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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 fU2qrcVu  
    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! ~^)^q8  
     
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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 6\d X  
    function z = zernfun(n,m,r,theta,nflag) T9y;OG  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. m)?5}ZwAH  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ://U^sFL  
    %   and angular frequency M, evaluated at positions (R,THETA) on the iy5R5L 2  
    %   unit circle.  N is a vector of positive integers (including 0), and QBE@(2G}C  
    %   M is a vector with the same number of elements as N.  Each element Xwu.AVsr  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) :_dICxaLZT  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, >N`6;gn*l  
    %   and THETA is a vector of angles.  R and THETA must have the same \94jrr  
    %   length.  The output Z is a matrix with one column for every (N,M) MXAEX2xmme  
    %   pair, and one row for every (R,THETA) pair. Il~01|3+m  
    % X.|Ygx  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike EH9Hpo  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), )xl6,bq3  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral nZvU 'k:  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 1-;?0en&0  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized zDBD.5R;  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ]= x 1`j  
    % * crw^e  
    %   The Zernike functions are an orthogonal basis on the unit circle. $G?(OWI}l`  
    %   They are used in disciplines such as astronomy, optics, and z)L}ECZh9  
    %   optometry to describe functions on a circular domain. r)l`  
    % ' lo.h""  
    %   The following table lists the first 15 Zernike functions. <4?*$  
    % r:l96^xs  
    %       n    m    Zernike function           Normalization pz}mF D&[  
    %       -------------------------------------------------- w{7 ji}  
    %       0    0    1                                 1 JAb$M{t  
    %       1    1    r * cos(theta)                    2 {K.rl%_|N  
    %       1   -1    r * sin(theta)                    2 u35q,u=I  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) *=nO  
    %       2    0    (2*r^2 - 1)                    sqrt(3) NtZ6$o<Y  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) t3F?>G#y  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) fNhT;Bux  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) *%- ?54B  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) @!H '+c  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) Sb<\-O14"  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) 7!d$M{0"  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ~Yl$I,  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) E[S':Q  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) }$)&{d G  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) ,Aa|Bd]b  
    %       -------------------------------------------------- _nX%#/{  
    % h(:<(o@<  
    %   Example 1: P>htQ  
    % i,OKf Xp  
    %       % Display the Zernike function Z(n=5,m=1) !kh{9I>M  
    %       x = -1:0.01:1; 1i,4".h?M  
    %       [X,Y] = meshgrid(x,x); 3q~Fl=|.o  
    %       [theta,r] = cart2pol(X,Y); jU$Y>S>l  
    %       idx = r<=1; k:0P+d  
    %       z = nan(size(X)); O)5 #Fcp(  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); [ -12]3  
    %       figure xii$e  
    %       pcolor(x,x,z), shading interp i[=C_+2  
    %       axis square, colorbar <d! 6[,W;  
    %       title('Zernike function Z_5^1(r,\theta)') hAa[[%wPhU  
    % 4I ,o&TK  
    %   Example 2: (t74a E pi  
    % uX0 Bp8P  
    %       % Display the first 10 Zernike functions [:pl-_.C  
    %       x = -1:0.01:1; ,kE=TR.|  
    %       [X,Y] = meshgrid(x,x); AF[>fMI  
    %       [theta,r] = cart2pol(X,Y); h ]}`@M"  
    %       idx = r<=1; q!2<=:f  
    %       z = nan(size(X)); YX `%A6  
    %       n = [0  1  1  2  2  2  3  3  3  3]; C9Wojo.  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; .;Z.F7{q  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; $[Q cEk  
    %       y = zernfun(n,m,r(idx),theta(idx)); 2fBYT4*P;  
    %       figure('Units','normalized') Ut;'Gk  
    %       for k = 1:10 w{P6i<J  
    %           z(idx) = y(:,k); Y UZKle  
    %           subplot(4,7,Nplot(k)) \*9Ua/H  
    %           pcolor(x,x,z), shading interp 4 m $sJ  
    %           set(gca,'XTick',[],'YTick',[]) "i''Ui\H  
    %           axis square XW:%vJu^`  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) -7 L  
    %       end '_E c_F  
    % 0%;M VMH  
    %   See also ZERNPOL, ZERNFUN2. C,='3^Nc  
    f-]><z  
    %   Paul Fricker 11/13/2006 a(!3Afi  
    LH.%\TMN$  
    \!7*(&yly  
    % Check and prepare the inputs: r4S=I   
    % ----------------------------- N4+g("  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) NCxn^$/+>9  
        error('zernfun:NMvectors','N and M must be vectors.') w%I8CU_}.  
