切换到宽版
  • 广告投放
  • 稿件投递
  • 繁體中文
    • 11441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 X/AA8QV o  
    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! 3:B4;  
     
    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 [*2|#KSCX  
    function z = zernfun(n,m,r,theta,nflag) %>)&QZig/  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. <cx,Z5W  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N (U@uJ  
    %   and angular frequency M, evaluated at positions (R,THETA) on the 63Dm{ 2i}F  
    %   unit circle.  N is a vector of positive integers (including 0), and ^[u*m%UB  
    %   M is a vector with the same number of elements as N.  Each element otSF8[  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) 0ofl,mXW  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, Jz Z9ua  
    %   and THETA is a vector of angles.  R and THETA must have the same =F>nqklc  
    %   length.  The output Z is a matrix with one column for every (N,M) :eR[lR^4*  
    %   pair, and one row for every (R,THETA) pair. "YQ%j+  
    % ,Y_[+  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike =^D{ZZw{  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), -mPrmapb3  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral g$eZT{{W  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, u*C"d1v=  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized _0c$SK  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. mzoNXf:x  
    % ja|XFs~  
    %   The Zernike functions are an orthogonal basis on the unit circle. ?ybX &V  
    %   They are used in disciplines such as astronomy, optics, and q oJ4w7  
    %   optometry to describe functions on a circular domain. 9CW8l0  
    % YkqauyV^  
    %   The following table lists the first 15 Zernike functions. i<]Y0_?s  
    % |Je+y;P7  
    %       n    m    Zernike function           Normalization 7IV:X _y  
    %       -------------------------------------------------- %G>|u/:U  
    %       0    0    1                                 1 ~!G&K`u  
    %       1    1    r * cos(theta)                    2 /qalj\ud  
    %       1   -1    r * sin(theta)                    2 VtJy0OGcRP  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) D8I)3cXa'  
    %       2    0    (2*r^2 - 1)                    sqrt(3) D_MNF =7  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) OJH:k~]0!  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) dS[="Set  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) %M_5C4&6  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) Q8sCI An{  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) GOeYw[Vh  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) /^>yDG T,0  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) gc6T`O-_;  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) ie+746tFW  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) w}jH,Ew  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) /Dn  
    %       -------------------------------------------------- 1n86Mp1.e  
    % D;l)&"|r?  
    %   Example 1: ;PrL)!  
    % At#'q>Dn  
    %       % Display the Zernike function Z(n=5,m=1) <(%cb.^c=N  
    %       x = -1:0.01:1; W%k0_Y/5  
    %       [X,Y] = meshgrid(x,x); m#oZu {  
    %       [theta,r] = cart2pol(X,Y); 9ywPWT[^  
    %       idx = r<=1; ,UD,)ZPf[  
    %       z = nan(size(X)); i%R2#F7I  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); {lhdropd  
    %       figure @Fl&@ $  
    %       pcolor(x,x,z), shading interp 5E#koy7 $s  
    %       axis square, colorbar 6c/Tm0[  
    %       title('Zernike function Z_5^1(r,\theta)') ;_^ "}  
    % B?xu!B,  
    %   Example 2: t/baze;V  
    % %Jr6pmc  
    %       % Display the first 10 Zernike functions ]GS@ub  
    %       x = -1:0.01:1; 1wqsGad+;  
    %       [X,Y] = meshgrid(x,x); X|WAUp?  
