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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 ;Wc4qJ.@  
    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! M\oTZ@  
     
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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 %XTcP2pRJ  
    function z = zernfun(n,m,r,theta,nflag) b;GD/UI  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. j' 0r'  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 17.x0 gW,  
    %   and angular frequency M, evaluated at positions (R,THETA) on the BZv+H=b  
    %   unit circle.  N is a vector of positive integers (including 0), and :_kAl? eJ  
    %   M is a vector with the same number of elements as N.  Each element N#C1-*[C  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) %\$;(#h  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, *&Lq!rFS  
    %   and THETA is a vector of angles.  R and THETA must have the same BV`-=wRC  
    %   length.  The output Z is a matrix with one column for every (N,M) x]|+\1  
    %   pair, and one row for every (R,THETA) pair. ]aryV?!6  
    % sZ<9A Xk-E  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike M$Zo.Bl$(  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), fV:4#j  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral *i{Y9f8  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, \C^;k%{LV  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized Wu6<\^A  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. b6(p  
    % dq1:s1  
    %   The Zernike functions are an orthogonal basis on the unit circle. {<>K]P~wD  
    %   They are used in disciplines such as astronomy, optics, and qFQ 8  
    %   optometry to describe functions on a circular domain. W5L iXM  
    % &sXRN &Fp  
    %   The following table lists the first 15 Zernike functions. h].~#*  
    % KInk^`C/H  
    %       n    m    Zernike function           Normalization D}C,![   
    %       -------------------------------------------------- -u!FOD/  
    %       0    0    1                                 1 C[!MS5  
    %       1    1    r * cos(theta)                    2 W1B)]IHc  
    %       1   -1    r * sin(theta)                    2 ORXm&z)  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) ig LMv+{  
    %       2    0    (2*r^2 - 1)                    sqrt(3) so$(_W3E,  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) _p-t<ytnh  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) K$K^=> I"o  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) *=V7@o  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) W|:lVAP.|}  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) me6OPc;:!  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) C;QAT  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) + b$=[nfG  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) \#-W <  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 65h @}9,U  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) 5]I|DHmu  
    %       -------------------------------------------------- RB* J=  
    % U7uKRv9  
    %   Example 1: C98]9  
    % 'bld,Do6  
    %       % Display the Zernike function Z(n=5,m=1) I+>%uShm  
    %       x = -1:0.01:1; W>VP'vn}  
    %       [X,Y] = meshgrid(x,x); "<_0A f]  
    %       [theta,r] = cart2pol(X,Y); l\M_-:I+4  
    %       idx = r<=1; @_:]J1jw7  
    %       z = nan(size(X)); ?m$a6'2-,J  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); 53-v|'9'  
    %       figure ac kqH+'  
    %       pcolor(x,x,z), shading interp "H -"  
    %       axis square, colorbar wn_b[tdxq  
    %       title('Zernike function Z_5^1(r,\theta)') #P]#9Ty:  
    % >9RD_QG7  
    %   Example 2: c|F[.;cR  
    % p~noM/*2r  
    %       % Display the first 10 Zernike functions 63`{.yZ*z  
    %       x = -1:0.01:1; o?1;<gs  
    %       [X,Y] = meshgrid(x,x); .s+aZwTMT  
    %       [theta,r] = cart2pol(X,Y); ~%?`P/.o  
    %       idx = r<=1; .q&'&~!_  
    %       z = nan(size(X));  (x^BKnZ  
    %       n = [0  1  1  2  2  2  3  3  3  3]; O+ }qQNe<  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; R4ht6Vm3g)  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; yaq'Lt`  
    %       y = zernfun(n,m,r(idx),theta(idx)); iyj+:t/  
    %       figure('Units','normalized') $zB[B;-!$  
    %       for k = 1:10 fDG0BNLY  
    %           z(idx) = y(:,k); 1]orUF&_  
    %           subplot(4,7,Nplot(k)) A,r*%&4~  
    %           pcolor(x,x,z), shading interp l;y7]DO  
    %           set(gca,'XTick',[],'YTick',[]) k} ]T;|h]  
    %           axis square hx/N1 x  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) K\XH4kic  
    %       end P/EM :  
    % |t; ~:A  
    %   See also ZERNPOL, ZERNFUN2.  /'31w9  
    6#IU*  
    %   Paul Fricker 11/13/2006 gX0R)spg  
    cZ)}LX  
    DjSbyXvrg  
    % Check and prepare the inputs: P!"&%d  
    % ----------------------------- \:'%9 x  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) yHxosxd<*  
        error('zernfun:NMvectors','N and M must be vectors.') ]4;PR("aU  
    end @+atBmt  
    fN'HE#W1Xa  
    if length(n)~=length(m) nLV9<M Zm  
        error('zernfun:NMlength','N and M must be the same length.') ooUk O  
    end WVY\&|)$  
    R(n^)^?  