    end %O Fj  
    $$~a=q,P[  
    if length(n)~=length(m) .hgH9$\  
        error('zernfun:NMlength','N and M must be the same length.') jRwa0Px(  
    end mQnL<0_<f  
    W%H]Uyt  
    n = n(:); 1::LN(`<  
    m = m(:); VB's  
    if any(mod(n-m,2)) i)8gCDc  
        error('zernfun:NMmultiplesof2', ... GM77Z.Y  
              'All N and M must differ by multiples of 2 (including 0).') .CvFE~  
    end +qZc} 7rJF  
    PgTDjEo  
    if any(m>n) n8Q* _?Z/  
        error('zernfun:MlessthanN', ... m/KjJ"s,  
              'Each M must be less than or equal to its corresponding N.') :Ip~)n9t  
    end T&!ZD2I  
    0hb/`[Q  
    if any( r>1 | r<0 ) *H?t;,\  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') ]p}#NPe5  
    end b<8q 92F  
    0+p 5/5  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) M!-q}5';  
        error('zernfun:RTHvector','R and THETA must be vectors.') }oV3EIH  
    end !2wETs?  
    )L|C'dJ<k`  
    r = r(:); h9U+ %=^O  
    theta = theta(:); ,Z?m`cx  
    length_r = length(r); 9Dy)nm^  
    if length_r~=length(theta) >Rr!rtc'x  
        error('zernfun:RTHlength', ... l-Fmn/V  
              'The number of R- and THETA-values must be equal.') cJ2y)`  
    end y3Y2 QC(  
    # UjEY9"M  
    % Check normalization: \y@ eBW  
    % -------------------- {GAsFnZk  
    if nargin==5 && ischar(nflag) ?${V{=)*X'  
        isnorm = strcmpi(nflag,'norm'); 4YBf ~Pp  
        if ~isnorm iq,ah"L  
            error('zernfun:normalization','Unrecognized normalization flag.') aQxe)  
        end <Ak:8&$O  
    else &bn*p.=G  
        isnorm = false; zv`zsqDJ  
    end FzA{U O  
    V;P1nL4L  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% W3"vTZJF  
    % Compute the Zernike Polynomials PVZEB  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% >J9IRAm}sc  
    j`{fB}  
    % Determine the required powers of r: Ia=&.,xub  
    % ----------------------------------- i_|h{JK)  
    m_abs = abs(m); Io2,% !D  
    rpowers = []; 5s#R`o %Z  
    for j = 1:length(n) CgN]dx* `  
        rpowers = [rpowers m_abs(j):2:n(j)]; PnI)n=(\  
    end pb~Ps#"Zg  
    rpowers = unique(rpowers); z9I1RX V  
    P Q6T| >  
    % Pre-compute the values of r raised to the required powers, )iT.A  
    % and compile them in a matrix: 8u4gx<;O  
    % ----------------------------- vM5k4%D  
    if rpowers(1)==0 [kVpzpGr  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); zUe#Wp[  
        rpowern = cat(2,rpowern{:}); aeLBaS  
        rpowern = [ones(length_r,1) rpowern]; 5T7_[{  
    else |:~("rA+v  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); b(g_.1[  
        rpowern = cat(2,rpowern{:}); hgF21Oj9  
    end U&w*Sb"  
    .%|OGl ?  
