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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 ?-tNRIPW@p  
    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! uQ7lC~  
     
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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 oB~V~c}8x  
    function z = zernfun(n,m,r,theta,nflag) lxr;AJ(  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. cBv"d ~  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 2e03m62*  
    %   and angular frequency M, evaluated at positions (R,THETA) on the B2|0.G|[j  
    %   unit circle.  N is a vector of positive integers (including 0), and ).A9>^6?{  
    %   M is a vector with the same number of elements as N.  Each element ayQeT  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) !~vx|_$#  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, %wI)uJ2  
    %   and THETA is a vector of angles.  R and THETA must have the same >Bu9D  
    %   length.  The output Z is a matrix with one column for every (N,M) f^ZhFu?  
    %   pair, and one row for every (R,THETA) pair. ^@{"a  
    % Pn6~66a6  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike OiS\tK?|GV  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), xGOVMo +  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral p1K]m>Y{?  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, <XtE|LG  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized j%Xa8$  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 6> z{xYat  
    % yz5! >|EB  
    %   The Zernike functions are an orthogonal basis on the unit circle. HFlExa u  
    %   They are used in disciplines such as astronomy, optics, and Tku6X/LF  
    %   optometry to describe functions on a circular domain. ~"<^4h  
    % ]>Gi_20*.  
    %   The following table lists the first 15 Zernike functions. I)s_f5'  
    % TdT`V f  
    %       n    m    Zernike function           Normalization x+;y0`oL  
    %       -------------------------------------------------- +l.LwA  
    %       0    0    1                                 1 WglpWp)  
    %       1    1    r * cos(theta)                    2 08D:2 z1z  
    %       1   -1    r * sin(theta)                    2 rHk,OC  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) Q'JK *.l  
    %       2    0    (2*r^2 - 1)                    sqrt(3) *'-t_F';  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) e+D]9wM8  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) "tK|/R+  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) 57 Bx-  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) |0?v4%g  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) >tx[UF@P@  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) {?2|rv)  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) !pkIaCxs  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) C&R U  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) (DS"*4ty  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) ~P"Agpx3u  
    %       -------------------------------------------------- {$i>\)  
    % 6x=w-32+ y  
    %   Example 1:  S~E@A.7  
    % 8lGM>(:o  
    %       % Display the Zernike function Z(n=5,m=1) 6-0sBB9=u  
    %       x = -1:0.01:1; ZoSyc--Bv  
    %       [X,Y] = meshgrid(x,x); 0fn*;f8{XJ  
    %       [theta,r] = cart2pol(X,Y); q-ko)]  
    %       idx = r<=1; W$()W)   
    %       z = nan(size(X)); ?6{g7S%  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); ?6hd(^  
    %       figure YD;d*E%t  
    %       pcolor(x,x,z), shading interp a]xGzv5  
    %       axis square, colorbar `b]wyP  
    %       title('Zernike function Z_5^1(r,\theta)') VZ =:`)  
    % K~I?i/P=z  
    %   Example 2: 6vR6=@(`>  
    % XWQ `]m)  
    %       % Display the first 10 Zernike functions `]]<.>R  
    %       x = -1:0.01:1; k?TZY|_  
    %       [X,Y] = meshgrid(x,x); x[Hx.G}5+  
    %       [theta,r] = cart2pol(X,Y); FfrC/"N  
    %       idx = r<=1; &v t)7[  
    %       z = nan(size(X)); /3K)$Er  
    %       n = [0  1  1  2  2  2  3  3  3  3]; 6M_:D  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; :z&kbG  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; H9_iTGBQ  
    %       y = zernfun(n,m,r(idx),theta(idx)); N N1}P'6Ha  
    %       figure('Units','normalized') $I>]61l%  
    %       for k = 1:10 b;5j awG  
    %           z(idx) = y(:,k); WFFQxd|Z  
    %           subplot(4,7,Nplot(k)) R@s7s%y=  
    %           pcolor(x,x,z), shading interp OKK Ko`RN  
    %           set(gca,'XTick',[],'YTick',[]) w,vnpdT  
    %           axis square )h&@}#A09  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) k ,+,,W  
    %       end [fV"tf;  
    % j BBl{  
    %   See also ZERNPOL, ZERNFUN2. 6$=>ckP  
    ~;H,cPvrEg  
    %   Paul Fricker 11/13/2006 Rvx 7}ZL!  
