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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 uXxc2}  
    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! 6.]x@=Wm  
     
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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 UL0%oJ#  
    function z = zernfun(n,m,r,theta,nflag) RfP>V/jy5  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. fFG, ^;7-O  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N n[zP}YRr  
    %   and angular frequency M, evaluated at positions (R,THETA) on the ]fH U/%  
    %   unit circle.  N is a vector of positive integers (including 0), and -eKi}e  
    %   M is a vector with the same number of elements as N.  Each element :r^c_Ui  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) 3JuWG\r)l  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, S"FIQ&n  
    %   and THETA is a vector of angles.  R and THETA must have the same 1i;-mYGaMn  
    %   length.  The output Z is a matrix with one column for every (N,M) <I.anIB:U  
    %   pair, and one row for every (R,THETA) pair. N 3IF j  
    % RhM]OJd'  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike `I$'Lp#5  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), \79KU   
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral 2#z6=M~A  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, t#s?:  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized q'kZ3 G   
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. _= RA-qZ"  
    % x\qS|q\N  
    %   The Zernike functions are an orthogonal basis on the unit circle. nZ?BC O  
    %   They are used in disciplines such as astronomy, optics, and M{Ss?G4H  
    %   optometry to describe functions on a circular domain. as\6XW$;Q  
    % v,t&t9}/  
    %   The following table lists the first 15 Zernike functions. !,}W|(P)  
    % A^+G w\  
    %       n    m    Zernike function           Normalization J[ 9yQ  
    %       -------------------------------------------------- =ogzq.+|  
    %       0    0    1                                 1 bH}6N>Fp  
    %       1    1    r * cos(theta)                    2 4&r+K`C0  
    %       1   -1    r * sin(theta)                    2 Kg0Vbzvb  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) V|.3Z\(  
    %       2    0    (2*r^2 - 1)                    sqrt(3) H\A!oB,sw  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) HC,YmO:df"  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) ODn6%fp%  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) JZ6{W  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) XGE:ZVpW  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) M7"I]$|\  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) /E'c y  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ^p#f B4z  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) f$a%&X6"-  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) td^2gjr^5  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) Q+/:5Z C  
    %       -------------------------------------------------- %)[mbb  
    % QF/A-[V  
    %   Example 1: h4C DZ  
    % 2XJn3wPi  
    %       % Display the Zernike function Z(n=5,m=1) w[w{~`([",  
    %       x = -1:0.01:1; ;2"#X2B  
    %       [X,Y] = meshgrid(x,x); YH33E~f  
    %       [theta,r] = cart2pol(X,Y); 55xv+|k  
    %       idx = r<=1; KE\p|Xi  
    %       z = nan(size(X)); |B&KT  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); V6l*!R  
    %       figure g ]|K@sm  
    %       pcolor(x,x,z), shading interp mIVnc`3s  
    %       axis square, colorbar @/}{Trmg/  
    %       title('Zernike function Z_5^1(r,\theta)') M0`nr}g  
    % 5Cxh >,k  
    %   Example 2: BCV<( @c  
    % WjZJQK  
    %       % Display the first 10 Zernike functions =T5vu~[J/e  
    %       x = -1:0.01:1; )&di c6r  
    %       [X,Y] = meshgrid(x,x); wH1 E7LY|R  
    %       [theta,r] = cart2pol(X,Y); xq_%|p}y  
    %       idx = r<=1; xlVQ[Mt  
    %       z = nan(size(X)); "?_adot5v  
    %       n = [0  1  1  2  2  2  3  3  3  3]; G)\s{qk  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; )@.bkzW  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; Iu6KW:x  
    %       y = zernfun(n,m,r(idx),theta(idx)); @AUx%:}0Y:  
    %       figure('Units','normalized') !=C4=xv  
    %       for k = 1:10 87%t=X  
