interp J:l%  
    %       axis square, colorbar :8Ugz~i  
    %       title('Zernike function Z_5^1(r,\theta)') R]N"P:wf@  
    % u(~(+1W  
    %   Example 2: F@1Eg  
    % !-tVt D  
    %       % Display the first 10 Zernike functions ^t P|8k  
    %       x = -1:0.01:1; 9j<7KSj  
    %       [X,Y] = meshgrid(x,x); ^]9.$$GU\A  
    %       [theta,r] = cart2pol(X,Y); 5 ~YaXh^  
    %       idx = r<=1; i}~U/.P   
    %       z = nan(size(X)); ><{Lh@{  
    %       n = [0  1  1  2  2  2  3  3  3  3]; c.uD%  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; }VqCyJu&{  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; vY]7oX+  
    %       y = zernfun(n,m,r(idx),theta(idx)); E2Ec`o  
    %       figure('Units','normalized') }T"&4Rvs2R  
    %       for k = 1:10 B1va]=([)W  
    %           z(idx) = y(:,k); rGQY  
    %           subplot(4,7,Nplot(k)) RC^k#+  
    %           pcolor(x,x,z), shading interp ^\w!D{Y7Q  
    %           set(gca,'XTick',[],'YTick',[]) r^`~GG!,Q  
    %           axis square y)T|1)  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 6f&qtJQ<A  
    %       end 4d%QJ7y  
    % F+/#ugI  
    %   See also ZERNPOL, ZERNFUN2. ),Igu  
    *_eY +\j  
    %   Paul Fricker 11/13/2006 4^k+wQU  
    zZiga q"  
    s[}cj+0  
    % Check and prepare the inputs: ~y1k2n  
    % ----------------------------- LuWY}ste  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) vpoJ{TPO  
        error('zernfun:NMvectors','N and M must be vectors.') '19kP.  
    end !gj_9"<  
    ]>,Lw=_[_  
    if length(n)~=length(m) +z+u=)I  
        error('zernfun:NMlength','N and M must be the same length.') v8\pOI}c  
    end v(^;%  
    Nh+XlgXG  
    n = n(:); EB8<!c ?  
    m = m(:); @O HsM?nW  
    if any(mod(n-m,2)) x n}HB  
        error('zernfun:NMmultiplesof2', ... J:0`*7  
              'All N and M must differ by multiples of 2 (including 0).') _nec6=S6(  
    end c@{M),C~E  
    -!X\xA/KN  
    if any(m>n) 1Wz5Iv#Ez  
        error('zernfun:MlessthanN', ... S=3H.D!f  
              'Each M must be less than or equal to its corresponding N.') +uMK_ds~  
    end K&;/hdS=F  
    %=5m!"F  
    if any( r>1 | r<0 ) DhT8Kh{  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') hc]5f3Z  
    end Q=#Wk$1.  
    ]9~6lx3/  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) aV G4D f  
        error('zernfun:RTHvector','R and THETA must be vectors.') x_#'6H\1ga  
    end %R?#Y1Tq;  
    zJG=9C?  
