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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 pbCj ^  
    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! G)|Xj70  
     
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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 T_)g/,5>  
    function z = zernfun(n,m,r,theta,nflag) M(^_/ 1Z  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. p4F%FS:`  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N z''ejq  
    %   and angular frequency M, evaluated at positions (R,THETA) on the $7QGi|W*k  
    %   unit circle.  N is a vector of positive integers (including 0), and oE.Ckz~*d  
    %   M is a vector with the same number of elements as N.  Each element ;J@U){R  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) A"B#t"  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, xfF;u9$;  
    %   and THETA is a vector of angles.  R and THETA must have the same GE8.{P  
    %   length.  The output Z is a matrix with one column for every (N,M) s=e`}4  
    %   pair, and one row for every (R,THETA) pair. m#$$xG  
    % 9u6VN]divB  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 0 <E2^  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), Z2U6<4?1%  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral n^q%_60H   
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, \0W0o5c$  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized 1{ H=The  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. \.aKxj5  
    % {?;qy\m]o  
    %   The Zernike functions are an orthogonal basis on the unit circle. P BVF'~f@j  
    %   They are used in disciplines such as astronomy, optics, and <NEz{1Z  
    %   optometry to describe functions on a circular domain. d,b]#fj  
    % yq?\.~ax  
    %   The following table lists the first 15 Zernike functions. '3w%K+eJY  
    % <vE|QxpR  
    %       n    m    Zernike function           Normalization 4(91T  
    %       -------------------------------------------------- ~,_@|,)  
    %       0    0    1                                 1 xHCdtloi?I  
    %       1    1    r * cos(theta)                    2 e^<'H  
    %       1   -1    r * sin(theta)                    2 'yosDT2{#  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) <PFF\NE9  
    %       2    0    (2*r^2 - 1)                    sqrt(3) ~ulcLvm:i  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) TI}a$I*  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) xk  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) Gshy$'_e  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) Bq;GO  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) +1a3^A\  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) cij8'( "+!  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) PqIskv+  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5)  &1f3e  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ?@z/#3b  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) !PA><F  
    %       -------------------------------------------------- !>"fDz<w`  
    % k*u6'IKi.4  
    %   Example 1: _s+G02/q1  
    % diNAT`|?#  
    %       % Display the Zernike function Z(n=5,m=1) b9ud8wLE[  
    %       x = -1:0.01:1; (&1.!R[X  
    %       [X,Y] = meshgrid(x,x); @tJ4^<`P{  
    %       [theta,r] = cart2pol(X,Y); r7sA;Y\  
    %       idx = r<=1; 2">de/jS  
    %       z = nan(size(X)); j 7^A%9  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); k+%&dEE|vH  
    %       figure S[gACEZ =  
    %       pcolor(x,x,z), shading interp W':b6}?  
    %       axis square, colorbar qDTdYf  
    %       title('Zernike function Z_5^1(r,\theta)') oB%_yy+  
    % u(fZ^  
    %   Example 2: @( \R@`#  
    % c:52pYf+  
    %       % Display the first 10 Zernike functions qco uZO  
    %       x = -1:0.01:1; 8{]nS8i  
    %       [X,Y] = meshgrid(x,x); o<J6KTLv  
    %       [theta,r] = cart2pol(X,Y); 6O/c%1VHA3  
    %       idx = r<=1; >gs_Bzy]  
    %       z = nan(size(X)); b\KbF/ T  
    %       n = [0  1  1  2  2  2  3  3  3  3]; mo3A*|U  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; |d z2Drc  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; BG8/  
    %       y = zernfun(n,m,r(idx),theta(idx)); 1hlU 6 =Y  
    %       figure('Units','normalized') k$ T  
    %       for k = 1:10 _Rb2jq(&0  
    %           z(idx) = y(:,k); ij|>hQC5i  
    %           subplot(4,7,Nplot(k)) {NQCe0S+p  
    %           pcolor(x,x,z), shading interp `|Hk+V  
    %           set(gca,'XTick',[],'YTick',[]) wx[m-\  
    %           axis square qp)Wt6 k?  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) o$Ylqb#  
    %       end o<iU;15  
    % !yVY[  
    %   See also ZERNPOL, ZERNFUN2. : 8j7}'  
    L&y"oAp<  
    %   Paul Fricker 11/13/2006 ?G,gPb  
    \EU^`o+  
    x@QNMK.7  
    % Check and prepare the inputs: FF#+d~$z  
