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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 @]j1:PN-  
    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! ^W ^OfY  
     
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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 _6Sp QW  
    function z = zernfun(n,m,r,theta,nflag) /uflpV|  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. q9"96({\@  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N Wr 4,YQM  
    %   and angular frequency M, evaluated at positions (R,THETA) on the l?e.9o2-  
    %   unit circle.  N is a vector of positive integers (including 0), and E GU2fA7x  
    %   M is a vector with the same number of elements as N.  Each element 7Q 3k 7  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) ?,z}%p  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, cuX)8+  
    %   and THETA is a vector of angles.  R and THETA must have the same Nn6%9PX_)  
    %   length.  The output Z is a matrix with one column for every (N,M) M`_0C38  
    %   pair, and one row for every (R,THETA) pair. O- wzz  
    % *dQSw)R  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike rI\FI0zIp_  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ,tFg4k[  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral &C}*w2]0S  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, dysS9a,  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized /ZX }Nc g  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. =X}J6|>X  
    % vM={V$D&  
    %   The Zernike functions are an orthogonal basis on the unit circle. UQsN'r\tS  
    %   They are used in disciplines such as astronomy, optics, and hrk r'3lv  
    %   optometry to describe functions on a circular domain. E .h*g8bXe  
    % }f ?y* H  
    %   The following table lists the first 15 Zernike functions. F59 TZI  
    % KNl$3nX  
    %       n    m    Zernike function           Normalization _`X:jj>  
    %       -------------------------------------------------- +{]j]OP  
    %       0    0    1                                 1 ^iA9%zp  
    %       1    1    r * cos(theta)                    2 }>\C{ClI  
    %       1   -1    r * sin(theta)                    2 [),ige  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) h[ ZN+M  
    %       2    0    (2*r^2 - 1)                    sqrt(3) &{:-]g\  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) +`4A$#$+y  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) sO Y:e/_F  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) Iu{V,U  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) 9r9NxKuAO  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) (7Qo  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) DU^loB+  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ceA9) {  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) SbZ6t$"  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) u*R_\*j@  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) MV"=19]  
    %       -------------------------------------------------- +ZYn? #IQ  
    % ]e3Ax(i)  
    %   Example 1: "@kaHIf[  
    % { w_e9Wbi  
    %       % Display the Zernike function Z(n=5,m=1) 4i bc  
    %       x = -1:0.01:1; K3C<{#r  
    %       [X,Y] = meshgrid(x,x); Cx"sw }  
    %       [theta,r] = cart2pol(X,Y); !>tL6+yj  
    %       idx = r<=1; ICCc./l|  
    %       z = nan(size(X)); }Jw,>}  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); =N@t'fOr  
    %       figure ~[: 2I  
    %       pcolor(x,x,z), shading interp yZ:qU({KhD  
    %       axis square, colorbar =Qq+4F)MD  
    %       title('Zernike function Z_5^1(r,\theta)') rQXzR  
    % U*:!W=XN  
    %   Example 2: :&Nbw  
    % 8L XHk l  
    %       % Display the first 10 Zernike functions <3iMRe  
    %       x = -1:0.01:1; E^PB)D(.  
    %       [X,Y] = meshgrid(x,x); ?%86/N>  
    %       [theta,r] = cart2pol(X,Y); ^.tg7%dJ  
    %       idx = r<=1; mOSv9w#,  
    %       z = nan(size(X)); 8MBAtVmy  
    %       n = [0  1  1  2  2  2  3  3  3  3]; ^8tEach  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; R]dg_Da  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; ^aQ"E9  
    %       y = zernfun(n,m,r(idx),theta(idx)); K,]=6 Rj  
    %       figure('Units','normalized') n%-0V>  
    %       for k = 1:10 =;k|*Ny  
    %           z(idx) = y(:,k); .hiSw  
    %           subplot(4,7,Nplot(k)) J1kM\8%b\  
    %           pcolor(x,x,z), shading interp !wNO8;(  
    %           set(gca,'XTick',[],'YTick',[]) e )ZUO_Q$  
    %           axis square fVwU e _Y  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) iE{&*.q_}>  
    %       end 2:R+tn(F  
    % .pq%?&  
    %   See also ZERNPOL, ZERNFUN2. 598i^z{~0%  
    f?b"iA(6  
    %   Paul Fricker 11/13/2006 'S~5"6r  
    \9d$@V  
    /xQPTT  
    % Check and prepare the inputs: JRFtsio*  
    % ----------------------------- g>sSS8R O  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) zQA`/&=Y  
        error('zernfun:NMvectors','N and M must be vectors.') Je@v8{][|  
    end P4?glh q#  
    }Lv;!  
