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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 m0 a<~  
    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! W,53|9b@  
     
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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 !J%m7 A  
    function z = zernfun(n,m,r,theta,nflag) f|cF [&wo  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. zB@@Gs>  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N BGSqfr1F  
    %   and angular frequency M, evaluated at positions (R,THETA) on the D,)^l@UP  
    %   unit circle.  N is a vector of positive integers (including 0), and xdV $dDCT  
    %   M is a vector with the same number of elements as N.  Each element {R{Io|   
    %   k of M must be a positive integer, with possible values M(k) = -N(k) LqOjVQxz  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, +'{@Xe}  
    %   and THETA is a vector of angles.  R and THETA must have the same S^/:O.X)c,  
    %   length.  The output Z is a matrix with one column for every (N,M) {z j<nu  
    %   pair, and one row for every (R,THETA) pair. xn`<g|"#  
    % 6lKM5,Oa  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike TXDb5ZCzM  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 9>1 $Jv3  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral Z"u|-RoBV  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, yS2[V,vS7  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized w*3DIVlxL  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. UB7H`)C}  
    % Pp9nilb_(  
    %   The Zernike functions are an orthogonal basis on the unit circle. Pqc +pE  
    %   They are used in disciplines such as astronomy, optics, and 4[$D3,A  
    %   optometry to describe functions on a circular domain. &8^1:CcE  
    % O:>9yZhV  
    %   The following table lists the first 15 Zernike functions. AWqc?K@   
    % oP0ZJK&;  
    %       n    m    Zernike function           Normalization n!>#o 1Qr  
    %       -------------------------------------------------- ^HM9'*&KJ  
    %       0    0    1                                 1 oO 8opS7F  
    %       1    1    r * cos(theta)                    2 $[NC$*N7  
    %       1   -1    r * sin(theta)                    2 ue~?xmZg  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) "k%B;!We)  
    %       2    0    (2*r^2 - 1)                    sqrt(3) /t<C_lLM  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) F]"Hs>  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) j& x=?jX  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) ncy?w e  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) A` iZ"?  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) )ZP-t!).G#  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) .!&S{;Vv?W  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) "~uo4n~H  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) ^^{gn3xJ  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) )U':NV2  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) >dTJ  
    %       -------------------------------------------------- nLfITr|5  
    % NxyrP**j  
    %   Example 1: UJX=lh.o  
    % ]F]!>dKA  
    %       % Display the Zernike function Z(n=5,m=1) w=txSF&Qr  
    %       x = -1:0.01:1; R Wd#)3  
    %       [X,Y] = meshgrid(x,x); )&$Zt(  
    %       [theta,r] = cart2pol(X,Y); tHj |_t  
    %       idx = r<=1; *k7vm%#ns  
    %       z = nan(size(X)); ,PyA$Z  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); ~{O9dEI  
    %       figure %N, P? ,U  
    %       pcolor(x,x,z), shading interp ;Npv 2yAab  
    %       axis square, colorbar \s[/{3  
    %       title('Zernike function Z_5^1(r,\theta)') r,` 59  
    % jP-=x(  
    %   Example 2: G\S>H  
    % 6a=Y_fma  
    %       % Display the first 10 Zernike functions %](H?'H  
    %       x = -1:0.01:1; ~D9VjXfL)  
    %       [X,Y] = meshgrid(x,x); t#p*{S 3u  
    %       [theta,r] = cart2pol(X,Y); Yom,{;Bv  
    %       idx = r<=1; mO UIGlv  
    %       z = nan(size(X)); >;;tX3(  
    %       n = [0  1  1  2  2  2  3  3  3  3]; 8#S}.|"?F  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; qC%[J:RwF  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; P 3CzX48^  
    %       y = zernfun(n,m,r(idx),theta(idx)); ``:AF:  
    %       figure('Units','normalized') ?xTh}Sky  
    %       for k = 1:10 R&Oqm hT!  