    % ----------------------------- w3"L5;oH  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) (X Oz0.W  
        error('zernfun:NMvectors','N and M must be vectors.') S6_:\Q  
    end _~MX~M3MB  
    |`Noj+T47I  
    if length(n)~=length(m) "/RMIS K[;  
        error('zernfun:NMlength','N and M must be the same length.') AD^I1 ]2f  
    end 'e' p`*  
    GB^`A  
    n = n(:); P$0c{B4I  
    m = m(:); ;x 2o|#`b  
    if any(mod(n-m,2)) lZ7 $DGe  
        error('zernfun:NMmultiplesof2', ... <G|i5/|7  
              'All N and M must differ by multiples of 2 (including 0).') r#2Fk &Z9  
    end JB].ht  
    z6l'v~\  
    if any(m>n) czU"  
        error('zernfun:MlessthanN', ... ;1PJS_@rX  
              'Each M must be less than or equal to its corresponding N.') 5-$D<}Z  
    end ;3wO1'=  
    enZZ+|h  
    if any( r>1 | r<0 ) p/RT*?<   
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') ZZZ9C#hK^9  
    end wBwTJCX  
    *Cf!p\7!  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) V" 8 G-dK  
        error('zernfun:RTHvector','R and THETA must be vectors.') ZAU#^bEQB  
    end KK3iui  
    "f_qG2A{  
    r = r(:); );VuZsmi  
    theta = theta(:); s[y.gR.(  
    length_r = length(r); D>7J[ Yxg-  
    if length_r~=length(theta) c`p '5qz  
        error('zernfun:RTHlength', ... t"YsIOT:O"  
              'The number of R- and THETA-values must be equal.') k_,& Q?GtU  
    end (DY[OIHI  
    ^iJyo&I  
    % Check normalization: *9$SFe|&n:  
    % -------------------- bKGX> %-  
    if nargin==5 && ischar(nflag) Y8]@y0(  
        isnorm = strcmpi(nflag,'norm'); ~gff{Nzk  
        if ~isnorm @`C'tfG/4  
            error('zernfun:normalization','Unrecognized normalization flag.') % g  
        end bTrusSAl  
    else z8awND  
        isnorm = false; j|wN7@Zc  
    end $.,B2}'  
    1n!:L!,`  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% '!`\!=j-`  
    % Compute the Zernike Polynomials [bP^RY:  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% V0_tk"  
    @WS77d~S  
    % Determine the required powers of r: 6Q [  
    % ----------------------------------- ]q{_i   
    m_abs = abs(m); 1J/'R37lP  
    rpowers = []; th[v"qD9G  
    for j = 1:length(n) Vi-Ph;6[  
        rpowers = [rpowers m_abs(j):2:n(j)]; UAhWJ$(C  
    end 6{ ]F#ig=  
    rpowers = unique(rpowers); @}g3\xLiK  
    fxPg"R!1i  
    % Pre-compute the values of r raised to the required powers, 3MNM<Ih  
    % and compile them in a matrix: 4xmJQ>/  
    % ----------------------------- 8I/3T  
    if rpowers(1)==0 ,P`NtTN-  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 5X)M)"rq;V  
        rpowern = cat(2,rpowern{:}); Dk^AnMx%_  
        rpowern = [ones(length_r,1) rpowern]; 5kTs7zJ^  
    else G/Sp/I<d  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); M=uT8JB  
        rpowern = cat(2,rpowern{:}); eN,9N]K  
    end }8Y! -qX  
    ,GYQ,9:  
    % Compute the values of the polynomials: .waw=C  
    % -------------------------------------- s__xBY  
    y = zeros(length_r,length(n)); \Dq'~ d  
    for j = 1:length(n) S \]O8#OX  
        s = 0:(n(j)-m_abs(j))/2; "4\  
        pows = n(j):-2:m_abs(j); EwN{|34C  
        for k = length(s):-1:1 h>\C2Q  
            p = (1-2*mod(s(k),2))* ... s<F*kLib  
                       prod(2:(n(j)-s(k)))/              ... d'ZNp2L  
                       prod(2:s(k))/                     ... j@z IJ  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... Mww^  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); /Rq\Mgb  
            idx = (pows(k)==rpowers); >pfeP"[(3  
            y(:,j) = y(:,j) + p*rpowern(:,idx); K9k!P8Rd  
        end ~h3G}EH  
         {V QGfN  
        if isnorm ]A=\P,D  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); OA3J(4!"W  
        end mEd2f^R  
    end 'l.tV7  
    % END: Compute the Zernike Polynomials W34xrm  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% H u;"TG  
    T(*,nJi~9  
    % Compute the Zernike functions: -/JEKw c  
    % ------------------------------ -|m3=#  
    idx_pos = m>0; +112{v=!i  
    idx_neg = m<0; '37 {$VHw  
    Mc@9ivwL#  
    z = y; ZDFq=)0C  
    if any(idx_pos) |?^<=%  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); JKM(fX+  
    end ?`U_|Yo  
    if any(idx_neg) 5 qfvHQ ~M  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ~o^|>]  
    end fAULuF  
    hI86WP9*  
    % 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) DA "V)  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. {:gx*4}q8  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated FTZ=u0  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive d*^JO4'  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, uBbQJvL  
    %   and THETA is a vector of angles.  R and THETA must have the same b\(f>g[  
    %   length.  The output Z is a matrix with one column for every P-value, L }*o8l`  
    %   and one row for every (R,THETA) pair. uy<3B>3~.  
