切换到宽版
  • 广告投放
  • 稿件投递
  • 繁體中文
    • 11261阅读
    • 9回复

    [求助]ansys分析后面型数据如何进行zernike多项式拟合? [复制链接]

    上一主题 下一主题
    离线niuhelen
     
    发帖
    19
    光币
    28
    光券
    0
    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 laqKP+G  
    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! Hk8:7"4Q  
     
    分享到
    离线phility
    发帖
    69
    光币
    11
    光券
    0
    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
    发帖
    69
    光币
    11
    光券
    0
    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
    发帖
    19
    光币
    28
    光券
    0
    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 B o.x  
    function z = zernfun(n,m,r,theta,nflag) -r]s #$  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. _)p@;vGV  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N +|r;t  
    %   and angular frequency M, evaluated at positions (R,THETA) on the m7z/@b[  
    %   unit circle.  N is a vector of positive integers (including 0), and ,W5pe#n  
    %   M is a vector with the same number of elements as N.  Each element Crh5^?  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) gWqmK/.U.0  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, jpZX5_o  
    %   and THETA is a vector of angles.  R and THETA must have the same aoz+g,1 //  
    %   length.  The output Z is a matrix with one column for every (N,M) ;gy_Qf2U  
    %   pair, and one row for every (R,THETA) pair. 6Bmv1n[X^h  
    % HI#}M|4n  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike -]~U_J]  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ;5ugnVXu  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral 5&v'aiWK  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, )NRY9\H  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized {}N*e"<O  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. @jN!j*Y H  
    % jWiZ!dtUZ  
    %   The Zernike functions are an orthogonal basis on the unit circle. (<s7X$(]e  
    %   They are used in disciplines such as astronomy, optics, and l Vo](#W  
    %   optometry to describe functions on a circular domain. 1Ls@|   
    % +VDwDJ)lG  
    %   The following table lists the first 15 Zernike functions. d"Y9go"Z  
    % -WE pBt7*  
    %       n    m    Zernike function           Normalization m/=,O_  
    %       -------------------------------------------------- (k6=o';y  
    %       0    0    1                                 1 4o9#B:N]J  
    %       1    1    r * cos(theta)                    2 35) ]R`f  
    %       1   -1    r * sin(theta)                    2 Hlp!6\gukp  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) eT[ ,k[#q  
    %       2    0    (2*r^2 - 1)                    sqrt(3) 6vro:`R ?  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) # Fw<R'c  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) ~e{AgY)  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8)  7.CzS  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) )M#~/~^f+  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) aWm0*W"(@  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) "Vho`x3  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) PDREwBX  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) /XEcA 5C<  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) zv  <,  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) 8II-'%S6q  
    %       -------------------------------------------------- DG O_fR5L  
    % g}{Rk>k  
    %   Example 1: ,(N&%  
    % |q^e&M<  
    %       % Display the Zernike function Z(n=5,m=1) }<uD[[FLB  
    %       x = -1:0.01:1; Lx8 ^V7 X  
    %       [X,Y] = meshgrid(x,x); [ 8N1tZ{`  
    %       [theta,r] = cart2pol(X,Y); RQ y|W}d_  
    %       idx = r<=1; o ]2=5;)  
    %       z = nan(size(X)); w:r0>  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); L7G':oA_`p  
    %       figure rs~RKTv-  
    %       pcolor(x,x,z), shading interp aN ). G1  
    %       axis square, colorbar 9Wb9g/L  
    %       title('Zernike function Z_5^1(r,\theta)') @NlnZfMu  
    % [Rs5hO  
    %   Example 2: Pw1V1v&> q  
    % 92]>"  
    %       % Display the first 10 Zernike functions yi"V'Us  
    %       x = -1:0.01:1; Z?oFee!4  
    %       [X,Y] = meshgrid(x,x); cm%QV?  