    %       [theta,r] = cart2pol(X,Y); Kb#}f/  
    %       idx = r<=1; N!e?K=}tL  
    %       z = nan(size(X)); QzQTE-SQ  
    %       n = [0  1  1  2  2  2  3  3  3  3];  :_qgpE<  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; w]{NaNIeq1  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; 7 vS]O$w<4  
    %       y = zernfun(n,m,r(idx),theta(idx)); 82X}@5o2  
    %       figure('Units','normalized') 2Q,8@2w;  
    %       for k = 1:10 R":nG7o  
    %           z(idx) = y(:,k); wghz[qe  
    %           subplot(4,7,Nplot(k)) Ass8c]H@  
    %           pcolor(x,x,z), shading interp 'CH|w~E  
    %           set(gca,'XTick',[],'YTick',[]) hOX$|0i  
    %           axis square jnK8 [och  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) M-2:$;D  
    %       end m_TZY_;  
    % cs?@Ri=g  
    %   See also ZERNPOL, ZERNFUN2. 'xdM>y#S  
    eqSCNYN  
    %   Paul Fricker 11/13/2006 lxRzyx  
    l.i"Z pik  
    `O5kI#m)L*  
    % Check and prepare the inputs: }[u9vZL  
    % ----------------------------- |f^/((:D  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Hy<4q^3$G  
        error('zernfun:NMvectors','N and M must be vectors.') <:u)C;  
    end W"rX$D [Le  
    N[j7^q7Xt  
    if length(n)~=length(m) ]u_^~  
        error('zernfun:NMlength','N and M must be the same length.') 2O|o%`?  
    end cz/mUU  
    E5lC'@Dcz  
    n = n(:); [|2uu."$  
    m = m(:); eB:obz  
    if any(mod(n-m,2)) -#b-@sD  
        error('zernfun:NMmultiplesof2', ... Y.?|[x0Wh  
              'All N and M must differ by multiples of 2 (including 0).') yKO84cSl  
    end =L$};ko  
    #[*e$C  
    if any(m>n) #ZIV>(Q\H  
        error('zernfun:MlessthanN', ... /h0<0b?i  
              'Each M must be less than or equal to its corresponding N.') W|T"'M_  
    end $2F*p#l(<Z  
    Uq/(xh,t5  
    if any( r>1 | r<0 ) @T1/S&F=  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') {Gs&u>>R"^  
    end {=7W;uL  
    L_jwM ^8  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) (J): >\a]  
        error('zernfun:RTHvector','R and THETA must be vectors.') Zg7~&vs$  
    end ~Xnq(}?ok  
    Jug1Va<^c  
    r = r(:); p]:~z|.Ba  
    theta = theta(:); >ofS'mp  
    length_r = length(r);  !+IxPn  
    if length_r~=length(theta) gtz!T2%  
        error('zernfun:RTHlength', ... Y ,?  
              'The number of R- and THETA-values must be equal.') 0-g,C=L  
    end SGH"m/ e  
    %|Vo Zx ^  
    % Check normalization: 0i$jtCCL(  
    % -------------------- ,u( g#T  
    if nargin==5 && ischar(nflag) <P( K,L?r  
        isnorm = strcmpi(nflag,'norm'); Xt'R@"H<V9  
        if ~isnorm %yQ-~T@  
            error('zernfun:normalization','Unrecognized normalization flag.') KbH#g>.oB  
        end ?4q6>ipx  
    else V/|Ln*rm  
        isnorm = false; M!=v"C#  
    end <HG~#oBRq  
    -z0,IYG }  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% < V"'j  
    % Compute the Zernike Polynomials K;-:C9@  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% " %|CD"@  
    +:It1`A~]  
    % Determine the required powers of r: Np|i Xwl1  
    % ----------------------------------- >S{1=N@Ev=  
    m_abs = abs(m); 622mNY  
    rpowers = []; v{=-#9-4 &  
    for j = 1:length(n) I]Wb\&$  
        rpowers = [rpowers m_abs(j):2:n(j)]; d[rxmEXht  
    end xzMa[D4(  
    rpowers = unique(rpowers); h&yaug,.  
    u[s+YGS  
    % Pre-compute the values of r raised to the required powers, jzEimKDE's  
    % and compile them in a matrix: \I,<G7!0  
    % ----------------------------- d2.eDEOsC  
    if rpowers(1)==0 5jy>)WqK  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); h+.^8fPR   
        rpowern = cat(2,rpowern{:}); /R k5n  
        rpowern = [ones(length_r,1) rpowern]; sj. eJX"z  
    else wGISb\rr  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); V}9wx%v  
        rpowern = cat(2,rpowern{:}); 5 qG7LO.  
