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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 正序阅读 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 /^eemx  
    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! ^?]H$e  
     
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    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)  ),,vu  
    q /?_djv  
    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。 5K{h)* *5  
    o7zfD94I  
    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)
    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niuhelen
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) V;$lgTs|'  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. rie1F,  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of SLW1]ZaG  
    %   order N and frequency M, evaluated at R.  N is a vector of yDPek*#^"q  
    %   positive integers (including 0), and M is a vector with the Z EW`?6  
    %   same number of elements as N.  Each element k of M must be a <R2bz1!h.  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) t^q/'9Ai&J  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is IySlu^a  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix  M`bK   
    %   with one column for every (N,M) pair, and one row for every t<4+CC2H  
    %   element in R. bp }~{]:b  
    % f.!cR3XgV  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- s;>jy/o0 s  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is ;lGjj9we>  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to *h`zV<j  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 8$1<N  
    %   for all [n,m]. kc}e},k  
    % AtSEKpKc  
    %   The radial Zernike polynomials are the radial portion of the 8kk$:8  
    %   Zernike functions, which are an orthogonal basis on the unit <MoWS9s!yb  
    %   circle.  The series representation of the radial Zernike 1Eh (U  
    %   polynomials is gP`8hNwR  
    % hW(Mf  
    %          (n-m)/2 K?) &8S  
    %            __ NI3_wV  
    %    m      \       s                                          n-2s < }G7#xg  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r Wfp[)MM;  
    %    n      s=0 {@k5e) Q  
    % d&F8nBIM5  
    %   The following table shows the first 12 polynomials.  IG 6yt  
    % ]"^U  
    %       n    m    Zernike polynomial    Normalization ;:f.a(~c  
    %       --------------------------------------------- +Ibcc8Qud  
    %       0    0    1                        sqrt(2) IF<pT)  
    %       1    1    r                           2 S-*4HV_l  
    %       2    0    2*r^2 - 1                sqrt(6) Pr9$( 6MX  
    %       2    2    r^2                      sqrt(6) PE0A`  
    %       3    1    3*r^3 - 2*r              sqrt(8) 4Z,MqG>  
    %       3    3    r^3                      sqrt(8) l@%MS\{  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) ckS.j)@.c  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) voEg[Gg4%I  
    %       4    4    r^4                      sqrt(10) O)n"a\LD  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12)  [td)v,  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) pO[ @2tF  
    %       5    5    r^5                      sqrt(12) P;C3{>G9  
    %       --------------------------------------------- CNwIM6t  
    % kZfa8w L]P  
    %   Example: Z_Qs^e$  
    % {1Z8cV   
    %       % Display three example Zernike radial polynomials ],V_"\ATD  
    %       r = 0:0.01:1; 2r4owB?  
    %       n = [3 2 5]; B&3oo   
    %       m = [1 2 1]; yY+)IU.  
    %       z = zernpol(n,m,r); yPs4S?<s  
    %       figure MEf`&<t  
    %       plot(r,z) %5Q5xw]w3  
    %       grid on >]s\%GO  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') QQ;<L"VW  
    % {xJq F4  
    %   See also ZERNFUN, ZERNFUN2. /P { Zo  
    >zx]% W  
    % A note on the algorithm. YW9r'{(D(I  
    % ------------------------ 7/C,<$Ep  
    % The radial Zernike polynomials are computed using the series P&I%!'<   
    % representation shown in the Help section above. For many special J'{69<`Dl  
    % functions, direct evaluation using the series representation can q=Xda0c  
