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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 Bo'v!bI7  
    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! 0< }BSv  
     
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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 7 y$a=+D i  
    function z = zernfun(n,m,r,theta,nflag) $\M];S=CY  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. _6g(C_m'T?  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N Jje!*?&8X  
    %   and angular frequency M, evaluated at positions (R,THETA) on the %36@1l-N  
    %   unit circle.  N is a vector of positive integers (including 0), and 8xkLfN|N=  
    %   M is a vector with the same number of elements as N.  Each element ,lFp4 C  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) s#(%u t  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, T8yMaC  
    %   and THETA is a vector of angles.  R and THETA must have the same !fjB oK+  
    %   length.  The output Z is a matrix with one column for every (N,M) 4=N(@mS  
    %   pair, and one row for every (R,THETA) pair. yM,Y8^  
    % jdx T662q  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike Iyb_5 UmpF  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), rZE+B25T~  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral {kr14 l*2  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, q1m{G1W n  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized S,Tc\}  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. Z9Z\2t  
    % R dNL f  
    %   The Zernike functions are an orthogonal basis on the unit circle. -=ZDfM  
    %   They are used in disciplines such as astronomy, optics, and 81w"*G5AM  
    %   optometry to describe functions on a circular domain.  M+:9U&>  
    % yhs:.h  
    %   The following table lists the first 15 Zernike functions. 7:<A_OLi  
    % ?/my G{E  
    %       n    m    Zernike function           Normalization 15r=d  
    %       -------------------------------------------------- 'K#ndCGJ$  
    %       0    0    1                                 1 e*U6^Xex  
    %       1    1    r * cos(theta)                    2 dcyHp>\)|  
    %       1   -1    r * sin(theta)                    2  T;V!>W37  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) 2u+!7D!w$  
    %       2    0    (2*r^2 - 1)                    sqrt(3) cv7:5P  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) *N"CV={No  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) vhcp[=e :  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) <XN=v!2;  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) RgZ9ZrE\  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) ml /S|`Drk  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) nd7g8P9p  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Ok fxX&n  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) m;t&P58f  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) K9y~ e  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) ,Q0H)// ~  
    %       -------------------------------------------------- d`=LZio  
    % j-.Y!$a%6  
    %   Example 1: ]hoq!:>M1  
    % l5\V4  
    %       % Display the Zernike function Z(n=5,m=1) Hmnxm gx  
    %       x = -1:0.01:1; < fV][W  
    %       [X,Y] = meshgrid(x,x); jL'`M%8O  
    %       [theta,r] = cart2pol(X,Y); \ Ce*5h  
    %       idx = r<=1; Vjw u:M  
    %       z = nan(size(X)); 9C0#K\  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); y*6/VSRkt4  
    %       figure xc\zRsY`  
    %       pcolor(x,x,z), shading interp ge<D}6GQ  
    %       axis square, colorbar <HzL%DX  
    %       title('Zernike function Z_5^1(r,\theta)') "Mhn?PTq  
    % (z?j{J  
    %   Example 2: JodD6 ;P  
    % xu%eg]  
    %       % Display the first 10 Zernike functions v+8Ybq  
    %       x = -1:0.01:1; Vzo< ma^  
    %       [X,Y] = meshgrid(x,x); 1@JusS0^K  
    %       [theta,r] = cart2pol(X,Y); ]5Dh<QY&.  
    %       idx = r<=1; Iy&,1CI"]  
    %       z = nan(size(X)); v^vi *c  
    %       n = [0  1  1  2  2  2  3  3  3  3]; \4^rb?B  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; R n]xxa'  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; yMTO5~U{  
    %       y = zernfun(n,m,r(idx),theta(idx)); :7mHPe }(  
    %       figure('Units','normalized') w( _42)v]g  
    %       for k = 1:10 Jazgn5  
    %           z(idx) = y(:,k); l;L_A@B<  
    %           subplot(4,7,Nplot(k)) k ~ByICE  
    %           pcolor(x,x,z), shading interp 0H]{,mVs  
    %           set(gca,'XTick',[],'YTick',[]) /jGV[_Q=P  
    %           axis square Wpi35JrC  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) |_>^vW1f  
    %       end U+@U/s%8  
    % y&-QLX L  
    %   See also ZERNPOL, ZERNFUN2. "WUS?Q  
    zsJermF,O  
    %   Paul Fricker 11/13/2006 _B&Lyg !J  
    ]JV'z<  
    nSC2wTH!1  
    % Check and prepare the inputs: "aCAA#$J  
    % ----------------------------- H;l_;c`  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) d Rnf  
        error('zernfun:NMvectors','N and M must be vectors.') Dfa3&# #{  
    end >m. .  