    n = n(:); V+I|1{@i0  
    m = m(:); `7/Y@}n  
    if any(mod(n-m,2)) H\XP\4#u  
        error('zernfun:NMmultiplesof2', ... 4)1s M=u  
              'All N and M must differ by multiples of 2 (including 0).') &QhX1dT+  
    end i hh/sPi  
    sZW^ !z  
    if any(m>n) $H+VA@_  
        error('zernfun:MlessthanN', ... u|4$+ QiD  
              'Each M must be less than or equal to its corresponding N.') %/9 EORdeH  
    end `'V4PUe  
    XS$OyW_Q  
    if any( r>1 | r<0 ) 7O, U?p  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') ;|UF)QGa2  
    end 7"8hC  
    ` AY_2>7  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ss5 m/i7  
        error('zernfun:RTHvector','R and THETA must be vectors.') i+gQE!  
    end J/}:x;Y  
    ,_"AT! r  
    r = r(:); {dmj/6Lc  
    theta = theta(:); ?s:d[To6  
    length_r = length(r); PssMTEf  
    if length_r~=length(theta) c+2FC@q{l  
        error('zernfun:RTHlength', ... H@ t'~ZO  
              'The number of R- and THETA-values must be equal.') W"Gkq!3u{  
    end `X3^fg  
    H"qOSf{  
    % Check normalization: yz0zFfiX  
    % -------------------- Yot?=T};3{  
    if nargin==5 && ischar(nflag) Uh][@35 p  
        isnorm = strcmpi(nflag,'norm'); e^O(e  
        if ~isnorm tO0!5#-VR  
            error('zernfun:normalization','Unrecognized normalization flag.')  =|9H  
        end S{Er?0wm.R  
    else (&!NC[n,  
        isnorm = false; rD*sl}  
    end qbv#I;  
    [ :zO}r:  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% lGVEpCS}  
    % Compute the Zernike Polynomials 4fe7U=#;Y  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% U*3uq7  
    bR V+>;L0@  
    % Determine the required powers of r: !%c'$f/  
    % ----------------------------------- Ox@sI:CT  
    m_abs = abs(m); 3\Xbmq8}  
    rpowers = []; vBog0KD);s  
    for j = 1:length(n) 7^g&)P  
        rpowers = [rpowers m_abs(j):2:n(j)]; &B|D;|7H  
    end {c (!;U  
    rpowers = unique(rpowers); CP6LHkM9  
    v'BZs   
    % Pre-compute the values of r raised to the required powers, ,u/aT5\_  
    % and compile them in a matrix: @WI2hHD  
    % ----------------------------- hiUD]5Kp  
    if rpowers(1)==0 +=:#wzK@  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ;g~TWy^o  
        rpowern = cat(2,rpowern{:}); 6,9o>zT%H  
        rpowern = [ones(length_r,1) rpowern]; /IsS;0K%L  
    else I}t#%/'YA  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); `(3/$%  
        rpowern = cat(2,rpowern{:}); . Z%{'CC  
    end lIProF0  
    0lv %`,  
    % Compute the values of the polynomials: W16,Alf:  
    % -------------------------------------- LU9A#  
    y = zeros(length_r,length(n)); 'z$Q rFW  
    for j = 1:length(n) HvVts\f  
        s = 0:(n(j)-m_abs(j))/2; CjiVnWSz<  
        pows = n(j):-2:m_abs(j); u{*SX k  
        for k = length(s):-1:1 YJo["Q  
            p = (1-2*mod(s(k),2))* ... phgm0D7  
                       prod(2:(n(j)-s(k)))/              ... VP6ZiQ|  
                       prod(2:s(k))/                     ... ,%)6jYHRw  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... yfm^?G|sW  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); ObiT-D?)g  
            idx = (pows(k)==rpowers); a|?4 )  
            y(:,j) = y(:,j) + p*rpowern(:,idx); h}xeChw]  
        end m o:D9  
         lg b?)=  
        if isnorm d.P\fPSD  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); Rb{U+/gq  
        end O/<K!;(@?  
