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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 Nk\/lK\  
    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! ,Igd<A=  
     
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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 X2C&q$8  
    function z = zernfun(n,m,r,theta,nflag) a.G;s2>  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. 5tu 4uYp;  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N CDDOm8  
    %   and angular frequency M, evaluated at positions (R,THETA) on the {edjvPlk  
    %   unit circle.  N is a vector of positive integers (including 0), and l 1Ns~  
    %   M is a vector with the same number of elements as N.  Each element #s]`jdc  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) ,wH]|`w  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, Xp_G9I,+  
    %   and THETA is a vector of angles.  R and THETA must have the same MN. $a9m  
    %   length.  The output Z is a matrix with one column for every (N,M) Jbqm?Fy4X  
    %   pair, and one row for every (R,THETA) pair. ^yVKW5x  
    % \m3ca-Y  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike {-e|x&-  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), !:<n]-U  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral 6 #Afj0  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ]c$)0O\O  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized kmF@u@5M  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ~BD 80s:f  
    % 20k@!BNq  
    %   The Zernike functions are an orthogonal basis on the unit circle. ^@n?&  
    %   They are used in disciplines such as astronomy, optics, and bZzB\FB~  
    %   optometry to describe functions on a circular domain. D eM/B5qw  
    % xe!6Pgcb  
    %   The following table lists the first 15 Zernike functions. C:@JLZB  
    % `l`)Cs;a  
    %       n    m    Zernike function           Normalization tU >?j1  
    %       -------------------------------------------------- {Z{!tR?+  
    %       0    0    1                                 1 rIZ^ix-N  
    %       1    1    r * cos(theta)                    2 :]k`;;vh  
    %       1   -1    r * sin(theta)                    2 $Z{Xt*  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) EnnE@BJ"  
    %       2    0    (2*r^2 - 1)                    sqrt(3) ^+'\ u;\  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) L<M H:  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) |$a!Zx94^  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) q, XRb  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) jxNnrIA  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) E [b6k&A  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) w{5v*SHl}`  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) x72T5.  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) tg' 2 v/  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) a!H t81gj  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) !JWZ}u M6  
    %       --------------------------------------------------  ]pP:  
    % !;s5\91  
    %   Example 1: ]B3\IT  
    % U *']7-  
    %       % Display the Zernike function Z(n=5,m=1) ^ woCwW8n  
    %       x = -1:0.01:1; l#k&&rI5x.  
    %       [X,Y] = meshgrid(x,x); |?/,ED+|>D  
    %       [theta,r] = cart2pol(X,Y); LyWgaf#/d  
    %       idx = r<=1; t }q \.  
    %       z = nan(size(X)); [$AOu0J  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx));  pu?D^h9/  
    %       figure TIre,s)_  
    %       pcolor(x,x,z), shading interp N=@Nn)  
    %       axis square, colorbar kcN#g- 0  
    %       title('Zernike function Z_5^1(r,\theta)') QC^ #ns&  
    % >%{H>?Hn  
    %   Example 2: p`2w\P3;)  
    % ^L"ENsOs  
    %       % Display the first 10 Zernike functions yV/A%y-P  
    %       x = -1:0.01:1; 5x/LHsr=m  
    %       [X,Y] = meshgrid(x,x); 6A,-?W'\  
    %       [theta,r] = cart2pol(X,Y); MclW!CmJ  
    %       idx = r<=1; o+I'nFtnI  
    %       z = nan(size(X)); Qvl3=[S  
    %       n = [0  1  1  2  2  2  3  3  3  3]; =#|K-X0d=  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; 1aBQ.-E-  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; <Gkmk?x`A  
    %       y = zernfun(n,m,r(idx),theta(idx)); 0\2#(^  
    %       figure('Units','normalized') -K*&I!  
    %       for k = 1:10 O[O[E}8#  
    %           z(idx) = y(:,k); bL9vjD'}  
    %           subplot(4,7,Nplot(k)) 0G}]d17ho  
    %           pcolor(x,x,z), shading interp '|^<|S_+K  
    %           set(gca,'XTick',[],'YTick',[]) 1]% ]"JbV  
    %           axis square Dj(!i1eQNZ  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) $:D-dUr1  
    %       end (Y>|P  
    % $>=?'wr  
    %   See also ZERNPOL, ZERNFUN2. BA(PWX`H  
    O{w'i|  
    %   Paul Fricker 11/13/2006 "Q <  
    ,3~[cE<4  
    PG*:3![2  
    % Check and prepare the inputs: (&^k''f  
    % ----------------------------- .R5(k'g?  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) W_%@nm\y  
        error('zernfun:NMvectors','N and M must be vectors.') K! I]0!:  
    end g886RhCe  
    t`oH7)nut  
    if length(n)~=length(m) ])3(@.  