    r = r(:); CwsC)]{/o  
    theta = theta(:); K<fB]44Y  
    length_r = length(r); 8:jakOeT  
    if length_r~=length(theta) Zmy:Etqi  
        error('zernfun:RTHlength', ... ,pa=OF  
              'The number of R- and THETA-values must be equal.') _OJ19Ry  
    end .%_=(C< E  
    q[%SF=~<k{  
    % Check normalization: |4F'Zu}g>  
    % -------------------- %^bN^Sq -  
    if nargin==5 && ischar(nflag) >{#QS"J#  
        isnorm = strcmpi(nflag,'norm'); 2UEjn>2  
        if ~isnorm "^5%g%  
            error('zernfun:normalization','Unrecognized normalization flag.') 6<9gVh<=w  
        end C^ Oy.s  
    else R9InUX"k  
        isnorm = false; 5Pd^Sew  
    end lNB<_SO  
    AmBLZ<f;  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% DTC OhUIV  
    % Compute the Zernike Polynomials <[tU.nh  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 9^^:Y3j  
    hmJa1fw=  
    % Determine the required powers of r: 9l}G{u9a  
    % ----------------------------------- %Q|Hvjk=E  
    m_abs = abs(m); !k8j8v&  
    rpowers = []; &U y Q<O>  
    for j = 1:length(n) VHx:3G  
        rpowers = [rpowers m_abs(j):2:n(j)]; Og(|bs!6  
    end "M=1Eb$6=  
    rpowers = unique(rpowers); ]gDX~]f[  
    P#qQde/y  
    % Pre-compute the values of r raised to the required powers,  @+!u{  
    % and compile them in a matrix: 9oxn-)6JC  
    % ----------------------------- $@<cZ4  
    if rpowers(1)==0 $WG<  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ^MUvd  
        rpowern = cat(2,rpowern{:}); xoN?[  
        rpowern = [ones(length_r,1) rpowern]; /U@Y2$TOF  
    else GqD_6cdh  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); Io7o*::6iw  
        rpowern = cat(2,rpowern{:}); +XL|bdK  
    end F'^?s= QX  
    48n7<M;I  
    % Compute the values of the polynomials: JI|MR#_u  
    % -------------------------------------- YF<U'EVU-  
    y = zeros(length_r,length(n)); y/!jC]!+c  
    for j = 1:length(n) j~k+d$a  
        s = 0:(n(j)-m_abs(j))/2; L] !M1\  
        pows = n(j):-2:m_abs(j); OsNJ;B  
        for k = length(s):-1:1 U t.#h="  
            p = (1-2*mod(s(k),2))* ... Mt%=z9OLq9  
                       prod(2:(n(j)-s(k)))/              ... aYcc2N%C  
                       prod(2:s(k))/                     ... GJn ~x  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... .X{U\{c|a  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); yPVK>em5  
            idx = (pows(k)==rpowers); 3Jw}MFFV  
            y(:,j) = y(:,j) + p*rpowern(:,idx); c_FnJ_++f  
        end K1m'20U  
         YQ7tZl;:t  
        if isnorm Rge\8H/z  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 287)\FU;3  
        end "UTAh6[3oD  
    end ZA'Qw2fF0  
    % END: Compute the Zernike Polynomials u]s}@(+.  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% n_Bi HMIU'  
    0'~Iv\s  
    % Compute the Zernike functions: Yo[Pu< zR  
    % ------------------------------ m$B)_WW  
    idx_pos = m>0; PR~9*#"v..  
    idx_neg = m<0; Q'Vejz/  
    +<w\K*  
    z = y; ><"0GPxrx  
    if any(idx_pos) +/DT#}JE  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); QW_W5|_  
    end -9^A,vX  
    if any(idx_neg) *e *V%w~75  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); )9z3T>QW  
    end pX]"^f1?O  
    ;B6m;[M+  
    % 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) -9~WtTaV.H  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. d0D*S?#8,C  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated eGQ -Ht,N  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive "*Gp@  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, N=~aj7B%  
    %   and THETA is a vector of angles.  R and THETA must have the same od RtJ[   
    %   length.  The output Z is a matrix with one column for every P-value, pW1(1M)[%Z  
    %   and one row for every (R,THETA) pair. (V9 ;  
    % <U~at+M  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike ;)rXQm  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) ,8Q&X~$rY  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) 9jW"83*5  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 i4H,Ggb  
    %   for all p.  :C9vs  
    % <_~e/+_.  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 j-9Zzgr  
    %   Zernike functions (order N<=7).  In some disciplines it is h #(J6ht  
    %   traditional to label the first 36 functions using a single mode :FX|9h  
    %   number P instead of separate numbers for the order N and azimuthal p~f=0K  
    %   frequency M. -8m3L  
    % ?yy,3:  
    %   Example: #MAXH7[  
    % ?OF9{$m3?  
