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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 Fpl<2eBg4  
    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! qky{]qNW  
     
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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)  v](7c2;  
    Yhb=^)@))  
    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。 w f,7  
    AFF7fK  
    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) N|EH`eu^i  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. qPK3"fzH  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of -o#HO_9  
    %   order N and frequency M, evaluated at R.  N is a vector of AF g*  
    %   positive integers (including 0), and M is a vector with the JV=d!Gi[C  
    %   same number of elements as N.  Each element k of M must be a UQgOtqL3  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) , |CT|2D>  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is &~ QQZ]q6  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix 6UXa 5t  
    %   with one column for every (N,M) pair, and one row for every In#V1[io  
    %   element in R. D^W6Cq5\  
    % x]&V7Y   
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- ;Oh4W<hH}  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is aT$q1!U`j2  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to 9JV(}v5[  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 x48Y#"'  
    %   for all [n,m]. 6?b 9~xRW  
    % ;Y5"[C9|  
    %   The radial Zernike polynomials are the radial portion of the L']EYK5  
    %   Zernike functions, which are an orthogonal basis on the unit 7^P!@o$v!  
    %   circle.  The series representation of the radial Zernike zA=gDuy3@  
    %   polynomials is }A3(g$8KR  
    % =|O`al  
    %          (n-m)/2 9!zUv:;  
    %            __ # 8 0DM  
    %    m      \       s                                          n-2s q$`{$RX  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r O'{UAb+-  
    %    n      s=0 /PH+K24v~  
    % i% 19|an  
    %   The following table shows the first 12 polynomials. -H5n>j0!{  
    % 2qLRcA=R  
    %       n    m    Zernike polynomial    Normalization fEf ",{I  
    %       --------------------------------------------- h4N!zj[  
    %       0    0    1                        sqrt(2) uF_gfjR[m  
    %       1    1    r                           2 rT9<_<  
    %       2    0    2*r^2 - 1                sqrt(6) )F4H'  
    %       2    2    r^2                      sqrt(6) xa#0y   
    %       3    1    3*r^3 - 2*r              sqrt(8) ;A7HEx  
    %       3    3    r^3                      sqrt(8) Aq@_^mq1A  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) Sr Z\]  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) 3CK4a,]Dm  
    %       4    4    r^4                      sqrt(10) Oaf!\ z}  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) >MZWm6M8  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) teH $hd-q  
    %       5    5    r^5                      sqrt(12) s1. YH?A;  
    %       --------------------------------------------- 0i/l2&x*k]  
    % ]CsF} wr'z  
    %   Example: E,&BP$B  
    % 0(\ybppx  
    %       % Display three example Zernike radial polynomials g N76  
    %       r = 0:0.01:1; 9r=@S  
    %       n = [3 2 5]; "W$,dWF  
    %       m = [1 2 1]; 0j\?zt?  
    %       z = zernpol(n,m,r); W}p>jP}  
    %       figure `p1szZD&  
    %       plot(r,z) :bFCnV`Q  
    %       grid on v1%rlP  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') )/kkvI()l  
    % ]4wyuP,up  
    %   See also ZERNFUN, ZERNFUN2. &^$dHr6v  
    v J9Uw  
    % A note on the algorithm. ~&B{"d  
    % ------------------------ N)|mA)S)  
    % The radial Zernike polynomials are computed using the series  w=5D>]  
    % representation shown in the Help section above. For many special u&Ts'j  
    % functions, direct evaluation using the series representation can ,DsqKXSU  
    % produce poor numerical results (floating point errors), because +>mbBu!7  
    % the summation often involves computing small differences between m`C c U`s  
    % large successive terms in the series. (In such cases, the functions U/'"w v1y  
    % are often evaluated using alternative methods such as recurrence GADbXp3  
    % relations: see the Legendre functions, for example). For the Zernike )\#w=P  
    % polynomials, however, this problem does not arise, because the +M-x*;.  