    %       [theta,r] = cart2pol(X,Y); t2BkQ8vr  
    %       idx = r<=1; mc?5,oz;pz  
    %       z = nan(size(X)); k <A>J-|  
    %       n = [0  1  1  2  2  2  3  3  3  3]; McNj TD  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; LV0g *ng  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; mdypZ1f_  
    %       y = zernfun(n,m,r(idx),theta(idx)); .oO_x>  
    %       figure('Units','normalized') :)g=AhBF  
    %       for k = 1:10 {K*l,U  
    %           z(idx) = y(:,k); #PVgx9T=_  
    %           subplot(4,7,Nplot(k)) 1jh^-d5  
    %           pcolor(x,x,z), shading interp ul(1)q^  
    %           set(gca,'XTick',[],'YTick',[]) 8 fVI33  
    %           axis square `dMOBYV  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) \x(J v Dt  
    %       end 0jrcXN~  
    % ',z'.t  
    %   See also ZERNPOL, ZERNFUN2. isj<lnQ  
    .P# c/SQp  
    %   Paul Fricker 11/13/2006 K~+y<z E  
    ?WG9}R[qE/  
    }z,4IHNn  
    % Check and prepare the inputs: |m"2B]"@  
    % ----------------------------- S!#7]wtbP  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) tJUMLn?  
        error('zernfun:NMvectors','N and M must be vectors.') @_FL,AC&m  
    end A_{QY&%m  
    Fw!5hR`,  
    if length(n)~=length(m) CP7Zin1S/w  
        error('zernfun:NMlength','N and M must be the same length.') -J:](p  
    end %HL@O]ftS  
    LdU, 32  
    n = n(:); ti`z:8n7  
    m = m(:); ~fAdOh  
    if any(mod(n-m,2)) yh]#V"W3  
        error('zernfun:NMmultiplesof2', ... }qmZ  
              'All N and M must differ by multiples of 2 (including 0).') [ \V]tpl!  
    end "h_n/}r=  
    Y%^&aacZ  
    if any(m>n) WWrD r  
        error('zernfun:MlessthanN', ... _&XT =SW}  
              'Each M must be less than or equal to its corresponding N.') >J3N,f  
    end aP cO9  
    ~Msee+ZZ :  
    if any( r>1 | r<0 ) =k2+VI  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') 7w @.)@5  
    end nDiD7:e7=  
    M7eO5  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) oE"!  
        error('zernfun:RTHvector','R and THETA must be vectors.') 6IPhy.8  
    end kkyn>Wxv  
    6%U1%;  
    r = r(:); I = qd\  
    theta = theta(:); ZA1?'  
    length_r = length(r);  +;Q &  
    if length_r~=length(theta) ^(N+s?  
        error('zernfun:RTHlength', ... }-V .upl  
              'The number of R- and THETA-values must be equal.') mmwwz  
    end BtBy.bR  
    k#JFDw\  
    % Check normalization: AjAmV hq  
    % -------------------- q_OIzZ@  
    if nargin==5 && ischar(nflag) WT'P[RU2  
        isnorm = strcmpi(nflag,'norm'); ,BW ^j.7  
        if ~isnorm +SrE  
            error('zernfun:normalization','Unrecognized normalization flag.') Gd%6lab  
        end }UXj|SY  
    else #n{wK+lz  
        isnorm = false; 15iCJ p  
    end OJ@';ZyT=  
    V/"0'H\"1  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% .oaW#f}0P  
    % Compute the Zernike Polynomials -R~;E[ {%  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  YDi_Gl$  
    a}M7"v9  
    % Determine the required powers of r: &5(|a"5+G  
    % ----------------------------------- s:*gjoL  
    m_abs = abs(m); z;#}u C  
    rpowers = []; V,|l&-  
    for j = 1:length(n) o7/_a/  
        rpowers = [rpowers m_abs(j):2:n(j)]; ;l4rg!r(S  
    end X2dTV}~i  
    rpowers = unique(rpowers); 7R7g$  
    =ub&@~E  
    % Pre-compute the values of r raised to the required powers, 73Mh65  
    % and compile them in a matrix: %dw-}1X  
    % ----------------------------- .N_0rPO,Kw  
    if rpowers(1)==0 ^._)HM  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); +_:Ih,-   
        rpowern = cat(2,rpowern{:}); 8Dhq_R'r  
        rpowern = [ones(length_r,1) rpowern]; LP@Q8{'  
    else H$(%FWzQ%  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 1_7x'5GdA  
        rpowern = cat(2,rpowern{:}); [ueT]%  
    end ~K:#a$!%,  
    # Sb1oLC  
    % Compute the values of the polynomials: .X_k[l9  
    % -------------------------------------- 3c@Cb`w@  
    y = zeros(length_r,length(n)); D*vrQ9&# 8  
    for j = 1:length(n) {(D$ Xb  
        s = 0:(n(j)-m_abs(j))/2; Tud[VS?99  
        pows = n(j):-2:m_abs(j); m`nv4i#o  
        for k = length(s):-1:1 lCWk)m8  
            p = (1-2*mod(s(k),2))* ... 8@6:UR.)  