    %           z(idx) = y(:,k); ^_b+o  
    %           subplot(4,7,Nplot(k)) q q}EXq^  
    %           pcolor(x,x,z), shading interp !}wJ+R ^2  
    %           set(gca,'XTick',[],'YTick',[]) Eq zS={Olj  
    %           axis square a5WVDh, cR  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) >B$ZKE  
    %       end ~Nf0 1,F  
    % \dj&4u3  
    %   See also ZERNPOL, ZERNFUN2. ! *\)7D  
    MfUG@  
    %   Paul Fricker 11/13/2006 N#{d_v^H?d  
    /km^IH  
    b Jt397  
    % Check and prepare the inputs: ]c{Zh?0  
    % ----------------------------- a9z|ef  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 5;{d*L  
        error('zernfun:NMvectors','N and M must be vectors.') ,Iq+v  
    end u2K{3+r`'  
    j &)Xi^^  
    if length(n)~=length(m) TF 6_4t6  
        error('zernfun:NMlength','N and M must be the same length.') x8%Q TTY  
    end a?6 r4u0  
    ]d?`3{h9LD  
    n = n(:); :~loy'  
    m = m(:); T/G1v;]  
    if any(mod(n-m,2)) Z"Z&X0O j  
        error('zernfun:NMmultiplesof2', ... $wU.GM$t~  
              'All N and M must differ by multiples of 2 (including 0).') p,}-8#K[  
    end & Sy0Of  
    B9|!8V  
    if any(m>n) '5wa"/ ?w  
        error('zernfun:MlessthanN', ... V1Dwh@iS  
              'Each M must be less than or equal to its corresponding N.') dA> t  
    end #6'oor X  
    K^tM$l\  
    if any( r>1 | r<0 ) {EbR =  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') G T#hqt'1x  
    end I z~#G6]M  
    N kp>yVj  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) tu6oa[s  
        error('zernfun:RTHvector','R and THETA must be vectors.') p3I{  
    end b!SGQv(^M  
     Y2vzK;  
    r = r(:); cv;&ff2%?  
    theta = theta(:); w[\*\'Vm0  
    length_r = length(r);  'vj45b  
    if length_r~=length(theta) leyhiL<  
        error('zernfun:RTHlength', ... t3u"2B7oG  
              'The number of R- and THETA-values must be equal.') HZCEr6}(  
    end Nkn0G _  
    I<|)uK7  
    % Check normalization: *#e%3N05_  
    % -------------------- Da1BxbDeI  
    if nargin==5 && ischar(nflag) o%X_V!B{V  
        isnorm = strcmpi(nflag,'norm'); 7CYu"+Ea  
        if ~isnorm R'qB-v.  
            error('zernfun:normalization','Unrecognized normalization flag.') %1SA!1>j  
        end 1i#uKKwE  
    else ;YNN)P%"  
        isnorm = false; 0!veLXeK!  
    end G/_#zIN`8M  
    2<>n8K  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% E4[ |=<  
    % Compute the Zernike Polynomials ZH/^``[.  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% /A}3kTp  
    "C.'_H!Ex  
    % Determine the required powers of r: kt%9PGw  
    % ----------------------------------- "o#"u[W ,  
    m_abs = abs(m); }$#e&&)n  
    rpowers = []; K CJ zE>  
    for j = 1:length(n) r4dG83qg  
        rpowers = [rpowers m_abs(j):2:n(j)]; pSkP8'  ?  
    end K`* 8 *k{  
    rpowers = unique(rpowers); &+6XdhX  
    #rMMOu9r2  
    % Pre-compute the values of r raised to the required powers, i0{pm q  
    % and compile them in a matrix: !1+L0,I6  
    % ----------------------------- ma@ws,H  
    if rpowers(1)==0  dKDtj:  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); GZ/.eYE  
        rpowern = cat(2,rpowern{:}); I? dh"*Js&  
        rpowern = [ones(length_r,1) rpowern]; y/mxdP w  
    else ur={+0 y  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); \D?6_ ,O  
        rpowern = cat(2,rpowern{:}); T[U&Y`3g  
    end {=IK(H  
    (ZQ{%-i?qR  
    % Compute the values of the polynomials: GV6!`@<  
    % -------------------------------------- WRZi^B8 @  
    y = zeros(length_r,length(n)); }cgEC-  
    for j = 1:length(n) WqqrfzlM  
        s = 0:(n(j)-m_abs(j))/2; n9gj{]%  
        pows = n(j):-2:m_abs(j); HKv:)h{ ?  
        for k = length(s):-1:1 tf|/_Y2  
            p = (1-2*mod(s(k),2))* ... $5r[YdnY<  
                       prod(2:(n(j)-s(k)))/              ... GBu&2}  
                       prod(2:s(k))/                     ... |!8[Vg^Wh  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... f3lFpS  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); ` B) ~  
            idx = (pows(k)==rpowers); {5 -4^|!  
            y(:,j) = y(:,j) + p*rpowern(:,idx); pA"x4\s   
        end T({:Y. A;  
         T9KzVxHp5  
        if isnorm Z/sB72K1  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); m46Q%hwV  
        end AR`X2m '  
    end K6@QZc5.!  
