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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 &KjMw:l  
    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! 8YFG*HSa  
     
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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 9i\RdJv.  
    function z = zernfun(n,m,r,theta,nflag) 3+Lwtb}XPF  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. ?{ )'O+s  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 3N_KNW  
    %   and angular frequency M, evaluated at positions (R,THETA) on the #&'S-XE+  
    %   unit circle.  N is a vector of positive integers (including 0), and LO_Xr j  
    %   M is a vector with the same number of elements as N.  Each element PEI$1,z  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) zDdo RK@  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, H1k)ya x4_  
    %   and THETA is a vector of angles.  R and THETA must have the same ww{k_'RRJ  
    %   length.  The output Z is a matrix with one column for every (N,M) LA6XTgcu  
    %   pair, and one row for every (R,THETA) pair. 4mDHAR%D  
    % g$uiwqNA%  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike Q#%LIkeq  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ec!e  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral Z;uKnJh  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 0XA\Ag\`G  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized i3)3. WK^  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. =78y* `L  
    % ~T@E")uR  
    %   The Zernike functions are an orthogonal basis on the unit circle. JPDxzp  
    %   They are used in disciplines such as astronomy, optics, and >&.N_,*  
    %   optometry to describe functions on a circular domain. "q?(rx;  
    % `:iMGq ZN  
    %   The following table lists the first 15 Zernike functions. j EbmW*   
    % %`bs<ZWT  
    %       n    m    Zernike function           Normalization Nf4@m|#  
    %       -------------------------------------------------- NuO@N r  
    %       0    0    1                                 1 12 )  
    %       1    1    r * cos(theta)                    2 =#2%[kGq  
    %       1   -1    r * sin(theta)                    2 tV=Qt[|@  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) >J9Qr#=H2  
    %       2    0    (2*r^2 - 1)                    sqrt(3) ,O:4[M!$w  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) a0ms9%Y;Q[  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) ]4t1dVD  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) >7WT4l)7!b  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) d[h=<?E5  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) OFo hyy(  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) !S<p"   
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ) P7oL.)  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) <\~@l^lU  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) S8v,' Cc  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) |Gq3pL<jkC  
    %       -------------------------------------------------- ~[!Tpq5  
    % -d?<t}a  
    %   Example 1: @u+LF]MY  
    % S>5w=RK   
    %       % Display the Zernike function Z(n=5,m=1) !v3d:n\W8  
    %       x = -1:0.01:1; 7 :\J2$P  
    %       [X,Y] = meshgrid(x,x); t,Tq3zB  
    %       [theta,r] = cart2pol(X,Y); /\fR6|tJ  
    %       idx = r<=1; M7qg\1L  
    %       z = nan(size(X)); d1yLDj?  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); ]#N8e?b,  
    %       figure =n.&N   
    %       pcolor(x,x,z), shading interp }G(#jOYk  
    %       axis square, colorbar k Jz^\Re  
    %       title('Zernike function Z_5^1(r,\theta)') vmxS^_I  
    % #pWy%U  
    %   Example 2: XFFm 'W6@  
    % +^J&x>5  
    %       % Display the first 10 Zernike functions h9d*N9!;M  
    %       x = -1:0.01:1; yodhDSO5i  
    %       [X,Y] = meshgrid(x,x); &#C|  
    %       [theta,r] = cart2pol(X,Y); yTc&C)Jba  
    %       idx = r<=1; Z{u]qI{l  
    %       z = nan(size(X)); 7yG%E  
    %       n = [0  1  1  2  2  2  3  3  3  3]; 3Q&@l49q  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; #x;d+Q@  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; C^?/9\  
    %       y = zernfun(n,m,r(idx),theta(idx)); -Nr*na^H9#  
    %       figure('Units','normalized') 7LaRFL.,kO  
    %       for k = 1:10 P{RGW.Ci@  
    %           z(idx) = y(:,k); P/S,dhs(  
    %           subplot(4,7,Nplot(k)) :S`12*_g"  
    %           pcolor(x,x,z), shading interp k-4z2qB  
    %           set(gca,'XTick',[],'YTick',[]) f !7fz~&Sh  
    %           axis square auB+g'l  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) uEsF 8  
    %       end [$6YPM>Ee  
    % fG?a"6~  
    %   See also ZERNPOL, ZERNFUN2. &/Gf@[  
    c*w0Jz>@.7  
    %   Paul Fricker 11/13/2006 CT\rx>[J.6  
    -{oZK{a1  
    %f\j)qw  
    % Check and prepare the inputs: AO-~dV  
    % ----------------------------- -f'&JwE0=  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) z3^gufOkQ  
        error('zernfun:NMvectors','N and M must be vectors.') F.Bij8\  
    end =q[+ e(,3  
    pgUjje>#  
    if length(n)~=length(m) nBd(p Oe  
        error('zernfun:NMlength','N and M must be the same length.')  >YdLB@  
    end Z@ec}`UO|u  
    6!6R3Za$  
    n = n(:); 2 9z@ !  
