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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 5fLCmLM`  
    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! <<`."RY#0  
     
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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 H`".L^  
    function z = zernfun(n,m,r,theta,nflag) `n_ Z  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. !^N/n5eoz  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N >!oN+8[~  
    %   and angular frequency M, evaluated at positions (R,THETA) on the Na`qAj}  
    %   unit circle.  N is a vector of positive integers (including 0), and ~{N|("nB  
    %   M is a vector with the same number of elements as N.  Each element "W1q}4_  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) s$]I@;_  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, T @z$g  
    %   and THETA is a vector of angles.  R and THETA must have the same [oYe/<3  
    %   length.  The output Z is a matrix with one column for every (N,M) w S;(u[W  
    %   pair, and one row for every (R,THETA) pair. qS7*.E~j|]  
    % sX=!o})0  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike crmnh4-  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), SC!IQ80H#D  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral 7IvCMb&%R  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, PffwNj/l  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized GRs;-Jt  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. d }]b  
    % E+Gea[c  
    %   The Zernike functions are an orthogonal basis on the unit circle. {1qEN_ERx  
    %   They are used in disciplines such as astronomy, optics, and pGOS'.K%t8  
    %   optometry to describe functions on a circular domain. S#""((U$  
    % ~PV>3c3l=  
    %   The following table lists the first 15 Zernike functions. 5= F-^  
    % CZ0 {*K:  
    %       n    m    Zernike function           Normalization :<jf}[w!  
    %       -------------------------------------------------- W6*(Y  
    %       0    0    1                                 1 (* 2"dd  
    %       1    1    r * cos(theta)                    2 1%+0OmV&  
    %       1   -1    r * sin(theta)                    2 KYeA=  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) lF#Kg !-l  
    %       2    0    (2*r^2 - 1)                    sqrt(3) ^yb_aCw  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) T^Z#x-Q  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) '}}DPoV  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) &"CS1P|  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) 2R_k$kHl  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) gVuN a)  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) a`{'u)@  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 5G2u(hx  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) =6 [!'K  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) mki=.l$O  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) `SU;TN0  
    %       -------------------------------------------------- ](W #Tj5-  
    % |UBJu `%  
    %   Example 1: -ss2X  
    % E+>;tLw3j  
    %       % Display the Zernike function Z(n=5,m=1) g-]td8}#  
    %       x = -1:0.01:1; Z-~^)lo  
    %       [X,Y] = meshgrid(x,x); }\irr9,  
    %       [theta,r] = cart2pol(X,Y);  ^@ux  
    %       idx = r<=1; )/=J=xw2  
    %       z = nan(size(X)); 2ru6 bIb;  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); !cq4+0{O;&  
    %       figure P_Z o}.{  
    %       pcolor(x,x,z), shading interp 9 V;m;sz  
    %       axis square, colorbar G(4k#jB  
    %       title('Zernike function Z_5^1(r,\theta)') Wqqo8Y~fq  
    % ?K]k(ZV_+Y  
    %   Example 2: R@EFG%|`_  
    % ]A\n>Z!;  
    %       % Display the first 10 Zernike functions _l  Jj6=  
    %       x = -1:0.01:1; 6z(_^CY  
    %       [X,Y] = meshgrid(x,x); |;].~7^  
    %       [theta,r] = cart2pol(X,Y); Z BYmAD  
    %       idx = r<=1; <>R7G)w F  
    %       z = nan(size(X)); M\]E;C'"U  
    %       n = [0  1  1  2  2  2  3  3  3  3]; Nn^el' S'  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; i0 R=P[  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; l==T3u r  
    %       y = zernfun(n,m,r(idx),theta(idx)); Hnaq+ _]  
    %       figure('Units','normalized')  Ne4A  
    %       for k = 1:10 6$z UFIk  
    %           z(idx) = y(:,k); d`xqs,0f  
    %           subplot(4,7,Nplot(k)) %1lLUgf3G/  
    %           pcolor(x,x,z), shading interp o 1b#q/  
    %           set(gca,'XTick',[],'YTick',[])  Wi|.Z/  
    %           axis square 9 (&!>z  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 4bKZ@r%  
    %       end O=mJ8W@  
    % 7j]@3D9[:p  
    %   See also ZERNPOL, ZERNFUN2. E6US  
    @3G3l|~>  
    %   Paul Fricker 11/13/2006 m:H )b{  
    z C``G<TB  
    6m{3GKaW~  
    % Check and prepare the inputs: %AJdtJ@0H  
    % ----------------------------- @!Pq"/  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) H@6  
        error('zernfun:NMvectors','N and M must be vectors.') WT0U)x( m5  
    end <k)rfv7  
    Zs4N0N{  
    if length(n)~=length(m) @B[V'|  
        error('zernfun:NMlength','N and M must be the same length.') L2:C6Sc  
    end ik]UzB  
    RS93_F8   
    n = n(:); #8(@a Y  
    m = m(:); 3j3AI 7c  
    if any(mod(n-m,2)) [m4<j  
        error('zernfun:NMmultiplesof2', ... CR _A{(  
              'All N and M must differ by multiples of 2 (including 0).') `,|7X]%b  
    end @Wv*`  
    n. T [a  
    if any(m>n) Io:xG6yG  
        error('zernfun:MlessthanN', ... D]0#A|n F  
              'Each M must be less than or equal to its corresponding N.') [`:\(( 8  
    end ;TR.UUT  
    .z9JoQ  
    if any( r>1 | r<0 )  g6~uf4;  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') c-3? D;  
    end "B\qp"N  
    'Kq%t M26!  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) {:"bX~<^  
        error('zernfun:RTHvector','R and THETA must be vectors.') 2yN~[, L  
    end : |Z*aI]9  
    1M+mH#?  