    % produce poor numerical results (floating point errors), because ;J [ed>v;3  
    % the summation often involves computing small differences between  KT'Ebb]  
    % large successive terms in the series. (In such cases, the functions WRIOjQ:  
    % are often evaluated using alternative methods such as recurrence RgTm^?Ex  
    % relations: see the Legendre functions, for example). For the Zernike }#'I,?_k  
    % polynomials, however, this problem does not arise, because the q+<<Ku(20  
    % polynomials are evaluated over the finite domain r = (0,1), and wwmHr!b:6  
    % because the coefficients for a given polynomial are generally all n@1;5)&k~  
    % of similar magnitude. n`v;S>aT  
    % F/8="dM  
    % ZERNPOL has been written using a vectorized implementation: multiple }enS'Fpf`  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] cl\Gh  
    % values can be passed as inputs) for a vector of points R.  To achieve -O&u;kh4g  
    % this vectorization most efficiently, the algorithm in ZERNPOL \Dn47V{7-  
    % involves pre-determining all the powers p of R that are required to ^lw0} i  
    % compute the outputs, and then compiling the {R^p} into a single &>%R)?SZh  
    % matrix.  This avoids any redundant computation of the R^p, and /i !3Fr"  
    % minimizes the sizes of certain intermediate variables. yLFZo"r  
    % ioJ~k[T  
    %   Paul Fricker 11/13/2006 \}:RG^*m  
    3P}^Wu  
    /Ko{S_3< I  
    % Check and prepare the inputs: H0dHW;U<1  
    % ----------------------------- xbTvv>'U  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) =P}BAJ  
        error('zernpol:NMvectors','N and M must be vectors.') 6cQ)*,Q  
    end 4hQ.RO  
    n7A %y2  
    if length(n)~=length(m) n "J+? ~9  
        error('zernpol:NMlength','N and M must be the same length.') GS*Mv{JJ  
    end #&$a7L}  
    XT "-   
    n = n(:); 9y$"[d27;+  
    m = m(:); W]TO%x{  
    length_n = length(n); Sd9%tO9mf  
    Lh_Q@>k  
    if any(mod(n-m,2)) s7 K](T4  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') I\y=uC  
    end n){F FM  
    rzk-_AFR  
    if any(m<0) WVMkLMg8d  
        error('zernpol:Mpositive','All M must be positive.') :~PzTUz  
    end )a;ou>u  
    xRiWg/Z~  
    if any(m>n) "TQ3{=j{  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') bMjE@S&  
    end _?m%i]~o  
    Fx]}<IudA^  
    if any( r>1 | r<0 ) B098/`r  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') ]ysEj3  
    end Ojwhcb^  
    M[}aQWT$v  
    if ~any(size(r)==1) "I n[= 2w  
        error('zernpol:Rvector','R must be a vector.') <<iwJ U%:  
    end &}."sGK  
    pf%B  
    r = r(:); c 0/vB  
    length_r = length(r); W6 y-~  
    ,T<q"d7-#  
    if nargin==4 P-25]-  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); iQ7S*s+l5O  
        if ~isnorm x./l27}6  
            error('zernpol:normalization','Unrecognized normalization flag.') 5p]Cwj<u  
        end NJqjW  
    else Zk .V   
        isnorm = false; 3;t{V$  
    end ynQ+yW74Z  
    s(u,mtG  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% }s>.Fh  
    % Compute the Zernike Polynomials 4 >2g&);B  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  |tVWmm^m  
    *41 2)zEy  
    % Determine the required powers of r: ~;ZT<eCIA  
    % ----------------------------------- ~bsL W:.'  
    rpowers = []; K'tckJ#%  
    for j = 1:length(n)  !L|PDGD  
        rpowers = [rpowers m(j):2:n(j)]; }]K^b1Fs5  
    end VQe@H8>3  
    rpowers = unique(rpowers);  >M-ZjT>  
    8-clL\bm  
    % Pre-compute the values of r raised to the required powers, ;mtv  
    % and compile them in a matrix: hY5tBL  
    % ----------------------------- 2z+-vT%  
    if rpowers(1)==0 [/'=M h  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); *v6 j7<H  
        rpowern = cat(2,rpowern{:}); 4Y!_tZ>  
        rpowern = [ones(length_r,1) rpowern]; k}tT l 2  
    else 9u%S<F"  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 3`3`iN!8\@  
        rpowern = cat(2,rpowern{:}); DwBKqhu  
    end -!JnyD   
    `j1(GQt  
    % Compute the values of the polynomials: 8*[Q{:'.  