    "\KBF  
    if length(n)~=length(m)  J}:.I>  
        error('zernfun:NMlength','N and M must be the same length.') ^B% =P  
    end +a 1iZ bh  
    ~rJG4U  
    n = n(:); #mA(x@:*  
    m = m(:); F_jHi0A  
    if any(mod(n-m,2)) T9H*]LxK  
        error('zernfun:NMmultiplesof2', ... P <+0sh  
              'All N and M must differ by multiples of 2 (including 0).') va'F '|  
    end 9S*"={}%  
    =@?[.`  
    if any(m>n) fzQR0  
        error('zernfun:MlessthanN', ... Zrr)<'!i  
              'Each M must be less than or equal to its corresponding N.') q*3keB;X  
    end ?! 6Itkg  
    W%-XN   
    if any( r>1 | r<0 ) '.(Gg%*\.  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') pX?3inQP%(  
    end Es%f@$0uy  
    JHt U"  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) x9 %=d  
        error('zernfun:RTHvector','R and THETA must be vectors.') %BP>,E/w  
    end  pUb1#=  
    Y}N\|*ye-  
    r = r(:); ~<m^  
    theta = theta(:); 0!_?\)X  
    length_r = length(r); !}#> ky!t  
    if length_r~=length(theta) f 7lj,GAZ  
        error('zernfun:RTHlength', ... _>R aw  
              'The number of R- and THETA-values must be equal.') ExS5RV@v'  
    end -HG .GA  
    nQjpJ /=  
    % Check normalization: Y \-W`  
    % -------------------- 9Yv:6@.F  
    if nargin==5 && ischar(nflag) *WQ?r&[_'  
        isnorm = strcmpi(nflag,'norm'); !m+Pd.4TaB  
        if ~isnorm :_~.Nt  
            error('zernfun:normalization','Unrecognized normalization flag.') ir_XU/ve  
        end 'z(Y9%+a  
    else &aLTy&8Fv  
        isnorm = false; 6*q1%rs:w  
    end d-D,Gx]>$  
    &>,;ye>A  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 8(L$a1#5W  
    % Compute the Zernike Polynomials d +D~NA[M  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 3ic /xy;}  
    %o0b~R  
    % Determine the required powers of r: w={q@. g%  
    % ----------------------------------- 3' i6<  
    m_abs = abs(m); =9GA LoGL  
    rpowers = []; %^IQ<   
    for j = 1:length(n) EfrQ~`\  
        rpowers = [rpowers m_abs(j):2:n(j)]; Y 3BJ@sqz  
    end qk2E>  
    rpowers = unique(rpowers); Q[biy{(b8  
    )4L2&e`k)(  
    % Pre-compute the values of r raised to the required powers, /Sw~<B!8N  
    % and compile them in a matrix: k&ci5MpN  
    % ----------------------------- !C#oZU]P  
    if rpowers(1)==0 1;ttwF>G7  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); aDF@A S  
        rpowern = cat(2,rpowern{:}); 'f\9'v  
        rpowern = [ones(length_r,1) rpowern]; 4>*=q*<V5E  
    else yV(#z2|  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); }=[p>3Dd  
        rpowern = cat(2,rpowern{:}); s6,~J F^  
    end *#T: _  
    .\R9tt}  
    % Compute the values of the polynomials: !p&<.H_  
    % -------------------------------------- J\L'HIs  
    y = zeros(length_r,length(n)); i1vz{Tc  
    for j = 1:length(n) >Ku4Il+36  
        s = 0:(n(j)-m_abs(j))/2; !kovrvM6F  
        pows = n(j):-2:m_abs(j); >G6kF!V  
        for k = length(s):-1:1 \,Y .5?  
