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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 e:G~P u`  
    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! BK!Yl\I<  
     
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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 U)dcemQY  
    function z = zernfun(n,m,r,theta,nflag) +  }"+  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. aQoB1 qd8  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N @Z/jaAjUC  
    %   and angular frequency M, evaluated at positions (R,THETA) on the +c8`N'~  
    %   unit circle.  N is a vector of positive integers (including 0), and 7#JnQ| ]  
    %   M is a vector with the same number of elements as N.  Each element ,X/j6\VBO  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) AYf}=t|  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, eX\v;~W*  
    %   and THETA is a vector of angles.  R and THETA must have the same r2:{r`ocM  
    %   length.  The output Z is a matrix with one column for every (N,M) ue8 @=}  
    %   pair, and one row for every (R,THETA) pair. -gGw_w?)(  
    % TX5??o  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike p7\LLJ y  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), xn, u$@F  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral !v2/sq$G  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ?Nt(sZ-  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized .7.1JT#@A7  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. qz- tXc ,  
    % ql9n`?Q  
    %   The Zernike functions are an orthogonal basis on the unit circle. 'n h^;  
    %   They are used in disciplines such as astronomy, optics, and JOuy_n  
    %   optometry to describe functions on a circular domain. Um/l{:S   
    % (pH)QG  
    %   The following table lists the first 15 Zernike functions. /@ em E0  
    % M? 8sy  
    %       n    m    Zernike function           Normalization '7oR|I  
    %       -------------------------------------------------- pYcs4f!?p  
    %       0    0    1                                 1 zsQ]U!*rD  
    %       1    1    r * cos(theta)                    2 cQ1[x>OcU  
    %       1   -1    r * sin(theta)                    2 QE/kR!r  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) l|+$4 Nb2  
    %       2    0    (2*r^2 - 1)                    sqrt(3) _L=-z*a\  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) ;):;H?WS|A  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) '-myOM7  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) T=/c0#Q|q  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) 8$c) ]Bv  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) e <+)IW:  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) nHF66,7t  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) A*BN  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) 'g <"@SS+  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) }bi hlyB&Q  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) !>'A2V~F  
    %       -------------------------------------------------- nt"\FZ*;3  
    % ku/vV+&O  
    %   Example 1: 6 JI8l`S  
    % ")9^  
    %       % Display the Zernike function Z(n=5,m=1) qbQdx Kk  
    %       x = -1:0.01:1; X\BFvSv8C  
    %       [X,Y] = meshgrid(x,x); u~,hT Y(%  
    %       [theta,r] = cart2pol(X,Y); 1OvoW Nx  
    %       idx = r<=1; ("(wap~<nD  
    %       z = nan(size(X)); 4-HBXG9#/  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); xrXfZ>$5bM  
    %       figure ^KD1dy3(  
    %       pcolor(x,x,z), shading interp P#3J@aRC  
    %       axis square, colorbar f#Ud=& >j  
    %       title('Zernike function Z_5^1(r,\theta)') 9e.v[K~  
    % qsEFf(9G  
    %   Example 2: Dy5&-yk  
    % },X.a@:  
    %       % Display the first 10 Zernike functions Mq\?J{E  
    %       x = -1:0.01:1; 7[ n |3  
    %       [X,Y] = meshgrid(x,x); )" Z|x  
    %       [theta,r] = cart2pol(X,Y); <iH   
    %       idx = r<=1; 16N |  
    %       z = nan(size(X)); !r6Yq,3  
    %       n = [0  1  1  2  2  2  3  3  3  3]; XFWE^*e=B  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; 'k}w|gNB  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; ltrti.&  