    %       % Display the first 16 Zernike functions ,WQg.neOA  
    %       x = -1:0.01:1; 55Ag<\7  
    %       [X,Y] = meshgrid(x,x); j-TRa,4bN  
    %       [theta,r] = cart2pol(X,Y); <Xl/U^B  
    %       idx = r<=1; =gQ^,x0R9  
    %       p = 0:15; ,f^ ICM  
    %       z = nan(size(X)); #VynADPs`o  
    %       y = zernfun2(p,r(idx),theta(idx)); ?z0N- A2C2  
    %       figure('Units','normalized') tzrvIVD  
    %       for k = 1:length(p) ]oxi~TwY^  
    %           z(idx) = y(:,k); 2VaKt4+`  
    %           subplot(4,4,k) zLybf:#  
    %           pcolor(x,x,z), shading interp =SV b k  
    %           set(gca,'XTick',[],'YTick',[]) jchq\q)_z  
    %           axis square *=zv:!  
    %           title(['Z_{' num2str(p(k)) '}']) lGpci  
    %       end O#?@' 1  
    % !Z<=PdI1Ys  
    %   See also ZERNPOL, ZERNFUN. uS7kkzt-x  
    yoG*c%3V?  
    %   Paul Fricker 11/13/2006 x4-_K%  
    {fa3"k_ke  
    ]Gf`nJDV  
    % Check and prepare the inputs: cUC!'+L  
    % ----------------------------- .X34[AXd  
    if min(size(p))~=1 MIdViS.g  
        error('zernfun2:Pvector','Input P must be vector.') ()e|BFL.  
    end #eIFRNRb)  
    ZY=a[K  
    if any(p)>35 AL;4-(KH  
        error('zernfun2:P36', ... ip!-~HNwJ  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... v"-K-AQjB  
               '(P = 0 to 35).']) v-DZW,  
    end % O u'+A  
    f"SK3hI$p  
    % Get the order and frequency corresonding to the function number: uF+0nv+  
    % ---------------------------------------------------------------- Dvm[W),(k  
    p = p(:); 8p_6RvG  
    n = ceil((-3+sqrt(9+8*p))/2); Ui.S)\B  
    m = 2*p - n.*(n+2); (9Q@I8}Iy  
    "/Pq/\,R|  
    % Pass the inputs to the function ZERNFUN: GQ2/3kt  
    % ---------------------------------------- Z}S7%m  
    switch nargin Z):Nd9  
        case 3 9qUkw&}H  
            z = zernfun(n,m,r,theta); v1z d[jqk  
        case 4 F}?4h Dt  
            z = zernfun(n,m,r,theta,nflag); <.&84c]/&  
        otherwise (dTQ,0  
            error('zernfun2:nargin','Incorrect number of inputs.') 0ZJj5<U  
    end nx{MUN7  
    lBGYZ--  
    % 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) :kFPPx?  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. K&eT*JW>  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of 8OiCldw:HN  
    %   order N and frequency M, evaluated at R.  N is a vector of O]' 2<;  
    %   positive integers (including 0), and M is a vector with the EjMVlZC>  
    %   same number of elements as N.  Each element k of M must be a :C2 @!W z  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) sNM ]bei  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is E^A S65%bL  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix [:uHe#L  
    %   with one column for every (N,M) pair, and one row for every e;g7Ek3n  
    %   element in R. v3]5`&3~  
    % *2pf> UzL  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- [S1 b\f#  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is c0Pj})-  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to -K3d u&j  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 YmOj.Q&  
    %   for all [n,m]. 5z =}o/?  
