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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 `]5XY8^kI  
    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! mwbkXy;8  
     
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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 ,nHz~Xi1t  
    function z = zernfun(n,m,r,theta,nflag) G)<k5U4  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. $S,Uoh  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N cK-!Evv  
    %   and angular frequency M, evaluated at positions (R,THETA) on the ,{oP`4\Lm  
    %   unit circle.  N is a vector of positive integers (including 0), and (O`=$e  
    %   M is a vector with the same number of elements as N.  Each element u'32nf?  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) -3 W 4  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, l}O`cC  
    %   and THETA is a vector of angles.  R and THETA must have the same i"e) LJz  
    %   length.  The output Z is a matrix with one column for every (N,M) `J-"S<c?_  
    %   pair, and one row for every (R,THETA) pair. ]/$tt@h  
    % |LNXu  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike '6/uc:zv  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), S0 yPg9v  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral t?0=;.D  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, YF:NRY[i  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized fA 3  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. zP9 HYS  
    % 6@I7UL >  
    %   The Zernike functions are an orthogonal basis on the unit circle. uWfse19  
    %   They are used in disciplines such as astronomy, optics, and -y/?w*Cx  
    %   optometry to describe functions on a circular domain. |f>y"T+1  
    % Y7{|EI+@  
    %   The following table lists the first 15 Zernike functions. sdO;vp^:b  
    % C*78ZwZ  
    %       n    m    Zernike function           Normalization yRgo1ow]  
    %       -------------------------------------------------- Gf%o|kX]  
    %       0    0    1                                 1 v5 9>  
    %       1    1    r * cos(theta)                    2 N %?o-IY  
    %       1   -1    r * sin(theta)                    2 Ffhbs D  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) S3J6P2P  
    %       2    0    (2*r^2 - 1)                    sqrt(3) jr9ZRHCU  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) +s S*EvF  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) tNUcmiY  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) 2i>xJMW  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) C$(t`G  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) F)%; gzs  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) {T^'&W>8G8  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 9 /zz@  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) NeK:[Q@je  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) jkdNisq37  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) w{r ->Phe  
    %       -------------------------------------------------- Tbwq_3f K  
    % t|y4kM  
    %   Example 1: .xk<7^ZD  
    % 7"[lWC!As5  
    %       % Display the Zernike function Z(n=5,m=1) UwM}!K7)G  
    %       x = -1:0.01:1; 9iOlR=-*  
    %       [X,Y] = meshgrid(x,x); .(hb8 rCM  
    %       [theta,r] = cart2pol(X,Y); 9M!_D?+P?  
    %       idx = r<=1; Xt7'clr  
    %       z = nan(size(X)); txgGL'  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); qB=pp!zQ  
    %       figure b1& {%.3[  
    %       pcolor(x,x,z), shading interp lZua"Ju  
    %       axis square, colorbar cj8r-Vu/N  
    %       title('Zernike function Z_5^1(r,\theta)') hZ#tB  
    % 5m bs0GL  
    %   Example 2: YVaQ3o|!  
    % ^twv0>vEo  
    %       % Display the first 10 Zernike functions $yc,D=*Isi  
    %       x = -1:0.01:1; :^*V[77  
    %       [X,Y] = meshgrid(x,x); @t2 Q5c  
    %       [theta,r] = cart2pol(X,Y); o|}%pc3  
    %       idx = r<=1; hKT:@l*  
    %       z = nan(size(X)); 6X jUb  
    %       n = [0  1  1  2  2  2  3  3  3  3]; +Va?wAnr  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; XnY}dsS O  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; 'w!gQ#De  
    %       y = zernfun(n,m,r(idx),theta(idx)); i&3 0n#  
    %       figure('Units','normalized') ^GAdl}  
    %       for k = 1:10 SB'YV#--  
    %           z(idx) = y(:,k); bOFLI#p&  
    %           subplot(4,7,Nplot(k)) E*I]v  
    %           pcolor(x,x,z), shading interp f|G7L5-  
    %           set(gca,'XTick',[],'YTick',[]) 87Uv+((H  
    %           axis square .;F+ QP0  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) \I'Zc]  
    %       end X@Bpjg  
    % 7E5Dz7  
    %   See also ZERNPOL, ZERNFUN2. b(yO  
    v-gT 3kJ  
    %   Paul Fricker 11/13/2006 ]Tl\9we  
    "@?|Vv,vn  
    e>>G4g  
    % Check and prepare the inputs:  bSR<d  
    % ----------------------------- bc4x"]!  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) (p?3#|^  
        error('zernfun:NMvectors','N and M must be vectors.') <  t (Pw  
    end ~76.S  
    ?xo,)``  
    if length(n)~=length(m) @r]s9~Lx9  
        error('zernfun:NMlength','N and M must be the same length.') yki k4MeB  
    end 5 muW*7  
    nMa^Eq#  
    n = n(:); vg.%.~!9  
    m = m(:); M$W#Q\<*#r  
    if any(mod(n-m,2)) .fsk DW  
        error('zernfun:NMmultiplesof2', ... bZ9NnSuH  
              'All N and M must differ by multiples of 2 (including 0).') j>Z]J'P  
    end LA?\~rh!  