    end *q1%IJ  
    % END: Compute the Zernike Polynomials V#`fs|e;y  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% _-#'j2  
    #cCL.p"]  
    % Compute the Zernike functions: Q_Gi]M9  
    % ------------------------------ dX)GPC-D7  
    idx_pos = m>0; X0n~-m"m  
    idx_neg = m<0; `3hSL R  
    W]5USFan  
    z = y; $t6e2=7  
    if any(idx_pos) R>(@Z M&  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); GO^_=EMR[  
    end /, !B2  
    if any(idx_neg) G^` 1]?  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Iwc{R8BV  
    end r}jGUe}d  
    n;:rf7hGY  
    % 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) 3dRr/Ilc  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. 6MxKl D7kl  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated ?A )hN8  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive YR;^hs?  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, DmOyBtj  
    %   and THETA is a vector of angles.  R and THETA must have the same !1G."fo  
    %   length.  The output Z is a matrix with one column for every P-value, ]TyisaT  
    %   and one row for every (R,THETA) pair. .({smN,B  
    % Ey4z.s'-l  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike P'O#I}Dmw<  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) 8{Fsm;UsY  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) HO' '&hz  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 /0eYMG+K=  
    %   for all p. J:kmqk!  
    % @, Wvvh  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 T0]*{k(FR  
    %   Zernike functions (order N<=7).  In some disciplines it is w&x!,yd;  
    %   traditional to label the first 36 functions using a single mode Iy6$7~  
    %   number P instead of separate numbers for the order N and azimuthal [V) L  
    %   frequency M. ~O1&@xX  
    % aN,M64F  
    %   Example: E]6z8juO6  
    % NMi45y(Y  
    %       % Display the first 16 Zernike functions j8sH#b7Z  
    %       x = -1:0.01:1; ^'ryNa;"  
    %       [X,Y] = meshgrid(x,x); w$u3W*EoU^  
    %       [theta,r] = cart2pol(X,Y); yOwA8^q  
    %       idx = r<=1; e A}%C.ZR  
    %       p = 0:15; -Fn  }4M  
    %       z = nan(size(X)); 4DOK4{4?5  
    %       y = zernfun2(p,r(idx),theta(idx)); zH*KYB  
    %       figure('Units','normalized') fks)+L'  
    %       for k = 1:length(p) EKz Ad  
    %           z(idx) = y(:,k); E~ a3r]V/  
    %           subplot(4,4,k) Y X_ gb/A  
    %           pcolor(x,x,z), shading interp mSo_} je(  
    %           set(gca,'XTick',[],'YTick',[]) t&(PN%icD  
    %           axis square ]7rj/l$ u  
    %           title(['Z_{' num2str(p(k)) '}']) hnznp1[#@  
    %       end +/ &_v^sC;  
    % H`geS  
    %   See also ZERNPOL, ZERNFUN. rgOfNVyJG<  
    %H+\>raLz  
    %   Paul Fricker 11/13/2006 - > J_ ~  
    Ii:>xuF&  
    np4+"  
    % Check and prepare the inputs: YQS5P#  
    % ----------------------------- %~QO8q_7  
    if min(size(p))~=1 x1BobhU~Zl  
        error('zernfun2:Pvector','Input P must be vector.') s-S }i{Z!  
    end )<xypDQ  
    yA3wtm/?  
    if any(p)>35 kMsnW}Nu  
        error('zernfun2:P36', ... ~M(5Ho  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... >pr=|$zk=  
               '(P = 0 to 35).']) XJ Iv1s\g  
    end -!\fpl{  
    _H^^y$+1  
    % Get the order and frequency corresonding to the function number: g38&P3/  
    % ---------------------------------------------------------------- 84{Q\c  
    p = p(:); UQ.7>Ug+8s  
    n = ceil((-3+sqrt(9+8*p))/2); 9RWkm%?  
