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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 2w["aVr =  
    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! GT0Of~?f  
     
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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 Exc9` 7%.  
    function z = zernfun(n,m,r,theta,nflag) G 8g<>d{j  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. L?WFm n  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N e=B|==E10M  
    %   and angular frequency M, evaluated at positions (R,THETA) on the 8~J(](QA  
    %   unit circle.  N is a vector of positive integers (including 0), and j g8fU  
    %   M is a vector with the same number of elements as N.  Each element VGpWg rmHk  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) M%2+y5  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, _qw?@478  
    %   and THETA is a vector of angles.  R and THETA must have the same { g/0x,-Z  
    %   length.  The output Z is a matrix with one column for every (N,M) -* WXMzr  
    %   pair, and one row for every (R,THETA) pair. &jslyQ#  
    % }BZ"S-hZ  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike Ji>o!  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), :6vm+5!  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral l49*<nkmq  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, <<+\X:,  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized /OLFcxEWh  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. >8WP0 Qx/  
    % IC1NKn<k  
    %   The Zernike functions are an orthogonal basis on the unit circle. VF?<{F  
    %   They are used in disciplines such as astronomy, optics, and ow_W%I=6  
    %   optometry to describe functions on a circular domain. {^CY..3 A  
    % lij.N) E  
    %   The following table lists the first 15 Zernike functions. -likj# Z  
    % DW5Y@;[  
    %       n    m    Zernike function           Normalization 5nT"rA  
    %       -------------------------------------------------- LBM ^9W  
    %       0    0    1                                 1 5-aj 2>=7  
    %       1    1    r * cos(theta)                    2 lQ" p !  
    %       1   -1    r * sin(theta)                    2 nqI@Y)  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) i;/5Y'KZ  
    %       2    0    (2*r^2 - 1)                    sqrt(3) Y9uC&/_C  
    %       2    2    r^2 * sin(2*theta)             sqrt(6)  gQ'zW  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) 9 7GV2]-M  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) &O9 |#YUq  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) 8$6Y{$&C  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) jcuB  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) %E#s\B,w  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) sz:g,}~h  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) mZSD(  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Sdt`i  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) A mNW0.}  
    %       -------------------------------------------------- ,l !Ta "  
    % [fAV5U  
    %   Example 1: wQ^EYKD  
    % tnH2sHby  
    %       % Display the Zernike function Z(n=5,m=1) "P 7nNa  
    %       x = -1:0.01:1; L^}_~PO N5  
    %       [X,Y] = meshgrid(x,x); ad*m%9Y1Q  
    %       [theta,r] = cart2pol(X,Y); _I@9HC 4  
    %       idx = r<=1; <0b)YJb4M  
    %       z = nan(size(X)); Y$Z x,  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); ,?>s>bHV  
    %       figure llcb~  
    %       pcolor(x,x,z), shading interp % tS,}ze  
    %       axis square, colorbar K#6P}tf  
    %       title('Zernike function Z_5^1(r,\theta)') /N=b\-]  
    % \-h%O jf4  
    %   Example 2: 8(pp2rlR  
    % K^1oDP  
    %       % Display the first 10 Zernike functions  gbF+WE  
    %       x = -1:0.01:1; \.MR""@y`{  
    %       [X,Y] = meshgrid(x,x); r_ I5. gK  
    %       [theta,r] = cart2pol(X,Y); ?zh9d%R  
    %       idx = r<=1; @.$|w>>T  
    %       z = nan(size(X)); /rWd=~[MO  
    %       n = [0  1  1  2  2  2  3  3  3  3]; ojaws+(& y  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; Q6PHpaj  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; '(U-(wTC'/  
    %       y = zernfun(n,m,r(idx),theta(idx)); X <f8,n  
    %       figure('Units','normalized') q!.byrod  
    %       for k = 1:10 .+PI}[g  
    %           z(idx) = y(:,k); .nrMfl_  
    %           subplot(4,7,Nplot(k)) \UPjf]&  
    %           pcolor(x,x,z), shading interp VCV"S>aVf  
    %           set(gca,'XTick',[],'YTick',[]) 6wBx;y |  
    %           axis square S0zD"T  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) @$ne{2J3  
    %       end g%sluT[#  
    % 8EW_V$>R  
    %   See also ZERNPOL, ZERNFUN2. @:+8?qcP  
    uxXBEq;  
    %   Paul Fricker 11/13/2006 a zCf  
    BF\XEm?!  
    '~5LY!H(pT  
    % Check and prepare the inputs: m8A#~i .  