    % END: Compute the Zernike Polynomials gR.zL>=_5e  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ;nji<  
    1d7oR`qr  
    % Compute the Zernike functions: s6OnHX\it7  
    % ------------------------------ Mr<2I  
    idx_pos = m>0; ~ 6 1?nu  
    idx_neg = m<0; o;{  
    p&B98c  
    z = y; e{:P!r aM  
    if any(idx_pos) H!4!1J.=xw  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); Zq2dCp%  
    end n*CH,fih:  
    if any(idx_neg) g)!B};AA  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ~;aSX1   
    end qCSJ=T;  
    yX$I<L<Suz  
    % 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) &1%W-&bc6  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. Z{EHV7  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated -. L)-%wIV  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive as)2ny!u  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, @SDsd^N{2P  
    %   and THETA is a vector of angles.  R and THETA must have the same xM9EO(u  
    %   length.  The output Z is a matrix with one column for every P-value, zPe4WE|  
    %   and one row for every (R,THETA) pair. $/}*HWVZ  
    % VE& ?Zd~  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike 'v* =}k  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) 'BpK(PlUh  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) k[6@\D-  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 AT<gV/1l  
    %   for all p. A5!j rSyv  
    % wA+J49  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 k.rP}76  
    %   Zernike functions (order N<=7).  In some disciplines it is [V  T&  
    %   traditional to label the first 36 functions using a single mode l\- 1W2  
    %   number P instead of separate numbers for the order N and azimuthal l:a+o gm3  
    %   frequency M. K%Mm'$fTw  
    % FviLlly6  
    %   Example: ik+qx~+`Qv  
    % n <6}  
    %       % Display the first 16 Zernike functions A-~#ydv  
    %       x = -1:0.01:1; L5(rP\B  
    %       [X,Y] = meshgrid(x,x); j?i Ur2  
    %       [theta,r] = cart2pol(X,Y); &9$0v"`H  
    %       idx = r<=1; LZMdW #,[  
    %       p = 0:15; )UI$ s"  
    %       z = nan(size(X)); [z]@ <99/  
    %       y = zernfun2(p,r(idx),theta(idx)); $yIcut7  
    %       figure('Units','normalized') }Y(Q7l  
    %       for k = 1:length(p) jj0@ez{3  
    %           z(idx) = y(:,k); O_ nk8  
    %           subplot(4,4,k) b,Ed}Ir  
    %           pcolor(x,x,z), shading interp }Jk.c~P)  
    %           set(gca,'XTick',[],'YTick',[]) u6'vzLmM  
    %           axis square Ms<^_\iPN  
    %           title(['Z_{' num2str(p(k)) '}']) 95_ ?F7}9  
    %       end 9r fR  
    % }; +'  
    %   See also ZERNPOL, ZERNFUN. 'X_iiR8n@p  
    : :uD%a zd  
    %   Paul Fricker 11/13/2006 %/,PY>:|  
    ?;0=>3p*0  
    4\pi<#X  
    % Check and prepare the inputs: ;Z-Cn.  
    % ----------------------------- ;*:d)'A  
    if min(size(p))~=1 &O#a==F!(  
        error('zernfun2:Pvector','Input P must be vector.') U; ?%rM6  
    end |H2{%!  
    kI<C\ *N  
    if any(p)>35 qlIC{:E0  
        error('zernfun2:P36', ... qDM/ 6xO  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... k)'hNk"x  
               '(P = 0 to 35).']) $G"PZ7  
    end K)]7e?:Wu  
    Y:FV+ SI  
    % Get the order and frequency corresonding to the function number: X8ev uN  
    % ---------------------------------------------------------------- U_ V0  
    p = p(:); N;F1Z-9  
    n = ceil((-3+sqrt(9+8*p))/2); VD,F?L!  