    % polynomials are evaluated over the finite domain r = (0,1), and |;3Ru vX?+  
    % because the coefficients for a given polynomial are generally all Q-o}Xnj*!L  
    % of similar magnitude. ; mnV)8:F  
    % 'X&sH/>r  
    % ZERNPOL has been written using a vectorized implementation: multiple lj0"2@z3"E  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] (PAkKY}  
    % values can be passed as inputs) for a vector of points R.  To achieve dx}) 1%  
    % this vectorization most efficiently, the algorithm in ZERNPOL !wy Qk  
    % involves pre-determining all the powers p of R that are required to ~Z-M?8:  
    % compute the outputs, and then compiling the {R^p} into a single 7pH`"$  
    % matrix.  This avoids any redundant computation of the R^p, and )jk X&7x  
    % minimizes the sizes of certain intermediate variables. 1Q1NircJ  
    % dU%Q=r8R  
    %   Paul Fricker 11/13/2006 IGz92&y  
    Im7<\ b@  
    e aLSq  
    % Check and prepare the inputs: JW[y  
    % ----------------------------- 6)63Yp(  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) >PdYQDyVS  
        error('zernpol:NMvectors','N and M must be vectors.') z%-Yz- G9  
    end P__JN\{9  
    QCB2&lN\&L  
    if length(n)~=length(m) L1=+x^WQ  
        error('zernpol:NMlength','N and M must be the same length.') xL8r'gV@  
    end 2z9\p%MX  
    |hBX"  
    n = n(:); h8@8Q w  
    m = m(:); I^erMQn[ z  
    length_n = length(n); q SR\=:$  
    C "XvspJ  
    if any(mod(n-m,2)) $D{ KXkrd  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') H]&a}WQ_  
    end G"w ?{W @  
    +oa\'.~?  
    if any(m<0)  1@Abs  
        error('zernpol:Mpositive','All M must be positive.') gz fs9e  
    end xCU^4DO3p  
    ZC}'! $r7  
    if any(m>n) Y_m/? [:  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') wh4ik`S 1  
    end 48;6C g  
    }  IJ  
    if any( r>1 | r<0 ) {A2EGUmF2  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') $|+q9 o\  
    end #ra"(/)  
    ]WlE9z7:8  
    if ~any(size(r)==1) HKu? J  
        error('zernpol:Rvector','R must be a vector.') Q9,H 0r-%  
    end k#mQLv  
    )I7~ <$w  
    r = r(:); 0>@D{_}s  
    length_r = length(r); /5cFa  
    q@K8,=/.#  
    if nargin==4 Ik[aiz  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); ?,G CR1|4  
        if ~isnorm hP1}Do  
            error('zernpol:normalization','Unrecognized normalization flag.') ~ *:{U   
        end 7{<:g!  
    else 1Rrp#E}  
        isnorm = false; * V;L|c  
    end g\9I&z~?  
    'a]4]d  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %eJolztKZ  
    % Compute the Zernike Polynomials #rZF4>c  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% U*Q1(C  
    F3BWi[Xh  
    % Determine the required powers of r: IQn|0$':Z  
    % ----------------------------------- h SGI  
    rpowers = []; VVY#g%(K  
    for j = 1:length(n) ODS8bD0!i  
        rpowers = [rpowers m(j):2:n(j)]; vo48\w7[  
    end &f12Q&jY7  
    rpowers = unique(rpowers); K@uUe3  
    ,3 !D(&  
    % Pre-compute the values of r raised to the required powers, \#1*r'V8  
    % and compile them in a matrix: P .I <.e  
    % ----------------------------- nR%ASUx:Y  
    if rpowers(1)==0 e,j2#wjor  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); fL3Px  
        rpowern = cat(2,rpowern{:}); CM$q{;y  
        rpowern = [ones(length_r,1) rpowern]; UO3QwZ4j;  
    else SePPI.n  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); j?!BHNs  
        rpowern = cat(2,rpowern{:}); 8sMDe'  
    end _<;;CI3w  
    -e#~CE-  
    % Compute the values of the polynomials: 9  Vn  
    % -------------------------------------- )8BGN'jyi  
    z = zeros(length_r,length_n); %V40I{1  
    for j = 1:length_n l,z# : k  
        s = 0:(n(j)-m(j))/2; )- 2sk@y  
        pows = n(j):-2:m(j); -)cau-(X  
        for k = length(s):-1:1 FE}!I  
            p = (1-2*mod(s(k),2))* ... 7d9kr?3(U  
                       prod(2:(n(j)-s(k)))/          ... K})=&<M0  
                       prod(2:s(k))/                 ... N0Y$QWr_$  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... \BIa:}9O  
                       prod(2:((n(j)+m(j))/2-s(k))); a/})X[2  
            idx = (pows(k)==rpowers); jZRf{  
            z(:,j) = z(:,j) + p*rpowern(:,idx); ~|W0+&):  
        end @UbH ;m  