    end |=38t8Ge&  
    I U 4[}x  
    % Compute the values of the polynomials: -mX _I{BJ  
    % -------------------------------------- Ks X@e)8u  
    y = zeros(length_r,length(n)); e@0wF59  
    for j = 1:length(n) A1%V<im@Z  
        s = 0:(n(j)-m_abs(j))/2; !M^pL|  
        pows = n(j):-2:m_abs(j); h{<^?=  
        for k = length(s):-1:1 giaO7Qh~  
            p = (1-2*mod(s(k),2))* ... W .Hv2r3  
                       prod(2:(n(j)-s(k)))/              ... g:;v]   
                       prod(2:s(k))/                     ... = "c _<?=[  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 2E2J=Do  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); {Fb)Z"8]  
            idx = (pows(k)==rpowers); (: ZOoL  
            y(:,j) = y(:,j) + p*rpowern(:,idx); #wM0p:<  
        end (eO0 Ic[c  
         v l{hE~  
        if isnorm J4lE7aFDA~  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); @[ :sP  
        end !k<+-Lf:2  
    end 1P2%n[y  
    % END: Compute the Zernike Polynomials B}P,sFghw  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% /B1< N}  
    %$]u6GKabi  
    % Compute the Zernike functions: gdCU1D\  
    % ------------------------------ YLobBtXc9  
    idx_pos = m>0; fEQ<L!'  
    idx_neg = m<0; 6Mk@,\1  
    R>gj"nB  
    z = y; 3<JZt.|  
    if any(idx_pos) 1uXtBk6  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); )[nzmL*w  
    end )b!q  
    if any(idx_neg) $AsM 9D<BE  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); D;d;:WT5  
    end y[r T5ed  
    2s 6Vy  
    % EOF zernfun
    离线niuhelen
    发帖
    19
    光币
    28
    光券
    0
    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) DK)qBxc8  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. bJBx~  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated Vn8Qsf1f  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive (1jkZ^7  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, v"$; aJ  
    %   and THETA is a vector of angles.  R and THETA must have the same PL8akA#  
    %   length.  The output Z is a matrix with one column for every P-value, s}?98?tYB  
    %   and one row for every (R,THETA) pair. mZM5aTQ3  
    % d1E~H]X4  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike 9Hc#[Ml  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) 9L&AbmIr  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) t}oxHEa V  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 `>#X,Lw$g  
    %   for all p. Nxt/R%(  
    % R jAeN#,?  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 {'r*Jb0  
    %   Zernike functions (order N<=7).  In some disciplines it is ^NnZYr.  
    %   traditional to label the first 36 functions using a single mode 9f"6Jw@F  
    %   number P instead of separate numbers for the order N and azimuthal ?tSY=DK\n  
    %   frequency M. Y":hb;&  
    % ZjI^0D8  
    %   Example: "?S#vUS+ 2  
    % 0cS.|\ZTA  
    %       % Display the first 16 Zernike functions foJ|Q\Z,T  
    %       x = -1:0.01:1; 1MV^~I8Dd  
    %       [X,Y] = meshgrid(x,x); T%]: tDa  
    %       [theta,r] = cart2pol(X,Y); _D+7w'8h  
    %       idx = r<=1; `\.n_nM  
    %       p = 0:15; `zsKc 6%  
    %       z = nan(size(X)); UHCx}LGe  
    %       y = zernfun2(p,r(idx),theta(idx)); 6U""TR!   
    %       figure('Units','normalized') c dGl[dQ/  
    %       for k = 1:length(p) "thu@~aC  
    %           z(idx) = y(:,k); H[G EAQO  
    %           subplot(4,4,k) 'Klz`)F  
    %           pcolor(x,x,z), shading interp n1;V2k{uV  
    %           set(gca,'XTick',[],'YTick',[]) T!*7G:\f"  
    %           axis square U>I#f  
    %           title(['Z_{' num2str(p(k)) '}']) i$NnHj|  
    %       end tx.YW9xD  
    % mC93 &0  
    %   See also ZERNPOL, ZERNFUN. SZ1C38bd,.  