    % Compute the values of the polynomials: kt;}]O2%R  
    % -------------------------------------- q] 2}UuM|U  
    y = zeros(length_r,length(n)); l_UXrnm/N  
    for j = 1:length(n) J,CJPUf&  
        s = 0:(n(j)-m_abs(j))/2; FRb&@(;  
        pows = n(j):-2:m_abs(j); ,)0/Ec  
        for k = length(s):-1:1 C~3@M<X  
            p = (1-2*mod(s(k),2))* ... V 22q*/iV  
                       prod(2:(n(j)-s(k)))/              ... L&+% Wd~  
                       prod(2:s(k))/                     ... I|Vk.,  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... qpluk!  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); [GcA.ABz  
            idx = (pows(k)==rpowers); XHU<4l:kl  
            y(:,j) = y(:,j) + p*rpowern(:,idx); t't^E,E .@  
        end -U/I'RDLEz  
          f'7 d4  
        if isnorm |6\FI?  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 7vB9K_wCI  
        end SQz$kIZR  
    end EKeBTb  
    % END: Compute the Zernike Polynomials 6)tB{:h&~0  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% &!3VqHQ`  
    Gnuo-8lb  
    % Compute the Zernike functions: eH"qI2A  
    % ------------------------------ g_-?h&W  
    idx_pos = m>0; #n6FQ$l8m  
    idx_neg = m<0; RPa?Nv?e  
    CDwFVR'_Af  
    z = y; wN/*|?`Z  
    if any(idx_pos) .j'@K+<45  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); s!nSE  
    end nN(D7wk  
    if any(idx_neg) N,'[:{GOY  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)');  0jip::x  
    end Z7m GC`>  
    y \mutm  
    % EOF zernfun
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) a V+o\fId  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. T9U2j-lA?  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated Bp=oTC G  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive TCEXa?,L  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, {8*d;[X50  
    %   and THETA is a vector of angles.  R and THETA must have the same !?us[f=g%  
    %   length.  The output Z is a matrix with one column for every P-value, o\=i0HR9  
    %   and one row for every (R,THETA) pair. T?p`Y| gl  
    % FJwZo}<6E  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike f2SU5e2  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) +UpMMh q  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) :<WQ;q  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 -KU)7V  
    %   for all p. fa*H cz  
    % vS24;:f  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 _L `N^I.  
    %   Zernike functions (order N<=7).  In some disciplines it is HqnKpZ  
    %   traditional to label the first 36 functions using a single mode 3Q!J9t5dc  
    %   number P instead of separate numbers for the order N and azimuthal kS\.  
    %   frequency M. |)72E[lL  
    % 7S~9E2N  
    %   Example: =p&'_a^$  
    % >`rNT|rg  
    %       % Display the first 16 Zernike functions +~i+k~{`H  
    %       x = -1:0.01:1; hB GGs  
    %       [X,Y] = meshgrid(x,x); !>Qc2&ZV  
    %       [theta,r] = cart2pol(X,Y); /i~^LITH  
    %       idx = r<=1; 8t*%q+Z  
    %       p = 0:15; ek;&<Z_ ]  
    %       z = nan(size(X)); ah!O&ECh  
    %       y = zernfun2(p,r(idx),theta(idx)); 5[j!\d}U  
    %       figure('Units','normalized') 0Z) ;.l^  
    %       for k = 1:length(p) %&=(,;d  
    %           z(idx) = y(:,k); ;KZtW  
    %           subplot(4,4,k) R{OE{8;  
    %           pcolor(x,x,z), shading interp Y +_5"LV  
    %           set(gca,'XTick',[],'YTick',[]) v(Zi;?c  
    %           axis square QSs$   
    %           title(['Z_{' num2str(p(k)) '}']) ?od}~G4s#  
    %       end 1f pS"_}  
    % mP$G9R  
    %   See also ZERNPOL, ZERNFUN. N5rG.6K  
    =`\,2Nb  
    %   Paul Fricker 11/13/2006 D`~{[cv)\  
    >&TnTv?I  
    moJT8tb  
    % Check and prepare the inputs: =[)N6XV3  
    % ----------------------------- g<T`F  
    if min(size(p))~=1 1-NX>E5  
        error('zernfun2:Pvector','Input P must be vector.') >K|GLP  
    end 7U[L\1zS  
    h3d\MYO)B  
    if any(p)>35 noUZ9M|hz  
        error('zernfun2:P36', ... +S5_J&~  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... #L IsL  
               '(P = 0 to 35).']) =Z>V}`n  
    end tId !C  
    hp z*jyh8  
    % Get the order and frequency corresonding to the function number: c>i*HN}Z|  
    % ---------------------------------------------------------------- ks#Z~6+3  
    p = p(:); n40MP5RxY  
    n = ceil((-3+sqrt(9+8*p))/2); t|U2 ws#  
    m = 2*p - n.*(n+2); i(f;'fb*  
    !E:Vn *k;  
    % Pass the inputs to the function ZERNFUN: Y\z\{JW  
    % ---------------------------------------- `w=H'"Zv  
    switch nargin J_[[BJ&}x  
        case 3 5f*'wA  
            z = zernfun(n,m,r,theta); L|1zHDxQ  
        case 4 Nb!6YY=Ez-  
            z = zernfun(n,m,r,theta,nflag); F3 l^^ Mc  
        otherwise j]l}K*8(  
            error('zernfun2:nargin','Incorrect number of inputs.') !>2\OSp!  