    %           z(idx) = y(:,k); \*_@`1m  
    %           subplot(4,7,Nplot(k)) #0+`dI_5/  
    %           pcolor(x,x,z), shading interp l/JE}Eg(  
    %           set(gca,'XTick',[],'YTick',[]) fnUR]5\tc  
    %           axis square rX*ATN  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) J01Y%W  
    %       end l{{wrU`  
    % *$KUnd-T  
    %   See also ZERNPOL, ZERNFUN2. YJ&K0 %R  
    !"dbK'jb^  
    %   Paul Fricker 11/13/2006 (j%d{y4  
    :LuzKCvBP  
    g]z[!&%Ahs  
    % Check and prepare the inputs: `xhiG9mz~  
    % ----------------------------- >}43xIRRCq  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 4[S0~O{r  
        error('zernfun:NMvectors','N and M must be vectors.') &tULSp@J  
    end f4s^$Q{Q  
    ;Ly(O'9  
    if length(n)~=length(m) MBbycI,  
        error('zernfun:NMlength','N and M must be the same length.') ^Fl6-|^~  
    end myVV5#{  
    9\/T #EP  
    n = n(:); WJ{hta  
    m = m(:); 86^xq#+Uw  
    if any(mod(n-m,2)) Rv)!p~V8  
        error('zernfun:NMmultiplesof2', ... ;?y*@ *2u  
              'All N and M must differ by multiples of 2 (including 0).') da[u@eNrnX  
    end Z(S=2r.  
    PC_#kz  
    if any(m>n) Y}bJN%M  
        error('zernfun:MlessthanN', ... ;JcOm&d/hk  
              'Each M must be less than or equal to its corresponding N.') 9q2 >_Mv  
    end +P7A`{Ae  
    G36}4  
    if any( r>1 | r<0 ) H(^O{JC]y!  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') _u`NIpXSP  
    end e#YQA  
    0,T'z,  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) pr|P#mc"J  
        error('zernfun:RTHvector','R and THETA must be vectors.') eB:OvOol*^  
    end m[7i<'+S  
    H<M ggs-  
    r = r(:); 6 1= ?(Iw  
    theta = theta(:); 'oZ/fUl|7  
    length_r = length(r); jhWNMu  
    if length_r~=length(theta) O?8^I<  
        error('zernfun:RTHlength', ... 8+&] q#W3  
              'The number of R- and THETA-values must be equal.') LF'M!C9|  
    end fq){?hk~O  
    jb' hqz  
    % Check normalization: y(K?mtQ   
    % -------------------- e!wS"[,  
    if nargin==5 && ischar(nflag) .wrNRU7s  
        isnorm = strcmpi(nflag,'norm'); Ojkbv  
        if ~isnorm PMJe6*(x/  
            error('zernfun:normalization','Unrecognized normalization flag.')  8@)/a  
        end w#Y<~W&  
    else }2.^n{Y  
        isnorm = false; ZhKYoPIq  
    end 1NO<K`  
    ny5 = =C{9  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% }b+tD3+  
    % Compute the Zernike Polynomials K?_4|  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% M%wj6!5  
    "h/{YjUS  
    % Determine the required powers of r: -{>Nrx|  
    % ----------------------------------- =nEl m*E  
    m_abs = abs(m); p~h= ]o'i  
    rpowers = []; Q{Gi**<  
    for j = 1:length(n) (9h{7<wD`  
        rpowers = [rpowers m_abs(j):2:n(j)]; C#X0Cn0ln  
    end K1Tq7/N  
    rpowers = unique(rpowers); YF=@nR$_~j  
    ;p"G<n  
    % Pre-compute the values of r raised to the required powers, 9n!<M)E  
    % and compile them in a matrix: E+$vIYq:W  
    % ----------------------------- qoBm!|q  
    if rpowers(1)==0 E[J7FgU)<S  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ,TD@s$2x  
        rpowern = cat(2,rpowern{:}); D"F5-s7  
        rpowern = [ones(length_r,1) rpowern]; f/ 9]o  
    else da3]#%i0  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); Y%$57,Bu n  
        rpowern = cat(2,rpowern{:}); vJ$#m_aa  
    end OGNjn9av  
    1Y410-.3w{  
    % Compute the values of the polynomials: {A4"KX(U  
    % -------------------------------------- raGov`  
    y = zeros(length_r,length(n)); 8=Di+r  
    for j = 1:length(n) H~+D2A  
        s = 0:(n(j)-m_abs(j))/2; hq/k}Y  
        pows = n(j):-2:m_abs(j); ]*pALT6  
        for k = length(s):-1:1  PA"xb3@I  
            p = (1-2*mod(s(k),2))* ... $Q1:>i@I|g  
                       prod(2:(n(j)-s(k)))/              ... oUEpzv,J  
                       prod(2:s(k))/                     ... GmN} +(  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 8 vB~1tl;  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); $%VFk53I  