    % -------------------------------------- blHJhB&8  
    z = zeros(length_r,length_n); Uc>$w?oA  
    for j = 1:length_n T 7EkRcb  
        s = 0:(n(j)-m(j))/2; JrhDqyk*  
        pows = n(j):-2:m(j); L kA_M'G  
        for k = length(s):-1:1 ?Gr2@,jlD  
            p = (1-2*mod(s(k),2))* ... kntM  
                       prod(2:(n(j)-s(k)))/          ... u U;]/  
                       prod(2:s(k))/                 ... e/* T,ZJ  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... `zmj iC  
                       prod(2:((n(j)+m(j))/2-s(k))); SZ)AO8&  
            idx = (pows(k)==rpowers); p#DJow  
            z(:,j) = z(:,j) + p*rpowern(:,idx); >56I`[)  
        end 0'Y'K6hG`  
         8n`O{8:fi  
        if isnorm p.50BcDg  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); ?r"QJa>  
        end /Q*o6G ys0  
    end [Kc"L+H\  
    /rQ[Ik$|  
    % EOF zernpol
    离线niuhelen
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) R<-u`uX nP  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. md.#n  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated =iZj&B X  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive h'D-e5i  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, RqnT*  
    %   and THETA is a vector of angles.  R and THETA must have the same q[rBu9  
    %   length.  The output Z is a matrix with one column for every P-value, ^P| K2at  
    %   and one row for every (R,THETA) pair. )08mG_&atL  
    % D;RZE  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike djGzJLH  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) wy)I6`v  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) |}YeQl  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 .&R j2d  
    %   for all p. COC6H'F  
    % 6gSo>F4=  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 "h>B`S  
    %   Zernike functions (order N<=7).  In some disciplines it is p~h)@  
    %   traditional to label the first 36 functions using a single mode rEl bzL"&<  
    %   number P instead of separate numbers for the order N and azimuthal 5*he  
    %   frequency M. Cc7YjsRW  
    % ,*Vt53@E  
    %   Example: @}Y,A~   
    % $2?10}mrx  
    %       % Display the first 16 Zernike functions k OycS  
    %       x = -1:0.01:1; fNkN  
    %       [X,Y] = meshgrid(x,x); @&7|Laa  
    %       [theta,r] = cart2pol(X,Y); SW^/\cJ^  
    %       idx = r<=1; X9lh@`3  
    %       p = 0:15; ^{vf|zZ _  
    %       z = nan(size(X)); in/ITy-  
    %       y = zernfun2(p,r(idx),theta(idx)); c,\!<4  
    %       figure('Units','normalized') iHp@R-g  
    %       for k = 1:length(p) ~sd+ch*  
    %           z(idx) = y(:,k); xK3 xiR  
    %           subplot(4,4,k) [^U#ic>cT  
    %           pcolor(x,x,z), shading interp m ^O9G?  
    %           set(gca,'XTick',[],'YTick',[]) `(RQh@H  
    %           axis square ;T6x$e  
    %           title(['Z_{' num2str(p(k)) '}']) yLo{^4a.  
    %       end $h({x~Oj9  
    % w# t[sI"IT  
    %   See also ZERNPOL, ZERNFUN. -]uN16\ F  
    {APsi7HYBr  
    %   Paul Fricker 11/13/2006 T2nbU6H  
    KvFGwq"X  
    ~}c`r4  
    % Check and prepare the inputs: qrFC4\q}  
    % ----------------------------- Pmj]"7Vd[  
    if min(size(p))~=1 .Q,IOCHk  
        error('zernfun2:Pvector','Input P must be vector.') &*}NN5Sv  
    end <By6%<JTn  
    5 W<\J  
    if any(p)>35 !Cm<K*c"&E  
        error('zernfun2:P36', ... 6~ *w~U  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... T:$zNX<f  
               '(P = 0 to 35).']) >bf29tr  
    end Q]';1#J\  
    He)<S?X-6  
    % Get the order and frequency corresonding to the function number: C] <K s  
    % ---------------------------------------------------------------- %SRUHx[D  
    p = p(:); _]"5]c&*3  
    n = ceil((-3+sqrt(9+8*p))/2); |g5B==KI  
    m = 2*p - n.*(n+2); KzQ\A!qG  
    oKIry 8'^N  
    % Pass the inputs to the function ZERNFUN: 't$(Ruw  
    % ---------------------------------------- |~mi6 lJ6  
    switch nargin #clPao?r  
        case 3 g)*[W>M  
            z = zernfun(n,m,r,theta); D6iHkDTg  
        case 4 $73j*@EQA  
            z = zernfun(n,m,r,theta,nflag); DiK@>$v  
        otherwise oYNP,8r^  
            error('zernfun2:nargin','Incorrect number of inputs.') 3K2`1+kBVG  
    end ^|@t2Rp@  
    )bZS0f-  
    % EOF zernfun2
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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 T]d9tX-  
    function z = zernfun(n,m,r,theta,nflag) [z$th  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. EnXNTat})  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 1BK-uv:  
    %   and angular frequency M, evaluated at positions (R,THETA) on the R]e?<,"X  
    %   unit circle.  N is a vector of positive integers (including 0), and fOEw]B#@  
    %   M is a vector with the same number of elements as N.  Each element &*\wr} a!  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) A+*M<W  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, X4LU/f<f  
    %   and THETA is a vector of angles.  R and THETA must have the same W'x/Kg,w-  
    %   length.  The output Z is a matrix with one column for every (N,M) 1w}%>e-S  
    %   pair, and one row for every (R,THETA) pair. e[f}Lxln  
    % '+LbFGrO3  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike uc]]zI6  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 45e-A{G~  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral m[6?v;w  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, .}Va~[0j  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized !t/I j~o  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. o.IJ4'}aN  
    %  nen(  
    %   The Zernike functions are an orthogonal basis on the unit circle. ygoA/*s  
    %   They are used in disciplines such as astronomy, optics, and T$[50~  
    %   optometry to describe functions on a circular domain. <7-:flQz~  
    % qyzmjV6J2  
    %   The following table lists the first 15 Zernike functions. |P!7T.  