    %       y = zernfun(n,m,r(idx),theta(idx)); H`k YDp  
    %       figure('Units','normalized') V:t{mu5j  
    %       for k = 1:10 e34g=]"  
    %           z(idx) = y(:,k); :RDk{^b)  
    %           subplot(4,7,Nplot(k)) t(vyi  
    %           pcolor(x,x,z), shading interp -`\n/"#X6i  
    %           set(gca,'XTick',[],'YTick',[]) 3 QXsr<  
    %           axis square nm_taER  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) YHY*dk*|C  
    %       end nxEC6Vh'  
    % g0QYBrp  
    %   See also ZERNPOL, ZERNFUN2. 'xG{q+jj'  
    ./zzuKO8XK  
    %   Paul Fricker 11/13/2006 ;FuST  
    KbciRRf!k  
    I d8MXdV  
    % Check and prepare the inputs: 4Q1R:Ra  
    % ----------------------------- X%og}Cfi  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 7wY0JS$fz  
        error('zernfun:NMvectors','N and M must be vectors.') iZ/iMDfC  
    end [5!{>L`  
    4Wvefq"  
    if length(n)~=length(m) `|&0j4(Pg  
        error('zernfun:NMlength','N and M must be the same length.') ,y-!h@(  
    end =eS?`|  
    cM,g, E}  
    n = n(:); 3me&isKL  
    m = m(:); `H9 +]TWj<  
    if any(mod(n-m,2)) . qf~t/o  
        error('zernfun:NMmultiplesof2', ... Xwu&K8q21  
              'All N and M must differ by multiples of 2 (including 0).') Z;tWV%F5  
    end Z<=L  
    BaUuDo/ZO  
    if any(m>n) U|QP] 6v  
        error('zernfun:MlessthanN', ... ;gAL_/_  
              'Each M must be less than or equal to its corresponding N.') 6wbH{}\ll  
    end vxi_Y\r=T  
    '~7zeZ'  
    if any( r>1 | r<0 ) AuM}L&`i^  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') A42!%>PB  
    end _d^d1Q}V  
    \J#&]o)Y  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) FI$ -."F  
        error('zernfun:RTHvector','R and THETA must be vectors.') xDPR^xY  
    end Hj`\Fm*A  
    7 _"G@h  
    r = r(:); $*:$-  
    theta = theta(:); 92C; a5s  
    length_r = length(r); 6f t6;*,  
    if length_r~=length(theta) .!+7|us8l\  
        error('zernfun:RTHlength', ... k}qCkm27  
              'The number of R- and THETA-values must be equal.') f<oU" WM  
    end 0`v-pL0|  
    zTPNQ0=|  
    % Check normalization: 'R- g:X\{  
    % -------------------- \"L0d1DK)  
    if nargin==5 && ischar(nflag) ZzQLbCV  
        isnorm = strcmpi(nflag,'norm'); x TH3g^E  
        if ~isnorm KW ZEi?  
            error('zernfun:normalization','Unrecognized normalization flag.') 3xdJ<Lrq  
        end k=d0%} `M(  
    else V[Sj+&e&  
        isnorm = false; sX}#L  
    end Wi,)a{  
    O.\\)8xA  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% <R~;|&o,$  
    % Compute the Zernike Polynomials !)`*e>]x  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% DSq?|H  
    p&4n"hC  
    % Determine the required powers of r: o=Mm=;H  
    % ----------------------------------- v046  
    m_abs = abs(m); ;n|%W,b-  
    rpowers = []; Hr7pcz/#l  
    for j = 1:length(n) r1}1lJ>7H  
        rpowers = [rpowers m_abs(j):2:n(j)]; 3Of!Ykf=  
    end ^K4?uABc  
    rpowers = unique(rpowers); C(8!("tU  
    m],.w M8  
    % Pre-compute the values of r raised to the required powers, Nz*,m'-1e  
    % and compile them in a matrix: f#7=N{wm  
    % ----------------------------- jaavh6h)  
    if rpowers(1)==0 lOe|]pQ.,  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); lF40n4}  
        rpowern = cat(2,rpowern{:}); ^j10 f$B  
        rpowern = [ones(length_r,1) rpowern]; ZSD7%gE<D  
    else f/\S:x-B  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ufw[Ei$I:  
        rpowern = cat(2,rpowern{:}); .yD 6$!6  
    end <$a-.C5  
    Y7I\<JG<  
    % Compute the values of the polynomials: d%nX;w,  
    % -------------------------------------- }*C  
    y = zeros(length_r,length(n)); X8R:9q_  
    for j = 1:length(n) %XZhSmlf  
        s = 0:(n(j)-m_abs(j))/2; Di}M\!-[  
        pows = n(j):-2:m_abs(j); !;d>}iE   
        for k = length(s):-1:1 7`^Y*:(  
            p = (1-2*mod(s(k),2))* ... 3)2{c  
                       prod(2:(n(j)-s(k)))/              ... _V0%JE'  
                       prod(2:s(k))/                     ... 6-g>(g   
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... cq3Z}Cp  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); .=d40m  