            idx = (pows(k)==rpowers); h\KQ{-Bl  
            y(:,j) = y(:,j) + p*rpowern(:,idx); &C3J6uCm+  
        end )`Tny]M  
         F ]\4<  
        if isnorm >Vc_.dR)E  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); AFL*a*  
        end .O'S@ %]  
    end o[^%0uVF  
    % END: Compute the Zernike Polynomials XU.ZYYZ=  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 3a9Oj'd1M  
    lyKV^7}  
    % Compute the Zernike functions: j& f-yc'i-  
    % ------------------------------ zt!mx{l'  
    idx_pos = m>0; +L*2 6ar6  
    idx_neg = m<0; PdJtJqA8h\  
    ,T_HE3K  
    z = y; {<&I4V@+  
    if any(idx_pos) wQ[~7 ,o  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); Z=DAA+T`  
    end V@<tIui$  
    if any(idx_neg) NFPW#-TF  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); lRnst-inlI  
    end q~.\NKc  
    A\lnH5A  
    % 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) z}yntY]n  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. GIsXv 2  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated U/Z!c\r  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive 5 |{0|mP  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, =El.uBz{  
    %   and THETA is a vector of angles.  R and THETA must have the same q .nsGbl  
    %   length.  The output Z is a matrix with one column for every P-value, A1Mr  
    %   and one row for every (R,THETA) pair. V:)k@W?P  
    % w<&Nn`V  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike ;2kiEATQ 1  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) fXvJ3w(  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) [oKc<o7)~"  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 jwyJ=W-  
    %   for all p. R*/%+  
    % {%^q8l4j  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 y _>HQs,:  
    %   Zernike functions (order N<=7).  In some disciplines it is SoS[yr  
    %   traditional to label the first 36 functions using a single mode lyfLkBF  
    %   number P instead of separate numbers for the order N and azimuthal  .VuZ=  
    %   frequency M. ,sP7/S)FR  
    % {HNGohZt  
    %   Example: 1wuLw Ad  
    % 'y}l9alF  
    %       % Display the first 16 Zernike functions Qr1%"^4  
    %       x = -1:0.01:1; }4ijLX>b  
    %       [X,Y] = meshgrid(x,x); B4pheKZ2  
    %       [theta,r] = cart2pol(X,Y); kM*f9x  
    %       idx = r<=1; p82&X+v/p  
    %       p = 0:15; 0!o&=Qh  
    %       z = nan(size(X)); 7=u\D  
    %       y = zernfun2(p,r(idx),theta(idx)); o~<37J3).  
    %       figure('Units','normalized') v$p<6^kJ  
    %       for k = 1:length(p) G#@o6r  
    %           z(idx) = y(:,k); Nj*J~&6G  
    %           subplot(4,4,k) ?Q="w5OOD  
    %           pcolor(x,x,z), shading interp  r75,mX  
    %           set(gca,'XTick',[],'YTick',[]) "X's>uM  
    %           axis square ;:vbOG#aSN  
    %           title(['Z_{' num2str(p(k)) '}']) ;gTdiwfgZ=  
    %       end v0jRoE#  
    % PbJn8o   
    %   See also ZERNPOL, ZERNFUN. K SDo)7`  
    {tk42}8k  
    %   Paul Fricker 11/13/2006 Dsw(ti`@  
    ]Hc `<P  
    aN}yS=(Ff  
    % Check and prepare the inputs: HZ5*PXg~  
    % ----------------------------- ;XSV}eLu  
    if min(size(p))~=1 `]_#_  
        error('zernfun2:Pvector','Input P must be vector.') o>311(:  
    end hvQOwA;e  
    R#1h.8  
    if any(p)>35 qm4 Ejc<  
        error('zernfun2:P36', ... &`0y<0z  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... lJpD>\$}@R  
               '(P = 0 to 35).']) QB'-`GwL  
    end Pan^@B=Q  
    L:IaJ?+?  
    % Get the order and frequency corresonding to the function number: b|u4h9  
    % ---------------------------------------------------------------- &Zs h-|N  
    p = p(:); _' Xt  
    n = ceil((-3+sqrt(9+8*p))/2); BR8z%R  
    m = 2*p - n.*(n+2); =7e~L 3 K  
    j0>S)Q  
    % Pass the inputs to the function ZERNFUN: %g^dB M#  
    % ---------------------------------------- |t1D8){!  