    r = r(:); avT>0b:  
    theta = theta(:); U"ZDt  
    length_r = length(r); h  qxe  
    if length_r~=length(theta) D,R/abYZH  
        error('zernfun:RTHlength', ... 6g!t1%Kb  
              'The number of R- and THETA-values must be equal.') 9SU;c l  
    end ed617J  
    /2YI!U@A  
    % Check normalization: U>{z*D  
    % -------------------- t[X'OK0W%3  
    if nargin==5 && ischar(nflag) Bp b_y;E  
        isnorm = strcmpi(nflag,'norm'); GB{%4)%6  
        if ~isnorm F &uU ,);  
            error('zernfun:normalization','Unrecognized normalization flag.') @NNN&%  
        end [WB8X,  
    else t<Og ?m}(  
        isnorm = false; Q!@" Y/  
    end |i|>-|`!  
    (llg!1  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% :lcoSJ  
    % Compute the Zernike Polynomials BK-{z).)  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {>syZZ,h  
    WyO10yvR  
    % Determine the required powers of r: hnyZXk1|  
    % ----------------------------------- T]0qd^\4w  
    m_abs = abs(m); iptzVr#b[  
    rpowers = []; z;Kyg}  
    for j = 1:length(n) TT>;!nb  
        rpowers = [rpowers m_abs(j):2:n(j)]; r% qgLP{v  
    end V RT| OUq  
    rpowers = unique(rpowers); "zYlddh  
    Y>IEB,w  
    % Pre-compute the values of r raised to the required powers, &'i>5Y  
    % and compile them in a matrix: &t`l,]PQ=6  
    % ----------------------------- w%`7,d u|  
    if rpowers(1)==0 teET nz_L  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); uN'e~X6  
        rpowern = cat(2,rpowern{:}); 0b4QcfB1[  
        rpowern = [ones(length_r,1) rpowern]; -MeGJX:^I  
    else 3>-^/  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); c!j$ -Ovm  
        rpowern = cat(2,rpowern{:}); V:yia^1  
    end yv&&x.!.Z  
    meE&, {  
    % Compute the values of the polynomials: q.~_vS%  
    % -------------------------------------- Ia[e 7  
    y = zeros(length_r,length(n)); r IY_1  
    for j = 1:length(n) )88z=5.  
        s = 0:(n(j)-m_abs(j))/2; eR =P  
        pows = n(j):-2:m_abs(j); }ob#LC,  
        for k = length(s):-1:1 <Knl6$B  
            p = (1-2*mod(s(k),2))* ... lor jMS  
                       prod(2:(n(j)-s(k)))/              ... yX/ 9jk  
                       prod(2:s(k))/                     ... `cCsJm$V"  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... w8c71C  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); 8|HuxE  
            idx = (pows(k)==rpowers); e'p'{]r<w  
            y(:,j) = y(:,j) + p*rpowern(:,idx); /0@'8f\I  
        end 7<=xc'*8t  
         F0qGkMs|f  
        if isnorm QT&2&#Z  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); R8sj>.I9j  
        end g>cp;co9g  
    end }[\l$sS  
    % END: Compute the Zernike Polynomials bU7n1pzW,o  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% P|l62!m<   
    I& DEF*  
    % Compute the Zernike functions: g Q6_]~4  
    % ------------------------------ F8S% \i  
    idx_pos = m>0; z;J"3kM  
    idx_neg = m<0; JgEPzHgx  
    6* (6>F5  
    z = y; iP)`yB5`  
    if any(idx_pos) ")}^\O m  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); AAb3Jf`UW  
    end (p>?0h9[  
    if any(idx_neg) I<Wp,E9G#  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); B<%cqz@  
    end Yw7txp`i  
    +`}QIp0  
    % EOF zernfun
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) 4z%::?  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. @cRR  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated v#c'p^T  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive {%Cb0Zh  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, zZp0g^;.?  