            idx = (pows(k)==rpowers); )~ &gBX  
            y(:,j) = y(:,j) + p*rpowern(:,idx); {X_I>)Wg  
        end fBz|-I:k +  
         :qj;f];|  
        if isnorm \1n (Jr.<  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); ` vFDO$K  
        end JL{fW>5y|  
    end $<&_9T#&w  
    % END: Compute the Zernike Polynomials )^"V}z t  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 3p?nQ O)L  
    ]%FP*YU4O  
    % Compute the Zernike functions: f4F%\ "  
    % ------------------------------ $d4&H/u^  
    idx_pos = m>0; F+ RE  
    idx_neg = m<0; qK2jJ3)>  
    C@zG(?X  
    z = y; PBFpV8P,  
    if any(idx_pos) SXO.|"M  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ue@W@pj  
    end ?U O aqcL  
    if any(idx_neg) 2Qh)/=8lM  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); _iEnS4$A8  
    end yJ ljCu)f  
    njO~^Hl7  
    % EOF zernfun
    离线niuhelen
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) XzqB=iX  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. HY?#r]Ryt  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated jt: *Y  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive ^6F, lS_t  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, =1(7T.t  
    %   and THETA is a vector of angles.  R and THETA must have the same v}D0t]  
    %   length.  The output Z is a matrix with one column for every P-value, 9ZatlI,  
    %   and one row for every (R,THETA) pair. 8O60pB;4  
    % h(J$-SUs  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike e>.^RtDF  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) ],~[^0  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) 9V&+xbR&  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 }|N88PN  
    %   for all p. }~ N\A  
    % 6gO(  8  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 XP:fL NpQ  
    %   Zernike functions (order N<=7).  In some disciplines it is @]8flb )T  
    %   traditional to label the first 36 functions using a single mode }$qrNbLJ  
    %   number P instead of separate numbers for the order N and azimuthal JKO*bbj  
    %   frequency M. yJO Jw o^  
    % *O@Zn  
    %   Example: j!oX\Y-:&  
    % S')DAx  
    %       % Display the first 16 Zernike functions D^P0X:T]  
    %       x = -1:0.01:1; YWDgRb  
    %       [X,Y] = meshgrid(x,x); 5L~lF8  
    %       [theta,r] = cart2pol(X,Y); (: k n)  
    %       idx = r<=1; 0dS(g&ZR  
    %       p = 0:15; N#)Klq87z  
    %       z = nan(size(X)); S1@r.z2L  
    %       y = zernfun2(p,r(idx),theta(idx)); Nq\)o{<1  
    %       figure('Units','normalized') Q=vo5)t   
    %       for k = 1:length(p) M8\/[R\  
    %           z(idx) = y(:,k); nN@ Ch  
    %           subplot(4,4,k) *zDDi(@vtK  
    %           pcolor(x,x,z), shading interp |O'*CCrCL  
    %           set(gca,'XTick',[],'YTick',[]) *n# =3D  
    %           axis square ad47 42  
    %           title(['Z_{' num2str(p(k)) '}']) 8fBhX,1  
    %       end PVdN)tG5  
    % ZqpK}I  
    %   See also ZERNPOL, ZERNFUN. |8[!`T*s  
    H.C*IL9  
    %   Paul Fricker 11/13/2006 V?) V2>]  
    w^ofH-R/  
    4}cxSl]jf!  
    % Check and prepare the inputs: _s5FYb#  
    % ----------------------------- a.JjbFL  
    if min(size(p))~=1 CyHHV  
        error('zernfun2:Pvector','Input P must be vector.') ZG#:3d*)  
    end 9n_Rk W5g  
    5 $$Cav  
    if any(p)>35 61&{I>~1  
        error('zernfun2:P36', ... Lc[TIX  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... I*(kv7(c0  
               '(P = 0 to 35).']) ;#IrHR*Bk  
    end Xo[cpcV  
    G,^ ?qbHg  
    % Get the order and frequency corresonding to the function number: W?P4oKsql*  
    % ---------------------------------------------------------------- rUyGTe(@h  
    p = p(:); k{b|w')  
    n = ceil((-3+sqrt(9+8*p))/2); +%Kk zdS'  
    m = 2*p - n.*(n+2); 5WxNH}{  
    S%Ky+0  
    % Pass the inputs to the function ZERNFUN: 1=sL[I7<  
    % ---------------------------------------- AFvv+ ss  
    switch nargin ot! m=s  
        case 3 vfx{:3fO  
            z = zernfun(n,m,r,theta); 3B,dL|q(@J  
        case 4 {}iS5[H]  
            z = zernfun(n,m,r,theta,nflag); 0.nkh6 ?  