    switch nargin J )oa:Q  
        case 3 V?kJYf(<  
            z = zernfun(n,m,r,theta); 5O#CdN-S  
        case 4 xqmP/1=NO  
            z = zernfun(n,m,r,theta,nflag); t(?m!Z?tb  
        otherwise ,oT?-PC$z  
            error('zernfun2:nargin','Incorrect number of inputs.') l"*qj#FD  
    end )Uw QsP  
    UN'hnqC  
    % 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) F[ N{7C3  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. #[ vmS  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of w")VcAq  
    %   order N and frequency M, evaluated at R.  N is a vector of .M!6${N);  
    %   positive integers (including 0), and M is a vector with the Us+|L|/  
    %   same number of elements as N.  Each element k of M must be a INby0S  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) CN#`m]l.  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is ,@tY D(Z  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix n,hHh=.Fu  
    %   with one column for every (N,M) pair, and one row for every oZHsCQ%  
    %   element in R. RWP`#(&/&  
    % ]sE~gro  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- 'uGn1|Pvy  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is s 4Lqam!  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to DPw"UY:  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 }\|$8~  
    %   for all [n,m]. 51;V#@CsQ  
    % \`;FL\1+W  
    %   The radial Zernike polynomials are the radial portion of the B_i@D?bTD  
    %   Zernike functions, which are an orthogonal basis on the unit <_=a1x  
    %   circle.  The series representation of the radial Zernike v AP)(I  
    %   polynomials is i=OPl  
    % }o0R`15dA  
    %          (n-m)/2 ,Bk mf|  
    %            __ vQ2kL`@  
    %    m      \       s                                          n-2s ~'m GGH2  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r *.K+"WS%  
    %    n      s=0 Pni  
    % U=\ZeYK.  
    %   The following table shows the first 12 polynomials. YK!nV ,  
    % Z)<ljW  
    %       n    m    Zernike polynomial    Normalization &%+}bt5  
    %       --------------------------------------------- cod__.  
    %       0    0    1                        sqrt(2) ~cbq5||  
    %       1    1    r                           2 U+CZv1  
    %       2    0    2*r^2 - 1                sqrt(6) ?wt%e;  
    %       2    2    r^2                      sqrt(6) }`E5I&r4  
    %       3    1    3*r^3 - 2*r              sqrt(8) ?M. n 9|}y  
    %       3    3    r^3                      sqrt(8) [wWip1OR  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) IeLG/ fB  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) w{f!t8C*s  
    %       4    4    r^4                      sqrt(10) ;XNe:g.CR  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) 2>p K  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) 2~Z P[wr  
    %       5    5    r^5                      sqrt(12) <e-9We."  
    %       --------------------------------------------- 0+jR,5 |  
    % 3%{A"^S=}  
    %   Example: E;.<'t>  
    % ? acm5dN  
    %       % Display three example Zernike radial polynomials HhDiGzOSi  
    %       r = 0:0.01:1; }-?_c#G 3  
    %       n = [3 2 5]; X<dQq`kZ  
    %       m = [1 2 1]; `% k9@k .  
    %       z = zernpol(n,m,r); (Gr8JpV  
    %       figure NNLZ38BV7  
    %       plot(r,z) izy7. (.a  
    %       grid on /6jt 5N&,  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') U??P  
    % vM!lL6T:  
    %   See also ZERNFUN, ZERNFUN2. qgg/_H:;w  
    nAPSs]D  
    % A note on the algorithm. ,*$L_itL  
    % ------------------------ 6SI`c+'@5  
    % The radial Zernike polynomials are computed using the series NBEcx>pma  
    % representation shown in the Help section above. For many special +EjH9;gx  
    % functions, direct evaluation using the series representation can JwG$lGNJ  
    % produce poor numerical results (floating point errors), because <^q4^Q[  
    % the summation often involves computing small differences between S L<P`H|  
    % large successive terms in the series. (In such cases, the functions 1DVu`<OXcH  
    % are often evaluated using alternative methods such as recurrence }tA77Cm)45  
    % relations: see the Legendre functions, for example). For the Zernike 8dgI&t  
    % polynomials, however, this problem does not arise, because the f1w&D ]|S+  
    % polynomials are evaluated over the finite domain r = (0,1), and *z I@Htp  
    % because the coefficients for a given polynomial are generally all <9z2:^  
    % of similar magnitude. ;#$ 67G$  
    % gw' uY$  
    % ZERNPOL has been written using a vectorized implementation: multiple B64L>7\>`  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] /.B7y(  
    % values can be passed as inputs) for a vector of points R.  To achieve tkX7yg>`  
    % this vectorization most efficiently, the algorithm in ZERNPOL @yiAi:v@  
    % involves pre-determining all the powers p of R that are required to kx&Xk0F_g  
    % compute the outputs, and then compiling the {R^p} into a single )d5H v2/0  
    % matrix.  This avoids any redundant computation of the R^p, and JAJo^}}{b  
    % minimizes the sizes of certain intermediate variables. C^9G \s'  
    %  2f>G   
    %   Paul Fricker 11/13/2006 ]S;^QZ  
    OXcQMVa 6  
    :EJ8^'0Q  
    % Check and prepare the inputs: 29{Ep   
    % -----------------------------  gP%S{<.?  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) I/4:SNha  
        error('zernpol:NMvectors','N and M must be vectors.') 9n4vuBgv  
    end dd1CuOd6(1  
    4M4Y2f BH  
    if length(n)~=length(m) =v^LShD2^  
        error('zernpol:NMlength','N and M must be the same length.') |R[@u=7s  
    end *+|D8xp  
    Xq:jp+WSG  
    n = n(:); #-vuY#gs  
    m = m(:); v"wxHro  
    length_n = length(n); 6[3oOO:uo  
    lh^-L+G:Ok  
    if any(mod(n-m,2)) jZwv !-:  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') );EW(7KeL  
    end z}?*1c  
    ;#-yyU  
    if any(m<0) tuH8!.  