    % ]E/^(T-O  
    %       n    m    Zernike function           Normalization *Ii_dpJ  
    %       -------------------------------------------------- J:g4ES-/   
    %       0    0    1                                 1 h= tzG KI  
    %       1    1    r * cos(theta)                    2 0;9X`z J  
    %       1   -1    r * sin(theta)                    2 %0 cFs'  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) i+rh&,  
    %       2    0    (2*r^2 - 1)                    sqrt(3) {/|RKV83  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) }7)iLfi  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) a`/\0~  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) r{oRN  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) H!N`hEEj>  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) v+\&8)W=  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) yhTC?sf<  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) JK.<(=y\  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) r xlKoa  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Z'hHXSXM  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) R3 Zg,YM  
    %       -------------------------------------------------- 2iX57-6Ub  
    % \3K%>   
    %   Example 1: \tCxz(vKz  
    % 2g0_[$[m  
    %       % Display the Zernike function Z(n=5,m=1) ~;)H |R5kV  
    %       x = -1:0.01:1; Pi/V3D) B  
    %       [X,Y] = meshgrid(x,x); $0[t<4K`yn  
    %       [theta,r] = cart2pol(X,Y); /&>vhpZ}  
    %       idx = r<=1; |[+/ ]Y  
    %       z = nan(size(X)); :<QmG3F  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); 1 r9.JS  
    %       figure DH IC:6EY  
    %       pcolor(x,x,z), shading interp =PM6:3aKh  
    %       axis square, colorbar ,#V }qSKUS  
    %       title('Zernike function Z_5^1(r,\theta)') j 3t,Cx  
    % c9/&A  
    %   Example 2: 9YQYg@+R  
    % ` zoC++hx  
    %       % Display the first 10 Zernike functions UlD]!5NO  
    %       x = -1:0.01:1; pP|LSr Y!  
    %       [X,Y] = meshgrid(x,x); ,^n5UA`PK  
    %       [theta,r] = cart2pol(X,Y); 96#aG h>  
    %       idx = r<=1; wSPwa,)7s  
    %       z = nan(size(X)); ^| r6>b  
    %       n = [0  1  1  2  2  2  3  3  3  3]; -Cc2|~n  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; /6@$^paB  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; o,yZ1"  
    %       y = zernfun(n,m,r(idx),theta(idx)); GOU>j "5}2  
    %       figure('Units','normalized') BenUyv1d  
    %       for k = 1:10 $ISx0l~  
    %           z(idx) = y(:,k); d`sIgll&n  
    %           subplot(4,7,Nplot(k)) .|c=]_{  
    %           pcolor(x,x,z), shading interp aB^`3J  
    %           set(gca,'XTick',[],'YTick',[]) njGZ#{"eC  
    %           axis square Y+Cqc.JBQ  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) I4Rd2G_  
    %       end Mh"vH0\Lj  
    % j<PpCL_8%  
    %   See also ZERNPOL, ZERNFUN2. zL=PxFw0  
    Wu@v%!0  
    %   Paul Fricker 11/13/2006 20`QA u)'  
    5c 69M5  
    Z"N}f ,  
    % Check and prepare the inputs: 9iM[3uyO  
    % ----------------------------- 3FsX3K,_X  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) xY@<<  
        error('zernfun:NMvectors','N and M must be vectors.') [W Ud9fUL  
    end 2'-o'z<  
    ,r,$x4*  
    if length(n)~=length(m) n0vhc;d  
        error('zernfun:NMlength','N and M must be the same length.') 19*D*dkBR  
    end V]6CHE:BS  
    H|s,;1#  
    n = n(:); 4%>2 >5  
    m = m(:); W;QU6z>  
    if any(mod(n-m,2)) 1+9}Xnxb  
        error('zernfun:NMmultiplesof2', ... I9hZ&ed16  
              'All N and M must differ by multiples of 2 (including 0).') fa 2hQJ02  
    end 8?G534*r@2  
    d#u*NwY}  
    if any(m>n) <4RP:2#  
        error('zernfun:MlessthanN', ... 2SJ|$VsLaE  
              'Each M must be less than or equal to its corresponding N.') a=AP*adx8  
    end n7iIY4gZ  
    PGJkQsp0  
    if any( r>1 | r<0 ) P*3PDa@  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') 0<u(!iL  
    end )5Ofr-Y  
    Qkx}A7sK  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) qqr]S^WW  
        error('zernfun:RTHvector','R and THETA must be vectors.') W7?f_E\>W  
    end !xz{X?  