            p = (1-2*mod(s(k),2))* ... msBoInhI  
                       prod(2:(n(j)-s(k)))/              ... }?s-$@$R  
                       prod(2:s(k))/                     ... P0l fK}  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ?+t;\  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); 8R MM97@1Q  
            idx = (pows(k)==rpowers); ,hn#DJ)  
            y(:,j) = y(:,j) + p*rpowern(:,idx); U>2KjZB  
        end Nk7y2[  
         }dkXRce*  
        if isnorm ~ WWhCRq  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 6!\V|  
        end lVvcrU  
    end D S U`(`  
    % END: Compute the Zernike Polynomials ip-X r|Bq  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ^Arv6kD,  
    q/EX`%U  
    % Compute the Zernike functions: 8^UF0>`'  
    % ------------------------------ )U %`7(bN  
    idx_pos = m>0; m!FuC=e  
    idx_neg = m<0; /wJ#-DZ  
    & kC  
    z = y; c4fH/-  
    if any(idx_pos) qp})4XTv  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); \CjJa(vV  
    end )'+[,z ;s  
    if any(idx_neg) Cbff:IP  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 32ki ?\P  
    end 5P!ZGbG  
    sX1DbEjj[o  
    % 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) Lf9hOMHx  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. &#PPXwmR  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated aO1^>hy  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive |4@cX<d.  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, }097[-g7  
    %   and THETA is a vector of angles.  R and THETA must have the same FyEKqYl  
    %   length.  The output Z is a matrix with one column for every P-value, +xYu@r%R  
    %   and one row for every (R,THETA) pair. BM!ZdoKrKt  
    % Mq0MtC6-  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike IWo'{pk  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) BE0l2[i?  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) SJiQg-+<Uf  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 sC3Vj(d!i  
    %   for all p. {!2K-7;  
    % PNm@mC_fh  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 dj0%?g>  
    %   Zernike functions (order N<=7).  In some disciplines it is Q:P)g#suc  
    %   traditional to label the first 36 functions using a single mode `3\aX|4@  
    %   number P instead of separate numbers for the order N and azimuthal NJBSVC b  
    %   frequency M. }d. X2?  
    % 2;Z 0pPR&  
    %   Example: }d%CZnY&7  
    % do8[wej<:  
    %       % Display the first 16 Zernike functions $@Vn+| Ix  
    %       x = -1:0.01:1; `rn/H;r!Z  
    %       [X,Y] = meshgrid(x,x); ZUI6VM  
    %       [theta,r] = cart2pol(X,Y); 4Fp[94 b  
    %       idx = r<=1; ta?NO{*  
    %       p = 0:15; N:lE{IvRJ  
    %       z = nan(size(X)); _<Ak M"  
    %       y = zernfun2(p,r(idx),theta(idx)); ?s2-iuMPd  
    %       figure('Units','normalized') &PJ;B)b  
    %       for k = 1:length(p) `NtW+v  
    %           z(idx) = y(:,k); |^1g*f y?  
    %           subplot(4,4,k) WOn53|GQK  
    %           pcolor(x,x,z), shading interp iZNS? ^U  
    %           set(gca,'XTick',[],'YTick',[]) D9+qT<ojN  
    %           axis square /l<(i+0  
    %           title(['Z_{' num2str(p(k)) '}']) D&FDPaJM  
    %       end 1'f_C<.0  
    % +2iD9X{$MX  
    %   See also ZERNPOL, ZERNFUN. ;a?<7LIx  
    v? ."`,e  
    %   Paul Fricker 11/13/2006 O|t>.<T?  
    f&CQn.K"  
    (?l ]}p^[  
    % Check and prepare the inputs: 5Y+YN1  
    % ----------------------------- 1 iox0  
    if min(size(p))~=1 !; >s.]  
        error('zernfun2:Pvector','Input P must be vector.') 1 *' /B  
    end $IQPB_:  
    "s|P,*Xf  
    if any(p)>35 6>]  
        error('zernfun2:P36', ... l 73% y  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... WVR/0l&bU  
               '(P = 0 to 35).']) (G F}c\=T7  
    end {}s/p9F4  
    VzXVy)d  
    % Get the order and frequency corresonding to the function number: ?t%{2a<X  
    % ---------------------------------------------------------------- Dn)yBA%  
    p = p(:); \Vme\Ke*v)  
    n = ceil((-3+sqrt(9+8*p))/2); vb[0H{TT2  
    m = 2*p - n.*(n+2); kTH"" h{  
    \:+\H0Bz  
    % Pass the inputs to the function ZERNFUN: 6 rnFXZ\  
    % ---------------------------------------- vD8pVR+  
    switch nargin Z5xQ -T`  
        case 3 t]SB .ja  
            z = zernfun(n,m,r,theta); ))AxU!*.  