    % OTl9MwW  
    %   The radial Zernike polynomials are the radial portion of the Wf^ sl  
    %   Zernike functions, which are an orthogonal basis on the unit @T:J<,  
    %   circle.  The series representation of the radial Zernike "YaT1` Kr  
    %   polynomials is @<`V q  
    % QO^V@"N  
    %          (n-m)/2 g9fYt&  
    %            __ T<"Bb[kH  
    %    m      \       s                                          n-2s at4JLbk  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r < Sgc6>)  
    %    n      s=0 eh> |m> JY  
    % {Wi*B(  
    %   The following table shows the first 12 polynomials. 3n)$\aBE  
    % P;o  {t  
    %       n    m    Zernike polynomial    Normalization ^RO<r}B u  
    %       --------------------------------------------- 1y\bJ  
    %       0    0    1                        sqrt(2) 1YQYZ^11  
    %       1    1    r                           2 W@ &a  
    %       2    0    2*r^2 - 1                sqrt(6) T0Xm}i  
    %       2    2    r^2                      sqrt(6) rZ|p{ym  
    %       3    1    3*r^3 - 2*r              sqrt(8) (qvH=VTwP  
    %       3    3    r^3                      sqrt(8) 3E^qh03(  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) \Oq8kJ=  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) q/@+.q  
    %       4    4    r^4                      sqrt(10) DxwR&S{  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) YoW)]n  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) ~ \<$H'  
    %       5    5    r^5                      sqrt(12) Li]96+C$}  
    %       --------------------------------------------- e4 ,SR(O>  
    % jU~ x^Y  
    %   Example: :L@n(bu RN  
    % !Ve0:$  
    %       % Display three example Zernike radial polynomials \8=)X})  
    %       r = 0:0.01:1; CgmAxcK  
    %       n = [3 2 5]; LWm1j:0  
    %       m = [1 2 1]; <g%A2 lI  
    %       z = zernpol(n,m,r); =Y81h-  
    %       figure / Ws>;0  
    %       plot(r,z) :\JCxS=EW  
    %       grid on =PciLh  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') kl]MP}wc  
    % /AAD Fa  
    %   See also ZERNFUN, ZERNFUN2. Zp3-Yo w2  
    v@tEHRadz  
    % A note on the algorithm. !p).3Kx0  
    % ------------------------ |T$?vIG[  
    % The radial Zernike polynomials are computed using the series 51b%uz  
    % representation shown in the Help section above. For many special NLY=o@<  
    % functions, direct evaluation using the series representation can Yj1|]i5b  
    % produce poor numerical results (floating point errors), because ""jW'%wR  
    % the summation often involves computing small differences between Qv5 fK  
    % large successive terms in the series. (In such cases, the functions N|$9v{ j_  
    % are often evaluated using alternative methods such as recurrence {'tfU  
    % relations: see the Legendre functions, for example). For the Zernike 4NW!{Vw ,  
    % polynomials, however, this problem does not arise, because the xi(1H1KN5B  
    % polynomials are evaluated over the finite domain r = (0,1), and Lv]%P.=[G  
    % because the coefficients for a given polynomial are generally all a`n)aXU l  
    % of similar magnitude. l[c '%M|N  
    % O*GF/ R8B  
    % ZERNPOL has been written using a vectorized implementation: multiple lUB?eQuN_  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] \.}T_,I  
    % values can be passed as inputs) for a vector of points R.  To achieve &TBFt;  
    % this vectorization most efficiently, the algorithm in ZERNPOL Y4 HN1  
    % involves pre-determining all the powers p of R that are required to j!>P7 8  
    % compute the outputs, and then compiling the {R^p} into a single E&zf<Y  
    % matrix.  This avoids any redundant computation of the R^p, and <+g77NL  
    % minimizes the sizes of certain intermediate variables. 05R"/r*  
    % yy=hCjQ)  
    %   Paul Fricker 11/13/2006 =~hb&  
    38p"lT  
    Hz GwO^tbK  
    % Check and prepare the inputs: =Q40]>bpx  
    % ----------------------------- &{.IUg  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) i,r:R g~  
        error('zernpol:NMvectors','N and M must be vectors.') ` = O  
    end =yZq]g6Q  
    fV|uKs(W  
    if length(n)~=length(m)  x)Bbo9J  
        error('zernpol:NMlength','N and M must be the same length.') 0>Snps3*Z  
    end 8#yu.\N.xt  
    h8asj0  
    n = n(:); Pd~=:4  
    m = m(:); Kh\ 7%>K#  
    length_n = length(n); uL^; i""  
    4T(d9y  
    if any(mod(n-m,2)) $ ubU"  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') F1stRZ1ZI  
    end GNMOHqg4  
    O|,9EOrP  
    if any(m<0) i`2SebDj'w  
        error('zernpol:Mpositive','All M must be positive.') ;7z6B|8  
    end +Pb:<WT}%  
    z qq  
    if any(m>n) Gn8 sB  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') uVn"L:_  
    end X6G{.Vh"  
    \s&Mz;:  
    if any( r>1 | r<0 ) t|>zke!'  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') }4YzP 4  
    end /%&  d:  
    ?Tc#[B  
    if ~any(size(r)==1) (0NffM1  
        error('zernpol:Rvector','R must be a vector.') losqc *|  
    end I@KM2 KMN  
    _eiqs  
    r = r(:); XG.[C>  
    length_r = length(r); wli cuY?  