    +xO3<u  
    p9u*l  
    % Check and prepare the inputs: X&LJ"ahK  
    % ----------------------------- |N% l at  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Xq03o#-p+  
        error('zernfun:NMvectors','N and M must be vectors.') #jG?{j3;?  
    end D&2NO/ R  
    adIrrK  
    if length(n)~=length(m) T 4p}5ew'  
        error('zernfun:NMlength','N and M must be the same length.') X' 5R4j  
    end n8=D zv0  
    &Tuj`DL  
    n = n(:); &*ocr&  
    m = m(:); !#W>x49}  
    if any(mod(n-m,2)) f^lcw  
        error('zernfun:NMmultiplesof2', ... f_[dFKoX  
              'All N and M must differ by multiples of 2 (including 0).')  Fpn*]x  
    end 8 b~  
    OG?7( UJ  
    if any(m>n) F0z7".)  
        error('zernfun:MlessthanN', ... [f6BA|   
              'Each M must be less than or equal to its corresponding N.') d~%7A5  
    end rf4f'cUa  
    Nr `R3(X  
    if any( r>1 | r<0 ) d;0]xG?%=  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') aK;OzB)  
    end =<p=?16 x  
    R2a99#J  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Xm>zT'B_tJ  
        error('zernfun:RTHvector','R and THETA must be vectors.') y$]<m+1  
    end 2&n6:"u|  
    ;<Hk Cd  
    r = r(:); !Md6Lh%-w  
    theta = theta(:); J( XDwt  
    length_r = length(r); IauLT;!X  
    if length_r~=length(theta) kBcTXl  
        error('zernfun:RTHlength', ... -sKtT 9o  
              'The number of R- and THETA-values must be equal.') oo &|(+"O_  
    end d]O:VghY\  
    h+j^VsP zB  
    % Check normalization: tJ K58m$  
    % -------------------- 0>td[f  
    if nargin==5 && ischar(nflag) d wG!]j>:_  
        isnorm = strcmpi(nflag,'norm'); s9CmR]C  
        if ~isnorm MooH`2Fd  
            error('zernfun:normalization','Unrecognized normalization flag.') Q~Mkf&s  
        end 1~K'r&  
    else U!r8}@  
        isnorm = false; )AkBo  
    end n:/!{.  
    hN!;Tny  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% b)KEB9w  
    % Compute the Zernike Polynomials )G^k$j  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% E9j<+Ik  
    Z+y'w#MZL  
    % Determine the required powers of r: WVpx  
    % ----------------------------------- s)]T"87H'_  
    m_abs = abs(m); Os$E,4,py  
    rpowers = []; OHBCanZZ,  
    for j = 1:length(n) HYGd :SeH  
        rpowers = [rpowers m_abs(j):2:n(j)]; |rk.t g9  
    end qK d ="PR}  
    rpowers = unique(rpowers); t :YZua  
    K=0xR*ll5  
    % Pre-compute the values of r raised to the required powers, $RY-yKmi  
    % and compile them in a matrix: lkTA"8d  
    % ----------------------------- "0jwCX Cu  
    if rpowers(1)==0 m=@xZw<  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); d}:- Q?  
        rpowern = cat(2,rpowern{:}); *izCXfW7  
        rpowern = [ones(length_r,1) rpowern]; TBPu&+3  
    else mJ<`/p?:  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); Ly8=SIZ   
        rpowern = cat(2,rpowern{:}); }M%3  
    end ^0|:  
    dFw+nGN  
    % Compute the values of the polynomials: LJPJENtFIs  
    % -------------------------------------- Fy<:iv0>t  
    y = zeros(length_r,length(n)); eo4z!@pRN  
    for j = 1:length(n) E-C]<{`O  
        s = 0:(n(j)-m_abs(j))/2; a5t&{ajJ  
        pows = n(j):-2:m_abs(j); |X:`o;Uma  
        for k = length(s):-1:1 zX*5yNd  
            p = (1-2*mod(s(k),2))* ... &}e>JgBe0  
                       prod(2:(n(j)-s(k)))/              ... iE"]S )  
                       prod(2:s(k))/                     ... h'&<A_C-7  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... z;oia!9z  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); 5)XUT`;'){  
            idx = (pows(k)==rpowers); 8e>B>'nH  
            y(:,j) = y(:,j) + p*rpowern(:,idx); ed',\+.uB  
        end _"Ym]y28li  
         .tG3g:  
        if isnorm i *:QbMb  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); )r{Wj*u  
        end e`={_R{N  
    end eEVB   
    % END: Compute the Zernike Polynomials jnOnV1I"  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% q&>fKSnKs  
    /}E2Rr?{  
    % Compute the Zernike functions: X:Wd%CHP  
    % ------------------------------ r&a} U6k(y  
    idx_pos = m>0; 2! ,ndLA  
    idx_neg = m<0; [XI:Yf  
    0;><@{'  
    z = y; EoPvF`T  
    if any(idx_pos) !J;Bm,Xn6  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); RRS)7fFm  
    end M| Gl&   
    if any(idx_neg) )cizd^{  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ?:`sE"  
    end q7KHx b  
    2_u+&7  
    % 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) _GxC|d  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. Sw E7U~  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated ,^e2ma|z  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive W"@'}y  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, h@O\j&#  
    %   and THETA is a vector of angles.  R and THETA must have the same ?m~1b_@A{  
    %   length.  The output Z is a matrix with one column for every P-value, T;BFO5G@  
    %   and one row for every (R,THETA) pair. M.dX;iM<  
    % }U'9 d#N  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike hzqgsmT)  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) N,XjZ26  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) VOr: G85*s  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 8%;Wyqdf]  
    %   for all p. }X8P5c!\  
    % iNkN'("  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 >CrrxiG  
    %   Zernike functions (order N<=7).  In some disciplines it is :I&y@@UG  
    %   traditional to label the first 36 functions using a single mode G|nBja8vm  
    %   number P instead of separate numbers for the order N and azimuthal BhAT@%  
    %   frequency M. (|AZO!  