    m = 2*p - n.*(n+2); mbsdiab#N  
    ,yWTk ql  
    % Pass the inputs to the function ZERNFUN: ZF51|b  
    % ---------------------------------------- uwj/]#`  
    switch nargin Oe'Nn250  
        case 3 '# "Z$  
            z = zernfun(n,m,r,theta); J@oGAa%3)  
        case 4 M`FsKK`  
            z = zernfun(n,m,r,theta,nflag); 5w gtc~  
        otherwise la8se=^  
            error('zernfun2:nargin','Incorrect number of inputs.') H#E   
    end R# 8D}5[&  
    ,vrdtL  
    % 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) ]RPv@z:V  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. L-Xd3RCD  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of && ecq   
    %   order N and frequency M, evaluated at R.  N is a vector of %pc0a^iB  
    %   positive integers (including 0), and M is a vector with the <.l5>mgkCw  
    %   same number of elements as N.  Each element k of M must be a 3a:(\:?z  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) BC(f1  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is ,'Y*e[  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix kmy?`P10(z  
    %   with one column for every (N,M) pair, and one row for every yZb@  
    %   element in R. u7^Z7; J  
    % cK(}B_D$  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- |O+R%'z'<  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is XC?H  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to A{>]M@QC2  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 8<^[xe  
    %   for all [n,m]. R{#-IH="  
    % ,<R/x[  
    %   The radial Zernike polynomials are the radial portion of the 3dcZ1Yrn  
    %   Zernike functions, which are an orthogonal basis on the unit n >xhT r<  
    %   circle.  The series representation of the radial Zernike Wxjk}&+pVa  
    %   polynomials is e:AB!k^xp$  
    % *W(b=u  
    %          (n-m)/2 bLCrh(<  
    %            __ ,PJl32  
    %    m      \       s                                          n-2s (-B0fqh=G  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r To}L%)  
    %    n      s=0 /p<mD-:.M  
    % i2N*3X~  
    %   The following table shows the first 12 polynomials. 2EG"xA5%  
    % $]|_xG-6{  
    %       n    m    Zernike polynomial    Normalization b7aAP*$  
    %       --------------------------------------------- /iy2j8: z  
    %       0    0    1                        sqrt(2) Bpo~x2p  
    %       1    1    r                           2 { zlq6z  
    %       2    0    2*r^2 - 1                sqrt(6) 9rn!U2  
    %       2    2    r^2                      sqrt(6) ]K XknEaxl  
    %       3    1    3*r^3 - 2*r              sqrt(8) sFSrMI#R  
    %       3    3    r^3                      sqrt(8) @faf  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) RZOk.~[v  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) d>@{!c-  
    %       4    4    r^4                      sqrt(10) e Yyl=YW  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) (niZN_qv  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) }mu8fm'  
    %       5    5    r^5                      sqrt(12) BAzc'x&<  
    %       --------------------------------------------- - /#3U{O  
    % +(PtOo.  
    %   Example: p"q-sMYl  
    % ai#EFo+#  
    %       % Display three example Zernike radial polynomials #g~~zwx/N  
    %       r = 0:0.01:1; uBl&|yvxB  
    %       n = [3 2 5]; 3AWB Y .  
    %       m = [1 2 1]; *eUL1m8Y  
    %       z = zernpol(n,m,r); )byQ=-< 1  
    %       figure oJ6 d:  
    %       plot(r,z) m6lNZb]  
    %       grid on d[TcA2nF  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') KC}B\~ +  
    % cTRCQ+W6:  
    %   See also ZERNFUN, ZERNFUN2. H#w?$?nIWu  
    5wGyM10  
    % A note on the algorithm. yQou8P=%  
    % ------------------------ dr'6N1B@  
    % The radial Zernike polynomials are computed using the series ;pAkdX&b  
    % representation shown in the Help section above. For many special B-@f.NO/s  
    % functions, direct evaluation using the series representation can `e`4[I  
    % produce poor numerical results (floating point errors), because pKr3(5~  
    % the summation often involves computing small differences between I62Yg p$K  
    % large successive terms in the series. (In such cases, the functions Qf=%%5+?8  
    % are often evaluated using alternative methods such as recurrence ZJR{c5TE  
    % relations: see the Legendre functions, for example). For the Zernike Yd/qcC(&  
    % polynomials, however, this problem does not arise, because the T,WWQm  
    % polynomials are evaluated over the finite domain r = (0,1), and t{?_]2vl  
    % because the coefficients for a given polynomial are generally all R L)'m  
    % of similar magnitude. K''b)v X4  
    % >!bYuVHA  
    % ZERNPOL has been written using a vectorized implementation: multiple M~"]h:m&'v  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] K =7(=Y{  
    % values can be passed as inputs) for a vector of points R.  To achieve Kl+*Sp!  