                       prod(2:(n(j)-s(k)))/              ... o6xl,T%  
                       prod(2:s(k))/                     ... DI!NP;E  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... G{+sC2  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); EZ1H0fm  
            idx = (pows(k)==rpowers); oF]0o`U&a  
            y(:,j) = y(:,j) + p*rpowern(:,idx); N(t1?R/e,  
        end 3t68cdFlz  
         K`(STvtM  
        if isnorm l= ~]MSwY  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); u6t.$a!5  
        end e_k1pox]l  
    end ,_u8y&<|I  
    % END: Compute the Zernike Polynomials 5y}}?6n+  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {-Yp~HQF  
    U+~0m!|4  
    % Compute the Zernike functions: #jA|04w  
    % ------------------------------ ],qG!,V  
    idx_pos = m>0; 1k{ E7eL  
    idx_neg = m<0; *ubLuC+b  
    ofcoNLX5c  
    z = y; +;:i,`Lmg  
    if any(idx_pos) 1ReO.Dd`R  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); aina6@S  
    end !?O:%QG  
    if any(idx_neg) BI4 p3-  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); q/70fR7{v  
    end :ozHuHJ#  
    7" Dw4}T  
    % EOF zernfun
    离线niuhelen
    发帖
    19
    光币
    28
    光券
    0
    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) &>L\unS  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. +Nc|cj  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated <JF78MD\  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive sl |S9Ix  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, t8]u#bx"?  
    %   and THETA is a vector of angles.  R and THETA must have the same fm&l 0  
    %   length.  The output Z is a matrix with one column for every P-value, 1m}'Y@I  
    %   and one row for every (R,THETA) pair. "Q2[A]4E  
    % <adu^5BI  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike uW Q`  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) }-: d*YtK  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) P*I\FV  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 N4)& K[  
    %   for all p. I>L lc Y  
    % 2r PKZ|  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 wpx,~`&  
    %   Zernike functions (order N<=7).  In some disciplines it is W=I%3F_C"R  
    %   traditional to label the first 36 functions using a single mode z7HC6{g%X  
    %   number P instead of separate numbers for the order N and azimuthal /\ ~{  
    %   frequency M. k(%RX _]C  
    % q_cqjly<  
    %   Example: ]y-r I  
    % d 'x;]#S  
    %       % Display the first 16 Zernike functions "pMXTRb  
    %       x = -1:0.01:1; 8Q#&=]W$  
    %       [X,Y] = meshgrid(x,x); uZ<Bfrc  
    %       [theta,r] = cart2pol(X,Y); >tib21*  
    %       idx = r<=1; eA{,=, v)  
    %       p = 0:15; m_\CK5T_  
    %       z = nan(size(X)); YJ rK oK}  
    %       y = zernfun2(p,r(idx),theta(idx)); //H+S q66  
    %       figure('Units','normalized') =wS:)%u  
    %       for k = 1:length(p) Og30&a!~F  
    %           z(idx) = y(:,k); #z~D1Zl  
    %           subplot(4,4,k) YwB 5Zqr  
    %           pcolor(x,x,z), shading interp .}Bb :*@  
    %           set(gca,'XTick',[],'YTick',[]) K8284A8v  
    %           axis square dn%/SJC  
    %           title(['Z_{' num2str(p(k)) '}']) ~aA+L-s|  
    %       end Haq23K  
    % _IT,>#ba  