    % ----------------------------- 94h]~GqNi  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) -.1y(k^4E  
        error('zernfun:NMvectors','N and M must be vectors.') gwLf'  
    end 7I&&bWB  
    kjIAep0rT  
    if length(n)~=length(m) uZNTHD  
        error('zernfun:NMlength','N and M must be the same length.') v\c>b:AofD  
    end %'b M){  
    ~-ia+A6GIV  
    n = n(:); <CS(c|7  
    m = m(:); 5 h-@|t  
    if any(mod(n-m,2)) ,|3MG",@@h  
        error('zernfun:NMmultiplesof2', ... `95r0t0hh\  
              'All N and M must differ by multiples of 2 (including 0).') &-;4.op  
    end PRx8I .  
    +9M^7/}H  
    if any(m>n) K*%9)hq  
        error('zernfun:MlessthanN', ... t_o['F  
              'Each M must be less than or equal to its corresponding N.') SEo'(-5  
    end sZjQ3*<-r  
    x3hB5p$q  
    if any( r>1 | r<0 ) 52%2R]G!  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') QX!-B  
    end UbXh,QEG*  
    Dt}JG6S  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) |t^E~HLm,  
        error('zernfun:RTHvector','R and THETA must be vectors.') caU0\VS  
    end %aHB"vi6  
    VrHv)lUr  
    r = r(:); >tYm+coS  
    theta = theta(:); y`VyQWW  
    length_r = length(r); Jb0`42  
    if length_r~=length(theta) 7r7YNn/?  
        error('zernfun:RTHlength', ... b+%f+zz*h  
              'The number of R- and THETA-values must be equal.') y=fx%~<> 8  
    end RmI]1S_=  
    uW=k K0E  
    % Check normalization: ^hG-~z<  
    % -------------------- )Lk639r  
    if nargin==5 && ischar(nflag) ERUz3mjA/  
        isnorm = strcmpi(nflag,'norm'); c?tBi9'Y]  
        if ~isnorm n&L+wqJ  
            error('zernfun:normalization','Unrecognized normalization flag.') lsJSYJG&  
        end |ax3sAg  
    else h:W;^\J:-  
        isnorm = false; 9Z|jxy  
    end s(5Y  
    [glLre^  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% u7Y WnD  
    % Compute the Zernike Polynomials ?h3Y)5xT  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ,g%0`SO  
    $ZUdT  
    % Determine the required powers of r: J28M@cn  
    % ----------------------------------- QCD .YFM  
    m_abs = abs(m); iNWw;_|1  
    rpowers = []; 7TgOK   
    for j = 1:length(n) K`yRr`pW  
        rpowers = [rpowers m_abs(j):2:n(j)]; $~~Jw]   
    end Ar%%}Gx /  
    rpowers = unique(rpowers); <C_jF  
    Lco~,OE  
    % Pre-compute the values of r raised to the required powers, Ye\rB\-  
    % and compile them in a matrix: rxVanDb=W  
    % ----------------------------- cpe+XvBuK  
    if rpowers(1)==0 4~ q5,^kgB  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ]jtK I4  
        rpowern = cat(2,rpowern{:}); e{h<g>7  
        rpowern = [ones(length_r,1) rpowern]; NiNM{[3oS  
    else =qoWCmg"&  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 7G:s2432  
        rpowern = cat(2,rpowern{:}); "$~':) V"  
    end dWM'fg  
    d:_t-ZZo  
    % Compute the values of the polynomials: sz5MH!/PJ  
    % -------------------------------------- OPetj.C/a  
    y = zeros(length_r,length(n)); aB*Bz]5;E  
    for j = 1:length(n) }HL]yDO  
        s = 0:(n(j)-m_abs(j))/2; m-%E-nr  
        pows = n(j):-2:m_abs(j); <>n0arAn  
        for k = length(s):-1:1 aFc1|.Nm  
            p = (1-2*mod(s(k),2))* ... $CxKuB(  
                       prod(2:(n(j)-s(k)))/              ... 5 z~1Dw  
                       prod(2:s(k))/                     ... d)"3K6s|5  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... -<c=US  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); j>*S5y.{  
            idx = (pows(k)==rpowers); 4qN{n#{+]  
            y(:,j) = y(:,j) + p*rpowern(:,idx); K#l:wH _  
        end @:;)~V  
         d4m=0G`  
        if isnorm `Y+J-EQ  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); )) Zf|86N  
        end z(o,m3@v  
    end IW)()*8;/  
    % END: Compute the Zernike Polynomials +y,T4^{  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% E_gD:PPU5  
    D_?K"E=fw  
    % Compute the Zernike functions: pny11C  
    % ------------------------------ `^91%f  
    idx_pos = m>0; V@\gS"Tu  
    idx_neg = m<0; Xk:OL,c  
    w _u\pa  
    z = y; NnO~dRx{  
    if any(idx_pos) 8{Q<N%Jnu  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); B6=ebM`q  
    end Bm.afsM;  
    if any(idx_neg) Q.bXM?V)  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); i}b${n o  
    end h-g+g#*  
    sD<a+Lw}x  
    % 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) #XK2Ien)Z  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. kTKq/G,Ft  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated b>_o xK  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive PxF <\pu&  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, 2Fy>.*,?  