    \l:g{GnoT  
    if any(m>n) ThlJhTh<%4  
        error('zernfun:MlessthanN', ... ^'fKey`  
              'Each M must be less than or equal to its corresponding N.') u#M)i30j  
    end sBb.Y k  
    +.lWck  
    if any( r>1 | r<0 ) 4 ufLP DH  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') 9sCk\`n  
    end ?R]y}6 P$  
    =.X?LWKY  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ^!<7#kX  
        error('zernfun:RTHvector','R and THETA must be vectors.') T"H )g  
    end $vLV< y07  
    v|I5Gz$qpa  
    r = r(:); 3NN'E$"3  
    theta = theta(:); 2E2}|: ||&  
    length_r = length(r); y,&M\3A  
    if length_r~=length(theta) =b!J)]  
        error('zernfun:RTHlength', ... @,4%8E5  
              'The number of R- and THETA-values must be equal.') SO<m(o)G2  
    end kN j3!u$  
    <gdgcvd  
    % Check normalization: K~8tN ,~&  
    % -------------------- dl6v <  
    if nargin==5 && ischar(nflag) daIL> c"  
        isnorm = strcmpi(nflag,'norm'); 8KtgSash  
        if ~isnorm MgQU6O<  
            error('zernfun:normalization','Unrecognized normalization flag.') ewrWSffe  
        end MXF"F:-Kn  
    else b_jZL'en  
        isnorm = false; j 3MciQ`  
    end R5eB,FN  
    _`_IUuj$E  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 8q [c  
    % Compute the Zernike Polynomials 3rdfg  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% p$nK@t}  
    2-V)>98  
    % Determine the required powers of r: XLmMK{gs  
    % ----------------------------------- %Sn6*\z  
    m_abs = abs(m); *fl{Y(_OO  
    rpowers = []; dA} 72D?  
    for j = 1:length(n) qX+gG",8  
        rpowers = [rpowers m_abs(j):2:n(j)]; ;:4P'FWm^  
    end v"r9|m~'  
    rpowers = unique(rpowers); T]6c9_  
    [9O~$! <%  
    % Pre-compute the values of r raised to the required powers, ,![Du::1  
    % and compile them in a matrix: ,=Nw(GI  
    % ----------------------------- `cP'~OT  
    if rpowers(1)==0  C5+`<  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); nF<y7XkO  
        rpowern = cat(2,rpowern{:}); #t@x6Vt  
        rpowern = [ones(length_r,1) rpowern]; M7DLs;sD  
    else %A62xnX  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); S&3X~jD(1  
        rpowern = cat(2,rpowern{:}); &QTeGn  
    end V(2,\+t  
    `(,*IK a  
    % Compute the values of the polynomials: ?7uK P}1|  
    % -------------------------------------- ~zxwg+:QO  
    y = zeros(length_r,length(n)); >&;>PZBPCO  
    for j = 1:length(n) H=&/Q  
        s = 0:(n(j)-m_abs(j))/2; icPp8EwH  
        pows = n(j):-2:m_abs(j); `pi-zE)  
        for k = length(s):-1:1 aZj J]~bO  
            p = (1-2*mod(s(k),2))* ... ;tp]^iB#  
                       prod(2:(n(j)-s(k)))/              ... QtY hg$K3  
                       prod(2:s(k))/                     ... 0\'Q&oTo  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 1BMB?I  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); -XVEV  
            idx = (pows(k)==rpowers); wb6L? t  
            y(:,j) = y(:,j) + p*rpowern(:,idx); @VC .>  
        end ,\lY Px\P[  
         VW9>xVd4  
        if isnorm a|QE *s.  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); xHJ8?bD p  
        end .Iw ur;/\  
    end :}@C9pqr2  
    % END: Compute the Zernike Polynomials dG\U)WA(p  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +Y>"/i. N  
    QqiJun_m  
    % Compute the Zernike functions: 7m:|u*ij2~  
    % ------------------------------ u C,"5C  
    idx_pos = m>0; 0]T.Lh$3  
    idx_neg = m<0; uu}`warW  
    R:'Ou:Mh  
    z = y; +@emX$cFV  
    if any(idx_pos) 'tb(J3ZP  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); qoC]#M$oo#  
    end EBoGJ_l  
    if any(idx_neg)  8]q  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); H2qf'  
    end ~+O`9&  
    jR{-  
    % EOF zernfun
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) bIgh@= 2  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. %35L=d[  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated CRH{E}>  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive 7~_I=-  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, w8O" =},  
    %   and THETA is a vector of angles.  R and THETA must have the same %*wzO9w4  
    %   length.  The output Z is a matrix with one column for every P-value, ;{vwBDV!'  