    m = m(:); iDCQqj`  
    if any(mod(n-m,2)) Vo%ikR #  
        error('zernfun:NMmultiplesof2', ... +Lr`-</VF  
              'All N and M must differ by multiples of 2 (including 0).') ( s+}l?  
    end ),,0T/69+9  
    Dz$dJF1 8  
    if any(m>n) G[d]t$f=  
        error('zernfun:MlessthanN', ... #|V)>")  
              'Each M must be less than or equal to its corresponding N.') 16ip:/5  
    end x=W5e ^0?  
    '^Q$:P{G?  
    if any( r>1 | r<0 ) e=!sMWx6  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') *I9O63  
    end %xX b5aY  
    f(EO|d^u  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 3z k},8fu  
        error('zernfun:RTHvector','R and THETA must be vectors.') {XXnMO4uR;  
    end U @}r?!)"f  
    Nah\4-75&  
    r = r(:); y :QnK0  
    theta = theta(:); i_y%HG  
    length_r = length(r); a0k/R<4  
    if length_r~=length(theta) d|sf2   
        error('zernfun:RTHlength', ... Nc^:v/(P  
              'The number of R- and THETA-values must be equal.') #A~7rH%hi  
    end JGYJ;j{E]  
    !Ks<%; rb  
    % Check normalization: |lIgvHgg  
    % -------------------- kb\\F:w(W  
    if nargin==5 && ischar(nflag) tt&{f <*  
        isnorm = strcmpi(nflag,'norm'); nwi8>MG  
        if ~isnorm 0 \1g-kc!v  
            error('zernfun:normalization','Unrecognized normalization flag.') /W{^hVkvC  
        end 9 H>J S  
    else z>*\nomOn=  
        isnorm = false; ; Yt'$D*CP  
    end _Q*,~ z~  
    ) '/xNR  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *"V) h I5  
    % Compute the Zernike Polynomials +WCV"m  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% <. V*]g/;  
    S:c d'68D  
    % Determine the required powers of r: S<I9`k G  
    % ----------------------------------- 0|mC k  
    m_abs = abs(m); aC3Qmo6?m  
    rpowers = []; =|V#~p*  
    for j = 1:length(n) CSzu $Hnq  
        rpowers = [rpowers m_abs(j):2:n(j)]; pWeD,!f  
    end m&- -$sr  
    rpowers = unique(rpowers); q^}iXE~  
    5_rx$avm  
    % Pre-compute the values of r raised to the required powers, k4J Tc2b  
    % and compile them in a matrix: C5TC@w1*  
    % ----------------------------- LoO"d'{  
    if rpowers(1)==0 G#.q%Up  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); q3u:Tpn4%  
        rpowern = cat(2,rpowern{:}); Go7 oj'"  
        rpowern = [ones(length_r,1) rpowern]; cZ ,}1?!  
    else VP }To  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); wYd{X 8$  
        rpowern = cat(2,rpowern{:}); C)&BtiUN/  
    end O~5*X f  
    P\$%p-G  
    % Compute the values of the polynomials: rDLgQ{Sea  
    % -------------------------------------- C:vVFU|4  
    y = zeros(length_r,length(n)); qKI)*o062  
    for j = 1:length(n) 'Z6x\p  
        s = 0:(n(j)-m_abs(j))/2; 1(WBvAPS  
        pows = n(j):-2:m_abs(j); ._6Q "JAB  
        for k = length(s):-1:1 K#x|/b'5d  
            p = (1-2*mod(s(k),2))* ... N}'2GBqfU4  
                       prod(2:(n(j)-s(k)))/              ... 15kkf~Z<t  
                       prod(2:s(k))/                     ... Hw,@oOh.  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... oUL4l=dj.  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); 7J|nqr`>t  
            idx = (pows(k)==rpowers); %vRCs]  
            y(:,j) = y(:,j) + p*rpowern(:,idx); +DYsBCVbag  
        end ]9}^}U1."  