    % *ezMS   
    %   Example: w+ ')wyB  
    % Z>g&%3j  
    %       % Display the first 16 Zernike functions .9ZK@xM&?  
    %       x = -1:0.01:1;  ]XlBV-@b  
    %       [X,Y] = meshgrid(x,x); {9|*au(K  
    %       [theta,r] = cart2pol(X,Y); |) ~-Wy  
    %       idx = r<=1; qm/>\4eLt  
    %       p = 0:15;  $L uU  
    %       z = nan(size(X)); $2*_7_Qb  
    %       y = zernfun2(p,r(idx),theta(idx)); qY%|Uo  
    %       figure('Units','normalized') 4=^Ha%l  
    %       for k = 1:length(p) gzhIOeY  
    %           z(idx) = y(:,k); ]m`:T  
    %           subplot(4,4,k) FsOJmWZ  
    %           pcolor(x,x,z), shading interp i75\<X  
    %           set(gca,'XTick',[],'YTick',[]) %k?/pRv$>  
    %           axis square #E Bd g  
    %           title(['Z_{' num2str(p(k)) '}']) 5V(#nz  
    %       end PJm@fK(j  
    % 71E~~$  
    %   See also ZERNPOL, ZERNFUN. _nz_.w0H9  
    v9@_ DlV\  
    %   Paul Fricker 11/13/2006 yR{rje*  
    tR9iFv_  
    I&i6-xp  
    % Check and prepare the inputs: $FT6c@&y  
    % ----------------------------- Jo[ &y,  
    if min(size(p))~=1 R*cef  
        error('zernfun2:Pvector','Input P must be vector.') Or$"f3gq  
    end QPp31o.!5  
    C[Q4OAFG  
    if any(p)>35 %$b}o7U"s  
        error('zernfun2:P36', ... &?p( UY7'"  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... bYQ@!  
               '(P = 0 to 35).']) &m36h`tM  
    end ktfxb <%  
    1jKj' 7/K  
    % Get the order and frequency corresonding to the function number: OB=bRLd.IR  
    % ---------------------------------------------------------------- CTg79 ITYk  
    p = p(:); P}Mu|AEG  
    n = ceil((-3+sqrt(9+8*p))/2); G2n. NW#d4  
    m = 2*p - n.*(n+2); '6\w4J(  
    46 0/eW\  
    % Pass the inputs to the function ZERNFUN: +|GHbwvp  
    % ---------------------------------------- KRd'!bG=1  
    switch nargin 89 m.,  
        case 3 v0&DD&mp  
            z = zernfun(n,m,r,theta); K ~-V([tWg  
        case 4 9i_@3OVl  
            z = zernfun(n,m,r,theta,nflag); >\'}&oi  
        otherwise u&={hJ&7  
            error('zernfun2:nargin','Incorrect number of inputs.') 4Hyp]07  
    end 3:$@DZT$  
    ;mD!8<~z.  