    %   See also ZERNPOL, ZERNFUN. oY+RG|j@  
    R`TM@aaS:  
    %   Paul Fricker 11/13/2006 e|+uLbN&;c  
    ks(PH6:]<  
    EMs$~CL4  
    % Check and prepare the inputs: #cjB <APY  
    % ----------------------------- El"XF?OgpP  
    if min(size(p))~=1 TN/I(pkt1B  
        error('zernfun2:Pvector','Input P must be vector.') {oz04KGsH  
    end mN@0lfk;  
    Lc<Gn y^  
    if any(p)>35 wx<5*8zP  
        error('zernfun2:P36', ... ='soSnT  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... pC#Z]_k  
               '(P = 0 to 35).']) <@;eN&  
    end 60X B  
    I&1Mh4yu  
    % Get the order and frequency corresonding to the function number: %pTbJaM\U  
    % ---------------------------------------------------------------- 5 0~L(<  
    p = p(:); He j0l^  
    n = ceil((-3+sqrt(9+8*p))/2); 6@Eip[e  
    m = 2*p - n.*(n+2); f"k/j?e*  
    ^z0[{1  
    % Pass the inputs to the function ZERNFUN: $2;YJjz(  
    % ---------------------------------------- j q1qj9KZ  
    switch nargin XUW~8P  
        case 3 ;]<$p[m  
            z = zernfun(n,m,r,theta); #;?z<  
        case 4 u7a4taM$d  
            z = zernfun(n,m,r,theta,nflag); Q?[k>fu0  
        otherwise ckhW?T>l  
            error('zernfun2:nargin','Incorrect number of inputs.') .>CqZN,^  
    end U%w-/!p  
    K})j5CJ/  
    % EOF zernfun2
    离线niuhelen
    发帖
    19
    光币
    28
    光券
    0
    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) 3R4-MK  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. A!iV iX &y  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of ~rn82an@G  
    %   order N and frequency M, evaluated at R.  N is a vector of 2psI\7UjA]  
    %   positive integers (including 0), and M is a vector with the Q&n  
    %   same number of elements as N.  Each element k of M must be a Qj 0@^LA  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) CXA)Zl5#  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is {u9VHAXCf  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix ; [dcbyu@  
    %   with one column for every (N,M) pair, and one row for every 4fpz;2%  
    %   element in R. oVmGZhkA@'  
    % =A=er1~%  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- WOgbz&S?J  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is 6S`eN\s  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to 7CwG(c/5  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 N],A&}30  
    %   for all [n,m]. $C^94$W  
    % ,ci tzh  
    %   The radial Zernike polynomials are the radial portion of the ] J:^$]  
    %   Zernike functions, which are an orthogonal basis on the unit 8 kd  
    %   circle.  The series representation of the radial Zernike nC[L"%E|se  
    %   polynomials is 6ng . =  
    % $?;aW^E  
    %          (n-m)/2 =xa`)#4(  
    %            __ % YU(,83(+  
    %    m      \       s                                          n-2s 5 QMu=/  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r A")B<BK  
    %    n      s=0 {\lu; b!  
    % KY4|C05 ,  
    %   The following table shows the first 12 polynomials. X}Fc0Oo  
    % ds7I .Q'  
    %       n    m    Zernike polynomial    Normalization xmq~:fcU=  
    %       --------------------------------------------- C=9|K`g5 R  
    %       0    0    1                        sqrt(2) qZA?M=NT?  