    % 5qnei\~  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike ,H7_eVLWR  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) 89&9VX^A  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) lubsLI  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 qB$-H' j:;  
    %   for all p. 8?nn4]P  
    % -t4:%-wv  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 cn} CI  
    %   Zernike functions (order N<=7).  In some disciplines it is 7He"IJ  
    %   traditional to label the first 36 functions using a single mode "rn  
    %   number P instead of separate numbers for the order N and azimuthal s=jmvvs_V}  
    %   frequency M. W]D YfR,  
    % fxcE1=a  
    %   Example: X <xM '  
    % 8`*5[ L~~/  
    %       % Display the first 16 Zernike functions 1-p#}VX  
    %       x = -1:0.01:1; #a}w&O";  
    %       [X,Y] = meshgrid(x,x); :5~Dca_iU4  
    %       [theta,r] = cart2pol(X,Y); { }/  
    %       idx = r<=1; )jL@GW  
    %       p = 0:15; g4WmUV#wp  
    %       z = nan(size(X)); aftt^h  
    %       y = zernfun2(p,r(idx),theta(idx)); ,5c7jZ5H  
    %       figure('Units','normalized') SdlO]y9E  
    %       for k = 1:length(p) ~},H+A!?  
    %           z(idx) = y(:,k); EcHZ mf  
    %           subplot(4,4,k) rd->@s|4mT  
    %           pcolor(x,x,z), shading interp pA.orx  
    %           set(gca,'XTick',[],'YTick',[]) ^N<aHFF  
    %           axis square (>0`e8v!  
    %           title(['Z_{' num2str(p(k)) '}']) wetu.aMp  
    %       end B@-\.m  
    % zRjbEL  
    %   See also ZERNPOL, ZERNFUN. t_Eivm-,B  
    a^&"gGg  
    %   Paul Fricker 11/13/2006 Jzf+"%lv  
    DL,R~  
    z!6_u@^-  
    % Check and prepare the inputs: 6E) T;R(@  
    % ----------------------------- _]*[TGap  
    if min(size(p))~=1 %t&Lq }e  
        error('zernfun2:Pvector','Input P must be vector.') oX)a6FXK>  
    end .'M.yE~5J  
    2Di~}*9&  
    if any(p)>35 AIOGa<^  
        error('zernfun2:P36', ... |iJz[%  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... RgoF4g+@  
               '(P = 0 to 35).']) i}LQ}35@  
    end vltE2mb  
    h:Gs9]Lvtv  
    % Get the order and frequency corresonding to the function number: ',hoe  
    % ---------------------------------------------------------------- -!+i ^r  
    p = p(:); ruagJS)+  
    n = ceil((-3+sqrt(9+8*p))/2); %. ((4 6)  
    m = 2*p - n.*(n+2); nycJZ}f:wP  
    ~*EipxhstJ  
    % Pass the inputs to the function ZERNFUN: bP$e1I3`  
    % ---------------------------------------- EUw4$Jt^p  
    switch nargin 6 T4"m  
        case 3 _\4r~=`HQ  
            z = zernfun(n,m,r,theta); 3SWDPy  
        case 4 K_U`T;Z\  
            z = zernfun(n,m,r,theta,nflag); iJ58RY  
        otherwise u*l>)_HD  
            error('zernfun2:nargin','Incorrect number of inputs.') '*Y mYU  
    end \|X 1  
    AIl`>ac  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) t`&mszd~T  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. miBCq l@x  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of ~+ae68{p  
    %   order N and frequency M, evaluated at R.  N is a vector of c5f57Z  
    %   positive integers (including 0), and M is a vector with the or ~@!  
