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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 UhDf6A`]  
    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! I@z@s}x>  
     
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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 E<tR8='F  
    function z = zernfun(n,m,r,theta,nflag) "(W;rl  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. w^zqYGxG)  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N Vb#a ,t  
    %   and angular frequency M, evaluated at positions (R,THETA) on the Kyk{:UnI  
    %   unit circle.  N is a vector of positive integers (including 0), and 6^J[SQ6P  
    %   M is a vector with the same number of elements as N.  Each element ,J+L_S+B~  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) 4Zu1G#(zP  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, d])ctxB  
    %   and THETA is a vector of angles.  R and THETA must have the same P-[})Z=  
    %   length.  The output Z is a matrix with one column for every (N,M) 8<0P Ssx  
    %   pair, and one row for every (R,THETA) pair. gi/k#3_m  
    % lr;ubBbT  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike *^g]QQ  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), .]KC*2  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral Q1|6;4L  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, &R.5t/x_  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized t oDi70o  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. gfN=0Xj4  
    %  WvF{`N  
    %   The Zernike functions are an orthogonal basis on the unit circle. aB (pdW4  
    %   They are used in disciplines such as astronomy, optics, and SXl~lYUL  
    %   optometry to describe functions on a circular domain. Q3=5q w^  
    % QPLWRZu@  
    %   The following table lists the first 15 Zernike functions. <X{w^ cT_Q  
    % E=,b;S-  
    %       n    m    Zernike function           Normalization Hicd -'  
    %       -------------------------------------------------- @+zWLq!1pB  
    %       0    0    1                                 1 3'6 UvAXFH  
    %       1    1    r * cos(theta)                    2 Go:(R {P  
    %       1   -1    r * sin(theta)                    2 j3%Wrt  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) t {1 [Ip  
    %       2    0    (2*r^2 - 1)                    sqrt(3) 2/t;}pw8  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) 4?@#w>(  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) [~|k;\2 +  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) 6J JA"] `  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) uUHWTyoO  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) s}Go")p<:  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) ]smu~t0\  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 5CcX'*P  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) w0nbL^f  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) .eVX/6,  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) eJ<P  
    %       -------------------------------------------------- iJ*Wsp  
    % 3k>#z%//  
    %   Example 1: :epB:r  
    % e~)4v  
    %       % Display the Zernike function Z(n=5,m=1) 5QXU"kWH  
    %       x = -1:0.01:1; QaEiPn~  
    %       [X,Y] = meshgrid(x,x); jCtk3No  
    %       [theta,r] = cart2pol(X,Y); Bx}"X?%S  
    %       idx = r<=1; +?3RC$jyw  
    %       z = nan(size(X)); `%#_y67v  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); OOIp)=4  
    %       figure A_ &IK;-go  
    %       pcolor(x,x,z), shading interp Uv.Xw}q  
    %       axis square, colorbar &-^*D%9  
    %       title('Zernike function Z_5^1(r,\theta)') WhH60/`  
    % x4g6Qze  
    %   Example 2: @V^.eVM\R  
    % O"TVxP:  
    %       % Display the first 10 Zernike functions .Xf_U.h$*@  
    %       x = -1:0.01:1; a9^})By&  
    %       [X,Y] = meshgrid(x,x);  Br s}  
    %       [theta,r] = cart2pol(X,Y); $,r%@'=&  
    %       idx = r<=1; S{2;PaK  
    %       z = nan(size(X)); +ru`Zw5,  
    %       n = [0  1  1  2  2  2  3  3  3  3]; 5z3WRg  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; KgD$P(J:[  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; x~Z7p)D_<  
    %       y = zernfun(n,m,r(idx),theta(idx)); 6?US<<MQ  
    %       figure('Units','normalized') -b+)Dp~$p  
    %       for k = 1:10 1#"wfiW  
    %           z(idx) = y(:,k); )q4nyT>M  
    %           subplot(4,7,Nplot(k)) AriV4 +  
    %           pcolor(x,x,z), shading interp GFbn>dY  
    %           set(gca,'XTick',[],'YTick',[]) I;_T_m4.q  