     b =R9@!  
    r = r(:); RZTC+ylj  
    theta = theta(:); 3R`eddenF  
    length_r = length(r); /<)kI(gf  
    if length_r~=length(theta) 'WcP+4c  
        error('zernfun:RTHlength', ... Tu7sA.73k  
              'The number of R- and THETA-values must be equal.') cnR18NK  
    end xF7q9'/F  
    xv~E wT)  
    % Check normalization: ?f4jqF~Fh  
    % -------------------- 2sYOO>  
    if nargin==5 && ischar(nflag) f]DO2 r  
        isnorm = strcmpi(nflag,'norm'); _aK4[*jnqh  
        if ~isnorm W'f)W4D$6  
            error('zernfun:normalization','Unrecognized normalization flag.') +=g9T`YbE  
        end #6F/:j;  
    else OuV f<@a  
        isnorm = false; ~.&2N Ur  
    end mH5[(?   
    V8+8?5'l  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% YZQF*fj  
    % Compute the Zernike Polynomials %G/j+Pf  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% UwVc!Lys  
    CK#SD|~:  
    % Determine the required powers of r: ;/)u/[KAv  
    % ----------------------------------- P"ATqQG%D  
    m_abs = abs(m); uH=^ILN.  
    rpowers = []; OVhtU+r  
    for j = 1:length(n) b,o@ m  
        rpowers = [rpowers m_abs(j):2:n(j)]; RZ GD5`n  
    end kbKGGn4u  
    rpowers = unique(rpowers); J>%uak<  
    _0 $W;8X  
    % Pre-compute the values of r raised to the required powers,  jgd^{!  
    % and compile them in a matrix: K f}h{X  
    % ----------------------------- +Qo]'xKr  
    if rpowers(1)==0 + -OnO7f  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); IPEJ7 n49  
        rpowern = cat(2,rpowern{:}); 3Q_L6Wj~  
        rpowern = [ones(length_r,1) rpowern]; -:NFF'  
    else J~(M%] &k^  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); C|H/x\?zRv  
        rpowern = cat(2,rpowern{:}); ;+Uc} =  
    end 8zWKKcf7t  
    -naoM  
    % Compute the values of the polynomials: Kta7xtu  
    % -------------------------------------- OF/DI)j3  
    y = zeros(length_r,length(n)); 2j( ]Bt:  
    for j = 1:length(n) (J,^)!g7  
        s = 0:(n(j)-m_abs(j))/2; /%9CR'%*c  
        pows = n(j):-2:m_abs(j); g_2EH  
        for k = length(s):-1:1 -lNT"9  
            p = (1-2*mod(s(k),2))* ... G$_=rHt_%  
                       prod(2:(n(j)-s(k)))/              ... TOvpv@?-  
                       prod(2:s(k))/                     ... R<FW?z*  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... Z9vJF.clO  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); f{j (H?5  
            idx = (pows(k)==rpowers); -&3mOn& (1  
            y(:,j) = y(:,j) + p*rpowern(:,idx); 3D*vNVI  
        end |uRZT3bGyj  
         p:@JCsH=  
        if isnorm -|aNHZr  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); !3 j@gi2  
        end @}B,l.Tj  
    end i!+Wv-  
    % END: Compute the Zernike Polynomials |E =8  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 'hn=X7  
    \yNe5  
    % Compute the Zernike functions: g (:%E  
    % ------------------------------ Wo[*P\8  
    idx_pos = m>0; _b(y"+k  
    idx_neg = m<0; *4oj' }  
    F^bzE5#  
    z = y; ^N`bA8  
    if any(idx_pos) eTrIN,4  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); d9>k5!  
    end f+o%N  
    if any(idx_neg) @+(TM5Ub  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ]BiLLDz(  
    end WUnmUW[/  
    JE$aYs<(TF  
    % EOF zernfun
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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
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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