        otherwise 0Bkz)4R  
            error('zernfun2:nargin','Incorrect number of inputs.') $?gKIv>g  
    end _K'Y`w']  
    :AqtPV'  
    % EOF zernfun2
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    function z = zernpol(n,m,r,nflag) (. YSs   
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. <ME>#,  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of f2SJ4"X  
    %   order N and frequency M, evaluated at R.  N is a vector of kX}sDvP3  
    %   positive integers (including 0), and M is a vector with the Jc]66   
    %   same number of elements as N.  Each element k of M must be a ~=[5X,Ta  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) 7,Z<PE  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is y\-iGKz{0  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix F9% +7Op^  
    %   with one column for every (N,M) pair, and one row for every ohJDu{V  
    %   element in R. @.}Y'`9L  
    % 8)pL0bg  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- S<'_{uz  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is /iQh'rp  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to B^GMncZO  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 a$^)~2U{  
    %   for all [n,m]. gYk5}E-  
    % JgKhrDx  
    %   The radial Zernike polynomials are the radial portion of the <u0}&/  
    %   Zernike functions, which are an orthogonal basis on the unit dvZlkMm   
    %   circle.  The series representation of the radial Zernike C|w<mryx  
    %   polynomials is 0nB[Udk?  
    % `D=`xSEYl  
    %          (n-m)/2 {+d)M  
    %            __ }fo_"bs@  
    %    m      \       s                                          n-2s /4;A.r`;  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r .;ofRx<  
    %    n      s=0 2g?q4e,  
    % Ef ?|0Gm  
    %   The following table shows the first 12 polynomials. 8+".r2*_iO  
    % y3QS! 3I  
    %       n    m    Zernike polynomial    Normalization Y hmveV  
    %       --------------------------------------------- D Y4!RjJ47  
    %       0    0    1                        sqrt(2) ,2 W=/,5A  
    %       1    1    r                           2 xWK/uE(  
    %       2    0    2*r^2 - 1                sqrt(6) T9]0/>  
    %       2    2    r^2                      sqrt(6) afD {w*[8  
    %       3    1    3*r^3 - 2*r              sqrt(8) jAy2C&aP  
    %       3    3    r^3                      sqrt(8) "XLtrAu{  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) >b5 ;I1o=y  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) :?FHqfN?_  
    %       4    4    r^4                      sqrt(10) z\8s |!  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) Pi9?l>  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) /cUu]#h  
    %       5    5    r^5                      sqrt(12) JEWc{)4QD  
    %       --------------------------------------------- #G`K<%{?f  
    % :& :P4Y1 E  
    %   Example: `"yxmo*0  
    % W+U0Y,N6  
    %       % Display three example Zernike radial polynomials XE2rx2k  
    %       r = 0:0.01:1; v#<{Y' K  
    %       n = [3 2 5]; yJ0 %6],^g  
    %       m = [1 2 1]; ^ACrWk~UY  
    %       z = zernpol(n,m,r); IO=$+c  
    %       figure -Eq[J k  
    %       plot(r,z) 0rI/$  
    %       grid on 6vps`k$,~  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') 2e-bt@0t  
    % wQd8/&mmk  
    %   See also ZERNFUN, ZERNFUN2. mv~?1aIKD  
    ONDO xXs  
    % A note on the algorithm. UpE +WzY  
    % ------------------------ q+p}U}L= k  
    % The radial Zernike polynomials are computed using the series )X|)X,~+-  
    % representation shown in the Help section above. For many special C8T0=o/-`  
    % functions, direct evaluation using the series representation can yZgWFf.X  
    % produce poor numerical results (floating point errors), because ']I!1>v$[  
    % the summation often involves computing small differences between mf{M-(6'  
    % large successive terms in the series. (In such cases, the functions }S?"mg& V  
    % are often evaluated using alternative methods such as recurrence xfI0P0+  
    % relations: see the Legendre functions, for example). For the Zernike rWDD$4y  
    % polynomials, however, this problem does not arise, because the *l"CIG'  
    % polynomials are evaluated over the finite domain r = (0,1), and 4cPZGZ{U  
    % because the coefficients for a given polynomial are generally all LW.j)wB]  
    % of similar magnitude. 4pQf*l8e  
    % Ok{1{EmP  
    % ZERNPOL has been written using a vectorized implementation: multiple EQd<!)HZ  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] Yo' Y-h#  
    % values can be passed as inputs) for a vector of points R.  To achieve lz<' L. .  
    % this vectorization most efficiently, the algorithm in ZERNPOL r<:d+5"  
    % involves pre-determining all the powers p of R that are required to yTK3eK  
    % compute the outputs, and then compiling the {R^p} into a single pmWy:0R  
    % matrix.  This avoids any redundant computation of the R^p, and gCiM\Qx  
    % minimizes the sizes of certain intermediate variables. |o9`h9i  
    % [+R_3'aK  
    %   Paul Fricker 11/13/2006 qhcx\eD:?  