    end c'#J{3d  
    X@AkA9'fq  
    % 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) wYMX1=  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. ^RAFmM#F  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of 0#/ 6P&6  
    %   order N and frequency M, evaluated at R.  N is a vector of c2mt<DtWW  
    %   positive integers (including 0), and M is a vector with the cA SHgm  
    %   same number of elements as N.  Each element k of M must be a Hh;6B!zb+  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) {BCj VmY  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is =egi?Ne  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix 5YH mp7c-z  
    %   with one column for every (N,M) pair, and one row for every LLY;IUK!R  
    %   element in R. *#^1rKGWK  
    % OHnjI> /  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- $(L7/M  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is 7c]Ai  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to MV d 3*  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 to|9)\  
    %   for all [n,m]. h}&IlDG  
    % >@Vr'kg+V  
    %   The radial Zernike polynomials are the radial portion of the Dj. +5f'  
    %   Zernike functions, which are an orthogonal basis on the unit XK-x*|  
    %   circle.  The series representation of the radial Zernike 6%INNIyAWa  
    %   polynomials is UBHQzc+,  
    % ;OJ0}\*iP8  
    %          (n-m)/2 @CI6$  
    %            __ A":b_!sW  
    %    m      \       s                                          n-2s W8h\ s {  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r 5g>kr< K  
    %    n      s=0 p}7&x[fTLk  
    % 3(*s|V"  
    %   The following table shows the first 12 polynomials. K/+C6Y?  
    % hBE>ea  
    %       n    m    Zernike polynomial    Normalization 5@%-=87S  
    %       ---------------------------------------------  ly%B!P|  
    %       0    0    1                        sqrt(2) U?j>28  
    %       1    1    r                           2 yZ0ZP  
    %       2    0    2*r^2 - 1                sqrt(6) emPm^M5/K  
    %       2    2    r^2                      sqrt(6) H^:|`T|,  
    %       3    1    3*r^3 - 2*r              sqrt(8) NT/B4'_@  
    %       3    3    r^3                      sqrt(8) 0%NI- Zyo  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) X2?_lZ[\  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) .LR>&N_U  
    %       4    4    r^4                      sqrt(10) &)jZ|Q~  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) AV3,4u  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) Z`c{LYP,y"  
    %       5    5    r^5                      sqrt(12) 6|cl`}g_j  
    %       --------------------------------------------- x.Ml~W[  
    % }3y\cv0ct  
    %   Example: :]Qx T8B  
    % NWK_(=n  
    %       % Display three example Zernike radial polynomials :?k=Yr  
    %       r = 0:0.01:1; Q 9<_:3  
    %       n = [3 2 5]; NYvj?>[y  
    %       m = [1 2 1]; iRHQRdij  
    %       z = zernpol(n,m,r); @2*6+w_Ae  
    %       figure }_;!E@  
    %       plot(r,z) fEv36xb2S  
    %       grid on ]X|G+[Ujv  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') &~f_1<  
    % S,RJ#.:F[t  
    %   See also ZERNFUN, ZERNFUN2. D07u?  
    YH9] T,  
    % A note on the algorithm. z1s"C[W2T  
    % ------------------------ ved Qwzh  
    % The radial Zernike polynomials are computed using the series 7b2<, .E  
    % representation shown in the Help section above. For many special |R/50axI  
    % functions, direct evaluation using the series representation can htym4\Z=  
    % produce poor numerical results (floating point errors), because ~U+'3.Wo  
    % the summation often involves computing small differences between lXKZNCL  
    % large successive terms in the series. (In such cases, the functions _/ZY&5N  
    % are often evaluated using alternative methods such as recurrence ;g]+MLV9  
    % relations: see the Legendre functions, for example). For the Zernike r'\TS U5!  
