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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 L AasmQ  
    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! ;+E]F8G9r  
     
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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 (?|M'gZ  
    function z = zernfun(n,m,r,theta,nflag) aV'bI  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. <giBL L!  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N \~ D(ww  
    %   and angular frequency M, evaluated at positions (R,THETA) on the xak)YOLRV  
    %   unit circle.  N is a vector of positive integers (including 0), and Y~I<Locv  
    %   M is a vector with the same number of elements as N.  Each element 7Bp7d/R-  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) 'E_~ |C  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, AEyvljv  
    %   and THETA is a vector of angles.  R and THETA must have the same uAn}qrqE9  
    %   length.  The output Z is a matrix with one column for every (N,M) 53])@Mmus  
    %   pair, and one row for every (R,THETA) pair. 'I]XX==_  
    % y/Xs+ {x  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike !RI _Uph  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), f jx`|MJ  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral R@o&c%K"  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, G\@pg;0|y  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized bE_8NA"2  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. tqGrhOt  
    % K;R H,o1  
    %   The Zernike functions are an orthogonal basis on the unit circle. ,|]J aZq  
    %   They are used in disciplines such as astronomy, optics, and jW'YQrj{<Y  
    %   optometry to describe functions on a circular domain. L^sjV/\oW  
    % $H)^o!  
    %   The following table lists the first 15 Zernike functions. _ %nz-I  
    % %!<Y  
    %       n    m    Zernike function           Normalization yajdRU  
    %       -------------------------------------------------- `L'g<VK;  
    %       0    0    1                                 1 3 _  
    %       1    1    r * cos(theta)                    2 3kn-tM  
    %       1   -1    r * sin(theta)                    2 sey,J5?  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) |?!i},Ki;  
    %       2    0    (2*r^2 - 1)                    sqrt(3) ;+9OzF ;  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) Oidf\%!mvR  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) o:Fq|?/e  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) T }#iXgyx  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) }s~c(sL?;  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) y}?|+/ dN  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) @Vm*b@  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) }t H$:Z  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) ~non_pJ  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) j6m;03<|  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) \ 2\{c1df  
    %       -------------------------------------------------- 2*: q$c  
    % n#(pT3&  
    %   Example 1: (\AN0_  
    % N,(!   
    %       % Display the Zernike function Z(n=5,m=1) 9wvlR6z;u  
    %       x = -1:0.01:1; /I%z7f91O  
    %       [X,Y] = meshgrid(x,x); kBo:)Vej4  
    %       [theta,r] = cart2pol(X,Y); :viW  
    %       idx = r<=1; $^]K611w9  
    %       z = nan(size(X)); 8dczC  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); s2<!Zb4  
    %       figure ]5ZXgz  
    %       pcolor(x,x,z), shading interp '~[8>Q>  
    %       axis square, colorbar M>AxVL  
    %       title('Zernike function Z_5^1(r,\theta)') `'YX>u/  
    % @>2pY_  
    %   Example 2: Vj*-E  
    % |+#Zuq  
    %       % Display the first 10 Zernike functions 6nx\|F  
    %       x = -1:0.01:1; ]fyfL|(;  
    %       [X,Y] = meshgrid(x,x); -k'<6op  
    %       [theta,r] = cart2pol(X,Y); jq+(2  
    %       idx = r<=1; z(|^fi(  
    %       z = nan(size(X)); xcB\Y:   
    %       n = [0  1  1  2  2  2  3  3  3  3]; Kj4/fB  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; jP+yN|  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; At Wv9  
    %       y = zernfun(n,m,r(idx),theta(idx)); lTx_E#^s  
    %       figure('Units','normalized') &,nv+>D  
    %       for k = 1:10 1!#N-^qk  
    %           z(idx) = y(:,k); .~]|gg~  
    %           subplot(4,7,Nplot(k)) 8w0~2-v.?V  
    %           pcolor(x,x,z), shading interp o@:"3s  
    %           set(gca,'XTick',[],'YTick',[]) ":!$Jnj,  
    %           axis square RZa/la*  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 1Viz`y)^  
    %       end ~ ld.I4  
    % qmrT d G  
    %   See also ZERNPOL, ZERNFUN2. SDnl^a  
    3c<aI =$^  
    %   Paul Fricker 11/13/2006 E>~R P^?Uz  
    ) c@gRb~  
    hkMeUxS  
    % Check and prepare the inputs: c./\sN@  
    % ----------------------------- =*\s`ox`  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) eM7@!CdA9q  
        error('zernfun:NMvectors','N and M must be vectors.') r.C6` a  
    end \6b~$\~B  
    aKI"<%PNn  
    if length(n)~=length(m) NRRJlY S  
        error('zernfun:NMlength','N and M must be the same length.')  }k^uup*{  
    end wi2`5G6|z  
    DX_ mrG  
    n = n(:); e" v%m 'G  
    m = m(:); bZu'5+(@  
    if any(mod(n-m,2)) YI0 wr1N  
        error('zernfun:NMmultiplesof2', ... v">?`8V  
              'All N and M must differ by multiples of 2 (including 0).') bC{4a_B  
    end cO?*(e1m=  
    @Z5q2Q  
    if any(m>n) wuqe{?  