    %   and THETA is a vector of angles.  R and THETA must have the same 79`OB##  
    %   length.  The output Z is a matrix with one column for every P-value, !LJEo>D  
    %   and one row for every (R,THETA) pair. JbX"K< nQ  
    % B8.a#@R  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike _@2}zT  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) # 0kVhx7%  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) EdcbWf7  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 /o L& <e  
    %   for all p. 8L1 vt Yz  
    % SbLm  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 *^ aEUp6&  
    %   Zernike functions (order N<=7).  In some disciplines it is !$n@-  
    %   traditional to label the first 36 functions using a single mode ] MP*5U>;  
    %   number P instead of separate numbers for the order N and azimuthal 3V uoDmG  
    %   frequency M. #z6[ 8B  
    % aW=c.Q.  
    %   Example: )+7|_7 !x  
    % X<8?>#  
    %       % Display the first 16 Zernike functions WoV"&9y  
    %       x = -1:0.01:1; r+:]lO  
    %       [X,Y] = meshgrid(x,x); {=I:K|&  
    %       [theta,r] = cart2pol(X,Y); 5}C.^J`  
    %       idx = r<=1; c!0u,6  
    %       p = 0:15; z][hlDv\j  
    %       z = nan(size(X)); .q90+9Ek=  
    %       y = zernfun2(p,r(idx),theta(idx)); FP=up#zl  
    %       figure('Units','normalized') %plu]^Vy  
    %       for k = 1:length(p) |VX )S!  
    %           z(idx) = y(:,k); #B2a?   
    %           subplot(4,4,k) I4XnJ[N%  
    %           pcolor(x,x,z), shading interp /O[<"Wcz  
    %           set(gca,'XTick',[],'YTick',[]) S2i*Li  
    %           axis square _"%hcCMw  
    %           title(['Z_{' num2str(p(k)) '}']) -.@dA'j[  
    %       end W{RZ@ 3ZY  
    % &L[i"1a  
    %   See also ZERNPOL, ZERNFUN. |l CS^bA3  
    LUs)"ZAi|  
    %   Paul Fricker 11/13/2006 %?`O .W  
    CP'b,}Dd?I  
    -=cxUDB  
    % Check and prepare the inputs: ~] &yHzp2  
    % ----------------------------- Kpg?' !I  
    if min(size(p))~=1 6o0}7T%6  
        error('zernfun2:Pvector','Input P must be vector.') efr9  
    end +0ALO%G;G"  
    `5Bv2 wlIV  
    if any(p)>35 >^6|^rc  
        error('zernfun2:P36', ... ;~-ZN?8   
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... a,|Hn  
               '(P = 0 to 35).']) @cr/&  
    end I A%ZCdA;  
    %*zV&H   
    % Get the order and frequency corresonding to the function number: ?6W v["%  
    % ---------------------------------------------------------------- v<@3&bot  
    p = p(:); `# U<'$  
    n = ceil((-3+sqrt(9+8*p))/2); Cnr=1E=  
    m = 2*p - n.*(n+2); < z#.J]  
    ds9U9t  
    % Pass the inputs to the function ZERNFUN: ExhK\J  
    % ---------------------------------------- }a !ny  
    switch nargin %4wEAi$I  
        case 3 0q28Ulv9  
            z = zernfun(n,m,r,theta); q`'m:{8  
        case 4 gR+P !Eow  
            z = zernfun(n,m,r,theta,nflag); Y\Z6u)  
        otherwise CcTdLq  
            error('zernfun2:nargin','Incorrect number of inputs.') 3$?nzKTW\  
    end @HzK)%@  
    9Af nMD  
    % 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) g(auB/0s  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. .sc80i4  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of @~Ys*]4UE  
    %   order N and frequency M, evaluated at R.  N is a vector of a"i(.(9$J  
    %   positive integers (including 0), and M is a vector with the g&S> Wq%L  
    %   same number of elements as N.  Each element k of M must be a dt@~8kS  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) !?R#e`}  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is k.7!)jL7  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix ?2_h.  