    % polynomials, however, this problem does not arise, because the ^0-=(JrC  
    % polynomials are evaluated over the finite domain r = (0,1), and  |?A-?-  
    % because the coefficients for a given polynomial are generally all D/UGN+  
    % of similar magnitude. h cXqg  
    % [Cp{i<C  
    % ZERNPOL has been written using a vectorized implementation: multiple = g}yA=.  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] zUqDX{I8  
    % values can be passed as inputs) for a vector of points R.  To achieve ht9b=1wd%s  
    % this vectorization most efficiently, the algorithm in ZERNPOL ?s33x#  
    % involves pre-determining all the powers p of R that are required to P$I\)Q H  
    % compute the outputs, and then compiling the {R^p} into a single ?9TogW>W  
    % matrix.  This avoids any redundant computation of the R^p, and  64fG,b  
    % minimizes the sizes of certain intermediate variables. -m/4\D  
    % K^ \9R  
    %   Paul Fricker 11/13/2006 sc60:IxgI  
    Dm#k-y  
    "QS7?=>*F  
    % Check and prepare the inputs: tO3 ;; %  
    % -----------------------------  U2$T}/@  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) '%N)(S`O7P  
        error('zernpol:NMvectors','N and M must be vectors.') R 0}%   
    end sf0U(XYQ^  
    yk2j&}M  
    if length(n)~=length(m) sN2l[Ous  
        error('zernpol:NMlength','N and M must be the same length.') {+Yo&F}n  
    end h[T3WE  
    VIzZmd  
    n = n(:); F}>`3//u  
    m = m(:); (xL=X%6a  
    length_n = length(n); |=s3a5sl  
    :f;|^(]"  
    if any(mod(n-m,2)) aDuanGC/V  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') gzF&7trN  
    end za7wNe(s  
    K#r` ^aUc  
    if any(m<0)  E"=$p $k  
        error('zernpol:Mpositive','All M must be positive.') Di*>PE@  
    end cDg27xOUi  
    plfB} p  
    if any(m>n) S# #W_OlrI  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') 6EY4@0%A  
    end 'Iu(lpF&  
    `2B+8,{%  
    if any( r>1 | r<0 ) *Y Ox`z!R  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') whCv9)x  
    end v0=~PN~E  
    1 <+^$QL  
    if ~any(size(r)==1) FhGbQJ?[3  
        error('zernpol:Rvector','R must be a vector.') { SV$fl;  
    end 1o%Hn"uG  
    zlE kP @)  
    r = r(:); 7(H/|2;-d8  
    length_r = length(r); t At+5H  
    bxs@_fH  
    if nargin==4 yFG&Ir  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); X*KT=q^?n  
        if ~isnorm GF&"nW9A  
            error('zernpol:normalization','Unrecognized normalization flag.') _qV_(TpS+  
        end #Z :r  
    else \#slZ;&s  
        isnorm = false; U*cj'`eqC  
    end RMXP)[  
    k:sh:G+=$d  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% R)/w   
    % Compute the Zernike Polynomials bPNsy@"6  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \XC1/LZQ  
    ("Zi,3"+  
    % Determine the required powers of r: *3|KbCX  
    % ----------------------------------- !SnpesTn  
    rpowers = []; Ax ^9J)C  
    for j = 1:length(n) ~&kV  
        rpowers = [rpowers m(j):2:n(j)]; PyYe>a;.  
    end #/T)9=m  
    rpowers = unique(rpowers); o&=m]hKpQl  
    *h Ur E  
    % Pre-compute the values of r raised to the required powers, HM/ q B^  
    % and compile them in a matrix: T~la,>p|}  
    % ----------------------------- pS0T>r  
    if rpowers(1)==0 Ab`Gb  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); YpJzRm{Ra  
        rpowern = cat(2,rpowern{:}); c c:xT0Y  
        rpowern = [ones(length_r,1) rpowern]; j2+&B9 (  
    else uJQeZEe  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); q6q= ,<T%S  
        rpowern = cat(2,rpowern{:}); b~r ?#2K  
    end }U9e#>e x  
    ;RXv%ML  
    % Compute the values of the polynomials: \a<E3 <  
    % -------------------------------------- pxgv(:Tw  
    z = zeros(length_r,length_n); N'4*L=Ut  
    for j = 1:length_n q+<TD#xoL  
        s = 0:(n(j)-m(j))/2; &f[[@EF7  
        pows = n(j):-2:m(j); ^-DK<jZ^  
        for k = length(s):-1:1 6`'^$wKs  
            p = (1-2*mod(s(k),2))* ... bkb}M)C  
                       prod(2:(n(j)-s(k)))/          ... rS=6d6@  
                       prod(2:s(k))/                 ... dpy,;nqzeN  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... s:%>H|-  
                       prod(2:((n(j)+m(j))/2-s(k))); _v-sb(* J  
            idx = (pows(k)==rpowers); *{uu_O  
            z(:,j) = z(:,j) + p*rpowern(:,idx); l! GPOmf9`  
        end s;bqUY?LD  
         jk~< si  
        if isnorm GE>&fG  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); K~uoZ~_gA  
        end bp }~{]:b  
    end fSj^/>  
    #]9yzyb_y  
    % 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)  xM"k qRZ  
    2dg+R)%  
    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。 j51Wod<[  
    0]p! Bscaf  
    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)