        error('zernfun:NMlength','N and M must be the same length.') uk=f /nT  
    end |fhYft  
    fNnX{Wq  
    n = n(:); V4>qR{5  
    m = m(:); %=EN 3>,  
    if any(mod(n-m,2)) 1Q>D^yPI[  
        error('zernfun:NMmultiplesof2', ... |';oIYs|$  
              'All N and M must differ by multiples of 2 (including 0).') s !XJ   
    end F\IJim-Rh  
    (`me}8  
    if any(m>n) 09L"~:rg  
        error('zernfun:MlessthanN', ... QK0-jYG^  
              'Each M must be less than or equal to its corresponding N.') +fRABY5C  
    end PRQEk.C  
    U+2U#v=<  
    if any( r>1 | r<0 ) o~J~-$T{  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') [,86||^  
    end @r=v*hu  
    <2,NWn.  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) +u\kTn  
        error('zernfun:RTHvector','R and THETA must be vectors.') w+W! dM  
    end aTU[H~dTU  
    g;mX{p_@  
    r = r(:); wpI_yp  
    theta = theta(:); jWjp0ii  
    length_r = length(r); c[<>e#s+;  
    if length_r~=length(theta) }{y(&Oy3Y  
        error('zernfun:RTHlength', ... CD:$22*]  
              'The number of R- and THETA-values must be equal.') YQ$EN>.eO  
    end XSoHh-  
    -J' 0qN!  
    % Check normalization: CEHtr90P  
    % -------------------- QpI\\Zt6  
    if nargin==5 && ischar(nflag) U *K6FWqiB  
        isnorm = strcmpi(nflag,'norm'); r~q 3nIe/,  
        if ~isnorm 2PTAIm Rq  
            error('zernfun:normalization','Unrecognized normalization flag.') ##r9/`A  
        end 6haw\ *  
    else Y6:b  
        isnorm = false; Xdl7'~k  
    end 3]wV 1<K  
    &@[pJ2  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% /C\tJs  
    % Compute the Zernike Polynomials E - +t[W  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -|g9__|@  
    VqL#w<A %  
    % Determine the required powers of r: e<+$E%"7hS  
    % ----------------------------------- -,a@bF:  
    m_abs = abs(m); J5"d|i  
    rpowers = []; f[fH1cu&`  
    for j = 1:length(n) NE5H\  
        rpowers = [rpowers m_abs(j):2:n(j)]; [x8_ax} w  
    end %Kzu&*9Hb  
    rpowers = unique(rpowers); Y5z5LG4  
    20Z=_},  
    % Pre-compute the values of r raised to the required powers, XmAu n  
    % and compile them in a matrix: ,,=VF(@G  
    % ----------------------------- B]#^&89wG)  
    if rpowers(1)==0 E]dc4US  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 1uco{JX<S  
        rpowern = cat(2,rpowern{:}); ifI0s)Pn  
        rpowern = [ones(length_r,1) rpowern]; !%Bhg?  
    else :`B70D8ku  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); D5"Xjo*  
        rpowern = cat(2,rpowern{:}); LMHii Os,  
    end 3-v&ktD&N'  
    1A}#j  
    % Compute the values of the polynomials: Bg.  
    % -------------------------------------- ?*L{xNC#  
    y = zeros(length_r,length(n)); 4QBPN@~t  
    for j = 1:length(n) }Uue}VOA  
        s = 0:(n(j)-m_abs(j))/2; ^y.|KA3[  
        pows = n(j):-2:m_abs(j); e:+[}I)  
        for k = length(s):-1:1 9Yhl q$;g  
            p = (1-2*mod(s(k),2))* ... szUJh9-  
                       prod(2:(n(j)-s(k)))/              ... h!J|4Q a  
                       prod(2:s(k))/                     ... Aaug0X  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... M3!4,_!~  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); ^GnR1.ux  
            idx = (pows(k)==rpowers); ?h)T\z  
            y(:,j) = y(:,j) + p*rpowern(:,idx); Go)}%[@w  
        end #`@5`;U>#  
         q+2v9K@  
        if isnorm I(uM`g  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); hdDL92JVg  
        end kgP6'`}E[  
    end d]vom@iI  
    % END: Compute the Zernike Polynomials )nlFyWXh.  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% t~%(Zu>S  
    *:?XbtIK u  
    % Compute the Zernike functions: "EBCf.3-  
    % ------------------------------ BVG.ZZR})  
    idx_pos = m>0; }poLH S/  
    idx_neg = m<0; KEjMxOv1  
    8Om4G]*|,  
    z = y; s\e b  
    if any(idx_pos) 7Qd boEa  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 4m!w<c0NL  
    end xbz O' C  
    if any(idx_neg) &2r[4  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); A "/|h].  