         $OaxetPH  
        if isnorm Wfsd$kN6{  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); [I( Yn  
        end j;EH[3  
    end lB   
    % END: Compute the Zernike Polynomials *~`BG5w  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% V=H:`n3k  
    5wC,:c[H7  
    % Compute the Zernike functions: kK.[v'[>&  
    % ------------------------------ && b;Wr  
    idx_pos = m>0; ,#j'~-5  
    idx_neg = m<0; sV]I]DR  
    [G"Va_A8  
    z = y; n]jw!;  
    if any(idx_pos) ,k}(]{ -  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); aqv'c j>  
    end 9<5S!?JL  
    if any(idx_neg) V}Ce3wgvA  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); &W*^&0AV  
    end b[~-b  
    {=ATRwUL  
    % 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) I5ZM U  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. mB"1QtD  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated t$rla _rbY  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive (QQkXlJ  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, T@a|*.V  
    %   and THETA is a vector of angles.  R and THETA must have the same \;JZt[  
    %   length.  The output Z is a matrix with one column for every P-value, S1U[{R?,  
    %   and one row for every (R,THETA) pair. ,( NN)Oj  
    % &_74h);2I:  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike w^N QLV S  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) w8a49Fv  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) Z{1B:aW  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 -_XTy!I  
    %   for all p. 5<YL^m{/L  
    % iU AY  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 8#/y`ul  
    %   Zernike functions (order N<=7).  In some disciplines it is -F MonM  
    %   traditional to label the first 36 functions using a single mode ],Yy)<e.  
    %   number P instead of separate numbers for the order N and azimuthal 13+. >  
    %   frequency M. JX{_,2*$  
    % ^9kx3Pw?8  
    %   Example: uaZHM@D  
    % n}c~+ 0`un  
    %       % Display the first 16 Zernike functions $=) Pky-~  
    %       x = -1:0.01:1; ~0@fK<C)O  
    %       [X,Y] = meshgrid(x,x); qw{`?1[+  
    %       [theta,r] = cart2pol(X,Y); ]J@-,FFC  
    %       idx = r<=1; #/{3qPN?@  
    %       p = 0:15; 73Hm:"Eqd  
    %       z = nan(size(X)); w^U{e xo  
    %       y = zernfun2(p,r(idx),theta(idx)); Ih OAMH1  
    %       figure('Units','normalized') h J0U-m  
    %       for k = 1:length(p) $m A2 AI  
    %           z(idx) = y(:,k); 2f62 0   
    %           subplot(4,4,k) 6@eF|GoP  
    %           pcolor(x,x,z), shading interp Ej1 <T,w_  
    %           set(gca,'XTick',[],'YTick',[]) bkS"]q)>  
    %           axis square 'b y+hXk  
    %           title(['Z_{' num2str(p(k)) '}']) CP0'pL=;  
    %       end |gIE$rt-~W  
    % @w`wJ*I4,  
    %   See also ZERNPOL, ZERNFUN. 9Vo*AK'&U  
    YoBe!-E  
    %   Paul Fricker 11/13/2006 ESYF4-d+  
    zTc*1(^  
    </u=<^ire  
    % Check and prepare the inputs: fP\q?X@]E  
    % ----------------------------- >H ?k0M`L  
    if min(size(p))~=1 hS&l4 \I'Z  
        error('zernfun2:Pvector','Input P must be vector.') D~#%^a+Aq_  
    end /(.:l +[w[  
    iaLZ|\`3a  
    if any(p)>35 6|~^P!&  
        error('zernfun2:P36', ... ?)186dp  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... zo8D"  
               '(P = 0 to 35).']) ~?FhQd\Q  
    end MX8|;t  
    /KNDo^P  
    % Get the order and frequency corresonding to the function number: v?Utz~lQ  
    % ---------------------------------------------------------------- .6xMLo,R  
    p = p(:); {_/o' 6  
    n = ceil((-3+sqrt(9+8*p))/2); Fm,` ]CO  
    m = 2*p - n.*(n+2); ixSr*+  
    kwL|gO1L  
    % Pass the inputs to the function ZERNFUN: Od)y4nr3~  
    % ---------------------------------------- E2d'P  
    switch nargin x~vNUyEN)  
        case 3 (zsv!U  
            z = zernfun(n,m,r,theta); ][ IOlR  
        case 4 &N._}ts  
            z = zernfun(n,m,r,theta,nflag); J=k=cFUX  
        otherwise }ML2-k  
            error('zernfun2:nargin','Incorrect number of inputs.') EK Vcz'w  
    end \2 e^x  
     tH44\~  
    % 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) dpw-a4o}  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. 0hV#]`9`gN  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of & UOxS W  
    %   order N and frequency M, evaluated at R.  N is a vector of 0B7G:X0  
    %   positive integers (including 0), and M is a vector with the YLGE{bS  