    %   with one column for every (N,M) pair, and one row for every ySI}Nm>&=  
    %   element in R. <)pPq+  
    % 9B![l=Gh  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- _u""v   
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is rizjH+  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to CDF;cM"td  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 eIy:5/s  
    %   for all [n,m]. o~9sO=-O  
    % EXF]y}n  
    %   The radial Zernike polynomials are the radial portion of the >0[:uu,'>  
    %   Zernike functions, which are an orthogonal basis on the unit TQ:h[6v  
    %   circle.  The series representation of the radial Zernike [m4M#Lg\0  
    %   polynomials is =E$bZe8  
    % Qn|8Ic` *  
    %          (n-m)/2 j D*<M/4  
    %            __ :ssj7wl :  
    %    m      \       s                                          n-2s ps?su`  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r *- ~GVe  
    %    n      s=0 am !ssF5s  
    % s?s ,wdp  
    %   The following table shows the first 12 polynomials. .%dGSDru  
    % `\|@w@f|;  
    %       n    m    Zernike polynomial    Normalization l]~9BPsR  
    %       --------------------------------------------- BeLqk3'/  
    %       0    0    1                        sqrt(2) B|V!=r1%  
    %       1    1    r                           2 3M(*q4A$"  
    %       2    0    2*r^2 - 1                sqrt(6) .#Nf0  
    %       2    2    r^2                      sqrt(6) ?w!8;xS8  
    %       3    1    3*r^3 - 2*r              sqrt(8) E<}sGzMc  
    %       3    3    r^3                      sqrt(8) E24SD'|)  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) `k*;%}X\  
    %       4    2    4*r^4 - 3*r^2            sqrt(10)  0Gc:+c7{  
    %       4    4    r^4                      sqrt(10) 9D &vxKE  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) Xn?.Od(  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) ~\2%h lA  
    %       5    5    r^5                      sqrt(12) _;0RW  
    %       --------------------------------------------- D5oYcGc  
    % 7QnWw0  
    %   Example: M%la@2SK=  
    % 6]Q#4  
    %       % Display three example Zernike radial polynomials aeI0;u  
    %       r = 0:0.01:1; hr&&b3W3p  
    %       n = [3 2 5]; (]\p'%A)  
    %       m = [1 2 1]; "f<gZsb  
    %       z = zernpol(n,m,r); q9p31b3  
    %       figure {tl{ j1d |  
    %       plot(r,z) /\<x8BJ  
    %       grid on bM5V=b_H  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') LLT6*up$  
    % ^fe,A=k~1  
    %   See also ZERNFUN, ZERNFUN2. KQb&7k .  
    :*h1ik4t  
    % A note on the algorithm. J)y g<*/3  
    % ------------------------ Hc@_@G  
    % The radial Zernike polynomials are computed using the series AG}j'   
    % representation shown in the Help section above. For many special }aVzr}!  
    % functions, direct evaluation using the series representation can S[q:b .  