        error('zernpol:Mpositive','All M must be positive.') @CGci lS=  
    end C1w~z4Qp  
    *Iy5 V7`KU  
    if any(m>n) 6&,n\EXF  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') K+2k}Hx6J  
    end R\DdU-k  
    @c 3GJ'"X  
    if any( r>1 | r<0 ) U`YPzZp_  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') Cg{V"B:  
    end )}ygzKEa  
    t!}QG"ma  
    if ~any(size(r)==1) 2stBW5v3  
        error('zernpol:Rvector','R must be a vector.') 8{DZew /  
    end j"G1D-S:  
    XS:W{tL!  
    r = r(:); 7b>FqW)%  
    length_r = length(r); |#_IAN  
    ^n! j"  
    if nargin==4 %DyukUJ  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); aqL#g18  
        if ~isnorm i/Zv@GF  
            error('zernpol:normalization','Unrecognized normalization flag.') GxH]  
        end GM]" $  
    else w5/`_m!  
        isnorm = false; u7PtGN0r%  
    end bcx,K b  
    </xz V<Pi  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ]oOSL=~c  
    % Compute the Zernike Polynomials )y~FeKh  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% RLy2d'DS  
    "&$ [@c  
    % Determine the required powers of r: <jt_<p +  
    % ----------------------------------- >WYiOXYv  
    rpowers = []; q,Oj  
    for j = 1:length(n) (RXOv"''=  
        rpowers = [rpowers m(j):2:n(j)]; ~rnbuIh  
    end 8{0=tOXx{  
    rpowers = unique(rpowers); >z{*>i,m1  
    'YQVf]4P  
    % Pre-compute the values of r raised to the required powers, \xCI8 *W  
    % and compile them in a matrix: @*Y"[\"$  
    % ----------------------------- 'gBGZ?^N!U  
    if rpowers(1)==0 g=ehAg  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); m-xnbTcQ  
        rpowern = cat(2,rpowern{:}); @=i- *U  
        rpowern = [ones(length_r,1) rpowern]; sxG8 jD  
    else uUhqj.::<Y  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 5OJ8o>BF  
        rpowern = cat(2,rpowern{:}); 0iKSUw ps  
    end cd&^ vQL8  
    4*`AYx(  
    % Compute the values of the polynomials: vJ +sdG  
    % -------------------------------------- %|"0p3  
    z = zeros(length_r,length_n); iX&eQ{LB  
    for j = 1:length_n VL =19[  
        s = 0:(n(j)-m(j))/2; ]VKM3[   
        pows = n(j):-2:m(j); ,0hk)Vvr3  
        for k = length(s):-1:1 xYmh{Vc8  
            p = (1-2*mod(s(k),2))* ... |_6V+/?"?`  
                       prod(2:(n(j)-s(k)))/          ... |\}&mBR  
                       prod(2:s(k))/                 ... s0zN#'o]  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... 0 _n Pq  
                       prod(2:((n(j)+m(j))/2-s(k))); @PQ% xcOC7  
            idx = (pows(k)==rpowers); kT@m*Etr{  
            z(:,j) = z(:,j) + p*rpowern(:,idx); y 4 wV]1  
        end hSN{jl{L`  
         .8 GX8[t  
        if isnorm &b__ /o  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); OfE>8*RI4  
        end QLPb5{>KDS  
    end KD<smwXjG  
    S3?Bl'  
    % 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)  *rVI[k L  
    5R6QZVc  
    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。 XO5E-Nh  
    zp\_5[qJ;  
    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)