    %   and one row for every (R,THETA) pair. wfgqgPo!v  
    % & _; y.!  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike 9O;cJ)tXY  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) )Im3'0l>  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) E2@`d6  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 wV(AT$  
    %   for all p. $ +;+:K  
    % N]NF\7(  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 N0i!l|G6  
    %   Zernike functions (order N<=7).  In some disciplines it is U{6oLqwq3Y  
    %   traditional to label the first 36 functions using a single mode ]r4bRK[1  
    %   number P instead of separate numbers for the order N and azimuthal 5W/{h q8}}  
    %   frequency M. BZK2$0  
    % y$X(S\W  
    %   Example: q\%cFB}  
    % tz26=8  
    %       % Display the first 16 Zernike functions ,LD m8   
    %       x = -1:0.01:1; UtnZNdl v  
    %       [X,Y] = meshgrid(x,x); !b8uLjd;  
    %       [theta,r] = cart2pol(X,Y);  qve ./  
    %       idx = r<=1; bu>qsU3  
    %       p = 0:15; C|MQ $~5:w  
    %       z = nan(size(X)); hoa7   
    %       y = zernfun2(p,r(idx),theta(idx));  Tc6:UF  
    %       figure('Units','normalized') #B8*gFZB  
    %       for k = 1:length(p) e  ^Ds  
    %           z(idx) = y(:,k); b_TS<,  
    %           subplot(4,4,k) Lxv6!?v|  
    %           pcolor(x,x,z), shading interp +oI3I~  
    %           set(gca,'XTick',[],'YTick',[]) Q8T`wd$D#  
    %           axis square <a4 TO8  
    %           title(['Z_{' num2str(p(k)) '}']) #]CFA9 z  
    %       end {:Aw_z:'  
    % 5G(3vRX|1  
    %   See also ZERNPOL, ZERNFUN. !gF9k8\Yr$  
    )=J5\3O*x  
    %   Paul Fricker 11/13/2006 )KE [!ofD  
    ~e `Bq>  
    .Obn&S  
    % Check and prepare the inputs: Z&Y=`GOI  
    % ----------------------------- mMSh2B  
    if min(size(p))~=1 Xd<t5{bD!  
        error('zernfun2:Pvector','Input P must be vector.') 7Ym(n8  
    end g:7,~}_}^  
    6sJw@Oa J  
    if any(p)>35 ~gE:-  
        error('zernfun2:P36', ... hoM|P8 }rh  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... xe4Oxo  
               '(P = 0 to 35).']) hA}~es=c  
    end |F[=b'?  