    %           axis square rs>,p)  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) #<&@-D8  
    %       end OraT$lV)_  
    % |mWSS'7fI  
    %   See also ZERNPOL, ZERNFUN2. >zJkG9a  
    =M@)q y  
    %   Paul Fricker 11/13/2006 <)O#Y76s  
    XZ$g~r  
    q2*)e/}H  
    % Check and prepare the inputs: SV ~QH&0'  
    % ----------------------------- }mZCQJ#`  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) .uX(-8n ~  
        error('zernfun:NMvectors','N and M must be vectors.') Z(4/;v <CT  
    end ';v2ld 9  
    GpXf).a@  
    if length(n)~=length(m) a>8] +@  
        error('zernfun:NMlength','N and M must be the same length.') k8}'@w  
    end JDnWBEV  
    p.4Sgeh#  
    n = n(:); ~KGE(o4p  
    m = m(:); u|ihUE!h  
    if any(mod(n-m,2)) *)\y52z  
        error('zernfun:NMmultiplesof2', ... y}U'8*,  
              'All N and M must differ by multiples of 2 (including 0).') (1er?4  
    end Eqny'44  
    &2@Rc?!6_P  
    if any(m>n) l&] %APL  
        error('zernfun:MlessthanN', ... SU7,uxF  
              'Each M must be less than or equal to its corresponding N.') HH(2  
    end zKYN5|17  
    ,T  3M  
    if any( r>1 | r<0 )  d*([!!i  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') X&49C:jN  
    end xQ?$H?5B<  
    k-s|gC4  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) oM#+Z qP  
        error('zernfun:RTHvector','R and THETA must be vectors.') \:n<&<aVSr  
    end *$('ous8  
    | z}VP-L  
    r = r(:); 5|bfrc  
    theta = theta(:); B=_5gZ4Y  
    length_r = length(r); vPy."/[u  
    if length_r~=length(theta) Opy{i#>  
        error('zernfun:RTHlength', ... ;uZq_^?:9&  
              'The number of R- and THETA-values must be equal.') 6_9@s*=d>  
    end 2ss*&BR.  
    gK *=T  
    % Check normalization: T`I4_x  
    % -------------------- r:U<cL T[9  
    if nargin==5 && ischar(nflag) pF~aR]Q  
        isnorm = strcmpi(nflag,'norm'); b|k(:b-G&.  
        if ~isnorm pwVGe|h%,  
            error('zernfun:normalization','Unrecognized normalization flag.') XK0lv8(  
        end /b4>0DXT5  
    else dt<P6pK-  
        isnorm = false; K7qR  
    end JkLpoe81  
    j{ri]?p  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% URr{J}5  
    % Compute the Zernike Polynomials O6q5qA  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% _t X1z ^  
    mI^S% HT  
    % Determine the required powers of r: { ux'9SA  
    % ----------------------------------- vhU $GG8  
    m_abs = abs(m); -7I %^u  
    rpowers = []; %wJ>V-\e  
    for j = 1:length(n) \:Hh'-77q  
        rpowers = [rpowers m_abs(j):2:n(j)]; j3 @Q  
    end `Z2-<:]6&a  
    rpowers = unique(rpowers); e&<=+\ul  
    2rf#Bq?7  
    % Pre-compute the values of r raised to the required powers, 8*]dA ft  
    % and compile them in a matrix: ~>%% kQt  
    % ----------------------------- xCu\jc)2  
    if rpowers(1)==0 Fcn@j#[J  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); B|AIl+y  
        rpowern = cat(2,rpowern{:}); 7u%OYt D E  
        rpowern = [ones(length_r,1) rpowern]; OR10IS  
    else ?Bd6<F -G  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); urD{'FQf  
        rpowern = cat(2,rpowern{:}); +5Y;JL<%/  
    end a7z% )i;Z  
    ]6WP;.[  
    % Compute the values of the polynomials: 2d OUY $4  
    % -------------------------------------- ~.S/<:`U  
    y = zeros(length_r,length(n)); -}>H3hr  
    for j = 1:length(n) Ht~YSQ~:y  
        s = 0:(n(j)-m_abs(j))/2; EuD$^#  
        pows = n(j):-2:m_abs(j); Ige*tOv2  
        for k = length(s):-1:1 Oh7wyQiV  
            p = (1-2*mod(s(k),2))* ... J>0RN/38o  
                       prod(2:(n(j)-s(k)))/              ... T'14OU2N{Y  
                       prod(2:s(k))/                     ... 6s:  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... '"V]>)  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); 7C@m(oK  
            idx = (pows(k)==rpowers); xI5zP? _v  
            y(:,j) = y(:,j) + p*rpowern(:,idx); ^%33&<mB}  
        end 2 3A)^j  
         2cv=7!K4Uv  
        if isnorm R+=Xr<`%U|  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); `S]DHxS  
        end /8>we`4  
    end TzV~I\a|  
    % END: Compute the Zernike Polynomials 4+N9Ylh  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% MBFn s/  
    [g lhru=+  
    % Compute the Zernike functions: |OBZSk1jp  
    % ------------------------------ KC-@2,c9V  
    idx_pos = m>0; ru*}lDJ  
    idx_neg = m<0; %wmbFj}  
    )KN]"<jB  
    z = y; ].x`Fq3  
    if any(idx_pos) l`EKL2n  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); k NUNh[  
    end TmgSV#G  
    if any(idx_neg) 212  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); $&0\BvS  