    %   and THETA is a vector of angles.  R and THETA must have the same eeIhed9  
    %   length.  The output Z is a matrix with one column for every P-value, .ELGWF`>  
    %   and one row for every (R,THETA) pair. dL:-Y.?0M  
    % k")R[)92b?  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike ]I-Z]m "  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) EQ2#/>  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) m[i+knYX  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 ~0?mBy!-O  
    %   for all p. ot @|!V  
    % Q*~LCtrI  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 -7m:91x  
    %   Zernike functions (order N<=7).  In some disciplines it is "b?v?V0%C  
    %   traditional to label the first 36 functions using a single mode [N1hWcfvd  
    %   number P instead of separate numbers for the order N and azimuthal mt9 .x  
    %   frequency M. m_hN*v Py  
    % l;af~ef)'  
    %   Example: W>!_|[a  
    % tc_286'x  
    %       % Display the first 16 Zernike functions pqUCqo!m\  
    %       x = -1:0.01:1; H|,Oswk~-  
    %       [X,Y] = meshgrid(x,x); 5>VY LI  
    %       [theta,r] = cart2pol(X,Y); %R1tJ(/  
    %       idx = r<=1; ~SQ?BoCI[  
    %       p = 0:15; f5F@^QXQ  
    %       z = nan(size(X)); 0MV>"aV  
    %       y = zernfun2(p,r(idx),theta(idx)); L{:9Cx!F  
    %       figure('Units','normalized') qNI, 62  
    %       for k = 1:length(p) KiRUvWqa  
    %           z(idx) = y(:,k);  pnMEB,)  
    %           subplot(4,4,k) ~i@Y|38C  
    %           pcolor(x,x,z), shading interp qe#P?[  
    %           set(gca,'XTick',[],'YTick',[]) g wz7krUTe  
    %           axis square HN_d{ 3  
    %           title(['Z_{' num2str(p(k)) '}']) x2@U.r"zo  
    %       end %cCs?ic  
    % 6)z?f4,  
    %   See also ZERNPOL, ZERNFUN. W-Fu-Cz=  
    /V=24\1Ky  
    %   Paul Fricker 11/13/2006 ~CtL9m3tO  
    J%V-Q>L  
    gWrgnlq  
    % Check and prepare the inputs: sBu=e7  
    % ----------------------------- "~=mG--I  
    if min(size(p))~=1 u583_k%  
        error('zernfun2:Pvector','Input P must be vector.') 'sLiu8G  
    end ,B08i o-  
    $|]" W=h  
    if any(p)>35 tBNoI  
        error('zernfun2:P36', ... Ad:TYpLD  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... FvN<<&B  
               '(P = 0 to 35).']) J~B<7O<?!1  
    end U`,0]"Qk  
    \|]Z8t7  
    % Get the order and frequency corresonding to the function number: kgy:Q'  
    % ---------------------------------------------------------------- XGJj3-eW {  
    p = p(:); klQC2drS  
    n = ceil((-3+sqrt(9+8*p))/2); L,SGT8lL  
    m = 2*p - n.*(n+2); kxp) ;  
    OS4q5;1#  
    % Pass the inputs to the function ZERNFUN: ONx( ]  
    % ---------------------------------------- rm4.aO~-F  
    switch nargin 6&DX] [G  
        case 3 $B kubWM  
            z = zernfun(n,m,r,theta); uA,>a>xYI  
        case 4 cZH-"  
            z = zernfun(n,m,r,theta,nflag); zS\E/.X2  
        otherwise -:Fe7c  
            error('zernfun2:nargin','Incorrect number of inputs.') LVc4CE f  
    end a^Tm u  
    #?*WPq  
    % EOF zernfun2
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    function z = zernpol(n,m,r,nflag) 2-=\~<)  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. 6'kS_Zu{<  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of 56e r`=ms  
    %   order N and frequency M, evaluated at R.  N is a vector of ?."&MZ  
    %   positive integers (including 0), and M is a vector with the 6(]tYcC  
    %   same number of elements as N.  Each element k of M must be a 5k6mmiaKk  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) Oy<5>2^P  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is ,ldI2 ]  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix =Zu^80/  
    %   with one column for every (N,M) pair, and one row for every QCH}-q)  
    %   element in R. <&&SX;  
    % U~} U\_  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- ~XyW&@  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is MZTx:EN!  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to R)M_|ca  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 )I^2k4Cg"  
    %   for all [n,m]. is1's[  
    % f;&]:2.j  
    %   The radial Zernike polynomials are the radial portion of the %V1Z~HC  
    %   Zernike functions, which are an orthogonal basis on the unit <V7>?U l  