        error('zernfun:MlessthanN', ... W}(A8g#6  
              'Each M must be less than or equal to its corresponding N.') I68u%fCv  
    end ;UdM8+^/V]  
    oF%m  
    if any( r>1 | r<0 ) 8_Oeui(i  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') vq$6e*A  
    end hRkCB  
    J/1kJ@5  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) @*;x1A-]V  
        error('zernfun:RTHvector','R and THETA must be vectors.') *!5CL'  
    end N?\X 2J1  
    )_#V>cvNG  
    r = r(:); +B ?qx Q  
    theta = theta(:); PRh C1#  
    length_r = length(r); {oQs*`=l>  
    if length_r~=length(theta) pbMANZU[  
        error('zernfun:RTHlength', ... :.:^\Q0  
              'The number of R- and THETA-values must be equal.') ]kj^T?&n.  
    end +){^HC\7h  
    JE.$]){  
    % Check normalization: P{Nvt/%  
    % -------------------- K?.~}82c  
    if nargin==5 && ischar(nflag) vs@d)$N  
        isnorm = strcmpi(nflag,'norm'); bZowc {!\  
        if ~isnorm !I7$e&Uz@  
            error('zernfun:normalization','Unrecognized normalization flag.') Ycr3$n]e  
        end h:Pfiw]  
    else F^dJ{<yX  
        isnorm = false; +t!]nE #  
    end y0%@^^-Ru  
    d4y#n=HnnV  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% :H}iL*  
    % Compute the Zernike Polynomials j0l,1=^>l  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% xm m,- u  
    /~LE1^1&U  
    % Determine the required powers of r: ing'' _  
    % ----------------------------------- P\"kr?jZP  
    m_abs = abs(m); \/Y(m4<P  
    rpowers = []; 1*O|[W  
    for j = 1:length(n) }7.A~h  
        rpowers = [rpowers m_abs(j):2:n(j)]; 5U 84 *RY  
    end NaR} 0  
    rpowers = unique(rpowers); \Ec<ch[)c  
    J""Cgf  
    % Pre-compute the values of r raised to the required powers, !LK xZ"  
    % and compile them in a matrix: E\iK_'#  
    % ----------------------------- -}7$;QK&a  
    if rpowers(1)==0 jCqz^5=$  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 1_Yx]%g<  
        rpowern = cat(2,rpowern{:}); v :pT(0N  
        rpowern = [ones(length_r,1) rpowern]; eMGJx"a  
    else I~7iIUD  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); pGie!2T E  
        rpowern = cat(2,rpowern{:}); 1AJ6NBC&c  
    end ;4 O[/;i  
    -%fQr5  
    % Compute the values of the polynomials: WwmYJl0  
    % -------------------------------------- yP58H{hQM8  
    y = zeros(length_r,length(n)); cAR `{%b  
    for j = 1:length(n) }Rh\JDiQ  
        s = 0:(n(j)-m_abs(j))/2; 6uE20O<z]  
        pows = n(j):-2:m_abs(j); :8 2T!  
        for k = length(s):-1:1 {B+}LL!  
            p = (1-2*mod(s(k),2))* ... kpgvAKyx  
                       prod(2:(n(j)-s(k)))/              ... 9p_?t'&>q  
                       prod(2:s(k))/                     ... p?gm=b#  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... L;V 8c  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); n Bm ]?  