    lx |5?P  
    % Get the order and frequency corresonding to the function number: r@U3sO#N  
    % ---------------------------------------------------------------- !8 -oR6/$%  
    p = p(:); |l0Ea  
    n = ceil((-3+sqrt(9+8*p))/2); /J3ZL[o?Q  
    m = 2*p - n.*(n+2); Aw5pd7qKL  
     R'}95S<  
    % Pass the inputs to the function ZERNFUN: qJPT%r  
    % ---------------------------------------- yF13Of^l./  
    switch nargin tz^/J=)"  
        case 3 m/B6[  
            z = zernfun(n,m,r,theta); 0Yl4eB-  
        case 4 )yG"^Ulu  
            z = zernfun(n,m,r,theta,nflag); ,](:<A)W&  
        otherwise ^/U27B  
            error('zernfun2:nargin','Incorrect number of inputs.') Vw tZLP36  
    end Bc7V)Y K  
    dY7'OAUyVl  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) uhyj5u)  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. KRAcnY;u  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of t.T UmJ  
    %   order N and frequency M, evaluated at R.  N is a vector of v+ 7kU=  
    %   positive integers (including 0), and M is a vector with the ?BA]7M(,4  
    %   same number of elements as N.  Each element k of M must be a fhPkEvJ  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) XfPFo6  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is K6/@]y%Wr  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix Q?b14]6im  
    %   with one column for every (N,M) pair, and one row for every :W#rhuzC  
    %   element in R. YB<*"HxM)}  
    % MP Z3D9  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- C+[%7vF1  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is )J]9 lW&y  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to [^CV>RuO  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 Y3.$G1{#0w  
    %   for all [n,m]. q6Rr.A  
    % :Z`:nq.a  
    %   The radial Zernike polynomials are the radial portion of the &|>S|  
    %   Zernike functions, which are an orthogonal basis on the unit m>USD? i  
    %   circle.  The series representation of the radial Zernike o#) {1<0vg  
    %   polynomials is 'c2W}$q  
    % **9x?s  
    %          (n-m)/2 :NJ_n6E  
    %            __ ]]7 mlQ  
    %    m      \       s                                          n-2s j',W 64  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r vgY3L  
    %    n      s=0 XO#/Fv!  
    % :w}{$v}#D;  
    %   The following table shows the first 12 polynomials. \(226^|j  
    % L,y6^J!  
    %       n    m    Zernike polynomial    Normalization sn7AR88M;  
    %       --------------------------------------------- QaUm1 i#  
    %       0    0    1                        sqrt(2) rpeJkG@+  
    %       1    1    r                           2 'iLpE7  
    %       2    0    2*r^2 - 1                sqrt(6) P96pm6H_;  
    %       2    2    r^2                      sqrt(6) 5T sUQc  
    %       3    1    3*r^3 - 2*r              sqrt(8) ]7-&V-Ct*  
    %       3    3    r^3                      sqrt(8) `:N# 'i  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) A-:O`RK  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) +"9hWb5  
    %       4    4    r^4                      sqrt(10) n]8<DX99Q0  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) z/i+EE  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) dJ$"l|$$  
    %       5    5    r^5                      sqrt(12) )`^p%k  
    %       --------------------------------------------- [MuEoWrq(}  
    % OL4z%mDZi  
    %   Example: s4&^D<  
    % U qG .:@T  
    %       % Display three example Zernike radial polynomials !9 fz(9  
    %       r = 0:0.01:1; z-M3  
    %       n = [3 2 5]; +P.+_7+:  
    %       m = [1 2 1];  hi g2  
    %       z = zernpol(n,m,r); xsWur(>]  
    %       figure WJI[9@^I~  
    %       plot(r,z) THmb6^  
    %       grid on t_mIOm)S%  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') 7 |DHplI  
    % UH3sH t  
    %   See also ZERNFUN, ZERNFUN2. aF03a-qw<  
    jxYc2  
    % A note on the algorithm. v\(2&*  
    % ------------------------ g{5A4|_7  
    % The radial Zernike polynomials are computed using the series f/CuE%7BR  
    % representation shown in the Help section above. For many special C6rg<tCH  
    % functions, direct evaluation using the series representation can Z7 E  
    % produce poor numerical results (floating point errors), because JN7k2]{  
    % the summation often involves computing small differences between 6uKTGc4  
    % large successive terms in the series. (In such cases, the functions 0uBl>A7qhn  
    % are often evaluated using alternative methods such as recurrence JxyB(  
    % relations: see the Legendre functions, for example). For the Zernike hYawU@R  
    % polynomials, however, this problem does not arise, because the 3sd"nR?aX  