    %   same number of elements as N.  Each element k of M must be a YnRO>`  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) HFZ'xp|3dn  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is @,TIw[p  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix XhHgXVVGG<  
    %   with one column for every (N,M) pair, and one row for every k#Sr;"  
    %   element in R. C| ~ A]wc=  
    % .i I{  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- >&KH!:OX|  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is rZJJ\ , |  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to  3Iv^  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 C2"^YRN,  
    %   for all [n,m]. uC^)#Y\"  
    % =g9n =spAn  
    %   The radial Zernike polynomials are the radial portion of the YWl#!"-  
    %   Zernike functions, which are an orthogonal basis on the unit 8qg%>ZU4d  
    %   circle.  The series representation of the radial Zernike SL-2^\R  
    %   polynomials is H.ksI;,  
    % :5,~CtF5 `  
    %          (n-m)/2  pE)NSZ  
    %            __ >5Yn`Fc5  
    %    m      \       s                                          n-2s Y~g{9 <!  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r RI&O@?+U  
    %    n      s=0 J W@6m  
    % X]p3?"7  
    %   The following table shows the first 12 polynomials. Fm0d0j  
    % 5 ix*wu`,  
    %       n    m    Zernike polynomial    Normalization PJC(:R(j  
    %       --------------------------------------------- W3tin3__  
    %       0    0    1                        sqrt(2) +ig%_QED[\  
    %       1    1    r                           2 k773h`;  
    %       2    0    2*r^2 - 1                sqrt(6) kg]6q T;Y  
    %       2    2    r^2                      sqrt(6) ly17FLJ].  
    %       3    1    3*r^3 - 2*r              sqrt(8) .Rk8qRB  
    %       3    3    r^3                      sqrt(8) I]v2-rB&-  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) z/ 1$G"  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) :}zyd;Rc  
    %       4    4    r^4                      sqrt(10) >,{s Fc  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) hi1Ial\Y  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) U]sAYp^$  
    %       5    5    r^5                      sqrt(12) QPDh!A3T  
    %       --------------------------------------------- pD%(Y^h?  
    % PlzM`g$A  
    %   Example: q>2bkcGY#  
    % g %f5hy  
    %       % Display three example Zernike radial polynomials OB(o OPH  
    %       r = 0:0.01:1; I/Vw2  
    %       n = [3 2 5]; [ ulub|  
    %       m = [1 2 1]; PR.3EL  
    %       z = zernpol(n,m,r); UPuoIfuqI  
    %       figure 3 }fOb  
    %       plot(r,z) mZR3Hl$  
    %       grid on 9;e!r DW,#  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest')  p@ ^G)x  
    % Z v@nK%#J  
    %   See also ZERNFUN, ZERNFUN2. 2CzhaO  
    SV>tw`2  
    % A note on the algorithm. 5[ zN M  
    % ------------------------ ;t{q]"? W  
    % The radial Zernike polynomials are computed using the series u1%URen[x  
    % representation shown in the Help section above. For many special ~(^P(  
    % functions, direct evaluation using the series representation can xak)YOLRV  
    % produce poor numerical results (floating point errors), because X/~uF 9a'<  
    % the summation often involves computing small differences between <=*f  
    % large successive terms in the series. (In such cases, the functions y!#-[K:  
    % are often evaluated using alternative methods such as recurrence oFWb.t9<  
    % relations: see the Legendre functions, for example). For the Zernike LORcf1X/  
    % polynomials, however, this problem does not arise, because the Z10Vx2B  
    % polynomials are evaluated over the finite domain r = (0,1), and 8z#Qp(he  
    % because the coefficients for a given polynomial are generally all  z% wh|q  
    % of similar magnitude. 4nsJZo#S/  
    % ~5N}P>4 *  
    % ZERNPOL has been written using a vectorized implementation: multiple WA`A/`taT  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M]  arYq$~U  
    % values can be passed as inputs) for a vector of points R.  To achieve ljKIxSvCFp  
    % this vectorization most efficiently, the algorithm in ZERNPOL qiNVaV\wr|  
    % involves pre-determining all the powers p of R that are required to JXB)'d0  
    % compute the outputs, and then compiling the {R^p} into a single l[/`kK  
    % matrix.  This avoids any redundant computation of the R^p, and ~#pATPW@(  
    % minimizes the sizes of certain intermediate variables. SGAzeymw  
    % *LEy# N  
    %   Paul Fricker 11/13/2006 *&NP?-E  
    RuPnWx!  