            idx = (pows(k)==rpowers); n/9afIN  
            y(:,j) = y(:,j) + p*rpowern(:,idx); h&4s%:_4  
        end a>j}@8[J  
         dIC\U  
        if isnorm ,dRaV</2  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); p]aEC+q  
        end o U=vl!\J  
    end FC0fe_U(F  
    % END: Compute the Zernike Polynomials A-Ba%Fv  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% O:?3B!wF  
    "#C2+SKM1  
    % Compute the Zernike functions: Sz5t~U=G  
    % ------------------------------ 1EU4/6!C  
    idx_pos = m>0; TPp]UG  
    idx_neg = m<0; GDLw_usV  
    8lQ}-8  
    z = y; rbvk.:"^w  
    if any(idx_pos) 'rhgM/I  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ,z&S;f.f  
    end rXB;#ypO  
    if any(idx_neg) ~& -h5=3  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); +&.zwniSS  
    end ^s)`UZ<C=  
    KZKE&bTx  
    % 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) `Gio 2gl9  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. d:)#-x*h7  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated aHN"I  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive ACyK#5E  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, Y4k2=w:D  
    %   and THETA is a vector of angles.  R and THETA must have the same `2Pa{g- .  
    %   length.  The output Z is a matrix with one column for every P-value, RU)(|;  
    %   and one row for every (R,THETA) pair. 3;>ls~4  
    % 8}|et~7!  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike C8m9H8Qm  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) I?rB7 *:  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) V\`Z|'WIQD  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 9q'&tU'a=c  
    %   for all p. ;e?M;-  
    % <Kt_ oxK,  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 / l$enexSt  
    %   Zernike functions (order N<=7).  In some disciplines it is ]PS`"o,pF$  
    %   traditional to label the first 36 functions using a single mode >~o- 6g  
    %   number P instead of separate numbers for the order N and azimuthal ! I:N<  
    %   frequency M. zsp%Cz7T  
    % l*l*5hA  
    %   Example: E@GYl85fI  
    % >pF*unC;  
    %       % Display the first 16 Zernike functions UF+Qx/4h0  
    %       x = -1:0.01:1; g~DuK|+  
    %       [X,Y] = meshgrid(x,x); 3^/w`(-{@  
    %       [theta,r] = cart2pol(X,Y); B|m)V9A%-  
    %       idx = r<=1; a0|hLqI  
    %       p = 0:15; L-|l$Ti"  
    %       z = nan(size(X)); 03~ ADj  
    %       y = zernfun2(p,r(idx),theta(idx)); JI>Y?1i0O  
    %       figure('Units','normalized') l5D)UO  
    %       for k = 1:length(p) g<Z :`00|  
    %           z(idx) = y(:,k); $7q3[skH  
    %           subplot(4,4,k) "<iH8MzZ  
    %           pcolor(x,x,z), shading interp s!IIvF  
    %           set(gca,'XTick',[],'YTick',[]) 'D21A8*N  
    %           axis square Go%Z^pF3CO  
    %           title(['Z_{' num2str(p(k)) '}']) d"XZlEV  
    %       end FCt<h/  
    % lEk@I"  
    %   See also ZERNPOL, ZERNFUN. |^Iox0A  
    7${<u0((!  
    %   Paul Fricker 11/13/2006 i.E2a)  
    W\l&wR  
    %0GwO%h},  
    % Check and prepare the inputs: P<vl+&*  
    % ----------------------------- ?oYO !  
    if min(size(p))~=1 Sp\TaUzg  
        error('zernfun2:Pvector','Input P must be vector.') q;68tEupR  
    end 6l\5J6x  
    Y D<3#Dr]  
    if any(p)>35 ]Q"T8drL  
        error('zernfun2:P36', ... kp*v:*  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... ]^ R':YE  
               '(P = 0 to 35).']) YdhV a!Y  
    end .`IhxE~mN  
    Y:DopKRD  
    % Get the order and frequency corresonding to the function number: W]po RTJ:  
    % ---------------------------------------------------------------- T]\1gs41  
    p = p(:); 5Cdn j  
    n = ceil((-3+sqrt(9+8*p))/2); Kg 6J:HD49  
    m = 2*p - n.*(n+2); &@lfr623  
    Cfi4~&  
    % Pass the inputs to the function ZERNFUN: T1C_L?L  
    % ---------------------------------------- s&$?m [w  
    switch nargin FtTq*[a  
        case 3 JSB+g;  
            z = zernfun(n,m,r,theta); "WKOlfPa  
        case 4 &. =8Q?  