    % polynomials are evaluated over the finite domain r = (0,1), and )7_"wD` z  
    % because the coefficients for a given polynomial are generally all `^lYw:xA  
    % of similar magnitude. m&|`x  
    % 2t $j  
    % ZERNPOL has been written using a vectorized implementation: multiple Bn>8&w/P  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] &+G"k~%  
    % values can be passed as inputs) for a vector of points R.  To achieve #s!'+|2n  
    % this vectorization most efficiently, the algorithm in ZERNPOL ayAo^q  
    % involves pre-determining all the powers p of R that are required to =J1rlnaaEL  
    % compute the outputs, and then compiling the {R^p} into a single ;NNe!}C  
    % matrix.  This avoids any redundant computation of the R^p, and znJ'iV f  
    % minimizes the sizes of certain intermediate variables. EI\9_}@,  
    % 7Pa@1']  
    %   Paul Fricker 11/13/2006   zxp`  
    ;E'"Ks[GH  
    LzYO$Ir:g  
    % Check and prepare the inputs: ak:c rrkx  
    % ----------------------------- ,^S@EDq  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) '= l[;Q^Q  
        error('zernpol:NMvectors','N and M must be vectors.') s: 3z'4oX  
    end +iI&c s  
    Q,80Hor#J  
    if length(n)~=length(m) j2 !3rI  
        error('zernpol:NMlength','N and M must be the same length.') 1T:Y0  
    end Iy }:F8F>g  
    o=nsy]'&  
    n = n(:); Xt#1Qs  
    m = m(:);  x]z2Z*  
    length_n = length(n); w |l1'   
    8/K!SpM*d  
    if any(mod(n-m,2)) x"~~l  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') `c'W-O/  
    end {W##^L~  
    +*_5tWAc  
    if any(m<0) ApjOj/  
        error('zernpol:Mpositive','All M must be positive.') DS<  }@  
    end ]^6c8sgnR  
    {aM<{_v  
    if any(m>n) )Z %T27r,^  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') UQ4% Xp  
    end Pzb|t+"$  
    Rar"B*b;$  
    if any( r>1 | r<0 ) +kFxi2L6  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') ,~?YBLw@c  
    end \cCV6A[  
    G}9=)  
    if ~any(size(r)==1) c5mZG7-  
        error('zernpol:Rvector','R must be a vector.') xzx$TUL  
    end w;l<[q?_  
    c*R/]Dn   
    r = r(:); ^gFqRbuS  
    length_r = length(r); $U/YR&vcw  
    :\=CRaA  
    if nargin==4 QFIL)'K  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); r 6Q Q  
        if ~isnorm %E_b'[8  
            error('zernpol:normalization','Unrecognized normalization flag.') 5 QO34t2  
        end N;+[`l  
    else K).n.:vYZ  
        isnorm = false; ]f1{n  
    end OaL\w D^  
    J "FC%\|  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% IVW1]y  
    % Compute the Zernike Polynomials _FwK-?4E-  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% "al `$%(  
    D0N9Ksq  
    % Determine the required powers of r: {f{ZHi|  
    % ----------------------------------- K x~|jq  
    rpowers = []; J sEa23  
    for j = 1:length(n) Xj*vh m%i  
        rpowers = [rpowers m(j):2:n(j)]; fJWC)E  
    end vpU#xm.K  
    rpowers = unique(rpowers); 5?Wto4j  
    Lq>&d,F06)  
    % Pre-compute the values of r raised to the required powers, pZU9^Z?~6  
    % and compile them in a matrix: ~l2aNVv;  
    % ----------------------------- C^ )*Dsp  
    if rpowers(1)==0 ko^\ HSXl  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); Ks\ NE=;5  
        rpowern = cat(2,rpowern{:}); ~g7l8H67  
        rpowern = [ones(length_r,1) rpowern]; f}D1|\7  
    else M,5"b+mX[~  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); !u]1 dxa  
        rpowern = cat(2,rpowern{:}); RV&^g*;E  
    end VS&TA>  
    KeNL0_ Pw  
    % Compute the values of the polynomials: K|oacOF9  
    % -------------------------------------- eu|j=mB  
    z = zeros(length_r,length_n); v<fnB  
    for j = 1:length_n 6K-5g/hL  
        s = 0:(n(j)-m(j))/2; +S))3 5N[  
        pows = n(j):-2:m(j); _`#3f1F@[  
        for k = length(s):-1:1 MfmACd^3$  
            p = (1-2*mod(s(k),2))* ... jV8q)=}*)  
                       prod(2:(n(j)-s(k)))/          ... ".dZn6"mI  
                       prod(2:s(k))/                 ... 2c/Ys4/H4]  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... 1bw{q.cmD  
                       prod(2:((n(j)+m(j))/2-s(k))); o!~Jzd.=h  
            idx = (pows(k)==rpowers); ltFq/M  
            z(:,j) = z(:,j) + p*rpowern(:,idx); A*|cdY]HP  
        end {hJXj,  
         U?!>Nd  
        if isnorm 0 u?{ \  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); ,hVvve,j}  
        end .I@CS>j  
    end dik9 >*"|o  
    * 9p |HX=  
    % 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)  kpob b  
    AH{^spD{7,  
    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。 >{Rb 3Z]  
    !D.0 (J  
    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)