    $L`7J$'^  
    %   Paul Fricker 11/13/2006 vu#:D1/BB  
    Jq; }q63:  
    f~U|flL^  
    % Check and prepare the inputs: '%~zu]f'  
    % ----------------------------- >:Oo[{)  
    if min(size(p))~=1 \Lc]6?,R  
        error('zernfun2:Pvector','Input P must be vector.') ahf$#UQLb  
    end P]x@h  
    &#)3v8  
    if any(p)>35 -0Q!:5EC  
        error('zernfun2:P36', ... |0bSxPXn!  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... ]O\6.>H  
               '(P = 0 to 35).']) +0a',`yc  
    end xFvSQ`sp  
    =kCpCpET  
    % Get the order and frequency corresonding to the function number: mee-Qq:}  
    % ---------------------------------------------------------------- 6D+k[oHZm  
    p = p(:); [+%*s3`c#  
    n = ceil((-3+sqrt(9+8*p))/2); ~/.&Z`ls  
    m = 2*p - n.*(n+2); +HcH]D;  
    Fb}9cpz{  
    % Pass the inputs to the function ZERNFUN: N0Y!  
    % ---------------------------------------- . =+7H`A  
    switch nargin O4#zsr:"  
        case 3 ov+qYBuFw  
            z = zernfun(n,m,r,theta); iN)@Cu7  
        case 4 bEE:6)]G  
            z = zernfun(n,m,r,theta,nflag); +pV3.VMH0  
        otherwise :L?zk"0C  
            error('zernfun2:nargin','Incorrect number of inputs.') 9 o18VJR  
    end Z*Y?"1ar  
    ht-6_]+ME  
    % 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) k^L (q\D  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. W)"q9(T?%  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of vB,N6~r>  
    %   order N and frequency M, evaluated at R.  N is a vector of COT;KC6 n  
    %   positive integers (including 0), and M is a vector with the ewLr+8  
    %   same number of elements as N.  Each element k of M must be a N;w1f"V}  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) qzLRA.#f^  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is oT_,k}LIX  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix [sad}@R7  
    %   with one column for every (N,M) pair, and one row for every 3646.i[D  
    %   element in R. ;L`'xFo>>  
    % a[u8x mH  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- N8vWwN[3  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is Nhf!;>  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to e 9:l  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 EbW7Av  
    %   for all [n,m]. (&B & V  
    % x|Ei_hI-  
    %   The radial Zernike polynomials are the radial portion of the J^W.TM&q$,  
    %   Zernike functions, which are an orthogonal basis on the unit ,aN/``j=  
    %   circle.  The series representation of the radial Zernike x?%vqg^r  
    %   polynomials is /yOd]N;$  
    % 'Hg(N?1"  
    %          (n-m)/2 <wuP*vI "h  
    %            __ kSJWQ  
    %    m      \       s                                          n-2s $""[( d?0  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r %mq]M  
    %    n      s=0 o0/03O  
    % A ssf f;  
    %   The following table shows the first 12 polynomials. n% *u;iG  
    % 0>'1|8+`(z  
    %       n    m    Zernike polynomial    Normalization m}XI?[!s  
    %       --------------------------------------------- l5R H~F  
    %       0    0    1                        sqrt(2) tSm|U<  
    %       1    1    r                           2 $'&5gFr9  
    %       2    0    2*r^2 - 1                sqrt(6) T#( s2  
    %       2    2    r^2                      sqrt(6) $+mmqc8  
    %       3    1    3*r^3 - 2*r              sqrt(8) ctCfLlK  