    %       1    1    r                           2 roL~r`f`  
    %       2    0    2*r^2 - 1                sqrt(6) hQl3F6-ud  
    %       2    2    r^2                      sqrt(6) 9\Yj`,i5  
    %       3    1    3*r^3 - 2*r              sqrt(8) 6,s@>8n  
    %       3    3    r^3                      sqrt(8) 2r[Q$GPM<  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) H={fY:%  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) W%~ S~wx  
    %       4    4    r^4                      sqrt(10) ~?[@KK  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) e2/&X;2  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) ::adT=  
    %       5    5    r^5                      sqrt(12) -+ $u  
    %       --------------------------------------------- wIi(p5*  
    % (lEWnf=2h  
    %   Example: <\Y>y+$3  
    % cWh Aj>?_Q  
    %       % Display three example Zernike radial polynomials eFZ`0V0  
    %       r = 0:0.01:1; u4+)lvt  
    %       n = [3 2 5]; {WFYNEQ[  
    %       m = [1 2 1]; |h6)p;`gc  
    %       z = zernpol(n,m,r); 0~n= |3*P  
    %       figure y>Nlj%XH  
    %       plot(r,z) ;~/  
    %       grid on ^$rt|]  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') \ m 2[  
    % #T !YFMh;  
    %   See also ZERNFUN, ZERNFUN2. p3sz32RX  
    tTQ>pg1{qh  
    % A note on the algorithm. M# S:'WN  
    % ------------------------ jY$|_o.4  
    % The radial Zernike polynomials are computed using the series S}*#$naK  
    % representation shown in the Help section above. For many special nLo:\I(  
    % functions, direct evaluation using the series representation can KX`MX5?x  
    % produce poor numerical results (floating point errors), because 63F0Za}h  
    % the summation often involves computing small differences between b/ ~&M+)  
    % large successive terms in the series. (In such cases, the functions HM ^rk  
    % are often evaluated using alternative methods such as recurrence &/a/V  
    % relations: see the Legendre functions, for example). For the Zernike !~>u\h  
    % polynomials, however, this problem does not arise, because the k]I<%  
    % polynomials are evaluated over the finite domain r = (0,1), and S{ fNeK  
    % because the coefficients for a given polynomial are generally all @8V8gV? zm  
    % of similar magnitude. @R`OAd y  
    % 9J l9\y9  
    % ZERNPOL has been written using a vectorized implementation: multiple )RA7Y}e|m  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] =o+t_.)N  
    % values can be passed as inputs) for a vector of points R.  To achieve c$ 1ez  
    % this vectorization most efficiently, the algorithm in ZERNPOL F+c*v#T  
    % involves pre-determining all the powers p of R that are required to /R F#B#9  
    % compute the outputs, and then compiling the {R^p} into a single Yckl,g_  
    % matrix.  This avoids any redundant computation of the R^p, and V{c n1Af  
    % minimizes the sizes of certain intermediate variables. .,tf[w 71  
    % Pf(z0o&  
    %   Paul Fricker 11/13/2006 xr.fZMOh4  
    IjNE1b$  
    Av+R~&h  
    % Check and prepare the inputs: CUY2eQJ{U  
    % ----------------------------- >f}rM20Vm  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) *3. ]  
        error('zernpol:NMvectors','N and M must be vectors.') FDpNM\SR1l  
    end y{"8VT)  
    h9SS o0]F  
    if length(n)~=length(m) MUVp8! *@  
        error('zernpol:NMlength','N and M must be the same length.') OG}0{?  
    end //| 9J(B]  
    'B6D&xn'%&  
    n = n(:); wK|&[m s  
    m = m(:); "64pVaT4  
    length_n = length(n); u3c e\  
    3}Uae#oy  
    if any(mod(n-m,2)) .X YSO  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') c69B[Vjb  
    end h*d&2>"0m?  