    m = 2*p - n.*(n+2); J=dJs k   
    5H9r=a  
    % Pass the inputs to the function ZERNFUN: g(| 6~}|o+  
    % ---------------------------------------- 8x[YZ@iM-  
    switch nargin {vE(l'  
        case 3 fkSwD(  
            z = zernfun(n,m,r,theta); L.=w?%:H=  
        case 4 )$Z=t-q  
            z = zernfun(n,m,r,theta,nflag); @EoZI~  
        otherwise E~kG2x{a  
            error('zernfun2:nargin','Incorrect number of inputs.') ^xZ e2@  
    end d;+[i  
    z~\t|Z]G,|  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) AE@NOM7u  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. Ap$y%6  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of ++Ww88820  
    %   order N and frequency M, evaluated at R.  N is a vector of dz[ bm< T7  
    %   positive integers (including 0), and M is a vector with the \sA*V%n  
    %   same number of elements as N.  Each element k of M must be a mw^7oO#  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) {w <+_++  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is 7zTqNnPnf  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix 1h7+@#<:a  
    %   with one column for every (N,M) pair, and one row for every  2Cg$,#H  
    %   element in R. Ac|5. ?|N  
    % LG]3hz9^9  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- z* <y5  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is ?tg  y|  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to *{o UWt  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 -]:G L>b  
    %   for all [n,m]. x#C@8Bxq=  
    % Ay{t254/  
    %   The radial Zernike polynomials are the radial portion of the M=]5WZO~A  
    %   Zernike functions, which are an orthogonal basis on the unit ggb |Ew  
    %   circle.  The series representation of the radial Zernike 1=2^90  
    %   polynomials is },[;O^Do^{  
    % yGp z,X4x  
    %          (n-m)/2 [4J6 iF  
    %            __ bY~@}gC**@  
    %    m      \       s                                          n-2s ,DnYtIERo  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r 8p1ziz`4>$  
    %    n      s=0 nIfCF,6,  
    % ,LOQDIyn  
    %   The following table shows the first 12 polynomials. +1 eCvt:,  
    % OJb*VtZz5R  
    %       n    m    Zernike polynomial    Normalization +{53a_q  
    %       --------------------------------------------- s0hBbL0DH  
    %       0    0    1                        sqrt(2) /( 6|{B  
    %       1    1    r                           2 -p-0;Hy  
    %       2    0    2*r^2 - 1                sqrt(6) Cz^Q5F`  
    %       2    2    r^2                      sqrt(6)  Zt E##p  
    %       3    1    3*r^3 - 2*r              sqrt(8) a1N!mQ^  
    %       3    3    r^3                      sqrt(8) B8I4[@m>w\  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) W2wpcc  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) _w ]4~V9  
    %       4    4    r^4                      sqrt(10) 1f (DU4h  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) N6Z{BLZ  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) s4T}Bs r  
    %       5    5    r^5                      sqrt(12) RD<75]**{  
    %       --------------------------------------------- Cpx+qQt0  
    % q\9d6u=Gm  
    %   Example: 4-v6=gz.  
    % 'q%%m/,VPQ  
    %       % Display three example Zernike radial polynomials ,,=apyr#&  
    %       r = 0:0.01:1; g2p"LWex-  
    %       n = [3 2 5]; +K6szGP  
    %       m = [1 2 1]; bZipm(e  
    %       z = zernpol(n,m,r); .+K S`  
    %       figure ZYtiMBJ  
    %       plot(r,z) >E"9*:.^a  
    %       grid on 0&fl#]oCE  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') %3Bpn=k>  
    % UU@fkk  
    %   See also ZERNFUN, ZERNFUN2. ?Hy+'sq[  
    VS/;aG$&y  
    % A note on the algorithm. ?$%%Mp(  
    % ------------------------ . \5$MIF  
    % The radial Zernike polynomials are computed using the series {)K](S ~  
    % representation shown in the Help section above. For many special 5^)_B;.f  
    % functions, direct evaluation using the series representation can rj  H`  
    % produce poor numerical results (floating point errors), because M1u{A^d.Z  
    % the summation often involves computing small differences between <`g3(?   