    end .!g  
    $"{I| UFC  
    % 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) `g,i `<  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. 0/b3]{skK  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated K55]W2I9  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive +bcJm  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, >Te h ?P  
    %   and THETA is a vector of angles.  R and THETA must have the same NAEAvXj  
    %   length.  The output Z is a matrix with one column for every P-value, zFO#oW,D  
    %   and one row for every (R,THETA) pair. T2MXwd&l  
    % JA6#qlylL  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike Vg8c}>7  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) N5@l[F7I  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) JcI~8;Z@Z~  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 d?1[xv;  
    %   for all p. sKGR28e  
    % $or8z2d1  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 >I *uo.OF  
    %   Zernike functions (order N<=7).  In some disciplines it is r>7Dg~)V  
    %   traditional to label the first 36 functions using a single mode S1d{! ` 3  
    %   number P instead of separate numbers for the order N and azimuthal kk7M$)>d  
    %   frequency M. FKkL%:?  
    % xSZ+6R|  
    %   Example: MDOP2y`2i  
    % '&Tq/;Ml  
    %       % Display the first 16 Zernike functions mu&%ph=  
    %       x = -1:0.01:1; aX(Y `g)|  
    %       [X,Y] = meshgrid(x,x); $}Ky6sBnvO  
    %       [theta,r] = cart2pol(X,Y); 5s=L5]]r_j  
    %       idx = r<=1; hG lRf_{  
    %       p = 0:15; > R2o7~  
    %       z = nan(size(X)); _J33u3v  
    %       y = zernfun2(p,r(idx),theta(idx)); `ouCQ]tKz  
    %       figure('Units','normalized') ]sV) '-  
    %       for k = 1:length(p) ];au! _o  
    %           z(idx) = y(:,k); J nf@u  
    %           subplot(4,4,k) aj@<4A=;  
    %           pcolor(x,x,z), shading interp E0<$zP}V}F  
    %           set(gca,'XTick',[],'YTick',[]) SW*Y u{  
    %           axis square 9|1J pb  
    %           title(['Z_{' num2str(p(k)) '}']) w2o5+G=  
    %       end gqQ"'SRw  
    % z Ud{9B$  
    %   See also ZERNPOL, ZERNFUN. tk,Vp3p  
    "gGv>]3  
    %   Paul Fricker 11/13/2006 ""u>5f  
    J:Ncy}AO  
    _16IP  
    % Check and prepare the inputs: |;(0]  
    % ----------------------------- @DA.$zn&  
    if min(size(p))~=1  wA7^   
        error('zernfun2:Pvector','Input P must be vector.') .3< sv  
    end Y SD|#0  
    CWS&f g%o{  
    if any(p)>35 -@yu 9=DT  
        error('zernfun2:P36', ... ,)7y? *D}  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... dSw%Qv*y  
               '(P = 0 to 35).']) qB44;!(  
    end Ym/y2B(  
    f%5 s8)  
    % Get the order and frequency corresonding to the function number: ^h\Y.  
    % ---------------------------------------------------------------- ':LV"c4 t  
    p = p(:); ;$$.L bb8  
    n = ceil((-3+sqrt(9+8*p))/2); X*Cvh|  
    m = 2*p - n.*(n+2); -/ h'uG  
    'r_NA!R  
    % Pass the inputs to the function ZERNFUN: bO\E)%zp  
    % ---------------------------------------- e!JC5Al7  
    switch nargin :~{x'`czJ  
        case 3 3X A8\Mg  
            z = zernfun(n,m,r,theta); ,CA3Q.y>|  
        case 4 a.!|A(zw  
            z = zernfun(n,m,r,theta,nflag); ~AbTbQ3  
        otherwise a2\r^fY/  
            error('zernfun2:nargin','Incorrect number of inputs.') -P7JaH/Q  
    end y( uE  
    w,v~  
    % 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) xU_Dg56z'&  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. HHU0Nku@ho  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of i`)h~V|G  
    %   order N and frequency M, evaluated at R.  N is a vector of ?YTngIa  
    %   positive integers (including 0), and M is a vector with the \6z_ ;  
    %   same number of elements as N.  Each element k of M must be a + IpC  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) DsZBhjCB  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is rfVHPMD0  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix .uGvmD <;x  
    %   with one column for every (N,M) pair, and one row for every Q4vl  
    %   element in R. zPKx: I3  
    % 2IGoAt>V  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- ohPCYt  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is Ug1n4X3FKn  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to _K5R?"H0  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 rbw5.NU  
    %   for all [n,m]. #ovmX  
    % 9;*-y$@  
    %   The radial Zernike polynomials are the radial portion of the sa26u`?  