    &5C%5C~ch  
    if any(m<0) k6G23p[9  
        error('zernpol:Mpositive','All M must be positive.') d4A}BTs1  
    end .>h|e_E  
    CDR^xo5 dP  
    if any(m>n) DF9Br D0{  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') !"p,9  
    end /m9t2,KB  
    D:%$a]_f  
    if any( r>1 | r<0 ) H6e ^" E  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') 85Ms*[g  
    end >TK`s@jdSV  
    Fda<cS]  
    if ~any(size(r)==1) RI-whA8+  
        error('zernpol:Rvector','R must be a vector.') 2t#9ih"9  
    end zg|yW6l)9  
    \/{qE hP  
    r = r(:); 0^{zq|%Q!  
    length_r = length(r); ];j8vts&  
    x{RTI#a.  
    if nargin==4 sHh2>f@x$  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); AE^&hH0^  
        if ~isnorm qdUlT*fw  
            error('zernpol:normalization','Unrecognized normalization flag.') 'VR5>r  
        end 'Y>!xm   
    else GTJ\APrH  
        isnorm = false; ${e(#bvGZ  
    end 5C{X$7u  
    LF{qI?LG  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @AVx4,!>[  
    % Compute the Zernike Polynomials d|DIq T~{W  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Zw"6-h4  
    /rJvw   
    % Determine the required powers of r: -hhE`Y  
    % ----------------------------------- 9-pd{Z~l  
    rpowers = []; QDxLy aL  
    for j = 1:length(n) p|Z"< I7p(  
        rpowers = [rpowers m(j):2:n(j)]; r_ r+&4n  
    end l m-ubzJN  
    rpowers = unique(rpowers); y$\K@B4  
    f{^n<\Jh  
    % Pre-compute the values of r raised to the required powers, WDgp(Av!  
    % and compile them in a matrix: ChGwG.-%L  
    % ----------------------------- 'KyT]OObS  
    if rpowers(1)==0 &t p5y}=n  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 5p"*n kF  
        rpowern = cat(2,rpowern{:}); ,3N8  
        rpowern = [ones(length_r,1) rpowern]; 8v(Xr}q,r  
    else GpxGDN3?  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); TvM{ QGN  
        rpowern = cat(2,rpowern{:}); ;|9VPv/  
    end EA?:GtH  
    r]8tl  
    % Compute the values of the polynomials: <*4=sX@  
    % -------------------------------------- y~U+MtSf#  
    z = zeros(length_r,length_n); o&I 0*~ sN  
    for j = 1:length_n 5Ko "-  
        s = 0:(n(j)-m(j))/2; EKwS~G.b!  
        pows = n(j):-2:m(j); s?OGB}  
        for k = length(s):-1:1 .%~ L  
            p = (1-2*mod(s(k),2))* ... "@`M>)*o  
                       prod(2:(n(j)-s(k)))/          ... q@Q|oB0W$)  
                       prod(2:s(k))/                 ... LnR3C:NO k  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... t*Lo;]P  
                       prod(2:((n(j)+m(j))/2-s(k))); ?e&CbVc4  
            idx = (pows(k)==rpowers); oJXZ}>>iT  
            z(:,j) = z(:,j) + p*rpowern(:,idx); L~vNW6#W  
        end ,{zvGZ|  
         Z AZQFr'*  
        if isnorm XXe7w3x{  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); S7N54X2JwL  
        end ) e;F@o3  
    end nJ2l$J<  
    B%'Np7  
    % EOF zernpol
    离线niuhelen
    发帖
    19
    光币
    28
    光券
    0
    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
    发帖
    59
    光币
    0
    光券
    0
    只看该作者 7楼 发表于: 2012-09-28
    我也正在找啊
    离线guapiqlh
    发帖
    856
    光币
    846
    光券
    0
    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
    发帖
    4352
    光币
    5478
    光券
    1
    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)  5Impv3qaZ  
    !Noabt  
    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。 &"L3U  
    g`1*p|  
    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)