         YH_mWN\Wu  
        if isnorm JCL+uEX4S  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); qG=?+em  
        end {VB n@^'s  
    end N)F&c!anh  
    1|]IWX|  
    % 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) nRP|Qt7>  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. =r w60B  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated % oPt],>  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive FU{$oCh/5  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, Yf=an`"  
    %   and THETA is a vector of angles.  R and THETA must have the same VR8 kY&  
    %   length.  The output Z is a matrix with one column for every P-value, vb o| q[z  
    %   and one row for every (R,THETA) pair. 8R3x74fL  
    % <7U\@si4  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike [uJfmrEH  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) 8OS@gpz  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) J$aE:g6'  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 n\-nBrVSf  
    %   for all p. fX ^h O+f  
    % {D6p?TL+  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 :\!D 6\o6  
    %   Zernike functions (order N<=7).  In some disciplines it is q=EHB5!q  
    %   traditional to label the first 36 functions using a single mode & bKl(,  
    %   number P instead of separate numbers for the order N and azimuthal {7'Evfn)  
    %   frequency M. @3^D[  
    % QLs9W& PG  
    %   Example: bv&#ay 7  
    % cEdf&*_-'I  
    %       % Display the first 16 Zernike functions [~aRA'qJ{V  
    %       x = -1:0.01:1; mp !S<m  
    %       [X,Y] = meshgrid(x,x); %>z4hH,  
    %       [theta,r] = cart2pol(X,Y); >/]` f8^  
    %       idx = r<=1; C`J>Gm  
    %       p = 0:15; 6#J>b[Q  
    %       z = nan(size(X)); YaBZ#$r  
    %       y = zernfun2(p,r(idx),theta(idx)); 2bs={p$}a  
    %       figure('Units','normalized') qG6?k}\\  
    %       for k = 1:length(p) NsPAWI|4  
    %           z(idx) = y(:,k); '3p7ee&  
    %           subplot(4,4,k) 6>yfm4o  
    %           pcolor(x,x,z), shading interp 8[k:FGp>  
    %           set(gca,'XTick',[],'YTick',[])  &x":  
    %           axis square Lhts4D/V7  
    %           title(['Z_{' num2str(p(k)) '}']) @QN(ouqQ  
    %       end /wR,P  
    % yd}1Mx  
    %   See also ZERNPOL, ZERNFUN. ~6Xr^An/Z  
    D2y[?RG  
    %   Paul Fricker 11/13/2006 K9HXy*y49  
    |3bCq(ZR\P  
    nxjP4d>  
    % Check and prepare the inputs: -MK9IO]i  
    % ----------------------------- <hV%OrBz-  
    if min(size(p))~=1 @^2?97i c  
        error('zernfun2:Pvector','Input P must be vector.') L0Ycf|[s,  
    end JK/gq}c  
    > u!# 4  
    if any(p)>35 NimW=X;c  
        error('zernfun2:P36', ... d:L|BkQ7*  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... hr/H vB  
               '(P = 0 to 35).']) tP. jJC~  
    end V0/PjD,jP  
    +_T`tmQ  
    % Get the order and frequency corresonding to the function number: SWLt5dV  
    % ---------------------------------------------------------------- {@&%Bq*&  
    p = p(:); +T/T\[  
    n = ceil((-3+sqrt(9+8*p))/2); -cONC9 =  
    m = 2*p - n.*(n+2); Mb^E  
    ;ztt*py  
    % Pass the inputs to the function ZERNFUN: }T~ }W8H  
    % ---------------------------------------- Xl %ax!/  
    switch nargin qRc Y(mb  
        case 3 !qe:M]C'l  
            z = zernfun(n,m,r,theta); BY5ODc$  
        case 4 ~-tKMc).X  
            z = zernfun(n,m,r,theta,nflag); fe4Ki  
        otherwise 6ec#3~ Y]  
            error('zernfun2:nargin','Incorrect number of inputs.') A-&XgOL  
    end ccY! OSae  
    1=Z, #r  
    % 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 $1uT`>%  
    function z = zernfun(n,m,r,theta,nflag) ".7\>8A#a  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. +GvPJI  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N  ae>B0#=  
    %   and angular frequency M, evaluated at positions (R,THETA) on the &e \UlM22  
    %   unit circle.  N is a vector of positive integers (including 0), and 'w8p[h (,  
    %   M is a vector with the same number of elements as N.  Each element '\% Kd+k  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) 4q)+nh~s  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, s4[PwD  
    %   and THETA is a vector of angles.  R and THETA must have the same _KJ!C!  