    % large successive terms in the series. (In such cases, the functions i</J@0}y  
    % are often evaluated using alternative methods such as recurrence @Z\~  
    % relations: see the Legendre functions, for example). For the Zernike xX@FWAj  
    % polynomials, however, this problem does not arise, because the oO=o|w|T  
    % polynomials are evaluated over the finite domain r = (0,1), and >xd<YwXZ  
    % because the coefficients for a given polynomial are generally all fnH3 CE  
    % of similar magnitude. uMFV% +I  
    %  . gT4_  
    % ZERNPOL has been written using a vectorized implementation: multiple N\R=cwk  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] F,v 7ifo#f  
    % values can be passed as inputs) for a vector of points R.  To achieve %cW;}Y[?P  
    % this vectorization most efficiently, the algorithm in ZERNPOL x0Bw{>Q  
    % involves pre-determining all the powers p of R that are required to Gq]d:-7l  
    % compute the outputs, and then compiling the {R^p} into a single bsO@2NP'  
    % matrix.  This avoids any redundant computation of the R^p, and }e=e",eAT  
    % minimizes the sizes of certain intermediate variables. T{ -2fp8r[  
    % d\Jji 6W  
    %   Paul Fricker 11/13/2006 g"y?nF.&F  
    <d@pmh  
    ^g!B.ll`  
    % Check and prepare the inputs: D@vMAW  
    % ----------------------------- zk>h u<_  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) kfj%  
        error('zernpol:NMvectors','N and M must be vectors.') &=-PRza%j  
    end !A[S6-18%-  
    &`@M8-m#F  
    if length(n)~=length(m) .s};F/(diD  
        error('zernpol:NMlength','N and M must be the same length.') F";FG 0  
    end ="B n=>  
    u7muaSy  
    n = n(:);  `$-lL"  
    m = m(:); "T*I|  
    length_n = length(n); ?~)Ak`=  
    R`Qp d3  
    if any(mod(n-m,2)) R$xY8+}V  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') %N{sD[^  
    end  ? ICDIn  
    4 =Fg!Eu<  
    if any(m<0) C ktX0  
        error('zernpol:Mpositive','All M must be positive.') _0]QS4a][c  
    end $Q4=37H+  
    eU~?p|Np  
    if any(m>n) 6_ ]8\n  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') .&(8(C  
    end  ^Fp=y,D  
    cQ,9Rnfl,  
    if any( r>1 | r<0 ) (C~dkR?  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') 2f`xHI/@fj  
    end !Mil?^  
    {_as!5l  
    if ~any(size(r)==1) oeGS  
        error('zernpol:Rvector','R must be a vector.') v Xf:~G]  
    end 08io<c,L  
    ^;64!BaK  
    r = r(:); l4Y1(  
    length_r = length(r); Y^5"qd|`  
    }s6G!v^2""  
    if nargin==4 pe#*I/)b  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); /h-6CR Ka  
        if ~isnorm !5E9sk{)  
            error('zernpol:normalization','Unrecognized normalization flag.') 4ac1m,Jlt  
        end )rbc;{.  
    else i;avwP<0  
        isnorm = false; lrn+d$!@  
    end 7%YYr^d  
    4 4<v9uSK  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% X?kPi&ru  
    % Compute the Zernike Polynomials :o<N!*pT  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% I1~G$)w#  
    ,0.|P`|w  
    % Determine the required powers of r: PAr|1i)mB  
    % ----------------------------------- uc\.oG;~q  
    rpowers = []; ?KCxrzf  
    for j = 1:length(n) ^ `E@/<w8  
        rpowers = [rpowers m(j):2:n(j)]; gb9[Meg'  
    end h^v9|~ZJ'7  
    rpowers = unique(rpowers); :SQ LfOQ  
    w. vY(s  
    % Pre-compute the values of r raised to the required powers, V-:`+&S{^  
    % and compile them in a matrix: #B\B(y  
    % ----------------------------- 9yDFHz w  
    if rpowers(1)==0 SCI1bMf  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); LEA;dSf  
        rpowern = cat(2,rpowern{:}); j]#wrm  
        rpowern = [ones(length_r,1) rpowern]; < )Alb\Z  
    else at=D&oy4"+  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); CR.bMF}  
        rpowern = cat(2,rpowern{:}); {QW-g  
    end E2-ojL[6  
     srvYAAE  
    % Compute the values of the polynomials: 6,a%&1_  
    % -------------------------------------- %OuX`w=  
    z = zeros(length_r,length_n); F1E. \l  
    for j = 1:length_n U~Xf=f_Q$  
        s = 0:(n(j)-m(j))/2; X+d&OcO=q  
        pows = n(j):-2:m(j); BjwMb&a;  
        for k = length(s):-1:1 FSFFk~  
            p = (1-2*mod(s(k),2))* ... Bmmb  
                       prod(2:(n(j)-s(k)))/          ...  JUmw$u  
                       prod(2:s(k))/                 ... xa' nJ"f;  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... H 'D#s;SlR  
                       prod(2:((n(j)+m(j))/2-s(k))); `~0P[>|+  
            idx = (pows(k)==rpowers); pEY>A_F  
            z(:,j) = z(:,j) + p*rpowern(:,idx); +tPx0>p;  
        end m\/>C|f\  
         P_v0))n{  
        if isnorm <( cM*kV  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); <&KLo>B^  
        end r+SEw ;  
    end jGJ.Pvc>i  
    Jk%'mEGE  
    % 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)  ]>VJ--fH  
    USnD7I/b  
    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。 *@\?}cX  
    d&[M8(  
    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)