    %   same number of elements as N.  Each element k of M must be a z1RHdu0;z  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) vIi&D;  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is x?6^EB|@  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix cJT_Qfxx  
    %   with one column for every (N,M) pair, and one row for every s!09cS  
    %   element in R. r_ 9"^Er  
    % !bK;/)  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- ; mV>k_AG  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is p^{yA"MQ  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to N<(rP1)`v  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 Y edF%  
    %   for all [n,m]. 4u p7 :?  
    % +CEt:KQ   
    %   The radial Zernike polynomials are the radial portion of the |L;Hd.l7^*  
    %   Zernike functions, which are an orthogonal basis on the unit 6EWCJ%_  
    %   circle.  The series representation of the radial Zernike KPK`C0mg@k  
    %   polynomials is WVyq$p/V  
    % Q\~#cLJ/  
    %          (n-m)/2 4`CO>Q  
    %            __ 8/"uS;yP  
    %    m      \       s                                          n-2s AnsJ3C  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r y}QqS/  
    %    n      s=0 50S*_4R  
    % T/L\|_:'  
    %   The following table shows the first 12 polynomials. @ bvWqMa  
    % Q Na*Y@i  
    %       n    m    Zernike polynomial    Normalization `EP-Qlm  
    %       --------------------------------------------- A?ESjMy(R  
    %       0    0    1                        sqrt(2) 1{xkAy0  
    %       1    1    r                           2 zS\m8[+]  
    %       2    0    2*r^2 - 1                sqrt(6) dZJU>o'BG  
    %       2    2    r^2                      sqrt(6) wGz_IL.D  
    %       3    1    3*r^3 - 2*r              sqrt(8) R;/LB^X]  
    %       3    3    r^3                      sqrt(8) yK2>ou  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) [di&N!Ao  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) fK4O N'[R:  
    %       4    4    r^4                      sqrt(10) DqH]FS?]  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) \Zk<|T61$  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) ijNI6_eU  
    %       5    5    r^5                      sqrt(12) K8iQ?  
    %       --------------------------------------------- 8/9YR(H3H  
    % Mb%[Qp60  
    %   Example: RCGpZyl  
    % VDy_s8Z#  
    %       % Display three example Zernike radial polynomials /3`fO^39Ta  
    %       r = 0:0.01:1; .w~L0(  
    %       n = [3 2 5]; vns Mh  
    %       m = [1 2 1]; zy9W{{:P(1  
    %       z = zernpol(n,m,r); ^\PNjj*C i  
    %       figure Sj'.)nz>  
    %       plot(r,z) OdJ=4 x>  
    %       grid on KU0;}GSNX}  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') b@1";+(27  
    % P$A'WEO'  
    %   See also ZERNFUN, ZERNFUN2. 0[OlJMVf  
    6<Zk%[7t  
    % A note on the algorithm. Eid~4a  
    % ------------------------ |i'w"Tz4  
    % The radial Zernike polynomials are computed using the series 0Szt^l7  
    % representation shown in the Help section above. For many special *5'l"YQ@1  
    % functions, direct evaluation using the series representation can %aJ8wYj*  
    % produce poor numerical results (floating point errors), because |fWR[\NU  
    % the summation often involves computing small differences between m3b?f B  
    % large successive terms in the series. (In such cases, the functions B\7 80p<  
    % are often evaluated using alternative methods such as recurrence h6gtO$A|p=  
    % relations: see the Legendre functions, for example). For the Zernike `XwKCI  
    % polynomials, however, this problem does not arise, because the fPsUIlI/A  
    % polynomials are evaluated over the finite domain r = (0,1), and [%7oq;^J  
    % because the coefficients for a given polynomial are generally all .`N&,&H  
    % of similar magnitude. -+.-Ab7  
    % R 9Y k9v  
    % ZERNPOL has been written using a vectorized implementation: multiple *&yt;|y  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] &uv7`VT  
    % values can be passed as inputs) for a vector of points R.  To achieve QcDtZg\  
    % this vectorization most efficiently, the algorithm in ZERNPOL WPNvZg9*c  