    %   length.  The output Z is a matrix with one column for every (N,M) 6FkBb !ASk  
    %   pair, and one row for every (R,THETA) pair. \P;2s<6i\  
    % &7r73~TXm  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ECk* H  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), n.7-$1  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral -oT3`d3  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, o/hj~;(]  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized LUzn7FZk  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. %j/}e>$"Nk  
    % WXQ+`OH7  
    %   The Zernike functions are an orthogonal basis on the unit circle. 6E{(_i  
    %   They are used in disciplines such as astronomy, optics, and P?hB`5X  
    %   optometry to describe functions on a circular domain. PJL [En*  
    % ] @uuB\u  
    %   The following table lists the first 15 Zernike functions. 4x-K0  
    % oc"7|YG  
    %       n    m    Zernike function           Normalization 97k}{tG  
    %       -------------------------------------------------- zG)vmysJf  
    %       0    0    1                                 1 _t>[gB,  
    %       1    1    r * cos(theta)                    2 Vt(s4  
    %       1   -1    r * sin(theta)                    2 uvl>Z= "  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) .Vrl:  
    %       2    0    (2*r^2 - 1)                    sqrt(3) snYyxi  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) Ot{~mMDp  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) C@WdPjxj  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) xEg@Y"NQ  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) 8GeJ%^0o}  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) 0"{-<Wot}  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) Mq='|0,  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) )|6OPR@(#/  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) _+OCI%=:  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) P9)L1l<3I  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) ~;}uYJ  
    %       -------------------------------------------------- -TS5g1  
    % W[:CCCDL  
    %   Example 1: q1Ad"rm  
    % |W*5<2Q9  
    %       % Display the Zernike function Z(n=5,m=1) S1#5oy2  
    %       x = -1:0.01:1; yN/g;bQ  
    %       [X,Y] = meshgrid(x,x); pM9M8d  
    %       [theta,r] = cart2pol(X,Y); ?}U?Q7vx@@  
    %       idx = r<=1; ZgfhNI\  
    %       z = nan(size(X)); B,] AfH  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); +g;{c+Kw:  
    %       figure 3Ww 37V>h  
    %       pcolor(x,x,z), shading interp &F8*>F^7  
    %       axis square, colorbar LqLhZBU9  
    %       title('Zernike function Z_5^1(r,\theta)') A 8g_BLj!e  
    % <}G/x*N  
    %   Example 2: yL %88,/  
    % g> m)XY  
    %       % Display the first 10 Zernike functions /VD[:sU7  
    %       x = -1:0.01:1; M)~sL1)  
    %       [X,Y] = meshgrid(x,x); 1a mEQ  
    %       [theta,r] = cart2pol(X,Y); r>gf&/Pl  
    %       idx = r<=1; U6.hH%\}@  
    %       z = nan(size(X)); s>)?MB*vb  
    %       n = [0  1  1  2  2  2  3  3  3  3]; N'CW Sf.e  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; o]WcODJdl  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; Z3U%Afl2{  
    %       y = zernfun(n,m,r(idx),theta(idx)); Vha,rIi  
    %       figure('Units','normalized') 4X!4S6JfB  
    %       for k = 1:10 Wt.['`c<  
    %           z(idx) = y(:,k); u$FL(m4  
    %           subplot(4,7,Nplot(k)) p W@Yr  
    %           pcolor(x,x,z), shading interp L)qUBp@MW  
    %           set(gca,'XTick',[],'YTick',[]) qHvU4v  
    %           axis square cG&@PO]+.  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) !<out4Mz"  
    %       end ?*.:*A  
    % z4(`>z2a  
    %   See also ZERNPOL, ZERNFUN2. raZkH8  
    =!)x`1j!S  
    %   Paul Fricker 11/13/2006 SLNq%7apx  
    KWM.e1(  
    UC j:]!P  
    % Check and prepare the inputs: m6'9Id-:L  
    % ----------------------------- mDk6@Gd@U  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) MS:,I?  
        error('zernfun:NMvectors','N and M must be vectors.') @urZ  
    end ky&wv+7  
    %`~+^{Wp  
    if length(n)~=length(m) t|s(V-Wq  
        error('zernfun:NMlength','N and M must be the same length.') V5p^]To!  