    if length(n)~=length(m) 23?rEhKe  
        error('zernfun:NMlength','N and M must be the same length.') & ~!Wym  
    end OZT.=^:A  
    {!`4iiF  
    n = n(:); "j-CZ\]U|  
    m = m(:); i!cCMh8  
    if any(mod(n-m,2)) 9kojLqCT  
        error('zernfun:NMmultiplesof2', ... nm+s{  
              'All N and M must differ by multiples of 2 (including 0).') m,S{p<-h  
    end G j1_!.T  
    z=FZiH  
    if any(m>n) \1`O_DF~o  
        error('zernfun:MlessthanN', ... ,47qw0=C  
              'Each M must be less than or equal to its corresponding N.') @KA4N`  
    end IAEAhqp  
    w*!aZ,P  
    if any( r>1 | r<0 ) ]d`VT)~vje  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') jIF |P-  
    end DN/YHSYK  
    uocGbi:V';  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) H1T.(M/"  
        error('zernfun:RTHvector','R and THETA must be vectors.') nd(S3rct&  
    end 6,uX,X5  
    qVPeB,kIz  
    r = r(:); {|\.i  
    theta = theta(:); 4~=l}H>&  
    length_r = length(r); ~v83pu1!2s  
    if length_r~=length(theta) B;WCTMy}  
        error('zernfun:RTHlength', ... 7Qsgys#/=  
              'The number of R- and THETA-values must be equal.') 5coZ|O&f8  
    end 0g\(+Qg^  
    v}(WaO#S  
    % Check normalization: Hef g[$m  
    % -------------------- [:V$y1  
    if nargin==5 && ischar(nflag) Ve=b16H  
        isnorm = strcmpi(nflag,'norm'); 1U\z5$V  
        if ~isnorm 2-b6gc7  
            error('zernfun:normalization','Unrecognized normalization flag.') v LZoa-w:  
        end Vg23!E  
    else ??T#QQ  
        isnorm = false; d %#b:(,  
    end `lPfb[b  
    $SE^S   
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% X jX2]  
    % Compute the Zernike Polynomials L-\GHu~)  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% D-4f.Tq4#  
    O~QB!<Q+  
    % Determine the required powers of r: = f i$}>\  
    % ----------------------------------- qw8Rlws%  
    m_abs = abs(m); g ci    
    rpowers = []; frQ{iUx  
    for j = 1:length(n) ]~nKK@Rw  
        rpowers = [rpowers m_abs(j):2:n(j)]; Rh |nP&6  
    end V> bCKtf&  
    rpowers = unique(rpowers); eY\y E"3  
    p$>l7?h  
    % Pre-compute the values of r raised to the required powers, [9 RR8  
    % and compile them in a matrix: =ruao'A  
    % ----------------------------- *:NQ&y*uj  
    if rpowers(1)==0 f {"?%Ku#  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ~nPtlrQa#*  
        rpowern = cat(2,rpowern{:}); Z<4AL\l 98  
        rpowern = [ones(length_r,1) rpowern]; 9mFE?J  
    else PuO&wI]:  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); j)GtEP<n#  
        rpowern = cat(2,rpowern{:}); Yuc> fFA  
    end (~en (  
    TU7' J  
    % Compute the values of the polynomials: ""D 4s  
    % -------------------------------------- <o= 8 FO  
    y = zeros(length_r,length(n)); H4JTGt1"  
    for j = 1:length(n) 4{l,  
        s = 0:(n(j)-m_abs(j))/2; (khL-F  
        pows = n(j):-2:m_abs(j); -tNUMi'  
        for k = length(s):-1:1 w-{c.x  
            p = (1-2*mod(s(k),2))* ... Ki~1qu:  
                       prod(2:(n(j)-s(k)))/              ... VQ{fne<  
                       prod(2:s(k))/                     ... ,{q;;b9  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 9k~8  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); FEVlZ<PW3I  
            idx = (pows(k)==rpowers); _7)n(1h[3b  
            y(:,j) = y(:,j) + p*rpowern(:,idx); +H2-ZXr  
        end Jq^T1_iqn  
         -)/$M(Pu"  
        if isnorm Y5d\d\e/  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); Ib0ZjX6  
        end ilva,WFa^  
    end `V3Fx{  
    % END: Compute the Zernike Polynomials +t:0SRSt  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 5P$4 =z91  
    pXK^Y'2C!  