        case 4 sUlf4<_zW  
            z = zernfun(n,m,r,theta,nflag); 6CFnE7TQf  
        otherwise ^mL X}E]  
            error('zernfun2:nargin','Incorrect number of inputs.') 7G+!9^  
    end Gy \ ]j  
    e.vt"eRB  
    % 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) s=:)!M.i  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. Ub\^3f  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of WSMpX -^e@  
    %   order N and frequency M, evaluated at R.  N is a vector of JOG- i  
    %   positive integers (including 0), and M is a vector with the Pd+*syOM  
    %   same number of elements as N.  Each element k of M must be a SZTn=\  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) VWzQXo  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is R ?s;L r  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix SZXSVz0j  
    %   with one column for every (N,M) pair, and one row for every j_5&w Znq  
    %   element in R. ?5CE<[  
    % ?#GTD?3d  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- F5X9)9S  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is YZ<z lU  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to :@)R@. -  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 `^#4okg]  
    %   for all [n,m]. <lR:^M[v5<  
    % 6am6'_{  
    %   The radial Zernike polynomials are the radial portion of the <pV8 +V)  
    %   Zernike functions, which are an orthogonal basis on the unit L~f~XgQ  
    %   circle.  The series representation of the radial Zernike ll0y@@Iy  
    %   polynomials is 9{4oz<U  
    % bM"?^\a&Q  
    %          (n-m)/2 `D|])^"{  
    %            __ Rd HCbk  
    %    m      \       s                                          n-2s l$1?@l$j  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r {96MfhkeBv  
    %    n      s=0 mKu,7nMvF  
    % t]0DT_iE  
    %   The following table shows the first 12 polynomials. ~ Rk.x +  
    % %0 {_b68x  
    %       n    m    Zernike polynomial    Normalization Z$INmo6  
    %       --------------------------------------------- w0;4O)H$O  
    %       0    0    1                        sqrt(2) Io*H}$Gf  
    %       1    1    r                           2 *lA+ -gkK*  
    %       2    0    2*r^2 - 1                sqrt(6) ##BbR  
    %       2    2    r^2                      sqrt(6) r+m.! +  
    %       3    1    3*r^3 - 2*r              sqrt(8) C-S>'\ |8  
    %       3    3    r^3                      sqrt(8)  &lU\9  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) aV7VbC  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) y;CX )!8  
    %       4    4    r^4                      sqrt(10) ;o'r@4^&$R  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) ]VQd *~ -  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) I5E =Ujc_  
    %       5    5    r^5                      sqrt(12) E9mu:T  
    %       --------------------------------------------- kh# QT_y  
    % PX/Y?DP  
    %   Example: *Sdx:G~gp  
    % N$e mS  
    %       % Display three example Zernike radial polynomials ]B;`Jf  
    %       r = 0:0.01:1; w>cqsTq  
    %       n = [3 2 5]; #8M?y*<I  
    %       m = [1 2 1]; hDTC~~J/  
    %       z = zernpol(n,m,r); x#3*C|A  
    %       figure #<==7X#  
    %       plot(r,z) ,x1OQ jtY  
    %       grid on qJT/4 8lf_  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') [/q Bvuun  
    % 9tv,,I;iU  
    %   See also ZERNFUN, ZERNFUN2. sgi5dQ  
    jZ-s6r2=  
    % A note on the algorithm. $.C-_L  
    % ------------------------ al}J^MJ  
    % The radial Zernike polynomials are computed using the series TW>GYGz  
    % representation shown in the Help section above. For many special $adZ|Q\  
    % functions, direct evaluation using the series representation can czIAx1R9  
    % produce poor numerical results (floating point errors), because &~+QPnI>Pm  
    % the summation often involves computing small differences between `XH0S`B  
    % large successive terms in the series. (In such cases, the functions b MD|  
    % are often evaluated using alternative methods such as recurrence izcaWt3 a  
    % relations: see the Legendre functions, for example). For the Zernike m-azd ~r[  
    % polynomials, however, this problem does not arise, because the Dq~;h \='  
    % polynomials are evaluated over the finite domain r = (0,1), and )aGSZ1`/  
    % because the coefficients for a given polynomial are generally all tnnGM,"ol  
    % of similar magnitude. o$</At  
    % R 39_!  
    % ZERNPOL has been written using a vectorized implementation: multiple 2Q%7J3I  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] FfM^2`xP  
    % values can be passed as inputs) for a vector of points R.  To achieve }NyQ<,+mq&  
    % this vectorization most efficiently, the algorithm in ZERNPOL h_#=f(.'j  
    % involves pre-determining all the powers p of R that are required to WtZI1`\qe  
    % compute the outputs, and then compiling the {R^p} into a single ;<Z6Y3>I8  
    % matrix.  This avoids any redundant computation of the R^p, and 7Q&-ObW  
    % minimizes the sizes of certain intermediate variables. TyIjDG6tM  
    % }~+,x#  
    %   Paul Fricker 11/13/2006 l90"1I A  
    tgG*k$8z  
    ^K"BQ~-w  
    % Check and prepare the inputs: DNq(\@x[!  