            z = zernfun(n,m,r,theta,nflag); 5M>h[Q"R  
        otherwise  DXf  
            error('zernfun2:nargin','Incorrect number of inputs.') K$Bv4_|x  
    end _%CM<z e  
    y_9\07va<  
    % 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) y hKH} kR  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. d dgDq0N1j  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of |Y Lja87  
    %   order N and frequency M, evaluated at R.  N is a vector of My>q%lF=fw  
    %   positive integers (including 0), and M is a vector with the 48 -j  
    %   same number of elements as N.  Each element k of M must be a ;87PP7~  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) x{;{fMN1  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is 7I ~O| Mw  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix B=OzP+  
    %   with one column for every (N,M) pair, and one row for every }-tJ.3Zw  
    %   element in R. ku,{NY f^Y  
    % V< F &\  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- /%cDX:7X  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is N)mZ!K44  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to b"$?(Y  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 H@-q NjM  
    %   for all [n,m]. JLm3qIC  
    % \HB fM&  
    %   The radial Zernike polynomials are the radial portion of the :Fhk$?/r  
    %   Zernike functions, which are an orthogonal basis on the unit ^1){ @(  
    %   circle.  The series representation of the radial Zernike YH58p&up  
    %   polynomials is _jX,1+M  
    % WdOxwsq"  
    %          (n-m)/2 wa,`BAKJ+F  
    %            __ Tbm ~@k(C  
    %    m      \       s                                          n-2s [C EV&B  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r .QP`Qn6(P  
    %    n      s=0 =+_nVO*  
    % /}1|'?P  
    %   The following table shows the first 12 polynomials. -o~zb-E  
    % j)/Vtf  
    %       n    m    Zernike polynomial    Normalization C]Fw*t   
    %       --------------------------------------------- oU )(/  
    %       0    0    1                        sqrt(2) [NE!  
    %       1    1    r                           2 %( OP  [  
    %       2    0    2*r^2 - 1                sqrt(6) 9U;  
    %       2    2    r^2                      sqrt(6) .=YV  
    %       3    1    3*r^3 - 2*r              sqrt(8) zx<t{e7  
    %       3    3    r^3                      sqrt(8) 8G)~#;x1  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) [G brKq(  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) (_#E17U)_  
    %       4    4    r^4                      sqrt(10) 346 z`5  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) 97 SS0J  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) T[OI/ WuK  
    %       5    5    r^5                      sqrt(12) 9+y&&;p  
    %       --------------------------------------------- Y 22Ai  
    % P8=|#yCi  
    %   Example: ]+`K\G ^X  
    % J jZB!Lg=  
    %       % Display three example Zernike radial polynomials U;3t{~Ym  
    %       r = 0:0.01:1;  gwIR3u  
    %       n = [3 2 5]; _886>^b@  
    %       m = [1 2 1];  #NyO'  
    %       z = zernpol(n,m,r); \ o2oQ3  
    %       figure kj2qX9 Ms  
    %       plot(r,z) KRGj6g+  
    %       grid on ZzK^ bNx)0  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') $|Ol?s  
    % W2h*t"5W  
    %   See also ZERNFUN, ZERNFUN2. fahQ^#&d`  
    9q0s  
    % A note on the algorithm. j+'ua=T3  
    % ------------------------ M p <r`PM2  
    % The radial Zernike polynomials are computed using the series F ]X<q uuL  
    % representation shown in the Help section above. For many special [3=Y 9P:  
    % functions, direct evaluation using the series representation can i<m) s$u  
    % produce poor numerical results (floating point errors), because q;R&valn  
    % the summation often involves computing small differences between b`%u}^B {  
    % large successive terms in the series. (In such cases, the functions 'r=2f6G>cP  
    % are often evaluated using alternative methods such as recurrence Wk^{Tn/]  
    % relations: see the Legendre functions, for example). For the Zernike fqbWD)L]  
    % polynomials, however, this problem does not arise, because the X`<z5W] !  
    % polynomials are evaluated over the finite domain r = (0,1), and ir}*E=*  
    % because the coefficients for a given polynomial are generally all _=x*yDPG}  
    % of similar magnitude. O*+HK1q7  
    % % dFz[b  
    % ZERNPOL has been written using a vectorized implementation: multiple bkR~>F]FAu  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] F%zMhX'AG  
    % values can be passed as inputs) for a vector of points R.  To achieve P;(@"gD8z5  
    % this vectorization most efficiently, the algorithm in ZERNPOL A/9<} m  
    % involves pre-determining all the powers p of R that are required to Hwd^C 2v  
    % compute the outputs, and then compiling the {R^p} into a single cl#XiyK>  
    % matrix.  This avoids any redundant computation of the R^p, and Lm!]m\LRZD  
    % minimizes the sizes of certain intermediate variables. _Cf:\Xs m  
    % k"7ZA>5jk  
    %   Paul Fricker 11/13/2006 c{`!$Z'k<  
    kqZRg>1A  
    UazK0{t<f  
    % Check and prepare the inputs: ~Ih` ayVq  
    % ----------------------------- 3,Z;J5VL4!  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) o *U-.&  
        error('zernpol:NMvectors','N and M must be vectors.') $#%R _G]  
    end r]}6iF.  