    % Compute the Zernike functions: { buy"X4  
    % ------------------------------ r(2uu  
    idx_pos = m>0; 4 N7^?  
    idx_neg = m<0; c{LO6dNg\z  
    s|B3~Q]  
    z = y; )tnh4WMh}  
    if any(idx_pos) ;]jNk'oa  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); lUiL\~Gq  
    end L z1ME(  
    if any(idx_neg) EUgs6[w 4  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 6B ?twh)  
    end 3 SGDy]  
    13=.H5  
    % 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) a+T.^koY  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. MO <3"@/,  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated q=qcm`ce  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive Q'mM3pq4r  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, = +?7''{>  
    %   and THETA is a vector of angles.  R and THETA must have the same d6sye^P  
    %   length.  The output Z is a matrix with one column for every P-value, m<qJcZk  
    %   and one row for every (R,THETA) pair. g!z&~Z:  
    % xLZG:^(I  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike 1\rz%E  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) (41|'eB\\  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) HuKc9U'7A  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 h@]XBv  
    %   for all p.  "{Eta  
    % v+=BCyT  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 Uwx E<=z  
    %   Zernike functions (order N<=7).  In some disciplines it is {Z5nGG  
    %   traditional to label the first 36 functions using a single mode 2oRg 2R}  
    %   number P instead of separate numbers for the order N and azimuthal fCobzDy  
    %   frequency M. yU}qOgXx  
    % ~Ti'FhN  
    %   Example: ["e3Ez  
    % 1!T1Y,w  
    %       % Display the first 16 Zernike functions 0f>5(ek  
    %       x = -1:0.01:1; "E?2xf|.  
    %       [X,Y] = meshgrid(x,x); pK'V9fD5J  
    %       [theta,r] = cart2pol(X,Y); oW Nh@C  
    %       idx = r<=1; hJ#xB6  
    %       p = 0:15; 2WVka  
    %       z = nan(size(X)); gH7|=W  
    %       y = zernfun2(p,r(idx),theta(idx)); EJ:%}HhA  
    %       figure('Units','normalized') 'B0{_RaTb  
    %       for k = 1:length(p) -JjM y X  
    %           z(idx) = y(:,k); q,eVjtF  
    %           subplot(4,4,k) 1.9}_4!  
    %           pcolor(x,x,z), shading interp :#?5X|Gz  
    %           set(gca,'XTick',[],'YTick',[]) <=0 u2~E  
    %           axis square W= qVc  
    %           title(['Z_{' num2str(p(k)) '}']) o/Q;f@  
    %       end $.rhRKs  
    % xzZ38xIhV  
    %   See also ZERNPOL, ZERNFUN. [ )dXIIM  
    C"T;Qp~B  
    %   Paul Fricker 11/13/2006 r_6ZO&  
    G&V/Gj8  
    Fv<F}h?6  
    % Check and prepare the inputs: ;Q*or2"!  
    % ----------------------------- #c?j\Y9nz  
    if min(size(p))~=1 :GP]P^M;G@  
        error('zernfun2:Pvector','Input P must be vector.') D"?fn<2  
    end 4X |(5q?  
    i,4>0o?  
    if any(p)>35 04l!:Tp,  
        error('zernfun2:P36', ... 9!}8UALD  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... {|:;]T"y  
               '(P = 0 to 35).']) D(RTVef  
    end 474SMx$  
    XkF%.hWo  
    % Get the order and frequency corresonding to the function number: QY?~ZwYB  
    % ---------------------------------------------------------------- Ix=}+K/  
    p = p(:); m(#LhlX  
    n = ceil((-3+sqrt(9+8*p))/2); H'HA+q  
    m = 2*p - n.*(n+2); b@f$nS B  
    T<p !5`B1  
    % Pass the inputs to the function ZERNFUN: ?>rW>U6:P  
    % ---------------------------------------- 4$S;(  
    switch nargin n}G|/v<  
        case 3 d0Qd$ .%A  
            z = zernfun(n,m,r,theta); VAf1" )pC  
        case 4 R$TB1w9]  
            z = zernfun(n,m,r,theta,nflag); "4+ WZR]  
        otherwise Slher0.Y  
            error('zernfun2:nargin','Incorrect number of inputs.') -pGE]nwDL  
    end @u]rWVy;\[  
    P5nO78  
    % 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) /bi[ e9R  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. 3#&7-o  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of DhT>']Z  
    %   order N and frequency M, evaluated at R.  N is a vector of "C SC  
    %   positive integers (including 0), and M is a vector with the K4;'/cS  
    %   same number of elements as N.  Each element k of M must be a $[&*Bj11Yg  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) f Tl<p&b  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is l q&wXi  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix FCuB\ Q  
    %   with one column for every (N,M) pair, and one row for every %$ Z7x\_  
    %   element in R. .5,(_p^  
    % A1#%`^W9  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- $!(pF  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is J}+6UlD  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to tj4VWJK  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 !Kj,9NX{U  
    %   for all [n,m]. jeX^}]x|%  
    % V<@ o<R  
    %   The radial Zernike polynomials are the radial portion of the i Ae<&Ms  
    %   Zernike functions, which are an orthogonal basis on the unit ;z:UN}  
    %   circle.  The series representation of the radial Zernike (B_\TdQ  
    %   polynomials is ?yR&/a  
    % 1ilBz9x*!  