    end @R<z=n"  
    <oi'yr  
    n = n(:); X"9N<)C  
    m = m(:); 6"NtVfui  
    if any(mod(n-m,2)) *>2e4j]  
        error('zernfun:NMmultiplesof2', ... 7rYBFSp  
              'All N and M must differ by multiples of 2 (including 0).') 5$Kd<ky  
    end `+0dz,  
    @t0T+T3  
    if any(m>n) 0$0 215  
        error('zernfun:MlessthanN', ... IVy<>xpt  
              'Each M must be less than or equal to its corresponding N.') HCCq9us  
    end #;5Q d'  
    $|@pY| f  
    if any( r>1 | r<0 ) )a5ON8?  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') bxzx@sF2l  
    end @eutp`xoT\  
    Jd?qvE>Pp  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 9_4(}|"N|  
        error('zernfun:RTHvector','R and THETA must be vectors.') 6Q J.=.>b  
    end =qbN?a/?2  
    L8H:, } 2  
    r = r(:); FS=LpvOG)  
    theta = theta(:); n).*=YLN  
    length_r = length(r); IuA4eDr^Y%  
    if length_r~=length(theta) ti$60Up  
        error('zernfun:RTHlength', ... m`):= ^nC  
              'The number of R- and THETA-values must be equal.') oRJ!TAbD  
    end 'Z:wEt!  
    o4OB xHKy  
    % Check normalization: 2(x| %  
    % -------------------- w^=(:`  
    if nargin==5 && ischar(nflag) f$9|qfW'$  
        isnorm = strcmpi(nflag,'norm'); *B \ @L  
        if ~isnorm 3,`M\#z%K  
            error('zernfun:normalization','Unrecognized normalization flag.') =v 'Aub  
        end +'+ Nr<  
    else CBNt _y  
        isnorm = false; 2b,edJVt?  
    end mq:WBSsV  
    %O f w"W  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  zY7M]Az  
    % Compute the Zernike Polynomials SAj#+_db  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ,qo^G0XO  
    [8q`~S%-]  
    % Determine the required powers of r: H "5,To  
    % ----------------------------------- 'n1$Y%t  
    m_abs = abs(m); cui%r!D  
    rpowers = []; k}I65 ^l#  
    for j = 1:length(n) (C1~>7L  
        rpowers = [rpowers m_abs(j):2:n(j)]; xWqV~NnE  
    end }Y|M+0   
    rpowers = unique(rpowers); Slj U=,  
    tIV{uVM[|D  
    % Pre-compute the values of r raised to the required powers, T8)X?>CIW  
    % and compile them in a matrix: mdQe)>  
    % ----------------------------- ~c] q:pU2  
    if rpowers(1)==0 !`4ie  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 2VUN  
        rpowern = cat(2,rpowern{:}); k.2GIc:5  
        rpowern = [ones(length_r,1) rpowern]; Q[aF"5h%  
    else eK5~gnv,  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ssS"X@VZ \  
        rpowern = cat(2,rpowern{:}); mPqK k  
    end h-sO7M0E]  
    c0HPS9N\  
    % Compute the values of the polynomials: jUl_ToX  
    % -------------------------------------- Nn-k hl|11  
    y = zeros(length_r,length(n)); Y2'HP)tfIw  
    for j = 1:length(n) ]Hq,Pr_+  
        s = 0:(n(j)-m_abs(j))/2; `B&=ya|bl  
        pows = n(j):-2:m_abs(j); M(:bM1AD`u  
        for k = length(s):-1:1 _?y3&4N)  
            p = (1-2*mod(s(k),2))* ... ZMr[:,Jp  
                       prod(2:(n(j)-s(k)))/              ... oM^vJ3  
                       prod(2:s(k))/                     ... mF!4*k  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... =R 4]Kf  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); {O).!  
            idx = (pows(k)==rpowers); EYZ&%.Sy5  
            y(:,j) = y(:,j) + p*rpowern(:,idx); 64 'QTF{D  
        end f2pA+j5[  
         *JZ9'|v_H  
        if isnorm tS5J{j>T  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); L[<Y6u>m!1  
        end S 1^t;{"  
    end 4p+Veo6B  
    % END: Compute the Zernike Polynomials "#gS?aS  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ZR0 OqSp]  
    ?(Tin80=r  
    % Compute the Zernike functions: )F\kGe  
    % ------------------------------ u|.|dv'mbp  
    idx_pos = m>0; @$L|   
    idx_neg = m<0; 6R8>w,  
    /*BK6hc  
    z = y; ?azLaAG  
    if any(idx_pos) ~Ym*QSD  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); {Y=k`t,  
    end &iq'V*+-\  
    if any(idx_neg) !FyO5`v  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); . XbDb  
    end n[qnrk*3 %  
    lKU{jWA  
    % 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