    %   circle.  The series representation of the radial Zernike a 4=N9X  
    %   polynomials is % r-V2)  
    % %]gn?`O  
    %          (n-m)/2 H;U)b{  
    %            __ iT;@bp  
    %    m      \       s                                          n-2s yC3yij<oR  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r tG8jFou  
    %    n      s=0 FuUD 61JHY  
    % aWwPvd3  
    %   The following table shows the first 12 polynomials. Rx*BwZ  
    % _(d.!qGz  
    %       n    m    Zernike polynomial    Normalization 4B4Z])$3  
    %       --------------------------------------------- i]=&  
    %       0    0    1                        sqrt(2) r6;$1 K*0  
    %       1    1    r                           2 }R)=S_j  
    %       2    0    2*r^2 - 1                sqrt(6) v?0r`<Mn  
    %       2    2    r^2                      sqrt(6) 7}GK%H-u  
    %       3    1    3*r^3 - 2*r              sqrt(8) U9&k;`  
    %       3    3    r^3                      sqrt(8) sK"9fU  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) "F3]X)}  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) c9\2YKo  
    %       4    4    r^4                      sqrt(10) :d0Y%vl  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) {TOmv  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) :-iMdtm  
    %       5    5    r^5                      sqrt(12) PN$X N<  
    %       --------------------------------------------- zW}[+el }  
    % 'DCFezdf3  
    %   Example: T1` |~Z?g-  
    % ( 7ws{)  
    %       % Display three example Zernike radial polynomials 8 F2|  
    %       r = 0:0.01:1; 2>]a)  
    %       n = [3 2 5]; p1D[YeF4  
    %       m = [1 2 1]; LuR,f"%2  
    %       z = zernpol(n,m,r); IcrL   
    %       figure &m {kHM  
    %       plot(r,z) V $|<  
    %       grid on Q\Gq|e*  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') 'JZJFE7Z  
    % fa&-. *  
    %   See also ZERNFUN, ZERNFUN2. ~1*A  
    Y7GF$}%UL  
    % A note on the algorithm. &3v&i*DG,I  
    % ------------------------ -/ x W  
    % The radial Zernike polynomials are computed using the series ,;<RW]r-P  
    % representation shown in the Help section above. For many special vLa#Y("  
    % functions, direct evaluation using the series representation can aup6?'G;  
    % produce poor numerical results (floating point errors), because tu>{  
    % the summation often involves computing small differences between `p0ypi3hn  
    % large successive terms in the series. (In such cases, the functions KtB!"yy#  
    % are often evaluated using alternative methods such as recurrence a`E*\O'd  
    % relations: see the Legendre functions, for example). For the Zernike wQ+dJ3b$  
    % polynomials, however, this problem does not arise, because the 3Wcy)y>2Ap  
    % polynomials are evaluated over the finite domain r = (0,1), and cBYfXI0`  
    % because the coefficients for a given polynomial are generally all G\/"}B:(  
    % of similar magnitude. D -IR!js ]  
    % ?X9]HlH  
    % ZERNPOL has been written using a vectorized implementation: multiple IN7<@OS7  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] T;\^#1  
    % values can be passed as inputs) for a vector of points R.  To achieve S'U@X  
    % this vectorization most efficiently, the algorithm in ZERNPOL m7=1%6FN3  
    % involves pre-determining all the powers p of R that are required to NQ|xM"MqD  
    % compute the outputs, and then compiling the {R^p} into a single r7]"?#  
    % matrix.  This avoids any redundant computation of the R^p, and 02JoA+  
    % minimizes the sizes of certain intermediate variables. t` 8!AhOgc  
    % W3&tJ8*3  
    %   Paul Fricker 11/13/2006 I\Glc=T*  
    )9=(|Lp  
    "R9Yb,tIN  
    % Check and prepare the inputs: /}5B&TZ=(3  
    % ----------------------------- b8BD8~;  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) i3mAfDF  
        error('zernpol:NMvectors','N and M must be vectors.') K{cbn1\,H  
    end rS*$rQCr=  
    :XV} c(+d  
    if length(n)~=length(m) Rt(J/%;  
        error('zernpol:NMlength','N and M must be the same length.') Zt@Z=r:&  
    end 68R1AqU_  
    MR'o{?{e`  
    n = n(:); XD-^w_  
    m = m(:); Y7U&Q:5'  
    length_n = length(n); zz_[S{v!#  
    fT Y/4(  
    if any(mod(n-m,2)) 6'OO-o  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') Q;h3v1GC\P  
    end \k.vN@K#  
    !:fv>FEI9  
    if any(m<0) *(GZ^QH.  