    %          (n-m)/2 o=?C&f{  
    %            __ $UCAhG$  
    %    m      \       s                                          n-2s w1"nffhO  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r yA(K=?sq  
    %    n      s=0 *B{j.{ p(  
    % ~-m"   
    %   The following table shows the first 12 polynomials. ^__Dd)(  
    % ICkp$u^  
    %       n    m    Zernike polynomial    Normalization J@X'PG< 6B  
    %       --------------------------------------------- 2e9es  
    %       0    0    1                        sqrt(2) y+6o{`0  
    %       1    1    r                           2 UE ,t8j  
    %       2    0    2*r^2 - 1                sqrt(6) ANSFdc  
    %       2    2    r^2                      sqrt(6) glXZZ=j  
    %       3    1    3*r^3 - 2*r              sqrt(8) .Pw\~X3!  
    %       3    3    r^3                      sqrt(8) ),!;| bh  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) $.v5~UGb{\  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) 7{qy7,Gp  
    %       4    4    r^4                      sqrt(10) .j>hI="b  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) C[Dav&=^F  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) x,S P'fcP  
    %       5    5    r^5                      sqrt(12) ) ^3avRsC  
    %       --------------------------------------------- hQHnwr  
    % _b.qkTWUB  
    %   Example: <_Q:'cx'  
    % z;wELz1L{  
    %       % Display three example Zernike radial polynomials snnbb0J  
    %       r = 0:0.01:1; 7=OQ8IM !  
    %       n = [3 2 5]; P*Tx14xe4  
    %       m = [1 2 1]; K/=_b<  
    %       z = zernpol(n,m,r); L^4-5`gj  
    %       figure i'wAE:Xe  
    %       plot(r,z) deixy. |  
    %       grid on JPWOPB'H  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') &F5@6nJ`  
    % (S`2[.j  
    %   See also ZERNFUN, ZERNFUN2. &0(  
    9>rPe1iv  
    % A note on the algorithm. T%n2$  
    % ------------------------ A7`1-#  
    % The radial Zernike polynomials are computed using the series ${nX:!)  
    % representation shown in the Help section above. For many special #\ n8M  
    % functions, direct evaluation using the series representation can e$uiJNS2  
    % produce poor numerical results (floating point errors), because @L:>!<  
    % the summation often involves computing small differences between -cm$[,b6  
    % large successive terms in the series. (In such cases, the functions SdwS= (e6  
    % are often evaluated using alternative methods such as recurrence ^e>Wo7r  
    % relations: see the Legendre functions, for example). For the Zernike Css l{B  
    % polynomials, however, this problem does not arise, because the N**g]T 0`  
    % polynomials are evaluated over the finite domain r = (0,1), and pOkLb #  
    % because the coefficients for a given polynomial are generally all J@ktyd(P  
    % of similar magnitude. IMl!,(6;  
    % Iu *^xn  
    % ZERNPOL has been written using a vectorized implementation: multiple MqA`yvQm  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] 13&0rLS  
    % values can be passed as inputs) for a vector of points R.  To achieve gxMfu?zk"  
    % this vectorization most efficiently, the algorithm in ZERNPOL d k<XzO~g  
    % involves pre-determining all the powers p of R that are required to Q\,o :ZU_  
    % compute the outputs, and then compiling the {R^p} into a single \VFHHi:I  
    % matrix.  This avoids any redundant computation of the R^p, and i^!ez5z  
    % minimizes the sizes of certain intermediate variables. V$rlA' +1v  
    % )& <=.q  
    %   Paul Fricker 11/13/2006 iTg;7~1pY  
    A &9(mB  
    !'*csg  
    % Check and prepare the inputs: O8W7<Wc |z  
    % ----------------------------- {?}*1,I  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) fQ=MJ7l  
        error('zernpol:NMvectors','N and M must be vectors.') e<#DdpX!H~  
    end "h7tnMS  
    z]bwnJfd  
    if length(n)~=length(m) F[!ckes<bB  
        error('zernpol:NMlength','N and M must be the same length.') kY&h~Q  
    end KB!|B.ChN(  
    Vax^8 -  
    n = n(:); b2b75}_A  
    m = m(:); Mf#83 <&K  
    length_n = length(n); )I-fU4?  