    end >02p,W6S>  
    8&SW Q  
    % 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) S^iT &;,  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. O-!Q~;3][  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated 3Xm> 3  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive 1[!7xA0j  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, kAs=5_?I  
    %   and THETA is a vector of angles.  R and THETA must have the same O*yA50Cn  
    %   length.  The output Z is a matrix with one column for every P-value, 0|ekwTx.  
    %   and one row for every (R,THETA) pair. j!"5, ~  
    % ?3gf)g=  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike "sT)<Wc  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) [WI'oy  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) :Sn4Pg `Q  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 +zK?1llt  
    %   for all p. yIg^iZD  
    % vXg^K}a#  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 a7aj:.wi  
    %   Zernike functions (order N<=7).  In some disciplines it is ?kS#g  
    %   traditional to label the first 36 functions using a single mode R&Ss ET.  
    %   number P instead of separate numbers for the order N and azimuthal uSv]1m_-]  
    %   frequency M. c4.2o<(Xt  
    % O*%5P5'p"{  
    %   Example: gHFQs](G.  
    % ^91Ae!)d  
    %       % Display the first 16 Zernike functions :i|Bz6Ht4  
    %       x = -1:0.01:1; )Myx(w"S  
    %       [X,Y] = meshgrid(x,x); q2/kegAT  
    %       [theta,r] = cart2pol(X,Y); qMw_`dC  
    %       idx = r<=1; _na/&J 6  
    %       p = 0:15; (gIFuOGi>  
    %       z = nan(size(X)); sQ+s3x1y  
    %       y = zernfun2(p,r(idx),theta(idx)); Tj}%G  
    %       figure('Units','normalized') 4'td6F  
    %       for k = 1:length(p) 53>(2 _/[r  
    %           z(idx) = y(:,k); YF>1 5{H  
    %           subplot(4,4,k) p0PK-e`@:  
    %           pcolor(x,x,z), shading interp bXA%|7*  
    %           set(gca,'XTick',[],'YTick',[]) RK p9[^/?  
    %           axis square 5n1`$T.WG  
    %           title(['Z_{' num2str(p(k)) '}']) = ?BhtW  
    %       end AR{$P6u!%|  
    % d[Zx [=h  
    %   See also ZERNPOL, ZERNFUN. Gl"hn  
    KGc!#C  
    %   Paul Fricker 11/13/2006 dH'02[;  
    !s:_>P`MQ  
    6n~)R  
    % Check and prepare the inputs: v+o6ZNX  
    % ----------------------------- eZD"!AT  
    if min(size(p))~=1 .m.Ga|;  
        error('zernfun2:Pvector','Input P must be vector.') Yhjv[9  
    end pH(X;OC 9S  
    Z?'?|vM  
    if any(p)>35 *j=58d`n  
        error('zernfun2:P36', ... ""Oir!4  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... D&i, `j  
               '(P = 0 to 35).']) |oSqy  
    end 3i\Np =  
    }R%*J  
    % Get the order and frequency corresonding to the function number: Z!*6;[]SfG  
    % ---------------------------------------------------------------- h50]%tp\  
    p = p(:); P4.)kK.3q|  
    n = ceil((-3+sqrt(9+8*p))/2); 0/1=2E ^,  
    m = 2*p - n.*(n+2); u6%\ZK._ \  
    A 8-a}0Gh  
    % Pass the inputs to the function ZERNFUN: R~eLEjezm  
    % ---------------------------------------- ]z#)XW3#i  
    switch nargin *!E~4z=  
        case 3 `P <#kt  
            z = zernfun(n,m,r,theta); ].2t7{64  
        case 4 "zkQu  
            z = zernfun(n,m,r,theta,nflag); `VvQems  
        otherwise 7SNdC8GZ~  
            error('zernfun2:nargin','Incorrect number of inputs.') Z9|A"[b  
    end q8n@fi6  
    bZ:xH48MY  
    % 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) m)  rVzL  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. [zXC\)&!  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of byMy- v;  
    %   order N and frequency M, evaluated at R.  N is a vector of cB36w$n8  
    %   positive integers (including 0), and M is a vector with the z]-m<#1  
    %   same number of elements as N.  Each element k of M must be a Lusd kc7  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) }fv7WhQ  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is H_Va$}8z  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix 6 VuyKt  