    % this vectorization most efficiently, the algorithm in ZERNPOL 0n(Q@O  
    % involves pre-determining all the powers p of R that are required to 0&5}[9?V'  
    % compute the outputs, and then compiling the {R^p} into a single JpSS[pOg  
    % matrix.  This avoids any redundant computation of the R^p, and i>!f|<  
    % minimizes the sizes of certain intermediate variables. f kP WGd  
    % ]'M4Unu#@  
    %   Paul Fricker 11/13/2006 @XmMD6{<  
    aQRZyE}  
    !knYD}Rxd  
    % Check and prepare the inputs: $f)Y !<bC  
    % ----------------------------- )dlt$VX  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) k] f 7 3r  
        error('zernpol:NMvectors','N and M must be vectors.') a,}{f]  
    end ](Sp0t  
    dF FB\|e;0  
    if length(n)~=length(m) JVXBm]  
        error('zernpol:NMlength','N and M must be the same length.') }>tUkXlhJ<  
    end 0K#dWc}"a  
    `8'|g8,wb0  
    n = n(:); 08.dV<P  
    m = m(:); B<0lif|  
    length_n = length(n); D ORFK  
    @``!P&h  
    if any(mod(n-m,2)) $6Ty~.RP5H  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') BF]b\/I  
    end 7J 0!v q  
    i 5_g z>  
    if any(m<0) L[O+9Yh  
        error('zernpol:Mpositive','All M must be positive.') ,u\M7,a^  
    end ueqR@i  
    P@*whjPmo  
    if any(m>n) vWj|[| <rX  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') IHB{US1G  
    end 5gEUE{S  
    OSq"q-Q  
    if any( r>1 | r<0 ) 2QBq  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') )IhI~,0Nmj  
    end q@^=im  
    xpSMbX{e  
    if ~any(size(r)==1) +  1v@L  
        error('zernpol:Rvector','R must be a vector.') /yH:ur  
    end l(<o,Uv[`  
    pX2 Ki^)]  
    r = r(:); Y> 7/>x6  
    length_r = length(r); rV1JJ.I  
    * .Kc-f4mP  
    if nargin==4 J#JZ^59lOS  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); O(!wDnhc  
        if ~isnorm &&>OhH`  
            error('zernpol:normalization','Unrecognized normalization flag.') .CmwR$u&  
        end FC)aR[  
    else ogQbST  
        isnorm = false; 'rz*mR8  
    end 8"p>_K=  
    M%6{A+(  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% tq1h1  
    % Compute the Zernike Polynomials `U?;9!|;6  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% X,gXgxP\  
    *S<>_R 8  
    % Determine the required powers of r: [kn`~hI  
    % ----------------------------------- C96|T>bk  
    rpowers = []; -6DfM,  
    for j = 1:length(n) Z*kg= hs^  
        rpowers = [rpowers m(j):2:n(j)]; w3B*%x)  
    end %>pglI  
    rpowers = unique(rpowers); pU}>}  
    kgYa0 e5  
    % Pre-compute the values of r raised to the required powers, *6=[Hmygi  
    % and compile them in a matrix: 44b'40  
    % ----------------------------- T!J\Dm-  
    if rpowers(1)==0 jaNkWTm :  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); )e Ub@Eu  
        rpowern = cat(2,rpowern{:}); X $cW!a  
        rpowern = [ones(length_r,1) rpowern]; K b{  
    else \N7 E!82  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 9 ?h)U|J?G  
        rpowern = cat(2,rpowern{:}); ?p6+?\H  
    end jJg 'Y:K9q  
    <A!v'Y  
    % Compute the values of the polynomials: |J~;yO SD  
    % -------------------------------------- ^<ayPV)+  
    z = zeros(length_r,length_n); 4qiG>^h9  
    for j = 1:length_n GHH1jJ_[7  
        s = 0:(n(j)-m(j))/2; I~#'76L[  
        pows = n(j):-2:m(j); %*:-4K  
        for k = length(s):-1:1 g+)T\_#u  
            p = (1-2*mod(s(k),2))* ... py@5]n%  
                       prod(2:(n(j)-s(k)))/          ... ,mjwQ6:Ny  
                       prod(2:s(k))/                 ... Qt!l-/flh  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... :? yv0Iu  
                       prod(2:((n(j)+m(j))/2-s(k))); FFP>Y*v(  
            idx = (pows(k)==rpowers); +&Sf$t 1  
            z(:,j) = z(:,j) + p*rpowern(:,idx); $t[`}I }  
        end E!jM&\Zj  
         RqH"+/wR  
        if isnorm K4A=lD+  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); vek9. 4! ]  
        end ])T*T$u  
    end eK4\v:oG1  
    l[rIjyL@  
    % 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)  Q Bc\=}  
    oQBfDD0  
    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。 J'sVT{@GS  
    >t.2!Z_RQ  
    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)