    *VkgQ`c  
    if any(mod(n-m,2)) 7RvUH-S[  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') P0-Fc@&Y  
    end U70]!EaT  
    T4;T6 9j;,  
    if any(m<0) ez9k4IO  
        error('zernpol:Mpositive','All M must be positive.') a3 >zoN  
    end sfVf@0g  
    9cv]y#  
    if any(m>n) M#@aB"@J>  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') .\qj;20W  
    end 7gS1~Q4\V2  
    1]T`n/d V  
    if any( r>1 | r<0 ) U#o'H @  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') M}hrO-C  
    end w_iamqe,  
    Bz`yfl2  
    if ~any(size(r)==1) fXQiNm[P  
        error('zernpol:Rvector','R must be a vector.') RP`2)/sMT  
    end 5b6s4ZyV  
    3?s ?XAh  
    r = r(:); Y3ZK%OyPR  
    length_r = length(r); :;!\vfZbU  
    da$BUAqU  
    if nargin==4 &wetzC )  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm');  oAZh~~tp  
        if ~isnorm ?oiKVL"7  
            error('zernpol:normalization','Unrecognized normalization flag.') 2n`Lg4=  
        end Sb:T*N0gS  
    else 0X(]7b&~R  
        isnorm = false; ^aRgMuU  
    end 7CB#YP?E  
    8)\M:s~7&  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% `4CWE_k  
    % Compute the Zernike Polynomials Kt.~aaG_  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% r++i=SQax  
    /LQ:Sv7  
    % Determine the required powers of r: i$-#dc2qY  
    % ----------------------------------- [[)_BmS5r  
    rpowers = []; ! qJI'+_  
    for j = 1:length(n) | H ;+1  
        rpowers = [rpowers m(j):2:n(j)]; G7* h{nE  
    end ER{3,0U  
    rpowers = unique(rpowers); T_OF7?  
    r5/R5Ga^  
    % Pre-compute the values of r raised to the required powers, y^FOsr  
    % and compile them in a matrix: S C_|A9  
    % ----------------------------- "L2m-e6  
    if rpowers(1)==0 *N/hc  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); qA/bg  
        rpowern = cat(2,rpowern{:}); ? 4)v`*  
        rpowern = [ones(length_r,1) rpowern]; u=qPzmywt  
    else q3'o|pp  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); / axTh  
        rpowern = cat(2,rpowern{:}); 1=Ilej1  
    end i.rU&yT%  
    M'1HA  
    % Compute the values of the polynomials: nb@"?<L!  
    % -------------------------------------- 27#8dV?  
    z = zeros(length_r,length_n); C 7n Kk/r  
    for j = 1:length_n }&G]0hCT!  
        s = 0:(n(j)-m(j))/2; Z-|li}lDr  
        pows = n(j):-2:m(j); E:VGji7s  
        for k = length(s):-1:1 +|C[-W7Sw  
            p = (1-2*mod(s(k),2))* ... Eqphd!\#6  
                       prod(2:(n(j)-s(k)))/          ... )XVh&'(r  
                       prod(2:s(k))/                 ... MMD<I6Iyv  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... SQKt}kDbM  
                       prod(2:((n(j)+m(j))/2-s(k))); hswTn`f  
            idx = (pows(k)==rpowers); A'"-m)1P  
            z(:,j) = z(:,j) + p*rpowern(:,idx); E5B8 Z?$a  
        end GF R!n1Hv  
         =[(1my7  
        if isnorm _F8T\f |  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); }h~'AM  
        end AQci,j"  
    end J`Oy.Qu)  
    A'DVJ9%xB  
    % 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)  d!w1t=2H  
    ?i/73H+;D3  
    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。 Xk 5oybDI  
    ![qRoYpbg8  
    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)