    %   with one column for every (N,M) pair, and one row for every u;!h   
    %   element in R. *SIYZE'  
    % DVMdRfA  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- R*0mCz^+h  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is uB3VCO.;_  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to mBb3Ta  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 JtsXMZz  
    %   for all [n,m]. VH<d[Mj  
    % 5k9 vYW5k  
    %   The radial Zernike polynomials are the radial portion of the >d&0a:  
    %   Zernike functions, which are an orthogonal basis on the unit 5S_fvW;  
    %   circle.  The series representation of the radial Zernike s6Dkh}:d  
    %   polynomials is :zq Un&k&  
    % %{pjC7j#  
    %          (n-m)/2  ;(J&%  
    %            __ u[PG/ploc  
    %    m      \       s                                          n-2s x X[WX#'f  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r -Eig#]Se3  
    %    n      s=0 VzIZT{  
    % 6({)O1Z  
    %   The following table shows the first 12 polynomials. z5 @i"%f  
    % p Zlt4  
    %       n    m    Zernike polynomial    Normalization 6 C O5:\  
    %       --------------------------------------------- ao=e{R)  
    %       0    0    1                        sqrt(2) rx 74v!  
    %       1    1    r                           2 _| cSXZ|  
    %       2    0    2*r^2 - 1                sqrt(6) +N7<[hE;  
    %       2    2    r^2                      sqrt(6) <8Tp]1z  
    %       3    1    3*r^3 - 2*r              sqrt(8) Lwx J:Kz.  
    %       3    3    r^3                      sqrt(8) esE!i0%  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) %'_:#!9  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) }9W[7V?  
    %       4    4    r^4                      sqrt(10) Ha/Qz'^S;  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) .VNz( s  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) MRK=\qjD  
    %       5    5    r^5                      sqrt(12) Y\WVkd(+G  
    %       --------------------------------------------- &JKQH  
    % j~V $q/7S  
    %   Example: n7G`b'  
    % a^|9rho<  
    %       % Display three example Zernike radial polynomials 4c{j9mh  
    %       r = 0:0.01:1; q~5zv4NX  
    %       n = [3 2 5]; MffCk!]  
    %       m = [1 2 1]; reArXmU<u  
    %       z = zernpol(n,m,r); X>Q44FV!  
    %       figure 'J-a2oiM(  
    %       plot(r,z) !OQ5AF$  
    %       grid on !G\gqkSL  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') :_:)S  
    % >5Lp;  
    %   See also ZERNFUN, ZERNFUN2. zv0sz])  
    zh0T3U0D  
    % A note on the algorithm. .w@B )f*  
    % ------------------------ !.q99DB  
    % The radial Zernike polynomials are computed using the series `''y,{Fs  
    % representation shown in the Help section above. For many special I= <eCv  
    % functions, direct evaluation using the series representation can 8@(?E[&O>  
    % produce poor numerical results (floating point errors), because #Y3-P  
    % the summation often involves computing small differences between 8! !h6dQgI  
    % large successive terms in the series. (In such cases, the functions f=Pn,.>tIz  
    % are often evaluated using alternative methods such as recurrence 94dd )/a  
    % relations: see the Legendre functions, for example). For the Zernike v0! 1W  
    % polynomials, however, this problem does not arise, because the , .~ k  
    % polynomials are evaluated over the finite domain r = (0,1), and 7RBEEE`)  
    % because the coefficients for a given polynomial are generally all ~/)]`w  
    % of similar magnitude. 60$;Q,]o  
    % !X$19"  
    % ZERNPOL has been written using a vectorized implementation: multiple R) dP=W*  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M]  $RRX-  
    % values can be passed as inputs) for a vector of points R.  To achieve R"JXWw  
    % this vectorization most efficiently, the algorithm in ZERNPOL CadIu x^  
    % involves pre-determining all the powers p of R that are required to AkW>*x  