        error('zernpol:Mpositive','All M must be positive.') 7\/5r.  
    end 7;LO2<|1  
    2L\3S ukj  
    if any(m>n) Y:x/!-  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') H5nS%D  
    end vz`@x45K  
    N dR ]  
    if any( r>1 | r<0 ) lQ*eH10H  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') #Ogt(5Sd  
    end 6J%iZ  
    2H[)1|]l  
    if ~any(size(r)==1) ;RN8\re  
        error('zernpol:Rvector','R must be a vector.') =^h~!ovj:  
    end o;`!kIQ  
    Jp;k+ "<q  
    r = r(:); 8&}~'4[b[$  
    length_r = length(r); b;9n'UX\  
    FWPW/oC  
    if nargin==4 <9@7,2  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); 0^_MN~s(X  
        if ~isnorm ;Gm>O7"|@  
            error('zernpol:normalization','Unrecognized normalization flag.') w;yx<1f  
        end H`<?<ak6'M  
    else C ?H{CP  
        isnorm = false;  pbB2wt  
    end a0d ,  
    t]m!ee8*X<  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% dc^Vc{26Z  
    % Compute the Zernike Polynomials aqQ o,5U>  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% <jd/t19DB  
    B.-A $/  
    % Determine the required powers of r: dl[ob,aCK  
    % ----------------------------------- w@N{ @tG  
    rpowers = []; o+)A'S  
    for j = 1:length(n) zg]Drm  
        rpowers = [rpowers m(j):2:n(j)];  t.3 \/  
    end %{ ~>n"  
    rpowers = unique(rpowers);  *q"G }  
    !Q[}s #g  
    % Pre-compute the values of r raised to the required powers, tSw>@FM  
    % and compile them in a matrix: O@`J_9  
    % ----------------------------- &d 3HB=x  
    if rpowers(1)==0 G:Hj;&'2  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); MTB@CP!u  
        rpowern = cat(2,rpowern{:});  [y{E  
        rpowern = [ones(length_r,1) rpowern]; _O 52ai><b  
    else x*mc -&N  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); #1l7FT?q  
        rpowern = cat(2,rpowern{:}); ?Y`zg`  
    end 0_V*B[V  
    cqg=8$RB  
    % Compute the values of the polynomials: ??TdrTS  
    % -------------------------------------- }=pOiILvD  
    z = zeros(length_r,length_n); cHn;}l!I  
    for j = 1:length_n S1C^+Sla]  
        s = 0:(n(j)-m(j))/2; 3LfTGO  
        pows = n(j):-2:m(j); 9 x [X<  
        for k = length(s):-1:1 FH -p!4+]  
            p = (1-2*mod(s(k),2))* ... tMG@K  
                       prod(2:(n(j)-s(k)))/          ... lQr6;D}+  
                       prod(2:s(k))/                 ... b3^R,6]x&  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... x(yX0 ,P/7  
                       prod(2:((n(j)+m(j))/2-s(k))); c9|a$^I6  
            idx = (pows(k)==rpowers); +G\0L_B  
            z(:,j) = z(:,j) + p*rpowern(:,idx); rIp'vy S\p  
        end 8 DL hk  
         ) :}Fu  
        if isnorm @fDQ^ 4  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); ;E2~L  
        end ;x RjQR  
    end Bb_}YU2#  
    RR'(9QJ$  
    % 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)  &CB.*\0  
    h,!G7V  
    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。 2<46jJYL'  
    ?.rH;:9To  
    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)