    \+Qd=,!i(  
    if length(n)~=length(m) gCYe ^KJ  
        error('zernpol:NMlength','N and M must be the same length.') VxOWv8}|  
    end w&#[g9G%  
    u*qI$?&  
    n = n(:); @) MG&X  
    m = m(:); A'}!'1  
    length_n = length(n); VJA/d2Oys  
    F@z%y'5 Z*  
    if any(mod(n-m,2)) *Q2 oc:6  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') Tw%1m  
    end b&,Z mDJh  
    N DI4EA~z  
    if any(m<0) 6ym$8^  
        error('zernpol:Mpositive','All M must be positive.') hX,RuI  
    end #v$wjqK5  
    3XUVUd~  
    if any(m>n) ?t}s3P!Q3w  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') Qa2p34Z/  
    end B(FM~TVZ  
    |gk4X%o6  
    if any( r>1 | r<0 ) Y$, ++wx  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') d/+s-g p  
    end `o9:6X?RA  
    T6?03cSE  
    if ~any(size(r)==1) E>#@ H  
        error('zernpol:Rvector','R must be a vector.') ~T9QpL1OJ  
    end [Djx@x  
    >^W6'Q$P<  
    r = r(:); Pdh`Gu1:3  
    length_r = length(r); &&jQ4@m}j  
    FP y}Wc*UA  
    if nargin==4 GM8>u O  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); M d Eds|D  
        if ~isnorm W}7Uh b  
            error('zernpol:normalization','Unrecognized normalization flag.') q$H@W. f  
        end li{<F{7  
    else 7F2:'3SQ  
        isnorm = false; lpB:lRM  
    end A#Ga!a  
    C\Ob!sv%H  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% RV]QVA*i  
    % Compute the Zernike Polynomials EHl~y=9  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% R:'&>.AUw  
    18kWnF]n=  
    % Determine the required powers of r: [(3 %$?[  
    % ----------------------------------- ncVt (!c,e  
    rpowers = []; ,9F3~Ryt(  
    for j = 1:length(n) V3|" v4  
        rpowers = [rpowers m(j):2:n(j)]; DqI"B  
    end mICx9oz]  
    rpowers = unique(rpowers); xVI"sBUu  
    C>-}BeY!  
    % Pre-compute the values of r raised to the required powers, V%t_,AT  
    % and compile them in a matrix: +wHa)A0MW  
    % ----------------------------- F }F{/  
    if rpowers(1)==0 ;$]a.9 -  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); VD!PF'  
        rpowern = cat(2,rpowern{:}); ]$.w I~J%  
        rpowern = [ones(length_r,1) rpowern]; |Ul4n@+2  
    else ::GW  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); [%/B"w Tt  
        rpowern = cat(2,rpowern{:}); vUL@i'0&o  
    end 7)>L#(N  
    JvCy&xrE;  
    % Compute the values of the polynomials: 23+JuXC6>  
    % -------------------------------------- tmeg=U7  
    z = zeros(length_r,length_n); !6#.%"{-  
    for j = 1:length_n 9Ns%<FRO@  
        s = 0:(n(j)-m(j))/2; @.dM1DN)  
        pows = n(j):-2:m(j); ?}uvpB1}  
        for k = length(s):-1:1 *y+K{ fM1  
            p = (1-2*mod(s(k),2))* ... ulEtZ#O{_  
                       prod(2:(n(j)-s(k)))/          ... g<w1d{Td  
                       prod(2:s(k))/                 ... <h~uGBS"  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... #!m^EqF1_  
                       prod(2:((n(j)+m(j))/2-s(k))); iH dX  
            idx = (pows(k)==rpowers); wXYT(R  
            z(:,j) = z(:,j) + p*rpowern(:,idx); R(}!gv}s  
        end =8]Ru(#Ig  
         DU5rB\!.~  
        if isnorm ;?-{Uk  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); plzwk>b_  
        end G!D~*B9 G  
    end UpiZd/K  
    v9gaRqi8  
    % 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)  yc@ :*Z  
    #r80FVwiD  
    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。 1);E!D[  
    q%Lw#f  
    07年就写过这方面的计算程序了。
    让光学不再神秘,让光学变得容易,快速实现客户关于光学的设想与愿望。