    % compute the outputs, and then compiling the {R^p} into a single 4ytdcb   
    % matrix.  This avoids any redundant computation of the R^p, and `{h)-Y``  
    % minimizes the sizes of certain intermediate variables. z,E`+a;  
    % 9kF0H a}J  
    %   Paul Fricker 11/13/2006 X=abaKl  
    vk X+{n  
    &g5PPQ18  
    % Check and prepare the inputs: "M-';;  
    % ----------------------------- 7%? bl  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 3imsIBr  
        error('zernpol:NMvectors','N and M must be vectors.') Ai[@2AyU  
    end -ZSN0Xk  
    |te=DCO  
    if length(n)~=length(m) .N.RpRz{f  
        error('zernpol:NMlength','N and M must be the same length.') .81Y/Gad_  
    end @~|;/OY>"  
     ^,ISz-4  
    n = n(:); XR7v\rd  
    m = m(:); v6=%KXSF  
    length_n = length(n); cAwqIihZ  
    52Lp_M  
    if any(mod(n-m,2)) u*I'c2m  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') D]*|Zmr+}  
    end bQq/~  
    $.d,>F6  
    if any(m<0) ]>Z9K@  
        error('zernpol:Mpositive','All M must be positive.') uI?Z_  
    end fR@Cg sw  
    ovM;6o  
    if any(m>n) 9D M,,h<`  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') r5nHYV&7  
    end Nr$78] o9  
    u<fZ.1  
    if any( r>1 | r<0 ) \ HUDZ2 s  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') WrDFbcH  
    end :n{{\SSIgX  
    `Ji WS  
    if ~any(size(r)==1) Udtz zka  
        error('zernpol:Rvector','R must be a vector.') ]N'% l]_$  
    end ~BuBma_   
    V~/-e- 9u  
    r = r(:); OOXSJE1  
    length_r = length(r); Xy K,  
    'V:MppQVZ.  
    if nargin==4 >FOCdlJ#  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); K`9~#Zx$  
        if ~isnorm =gR/ t@Ld  
            error('zernpol:normalization','Unrecognized normalization flag.') hR7uAk_?  
        end U*Y]cohh  
    else e<1Ewml(]  
        isnorm = false; |36%B7H  
    end 0%L:jq{5  
    GfK%UZ$C  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% X,3\c:  
    % Compute the Zernike Polynomials 579D  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 9,_~qWw  
    >]ux3F3\  
    % Determine the required powers of r: XK/l1E3N  
    % ----------------------------------- w8Z#]kRv  
    rpowers = []; TS+jDs  
    for j = 1:length(n) >I~Q[  
        rpowers = [rpowers m(j):2:n(j)]; rm3/R<  
    end 5,^DT15a4P  
    rpowers = unique(rpowers); )mOM!I7D@  
    l\V1c90m  
    % Pre-compute the values of r raised to the required powers,  {p/Yz#  
    % and compile them in a matrix: 9%NsW3|  
    % ----------------------------- 0vSPeZ  
    if rpowers(1)==0 K*DH_\SPK  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ;-py h(  
        rpowern = cat(2,rpowern{:}); 0<@['W}G  
        rpowern = [ones(length_r,1) rpowern]; qQDe'f~  
    else t(roj@!x_o  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); )=K8mt0qob  
        rpowern = cat(2,rpowern{:}); 1DAU *^-  
    end ETU-6qFtO  
    A. tGr(r  
    % Compute the values of the polynomials: c\rP -"C  
    % -------------------------------------- ?K2EK'-q  
    z = zeros(length_r,length_n); ,ps?@lD  
    for j = 1:length_n lv!j  
        s = 0:(n(j)-m(j))/2; r`Fs"n#^-4  
        pows = n(j):-2:m(j); oVHe<zE.  
        for k = length(s):-1:1 ZLKbF9lo  
            p = (1-2*mod(s(k),2))* ... IZ>l  
                       prod(2:(n(j)-s(k)))/          ... VV$#<D<)  
                       prod(2:s(k))/                 ... $X Uck[  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... ju[y-am$/  
                       prod(2:((n(j)+m(j))/2-s(k))); x!s=Nola  
            idx = (pows(k)==rpowers); u5rvrn ]  
            z(:,j) = z(:,j) + p*rpowern(:,idx); %`5K8eB  
        end af @a /  
         :qj^RcmVPL  
        if isnorm &P}t<;  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); <aaT,J8%[  
        end hVB(*WA^D  
    end _qf~ hhi  
    U%@C<o "  
    % 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)  <ycR/X  
    Y1ca=ewFx  
    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。 )~"0d;6_  
    }$uwAevP{y  
    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)