    %   Zernike functions, which are an orthogonal basis on the unit ]gHi5]\NC  
    %   circle.  The series representation of the radial Zernike eVy>  
    %   polynomials is m5/d=k0l  
    % eAPNF?0yh  
    %          (n-m)/2 S[zX@3eZV  
    %            __ qB0F9[U  
    %    m      \       s                                          n-2s +.u)\'r;h  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r i G%h-  
    %    n      s=0 QSxR@hC  
    % Xbp~cn  
    %   The following table shows the first 12 polynomials. tDk!]  
    % }KZt7)  
    %       n    m    Zernike polynomial    Normalization ,4&?`Q  
    %       --------------------------------------------- ][IEzeI_LN  
    %       0    0    1                        sqrt(2) d@?++z  
    %       1    1    r                           2 [_pw|BGp  
    %       2    0    2*r^2 - 1                sqrt(6) Jiv%Opo/|  
    %       2    2    r^2                      sqrt(6) [m9Iz!E  
    %       3    1    3*r^3 - 2*r              sqrt(8) qQ%RnD9  
    %       3    3    r^3                      sqrt(8) >A RZ=x[  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) th?w&;L  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) 5UgxuuP4  
    %       4    4    r^4                      sqrt(10) ev}ugRxt|k  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) } qf=5v  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) AJ0 ;wx  
    %       5    5    r^5                      sqrt(12) 3?+CP-T-j  
    %       --------------------------------------------- PS=N]e7k'  
    % TOe=6 Z5h  
    %   Example: [7btoo|P]  
    % m@Vz42g~+  
    %       % Display three example Zernike radial polynomials 5@kNvi  
    %       r = 0:0.01:1; <V~B8C!)  
    %       n = [3 2 5]; @g{FNXY$m  
    %       m = [1 2 1]; $gv3Up"U  
    %       z = zernpol(n,m,r); y| 7sh  
    %       figure Hv~& RZpe  
    %       plot(r,z) DNGXp5I  
    %       grid on Gz,?e]ZV  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') 5>e#SW  
    % R iPxz=kr  
    %   See also ZERNFUN, ZERNFUN2. ; m]KKB  
    $:i%\7=  
    % A note on the algorithm. Sz_{#-  
    % ------------------------ t6+c"=P#  
    % The radial Zernike polynomials are computed using the series KS3>c7  
    % representation shown in the Help section above. For many special 9[5qN!P;y  
    % functions, direct evaluation using the series representation can fK %${   
    % produce poor numerical results (floating point errors), because ZgzjRa++  
    % the summation often involves computing small differences between qq,#bRe  
    % large successive terms in the series. (In such cases, the functions  `u 't  
    % are often evaluated using alternative methods such as recurrence +'ZJ]  
    % relations: see the Legendre functions, for example). For the Zernike `Pcbc\"*y  
    % polynomials, however, this problem does not arise, because the D["~G v  
    % polynomials are evaluated over the finite domain r = (0,1), and RI[=N:C^  
    % because the coefficients for a given polynomial are generally all .T63:  
    % of similar magnitude. aJ{-m@/ 5  
    % .yF@Ow  
    % ZERNPOL has been written using a vectorized implementation: multiple zarxv| }$  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] Ki,SFww8r  
    % values can be passed as inputs) for a vector of points R.  To achieve Y_gMoo  
    % this vectorization most efficiently, the algorithm in ZERNPOL R_7[7 /a  
    % involves pre-determining all the powers p of R that are required to ZR," w  
    % compute the outputs, and then compiling the {R^p} into a single ILU7Yhk  
    % matrix.  This avoids any redundant computation of the R^p, and x%!Ea{ s  
    % minimizes the sizes of certain intermediate variables. C+m%_6<  
    % 5Qh$>R4!"  