    %       3    3    r^3                      sqrt(8) ^fx9R 5E$:  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) X23TS`  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) A>PM'$"sT  
    %       4    4    r^4                      sqrt(10) [$V_qFv{  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) _ x7Vyy5  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) }r i"u;.R  
    %       5    5    r^5                      sqrt(12) _Y0o\0B  
    %       --------------------------------------------- 3!d|K%J  
    % eg}|%GG  
    %   Example: :&a|8Wi[W  
    % (YR] X_  
    %       % Display three example Zernike radial polynomials ]y(#]Tw\  
    %       r = 0:0.01:1; T&!>lqU!J  
    %       n = [3 2 5]; U{;i864:}  
    %       m = [1 2 1]; Og,,s{\  
    %       z = zernpol(n,m,r); ML R3 A s  
    %       figure 3QF!fll^  
    %       plot(r,z) H7{ 6t(0j  
    %       grid on /,;9hx  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') x3vz4m[  
    % CSD8?k]2  
    %   See also ZERNFUN, ZERNFUN2. R~tv?hP  
    $zdJ\UX  
    % A note on the algorithm. r)b`3=  
    % ------------------------ IUZsLNW  
    % The radial Zernike polynomials are computed using the series X-ml0 =M[  
    % representation shown in the Help section above. For many special wRuJein#  
    % functions, direct evaluation using the series representation can R sujKh/  
    % produce poor numerical results (floating point errors), because F"bbU/5  
    % the summation often involves computing small differences between sQ`8L+oY  
    % large successive terms in the series. (In such cases, the functions {g);HnmPN  
    % are often evaluated using alternative methods such as recurrence 1di?@F2f  
    % relations: see the Legendre functions, for example). For the Zernike 1LE8,Gm&  
    % polynomials, however, this problem does not arise, because the 2{ptV\f]D  
    % polynomials are evaluated over the finite domain r = (0,1), and yEz2F3[ S  
    % because the coefficients for a given polynomial are generally all GWhb@K  
    % of similar magnitude. Fc nR}TE  
    % 9@S icqx   
    % ZERNPOL has been written using a vectorized implementation: multiple r` 3)sc  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] ?5->F/f&  
    % values can be passed as inputs) for a vector of points R.  To achieve ikY=}  
    % this vectorization most efficiently, the algorithm in ZERNPOL PvA%c<z  
    % involves pre-determining all the powers p of R that are required to 3rWqt  
    % compute the outputs, and then compiling the {R^p} into a single MjLyB^ M  
    % matrix.  This avoids any redundant computation of the R^p, and T? =jKLPC  
    % minimizes the sizes of certain intermediate variables. CUYp(GU  
    % .AV--oA~  
    %   Paul Fricker 11/13/2006 u]oS91  
    CjO/q)vV  
    Jw86P=  
    % Check and prepare the inputs: 21;n0E  
    % ----------------------------- )kd PAw  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) H!xBFiOH$n  
        error('zernpol:NMvectors','N and M must be vectors.') &QE* V  
    end =,(Ba'  
    O'p7^"M  
    if length(n)~=length(m) %i^%D  
        error('zernpol:NMlength','N and M must be the same length.') $x 2t0@  
    end F 6SIhf.;  
    d$>1 2>>  
    n = n(:); Maq{H`  
    m = m(:); ]{;K|rCR-  
    length_n = length(n); y[}BFUy  
    {UwJg  
    if any(mod(n-m,2)) H{+U; 6b  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') "Pu P J|  
    end U"ga0X5  
    zS?L3*u  
    if any(m<0) tnTr &o#  
        error('zernpol:Mpositive','All M must be positive.') ;)o%2#I  