    % involves pre-determining all the powers p of R that are required to Fm.IRu<\`  
    % compute the outputs, and then compiling the {R^p} into a single FkIT/H  
    % matrix.  This avoids any redundant computation of the R^p, and WO6;K]  
    % minimizes the sizes of certain intermediate variables. t.m C q 4{  
    % bMF`KRP2  
    %   Paul Fricker 11/13/2006 r)t-_p37  
    {nmBIk2v  
    !xZ`()D#  
    % Check and prepare the inputs: N]@e7P'9F  
    % ----------------------------- ig,v6lqhM  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) E@$HO_;&  
        error('zernpol:NMvectors','N and M must be vectors.')  'x\{sv  
    end J"RmV@|  
    <)9E.h  
    if length(n)~=length(m) R a?0jcSQ$  
        error('zernpol:NMlength','N and M must be the same length.') Q" an6ht|  
    end ~ f>km|Q{u  
    H;eOrX {GT  
    n = n(:); 9 l9|w4YJs  
    m = m(:); ZvO,1B  
    length_n = length(n); ) bGzsb1\  
    oT27BK26?h  
    if any(mod(n-m,2)) d#G H4+C  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') rY&Y58./  
    end s;-%Dfn  
    |#!P!p}  
    if any(m<0) eMC0 )B  
        error('zernpol:Mpositive','All M must be positive.') #>\+6W17U  
    end 0?nm`9v6  
    -( ,iwF b  
    if any(m>n) ]):kMRv  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') DN;An0 {MK  
    end  Enj],I  
     =:-x;  
    if any( r>1 | r<0 ) &-0 eWwMW  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') HN tl>H  
    end S7 Tem:/  
    D#,P-0+%  
    if ~any(size(r)==1) w_!]_6%{b  
        error('zernpol:Rvector','R must be a vector.') +b]+5!  
    end *aF<#m v  
    6+[7UH~pm^  
    r = r(:); 9>"To  
    length_r = length(r); 7EAkY`Op  
    "Aq-H g  
    if nargin==4 lE?F Wt  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); 4^O'K;$leD  
        if ~isnorm lx&ME#~  
            error('zernpol:normalization','Unrecognized normalization flag.') qrmJJSJ  
        end U0:tE>3`  
    else +wwK#ocw  
        isnorm = false; 7BhRt8FSD+  
    end IuQY~!  
    T;%ceLD  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% M6J/S  
    % Compute the Zernike Polynomials ~^Y(f'{  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {6yiD  
    \;>idbV  
    % Determine the required powers of r: 8HyK;+ZkVd  
    % ----------------------------------- vK?{Z^J][  
    rpowers = []; qeyBZ8BG  
    for j = 1:length(n) zV }-_u.  
        rpowers = [rpowers m(j):2:n(j)]; v5 yOh5  
    end ZdD]l*.\i  
    rpowers = unique(rpowers); y^oSVj  
    C9q`x2  
    % Pre-compute the values of r raised to the required powers, Tl!}9/Q5E:  
    % and compile them in a matrix: hfGA7P"  
    % ----------------------------- VlVd"jW  
    if rpowers(1)==0 In)#`E` g.  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); "yI)F~A  
        rpowern = cat(2,rpowern{:}); 46dh@&U  
        rpowern = [ones(length_r,1) rpowern]; Z;_WU  
    else /EOtK|E  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ;!k1LfN  
        rpowern = cat(2,rpowern{:}); uL!{xuN  
    end >4.{|0%ut  
    he/UvMu  
    % Compute the values of the polynomials: S) [`Bm  
    % -------------------------------------- SZCF db  
    z = zeros(length_r,length_n); sY t8NsQ  
    for j = 1:length_n b "4W` A  
        s = 0:(n(j)-m(j))/2; Vl!Z|}z  
        pows = n(j):-2:m(j); /R< Q~G|\  
        for k = length(s):-1:1 J}coWjw`q  
            p = (1-2*mod(s(k),2))* ... R4"g? e  
                       prod(2:(n(j)-s(k)))/          ... kg$<^:uX  
                       prod(2:s(k))/                 ... AG#5_0]P~  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... ^z$-NSlI  
                       prod(2:((n(j)+m(j))/2-s(k))); 5M~\'\;  
            idx = (pows(k)==rpowers); $H/3t?6h`  
            z(:,j) = z(:,j) + p*rpowern(:,idx); WZ'3  
        end bf `4GD(  
         HzM^Zn57%  
        if isnorm w*ig[{ I  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); 'Z-jj2t}  
        end h]<Ld9  
    end EeKEw Sg  
    laqW {sX^5  
    % 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)  m\__Fl  
    GNA:|x  
    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。 ^TJn&k  
    bBc<yaN  
    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)