    %   Paul Fricker 11/13/2006 9*&c2jh  
    +I$,Y~&`>  
    vh/&KTe?:  
    % Check and prepare the inputs: e2><Y<  
    % ----------------------------- 4m:D8&D_M  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) @CTSvTt$  
        error('zernpol:NMvectors','N and M must be vectors.') )/y7Fh  
    end 'xP&u<(F  
    a7fFp 9l!  
    if length(n)~=length(m) F{*h~7D-|  
        error('zernpol:NMlength','N and M must be the same length.') (2J\o  
    end =.48^$LWx  
    x_+-TC4IXn  
    n = n(:); vH?rln  
    m = m(:); }mYxI^n  
    length_n = length(n); ixY[ HDPq  
    1`Ig A0V`"  
    if any(mod(n-m,2)) K7-z.WTUR  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') 3-PqUJT$   
    end 0z =?}xr  
    !0Mx Bem  
    if any(m<0) +L,V_z  
        error('zernpol:Mpositive','All M must be positive.') GyZpdp!  
    end LsI8T uv  
    nf0]<x2  
    if any(m>n) N*`qsv 0  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') }`~n$OVx  
    end N;q)r  
    ;Zy[2M  
    if any( r>1 | r<0 ) L+X:M/)  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') Due@ '  
    end F+SqJSa  
    A`:a T{j  
    if ~any(size(r)==1) I !J'  
        error('zernpol:Rvector','R must be a vector.') 0g`$Dap  
    end FPE%h =sw  
    w$DHMpW'  
    r = r(:); mz|p=[lR|  
    length_r = length(r); ^?8/9 o  
    3OB=D{$V  
    if nargin==4 zMXQfR   
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); $3 =S\jyfK  
        if ~isnorm TYKs2+S6  
            error('zernpol:normalization','Unrecognized normalization flag.') o* ~aB_  
        end N XCvS0/h  
    else bP Q=88*  
        isnorm = false; ]SmN}Iq1  
    end gkmV; 0  
    +^DDWVp  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% f.Y [2b  
    % Compute the Zernike Polynomials ]5r@`%9  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 4D}hYk$eP0  
    \2^o,1r/  
    % Determine the required powers of r: 4Ql9VM%y  
    % ----------------------------------- ij,Rq`}l  
    rpowers = []; pft-.1py  
    for j = 1:length(n) jVPX]8  
        rpowers = [rpowers m(j):2:n(j)]; EO`eg]  
    end  b]gVZ-  
    rpowers = unique(rpowers); bE;c&g  
    q5G`q&O5  
    % Pre-compute the values of r raised to the required powers, DF>3)oTF  
    % and compile them in a matrix: Sh2BU3  
    % ----------------------------- }P'c8$  
    if rpowers(1)==0 b_X&>^4Dkl  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); *XOLuPL>6)  
        rpowern = cat(2,rpowern{:}); ^ -4~pDv^  
        rpowern = [ones(length_r,1) rpowern]; 7$*X   
    else EsS$th)d  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); (6Ciqf8  
        rpowern = cat(2,rpowern{:}); nb.|^O?  
    end j5[Y0)pV\  
    O 6ph_$nt.  
    % Compute the values of the polynomials: Q5b9q$L$  
    % -------------------------------------- q B IekQT  
    z = zeros(length_r,length_n); ed2r<H$  
    for j = 1:length_n >6R3KJe  
        s = 0:(n(j)-m(j))/2; uBl&{$<  
        pows = n(j):-2:m(j); #W&o]FAA3y  
        for k = length(s):-1:1 #jh5%@  
            p = (1-2*mod(s(k),2))* ... #aQQd8   
                       prod(2:(n(j)-s(k)))/          ... |BUgsE  
                       prod(2:s(k))/                 ... .DI?-=p|_#  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... Eq%f`Qg+1E  
                       prod(2:((n(j)+m(j))/2-s(k))); wB bCGU  
            idx = (pows(k)==rpowers); &f}w&k2yj  
            z(:,j) = z(:,j) + p*rpowern(:,idx); /,_m\ JkwL  
        end 58d[>0Xa[g  
         tpb lm|sW  
        if isnorm \,fa"^8  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); 7 =D,D+f  
        end jfiUf1Mj  
    end ?;y-skh  
    v;`>pCal  
    % 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)  PGJh>[ s  
    ZPmqoR[  
    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。 x 4+WZYv3  
    ^x\VMd3*w  
    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)