    % 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) ^f(El(w  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. iV@\v0k  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of $b^niL  
    %   order N and frequency M, evaluated at R.  N is a vector of uJ1oo| sn  
    %   positive integers (including 0), and M is a vector with the tycVcr \(  
    %   same number of elements as N.  Each element k of M must be a 6 AY~>p  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) pXQ$n:e  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is lt5Knz2G,Z  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix J7EWaXGbz  
    %   with one column for every (N,M) pair, and one row for every X?r$o>db  
    %   element in R. . Z9c.E{  
    % cF9ZnT.  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- .hf%L1N%F  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is ]-heG'y]{  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to 8c%N+E]  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 y{P~!Yn|  
    %   for all [n,m]. }HbUB$5  
    % ':\bn:;  
    %   The radial Zernike polynomials are the radial portion of the ??hKsjNAm0  
    %   Zernike functions, which are an orthogonal basis on the unit mMu+MXTk<  
    %   circle.  The series representation of the radial Zernike Yu[MNX ;G  
    %   polynomials is ]E.FBGT  
    % X,8Zn06M  
    %          (n-m)/2 nhm#_3!6A  
    %            __ uE,T Ea9;  
    %    m      \       s                                          n-2s ,(hP /<  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r 51AA,"2[_  
    %    n      s=0 %vG;'_gM B  
    % YWANBM(v+  
    %   The following table shows the first 12 polynomials. X2np.9hie  
    % }LWrtmc  
    %       n    m    Zernike polynomial    Normalization Vd) %qw  
    %       --------------------------------------------- |`I9K#w3  
    %       0    0    1                        sqrt(2) jW| ,5,43  
    %       1    1    r                           2 3c`  
    %       2    0    2*r^2 - 1                sqrt(6) op&j4R  
    %       2    2    r^2                      sqrt(6) I.2>d_^<  
    %       3    1    3*r^3 - 2*r              sqrt(8) \D%n8O  
    %       3    3    r^3                      sqrt(8) {@" F/G+  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) $S/WAw,/  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) 5!6iAS+I  
    %       4    4    r^4                      sqrt(10) dleLX%P  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) ^zG!Z:E  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) S~g "  
    %       5    5    r^5                      sqrt(12) >;xkiO>Y  
    %       --------------------------------------------- \w$e|[~  
    % Kt"4<'  
    %   Example: --`W1!jI@  
    % N45@)s!F9j  
    %       % Display three example Zernike radial polynomials vL;=qk TCQ  
    %       r = 0:0.01:1; V<n#%!M5gV  
    %       n = [3 2 5]; ZGd7e.u=  
    %       m = [1 2 1]; Yka>r9wr  
    %       z = zernpol(n,m,r); VhgcvS@V  
    %       figure X5Y. o&  
    %       plot(r,z) LXc;`]  
    %       grid on ,;=is.h9  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') 6k1_dRu  
    % 4`"Q!T_'  
    %   See also ZERNFUN, ZERNFUN2. 7:C2xC  
    {]y!2r  
    % A note on the algorithm. cgQ2Wo7tCq  
    % ------------------------ Fow{-cs_p  
    % The radial Zernike polynomials are computed using the series 'EU|w,GL}  
    % representation shown in the Help section above. For many special w*4sT+ P  
    % functions, direct evaluation using the series representation can *+ O  
    % produce poor numerical results (floating point errors), because <ou=f'  
    % the summation often involves computing small differences between {~=gKZ:-@  
    % large successive terms in the series. (In such cases, the functions Aacj?   
    % are often evaluated using alternative methods such as recurrence r?Pk}Q  
    % relations: see the Legendre functions, for example). For the Zernike #W L>ha v  
    % polynomials, however, this problem does not arise, because the KZ/2W9r_,  
    % polynomials are evaluated over the finite domain r = (0,1), and 0n)UvJ  
    % because the coefficients for a given polynomial are generally all =vKSvQP@)  
    % of similar magnitude. ).r04)/  
    % *&F~<HC2+  
    % ZERNPOL has been written using a vectorized implementation: multiple .YIb ny1  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] !wjD6 NK  
    % values can be passed as inputs) for a vector of points R.  To achieve vqwSOh|P9  
    % this vectorization most efficiently, the algorithm in ZERNPOL P76QHBbl  
    % involves pre-determining all the powers p of R that are required to +]# p m9  
    % compute the outputs, and then compiling the {R^p} into a single *C^`+*}OE$  
    % matrix.  This avoids any redundant computation of the R^p, and kQtnT7  
    % minimizes the sizes of certain intermediate variables. f{R/rb&iB  
    % Snas:#B!  