    % produce poor numerical results (floating point errors), because rr2 !H%:  
    % the summation often involves computing small differences between P7zUf  
    % large successive terms in the series. (In such cases, the functions [<{r~YFjWW  
    % are often evaluated using alternative methods such as recurrence NOwd'iU  
    % relations: see the Legendre functions, for example). For the Zernike 9G2rVk  
    % polynomials, however, this problem does not arise, because the q2J |koT  
    % polynomials are evaluated over the finite domain r = (0,1), and kr~n5WiAZ  
    % because the coefficients for a given polynomial are generally all ;,6C&|n]w  
    % of similar magnitude. Dn J `]r  
    % y\uBVa<B  
    % ZERNPOL has been written using a vectorized implementation: multiple 8f[ztT0`g  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] n"aF#HR?0d  
    % values can be passed as inputs) for a vector of points R.  To achieve XEbVsw  
    % this vectorization most efficiently, the algorithm in ZERNPOL QEbf]U=  
    % involves pre-determining all the powers p of R that are required to 7S 8X)  
    % compute the outputs, and then compiling the {R^p} into a single ]UEA"^  
    % matrix.  This avoids any redundant computation of the R^p, and gED|2%BXb  
    % minimizes the sizes of certain intermediate variables. 8x)i{>#i  
    % %AtT(G(n  
    %   Paul Fricker 11/13/2006 WkE="E}  
    yA"?Hv\o;  
    ) 9 2(C  
    % Check and prepare the inputs: IO9|o!&>  
    % ----------------------------- n.a2%,|v  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Yb`b /BMR  
        error('zernpol:NMvectors','N and M must be vectors.') z9OpMA  
    end F(."nUrf  
    dmXfz D  
    if length(n)~=length(m) LTzdg >\oJ  
        error('zernpol:NMlength','N and M must be the same length.') Ssz;d&93  
    end xg7KU&  
    C P&u  
    n = n(:); xR%NiYNQz  
    m = m(:); r<n:o7  
    length_n = length(n); ,{S $&g*  
    13Z,;YW  
    if any(mod(n-m,2)) K-\wx5#l/  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') cf$ hIB)Oi  
    end VVAcbAGJ  
    aXqig&:  
    if any(m<0) d9U)O6=  
        error('zernpol:Mpositive','All M must be positive.') <'$>&^!^  
    end R= mT J'y  
    R31Z(vY  
    if any(m>n) )P b$  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') GVlT+Rs7  
    end YJHb\Cf.  
    $ -<(geI  
    if any( r>1 | r<0 ) +x`pWH]2  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') 1;c>#20  
    end "|dhmV[;  
    '6){~ee S  
    if ~any(size(r)==1) U`EOun ,  
        error('zernpol:Rvector','R must be a vector.') xrBM`Bj0@  
    end bcy  
    %5zIh[!1$  
    r = r(:); |O>e=HC#q8  
    length_r = length(r); k*6"!J%A  
    y7!&  
    if nargin==4 .'H$|"( v  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); gy,)% {,G  
        if ~isnorm n!~ $Z/  
            error('zernpol:normalization','Unrecognized normalization flag.') >lqo73gM9  
        end C@:X9NU  
    else |y0k}ed  
        isnorm = false; Ad-5Zn c5  
    end T6\]*mlr  
    VK*`&D<P  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% i?M-~EKu  
    % Compute the Zernike Polynomials Tq )hAZ  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% <Fx%P:d  
    +2eri_p  
    % Determine the required powers of r: NrXIaN  
    % ----------------------------------- gfIS  
    rpowers = []; c u";rnj  
    for j = 1:length(n) Da8gOZ  
        rpowers = [rpowers m(j):2:n(j)]; .xT{Rz  
    end 6la'\l#  
    rpowers = unique(rpowers); yFmy  
    qyVARy  
    % Pre-compute the values of r raised to the required powers, Iq,h}7C8'  
    % and compile them in a matrix: }ff^^7_  
    % ----------------------------- H{N},B  
    if rpowers(1)==0 PknKzrEG:>  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ~4FzA,,  
        rpowern = cat(2,rpowern{:}); 2BF455e   
        rpowern = [ones(length_r,1) rpowern]; z?Z"*z  
    else 0',buJncV  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); s1::\&`za  
        rpowern = cat(2,rpowern{:}); &{<hY|%  
    end b3[!1i  
    :5j+^/   
    % Compute the values of the polynomials: JM/\n 4ea:  
    % -------------------------------------- B?^~1Ua9Zv  
    z = zeros(length_r,length_n); Vrlqje_Q  
    for j = 1:length_n oidK_mU9q  
        s = 0:(n(j)-m(j))/2; =Gzs+6A8  
        pows = n(j):-2:m(j); t&w.Wc X)  
        for k = length(s):-1:1 8pr toCB  
            p = (1-2*mod(s(k),2))* ... H$WD7/?j  
                       prod(2:(n(j)-s(k)))/          ... 2-Q5l*  
                       prod(2:s(k))/                 ... FF^h(Ea  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... xgkCN$zQ`  
                       prod(2:((n(j)+m(j))/2-s(k))); ey9hrRMR  
            idx = (pows(k)==rpowers); VR1]CN"G  
            z(:,j) = z(:,j) + p*rpowern(:,idx); CC3 i@  
        end .nVY" C&  
         t$t'{*t( T  
        if isnorm "bRjY?D  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); GKF!GbGR@  
        end F[jqJzCz  
    end `\"<%CCe  
    3 i>NKS  
    % 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)  [z;}^3b  
    C4-%|+Q i  
    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。 u-.nR}DM_  
    x @9rc,by  
    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)