    .e~"+Pe6b  
    % Check and prepare the inputs: ` =>}*GS  
    % ----------------------------- dvB=Zk]m  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) #CC5+  
        error('zernpol:NMvectors','N and M must be vectors.') =>$)F 4LW  
    end 6X \g7bg  
    n=.P46|  
    if length(n)~=length(m) xa~]t<2  
        error('zernpol:NMlength','N and M must be the same length.') o:Fq|?/e  
    end T }#iXgyx  
    }s~c(sL?;  
    n = n(:); Q\$3l'W  
    m = m(:); }Pcm'o_wT  
    length_n = length(n); rW{!8FhI  
    .IeO+RDQ  
    if any(mod(n-m,2)) M-o'`e'  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') 'R 7 \  
    end -> cL)  
    FZHA19Kb  
    if any(m<0) z#*.9/y\^R  
        error('zernpol:Mpositive','All M must be positive.') [l23b{  
    end #{cy(&cz  
    r_T)| ||v  
    if any(m>n) @-$8)?`q  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') U$OZkHA[  
    end 3!CUJs/W  
    UqP{Cyy{  
    if any( r>1 | r<0 ) =oBpS=<7  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.')  ;l$$!PJ  
    end |mEWN/@C  
    MEDh  
    if ~any(size(r)==1) %* "+kw Z  
        error('zernpol:Rvector','R must be a vector.') ~>u .d  
    end bZnOX*y]  
    -#v~;Ci  
    r = r(:); B~]Kqp7yU  
    length_r = length(r); }3(!kW  
    XM$ ~HG  
    if nargin==4 oZ'a}kF  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); y* +y&  
        if ~isnorm /R# zu_i  
            error('zernpol:normalization','Unrecognized normalization flag.') /"{d2  
        end 2\xw2VQ@P  
    else 4EB\R"rWXf  
        isnorm = false; @*6fEG{,q  
    end ^m>4<~/  
    1QoW/X'>.  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 8Y($ F2  
    % Compute the Zernike Polynomials n]4E>/\  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% nS+Rbhs  
    UC!mp?   
    % Determine the required powers of r: |L2>|4  
    % ----------------------------------- ?cV,lak  
    rpowers = []; {;yO3];Hqw  
    for j = 1:length(n) <FT7QO$I  
        rpowers = [rpowers m(j):2:n(j)]; R<|\Z@z  
    end a'J0}j!  
    rpowers = unique(rpowers); pjeNBSu6  
    E7Cobpm  
    % Pre-compute the values of r raised to the required powers, U&^q#['  
    % and compile them in a matrix: kCBtK?g  
    % ----------------------------- q W(@p`  
    if rpowers(1)==0 QS#@xhH  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); T ,lM(2S[  
        rpowern = cat(2,rpowern{:}); =2R4Z8G  
        rpowern = [ones(length_r,1) rpowern]; Bx?3E^!T  
    else xGd60"w2  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); "Y&I#&$b\  
        rpowern = cat(2,rpowern{:}); o@meogkL  
    end =ZgueUz,  
    =KE7NXu]-  
    % Compute the values of the polynomials: vrs  
    % -------------------------------------- "hIYf7r##  
    z = zeros(length_r,length_n); q<YM,%mgj  
    for j = 1:length_n Oa[  
        s = 0:(n(j)-m(j))/2; ",#.?vT`  
        pows = n(j):-2:m(j); -]N2V'QB  
        for k = length(s):-1:1 k/K)nH@)  
            p = (1-2*mod(s(k),2))* ... (NJ{>@&  
                       prod(2:(n(j)-s(k)))/          ... jPh<VVQ$@  
                       prod(2:s(k))/                 ... 5y#,z`S  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... (.J/Ql0Y  
                       prod(2:((n(j)+m(j))/2-s(k))); k8gH#ENNK  
            idx = (pows(k)==rpowers); O NabL.CV  
            z(:,j) = z(:,j) + p*rpowern(:,idx); [,=?e  
        end eg[EFI.h  
         CK_dEh2c  
        if isnorm >M<3!?fW)  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); (Y1*Bs[l  
        end 4_#$k{  
    end g"-j/ c   
    ~f<'] zXv  
    % 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)  9=>q0D2  
    ZvRa"j  
    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。 G+dq */  
    0? {ADQz  
    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)