    %   Paul Fricker 11/13/2006 Y &*nj`n  
    5'"9)#Ve  
    eJZt&|7N  
    % Check and prepare the inputs: ZOHGGO]1M  
    % ----------------------------- #xxs^Kbqa#  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) J|o )c~  
        error('zernpol:NMvectors','N and M must be vectors.') }{)>aJ  
    end {(`xA,El  
    =q*j". <  
    if length(n)~=length(m) iD!]I$  
        error('zernpol:NMlength','N and M must be the same length.') ljO t~@Ea  
    end (fnp\j3w  
    C5'#0}6i  
    n = n(:); _O>8jH!#  
    m = m(:); kT{d pGU9  
    length_n = length(n); gA DF  
    RD'i(szi?  
    if any(mod(n-m,2)) 25jgM!QBXF  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') Ceak8#|4  
    end =xsTVT;sj  
    1mz72K  
    if any(m<0) mA']*)L1  
        error('zernpol:Mpositive','All M must be positive.') vBjrI*0  
    end X/`M'8v.%  
    xy1R_*.F^T  
    if any(m>n) 6"Rw&3D?  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') 7 a_99? J  
    end JG!@(lr  
    i6D66E  
    if any( r>1 | r<0 ) -85W/%  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') T0P_&E@X  
    end p v*n.U6  
    H~r":A'"*  
    if ~any(size(r)==1) "iTi+UZxe  
        error('zernpol:Rvector','R must be a vector.') !%%(o%bi~  
    end :Z5Twb3h  
    <;nhb  
    r = r(:); E?[]N[0Kl  
    length_r = length(r); d;O4)8 >  
    YA%0{Tdxz  
    if nargin==4 "AUHe6Yv  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); 5]yby"Z?}  
        if ~isnorm \J,pV  
            error('zernpol:normalization','Unrecognized normalization flag.') mR6hnKa_53  
        end $^j#z^7  
    else uiIS4S_  
        isnorm = false; El#"vIg(\  
    end "s5[w+,R  
    ae|j#!~oi  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Z1ZjQt#~+  
    % Compute the Zernike Polynomials i-*ZW:  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *3]_Huw<  
    N.@@ebuE  
    % Determine the required powers of r: <m X EX`?  
    % ----------------------------------- g<$q#l~4xH  
    rpowers = []; R;EdYbiF b  
    for j = 1:length(n) .7|Iausv  
        rpowers = [rpowers m(j):2:n(j)]; s'JbG&T[J  
    end  c 1o8   
    rpowers = unique(rpowers); 8rY[Q(]  
    Cmj+>$')0  
    % Pre-compute the values of r raised to the required powers, (N :vDq'  
    % and compile them in a matrix: @J UCXm  
    % ----------------------------- :M _N  
    if rpowers(1)==0 *$yU|,  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); G=l-S\0@  
        rpowern = cat(2,rpowern{:}); w=feXA3-S  
        rpowern = [ones(length_r,1) rpowern]; OT{cP3;0*o  
    else ^'ac |+  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); * faG0le  
        rpowern = cat(2,rpowern{:}); @K=C`N_22  
    end  -#<AbT  
    KO3X)D<3  
    % Compute the values of the polynomials: zqt{oN_  
    % -------------------------------------- Q.Xs%{B  
    z = zeros(length_r,length_n); w9]HJ3qi  
    for j = 1:length_n qbD>)}:1  
        s = 0:(n(j)-m(j))/2; BH\!yxK  
        pows = n(j):-2:m(j); %v)O!HC}  
        for k = length(s):-1:1 xka&,`z  
            p = (1-2*mod(s(k),2))* ... ^2)<H7p  
                       prod(2:(n(j)-s(k)))/          ... 7w51UmO  
                       prod(2:s(k))/                 ... Bx 9v2x.  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... ]wm<$+@  
                       prod(2:((n(j)+m(j))/2-s(k))); >7,?X_:A-1  
            idx = (pows(k)==rpowers); ^Cy=L]  
            z(:,j) = z(:,j) + p*rpowern(:,idx); B3g # )  
        end *r(Qy0(  
         ;(r,;S_`0  
        if isnorm )/cf%  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); s&7TARd  
        end l#bE_PD;  
    end JBk >|q"  
    B|SE |  
    % EOF zernpol
    离线niuhelen
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    只看该作者 6楼 发表于: 2011-03-12
    这三个文件,我不知道该怎样把我的面型节点的坐标及轴向位移用起来,还烦请指点一下啊,谢谢啦!
    离线li_xin_feng
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    只看该作者 7楼 发表于: 2012-09-28
    我也正在找啊
    离线guapiqlh
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    只看该作者 8楼 发表于: 2014-03-04
    我也一直想了解这个多项式的应用,还没用过呢
    离线phoenixzqy
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    只看该作者 9楼 发表于: 2014-04-22
    回 guapiqlh 的帖子
    guapiqlh:我也一直想了解这个多项式的应用,还没用过呢 (2014-03-04 11:35)  q0DRT4K  
    J=&}$  
    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。 4)8e0L*[B?  
    ?C- ju8]|  
    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)