    G/(,,T}eG  
    _(8#  
    % Check and prepare the inputs: "M[&4'OM  
    % ----------------------------- GQhy4ji'z  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) _xm<zy{`S  
        error('zernpol:NMvectors','N and M must be vectors.') x0ipk}  
    end _ A# lyp  
    6S_mfWsi  
    if length(n)~=length(m) Sa[lYMuB  
        error('zernpol:NMlength','N and M must be the same length.') !y/e Fx  
    end ZN;ondp4  
    `O0Qtq.  
    n = n(:); |?Edk7`  
    m = m(:); oe|;>0yf  
    length_n = length(n); )R'%SLw  
    'Q :%s  
    if any(mod(n-m,2)) ty"L&$bf  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') k+P3z&e  
    end 2YY4 XHQS  
    @{_X@Wv4iV  
    if any(m<0) lMu-,Z="  
        error('zernpol:Mpositive','All M must be positive.') ji\LC%U-  
    end ^gp]tAf  
    N wNxO  
    if any(m>n) -=gI_wLbM  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') f+s)A(?3  
    end rCczQ71W  
    >4kQ9lXL  
    if any( r>1 | r<0 ) V59!}kel1%  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') $t}W,?   
    end L?j<KW  
    oi,KA  
    if ~any(size(r)==1) u pUJF`3  
        error('zernpol:Rvector','R must be a vector.') j n SZ@u  
    end V?"U)Y@Y  
    WoGnJ0N q  
    r = r(:); ?Sa,n^b*H  
    length_r = length(r); C R?}*  
    $b{8 $<;9  
    if nargin==4 t1mG]  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); \ctzv``/n  
        if ~isnorm W5Pur lu?  
            error('zernpol:normalization','Unrecognized normalization flag.') 2}/r>]9^-  
        end >#;;g2UV  
    else c!wRq4  
        isnorm = false; ~J+ qIZge  
    end VBDb K|  
    C6a-  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *!BQ1 ] G  
    % Compute the Zernike Polynomials =FIZh}JD  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +$(y2F7|u-  
    kJOZ;X=9/  
    % Determine the required powers of r: !A% vR\  
    % ----------------------------------- >.od(Fh{l|  
    rpowers = []; |IzL4>m:;  
    for j = 1:length(n) ~p n$'1Q  
        rpowers = [rpowers m(j):2:n(j)]; 0]'  2i  
    end -UzWLVB^  
    rpowers = unique(rpowers); Nb_Glf  
    MMET^SO  
    % Pre-compute the values of r raised to the required powers, DO*6gzW  
    % and compile them in a matrix: sg}<()  
    % ----------------------------- W1 xPK*  
    if rpowers(1)==0 Lk#)VGk:  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); b`S9#`  
        rpowern = cat(2,rpowern{:}); hslT49m>  
        rpowern = [ones(length_r,1) rpowern]; t5K#nRd Z:  
    else +`Nu0y!rj  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); Z"w}`&TC$^  
        rpowern = cat(2,rpowern{:}); (,+#H]L  
    end |P|2E~[r  
    t!J>853  
    % Compute the values of the polynomials: Sw-2vnSdM  
    % -------------------------------------- dJ])`S  
    z = zeros(length_r,length_n); aCQ[Uc<B:  
    for j = 1:length_n S\t!7Xs%*U  
        s = 0:(n(j)-m(j))/2; <'sm($.2  
        pows = n(j):-2:m(j); >Jn`RsuV  
        for k = length(s):-1:1 1'TS!/ll];  
            p = (1-2*mod(s(k),2))* ... b 1Wz  
                       prod(2:(n(j)-s(k)))/          ... UCj+V@{  
                       prod(2:s(k))/                 ... R N@)nc_  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... `}sFT:1&  
                       prod(2:((n(j)+m(j))/2-s(k))); b.[9Adi >  
            idx = (pows(k)==rpowers); _]Ob)RUVH  
            z(:,j) = z(:,j) + p*rpowern(:,idx); G@jx&#v  
        end 06.8m;{N  
         OT|0_d?bD  
        if isnorm )*uotV  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); [U^Cz{G  
        end MzKl=G  
    end 4o@:+T:1  
    i-4L{T\K  
    % 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)  WNQ<XB qAw  
    S QGYH  
    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。 &,{YfAxQ`  
    O.xtY @'"  
    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)