    end wlkS+$<  
    ]P 2M  
    if any(m>n) .&L#%C  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') AA@J~qd u  
    end D@f%&|IZ  
    M.t5,NJ  
    if any( r>1 | r<0 ) L1aN"KGMF  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') ZM5[ o m  
    end T$'Ja'9Kj  
    @jjp\~  
    if ~any(size(r)==1) 6C   
        error('zernpol:Rvector','R must be a vector.') n2T vPt\  
    end fEM8/bhq  
    tFb49zbk  
    r = r(:); *WOA",gZ  
    length_r = length(r); 6g<JPc  
    ;}:"[B3$  
    if nargin==4 ku\_M  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); E|ZY2&J`4  
        if ~isnorm *BSL=8G{  
            error('zernpol:normalization','Unrecognized normalization flag.') psYfz)1;  
        end ;;UvK v  
    else B_:K.]DK`  
        isnorm = false; \24neD4cM@  
    end JSO>rpO  
    kkqrl JO|  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% uD<*g(R  
    % Compute the Zernike Polynomials agt7b@-5=  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% koaH31Q  
    )0/ D Y  
    % Determine the required powers of r: @aBZ|8  
    % ----------------------------------- d<#Xqc  
    rpowers = []; G;, 2cu K  
    for j = 1:length(n) T0Y=g n  
        rpowers = [rpowers m(j):2:n(j)]; o.sa ?*  
    end A*@!tz<  
    rpowers = unique(rpowers); 0Wk}d(f  
    G&:YgwG  
    % Pre-compute the values of r raised to the required powers, 9t;aJFI  
    % and compile them in a matrix: Lw-)ijBW  
    % ----------------------------- EjvxfqPv  
    if rpowers(1)==0 hcM 0?=  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); e}aD <E G  
        rpowern = cat(2,rpowern{:}); L(}T-.,Slr  
        rpowern = [ones(length_r,1) rpowern]; I'J=I{p*  
    else [NJ!  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); pNE!waR>  
        rpowern = cat(2,rpowern{:}); 011 _(v  
    end Pbz-I3+66  
    Rqu;;VI[  
    % Compute the values of the polynomials: nuXaZRH  
    % -------------------------------------- ou@Dd4  
    z = zeros(length_r,length_n); wgI$'tI  
    for j = 1:length_n AnIENJ  
        s = 0:(n(j)-m(j))/2; U9kt7#@FDK  
        pows = n(j):-2:m(j); >b<br  
        for k = length(s):-1:1 pH)V:BmJ  
            p = (1-2*mod(s(k),2))* ... 2<U5d`  
                       prod(2:(n(j)-s(k)))/          ... {8b6A~/  
                       prod(2:s(k))/                 ... 6rdm=8WFA  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... `/0X].s#o  
                       prod(2:((n(j)+m(j))/2-s(k))); .wYx_  
            idx = (pows(k)==rpowers); llQDZ}T  
            z(:,j) = z(:,j) + p*rpowern(:,idx); YAd.i@^  
        end ^d-`?zb  
         ;J2=6np  
        if isnorm 7nfQ=?XNK  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); Ma wio5  
        end {Pu\KRU  
    end 4B!]%Mw;c  
    9%iqequ  
    % EOF zernpol
    离线niuhelen
    发帖
    19
    光币
    28
    光券
    0
    只看该作者 6楼 发表于: 2011-03-12
    这三个文件,我不知道该怎样把我的面型节点的坐标及轴向位移用起来,还烦请指点一下啊,谢谢啦!
    离线li_xin_feng
    发帖
    59
    光币
    0
    光券
    0
    只看该作者 7楼 发表于: 2012-09-28
    我也正在找啊
    离线guapiqlh
    发帖
    856
    光币
    846
    光券
    0
    只看该作者 8楼 发表于: 2014-03-04
    我也一直想了解这个多项式的应用,还没用过呢
    离线phoenixzqy
    发帖
    4352
    光币
    5478
    光券
    1
    只看该作者 9楼 发表于: 2014-04-22
    回 guapiqlh 的帖子
    guapiqlh:我也一直想了解这个多项式的应用,还没用过呢 (2014-03-04 11:35)  O BF5Tl4  
    #oJ9BgDry  
    数值分析方法看一下就行了。其实就是正交多项式的应用。zernike也只不过是正交多项式的一种。 6TlkPM$~2  
    2px l!  
    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)