    % ----------------------------- 2%fIe   
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) O%kUj&h^  
        error('zernpol:NMvectors','N and M must be vectors.') 1}a4AGAp  
    end jG7PT66>;  
    KWY_eY_|  
    if length(n)~=length(m) =W3 K6w  
        error('zernpol:NMlength','N and M must be the same length.') <9ucpV  
    end <$e|'}>A  
    24#qg '  
    n = n(:); =w+8q1!o  
    m = m(:); ?nW>' z  
    length_n = length(n);  <EU R:  
    I)'bf/6?  
    if any(mod(n-m,2)) ?MRY*[$  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') ]7Vg9&1`  
    end *p $0(bz  
    cw!,.o%cD  
    if any(m<0) *KvD$(ny  
        error('zernpol:Mpositive','All M must be positive.') uRko[W(  
    end {7goYzQsi%  
    c$V5E t  
    if any(m>n) 0i_:J  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') 9(TGkz(NA  
    end i$E [@  
    Q"qI'*Kgt  
    if any( r>1 | r<0 ) #_35bg4h{  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') W#<1504ip  
    end r+Ki`HD%  
    `RnWh9  
    if ~any(size(r)==1) WChP,hw  
        error('zernpol:Rvector','R must be a vector.') V+Tv:a  
    end nFn!6,>E  
    acl<dY6  
    r = r(:); tsc `u>  
    length_r = length(r); p?Azn>qBa  
    "9s_[e  
    if nargin==4 'vBZh1`p  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); 2HFn\kjj.s  
        if ~isnorm 12n:)yQy  
            error('zernpol:normalization','Unrecognized normalization flag.') qazA,|L!  
        end /J#(8p  
    else 2 DW @}[G  
        isnorm = false; TsTc3  
    end o]oiJvOr  
    Kn~Rck| ]  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% =D/zC'l  
    % Compute the Zernike Polynomials tON>wmN  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% R7bG!1SHl  
    lDYgt UKG  
    % Determine the required powers of r: 5X!-Hj  
    % ----------------------------------- |HK:\)L%  
    rpowers = []; _HUbE /  
    for j = 1:length(n) P'Rw/c o  
        rpowers = [rpowers m(j):2:n(j)]; sApix=Lr  
    end HK!ecQ^+  
    rpowers = unique(rpowers); u;_~{VJ-  
    O]u'7nO{{  
    % Pre-compute the values of r raised to the required powers, y'_8b=*  
    % and compile them in a matrix: -@#w)  
    % ----------------------------- .hat!Tt9  
    if rpowers(1)==0 ]1!" q40)]  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); *1dDs^D#|  
        rpowern = cat(2,rpowern{:}); 8?+|4:#=*J  
        rpowern = [ones(length_r,1) rpowern]; eQbHf  
    else A8Ju+  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ]b4IO4T  
        rpowern = cat(2,rpowern{:}); 6z9 '|;,4  
    end P"w\hF  
    Rg?6eN  
    % Compute the values of the polynomials: Z4] n<~o  
    % -------------------------------------- P3_.U8g$r  
    z = zeros(length_r,length_n); <sH}X$/  
    for j = 1:length_n \Rny*px  
        s = 0:(n(j)-m(j))/2; L80(9Y^xn  
        pows = n(j):-2:m(j); cl~Yx 4  
        for k = length(s):-1:1 e,U:H~+]  
            p = (1-2*mod(s(k),2))* ... 11=$] K>  
                       prod(2:(n(j)-s(k)))/          ... &~,4$& _  
                       prod(2:s(k))/                 ... z K<af  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... x)::^'74  
                       prod(2:((n(j)+m(j))/2-s(k))); c$g@3gL  
            idx = (pows(k)==rpowers); A'|!O:s   
            z(:,j) = z(:,j) + p*rpowern(:,idx); W7>2&$  
        end 9@ tp#  
         Zl9@E;|=  
        if isnorm w_(3{P[Iz  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); HxG8 'G  
        end isZ5s\  
    end %nZl`<M  
    ":Wq<Z'  
    % 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)  .8hI ad  
    xNP_>Qa~  
    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。 5En6f`nR{  
    1v o)]ff  
    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)