    Jr !BDg  
    if nargin==4 ^f! M"@  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); {h+8^   
        if ~isnorm Pz2 b  
            error('zernpol:normalization','Unrecognized normalization flag.') MT&aH~YB  
        end =tP9n;D  
    else T ?[28|  
        isnorm = false; 8D )nM|  
    end $CEdJ+0z  
    9i5?J]o^  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +-<G(^  
    % Compute the Zernike Polynomials T o["o!(;z  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% $v@$C4  
    (kLaXayn  
    % Determine the required powers of r: $os]$5(  
    % ----------------------------------- *lSu=dk+  
    rpowers = []; (+|+ELfqW  
    for j = 1:length(n) V8M()7uJ  
        rpowers = [rpowers m(j):2:n(j)]; 3@V?L:J  
    end 27D*FItc  
    rpowers = unique(rpowers); ,- AF8BP  
    Da(k>vR@4  
    % Pre-compute the values of r raised to the required powers, z{L'7  
    % and compile them in a matrix: "#O9ij  
    % ----------------------------- %b^4XTz  
    if rpowers(1)==0 6j|Ncv  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); !XtG6ON=  
        rpowern = cat(2,rpowern{:}); `yv?PlKL  
        rpowern = [ones(length_r,1) rpowern]; $NVVurXa  
    else fY00  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); W Ej{2+  
        rpowern = cat(2,rpowern{:}); G]ek-[-  
    end I8gNg Z  
    vkE`T5??  
    % Compute the values of the polynomials: "b hK %N;  
    % -------------------------------------- Y#9W]78He  
    z = zeros(length_r,length_n); "'z}oS  
    for j = 1:length_n -MTk9<qnT  
        s = 0:(n(j)-m(j))/2; -iR2UE@M  
        pows = n(j):-2:m(j); J 8%gC  
        for k = length(s):-1:1 |E1U$,s~u  
            p = (1-2*mod(s(k),2))* ... xT+_JT65  
                       prod(2:(n(j)-s(k)))/          ... |jiIx5qr  
                       prod(2:s(k))/                 ... ;A,X,f  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... *K|~]r(F?  
                       prod(2:((n(j)+m(j))/2-s(k))); 3*h"B$g!  
            idx = (pows(k)==rpowers); Gm B&TD m  
            z(:,j) = z(:,j) + p*rpowern(:,idx); Z<.&fZ^jS  
        end V~dhTdQ5}  
         Ok_}d&A  
        if isnorm 7%  D4  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); B"_O!  
        end PB@IPnB-  
    end gE6'A  
    ,H{9`a#+:  
    % EOF zernpol
    离线niuhelen
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    只看该作者 6楼 发表于: 2011-03-12
    这三个文件,我不知道该怎样把我的面型节点的坐标及轴向位移用起来,还烦请指点一下啊,谢谢啦!
    离线li_xin_feng
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    只看该作者 7楼 发表于: 2012-09-28
    我也正在找啊
    离线guapiqlh
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    只看该作者 8楼 发表于: 2014-03-04
    我也一直想了解这个多项式的应用,还没用过呢
    离线phoenixzqy
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    只看该作者 9楼 发表于: 2014-04-22
    回 guapiqlh 的帖子
    guapiqlh:我也一直想了解这个多项式的应用,还没用过呢 (2014-03-04 11:35)  m FTuqujO  
    g8.z?Ia#5Z  
    数值分析方法看一下就行了。其实就是正交多项式的应用。zernike也只不过是正交多项式的一种。 zwV!6xG  
    h,$CJdDY]  
    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)