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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 )U:m:cr<  
    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! !ons]^km  
     
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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)  +iRh  
    t-bB>q#3>  
    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。 5<Nx^D  
    o]oum,Q  
    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) ri-b=|h2j  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. @_}P-h  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of G2: agqL/  
    %   order N and frequency M, evaluated at R.  N is a vector of NyNXP_8  
    %   positive integers (including 0), and M is a vector with the p9{mS7R9T  
    %   same number of elements as N.  Each element k of M must be a <x>M o   
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) =fFP5e ['  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is d5:c^`  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix IyG}H}  
    %   with one column for every (N,M) pair, and one row for every > /caXvS  
    %   element in R. i?^L/b`H  
    % J<jy2@"tXo  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- n,WqyNt*  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is B \2 SH%\  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to ; kI134i=  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 >}6%#CAf  
    %   for all [n,m]. 4 "'~NvO  
    % a<bwzX|.  
    %   The radial Zernike polynomials are the radial portion of the u.xnOcOH!  
    %   Zernike functions, which are an orthogonal basis on the unit ?^\|-Gr  
    %   circle.  The series representation of the radial Zernike /h|#J  
    %   polynomials is ]Er$*7f  
    % -PR N:'T  
    %          (n-m)/2 ~2-1 j  
    %            __ nZYBE030  
    %    m      \       s                                          n-2s </*6wpN  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r kMN~Y  
    %    n      s=0 P( 8OQL:  
    % { 9q4)R}G  
    %   The following table shows the first 12 polynomials. |aq"#Ml)  
    % -6B4sZpzD  
    %       n    m    Zernike polynomial    Normalization r\^b(rNe  
    %       --------------------------------------------- *(DV\.l`  
    %       0    0    1                        sqrt(2) c9h6C  
    %       1    1    r                           2 iGB}Il)  
    %       2    0    2*r^2 - 1                sqrt(6) $1`2 kM5  
    %       2    2    r^2                      sqrt(6) z-)O9PV  
    %       3    1    3*r^3 - 2*r              sqrt(8) SO0PF|{\r  
    %       3    3    r^3                      sqrt(8) g]0_5?i  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) *gWwALGo5  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) r* Ca}Z  
    %       4    4    r^4                      sqrt(10) xU`p|(SS-  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) :"/d|i`T  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) }&D32\  
    %       5    5    r^5                      sqrt(12) #AQV(;r7@  
    %       --------------------------------------------- Ds:'Lb  
    % oNF6<A(@$  
    %   Example: Ig>(m49d  
    % }*]-jWt1J\  
    %       % Display three example Zernike radial polynomials 1iF1GkLEq  
    %       r = 0:0.01:1; ~Z' ?LV<t  
    %       n = [3 2 5]; 3h`f  6  
    %       m = [1 2 1]; P~X2^bw  
    %       z = zernpol(n,m,r); R4:b{)=O  
    %       figure S30%)<W  
    %       plot(r,z) |&i<bqLw:  
    %       grid on t"oeQ*d%  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') .%  
    % kE1TP]|  
    %   See also ZERNFUN, ZERNFUN2. U&qZ"  
    j1T#yt J  
    % A note on the algorithm. m ~$v;?i  
    % ------------------------ aK^q_ghh[  
    % The radial Zernike polynomials are computed using the series j?4qO]_Wx+  
    % representation shown in the Help section above. For many special X#^[<5  
    % functions, direct evaluation using the series representation can z<' u1l3  
    % produce poor numerical results (floating point errors), because |P?*5xPB  
    % the summation often involves computing small differences between nAlQ7 '  
    % large successive terms in the series. (In such cases, the functions :Zw2'IV  
    % are often evaluated using alternative methods such as recurrence >i?oC^QM  
    % relations: see the Legendre functions, for example). For the Zernike [(7S.5I  
    % polynomials, however, this problem does not arise, because the FGq [ \B  
    % polynomials are evaluated over the finite domain r = (0,1), and .HABNPNg(  
    % because the coefficients for a given polynomial are generally all 7s^'d,P  
    % of similar magnitude. U|R_OLWAg  
    % KF:78C  
    % ZERNPOL has been written using a vectorized implementation: multiple ~*];pV]A[  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] BnF^u5kv%  
    % values can be passed as inputs) for a vector of points R.  To achieve & wDs6xq  
    % this vectorization most efficiently, the algorithm in ZERNPOL X%x*f3[  
    % involves pre-determining all the powers p of R that are required to g *+>H1}  
    % compute the outputs, and then compiling the {R^p} into a single sc#qwQ#  
    % matrix.  This avoids any redundant computation of the R^p, and 5*u+q2\F  
    % minimizes the sizes of certain intermediate variables. \1M4Dl5!  
    % 'PW5ux@`<  
    %   Paul Fricker 11/13/2006 `C'H.g\>2Q  
    U- k`s[dv  
    +X 88;-  
    % Check and prepare the inputs: &s>Jb?_5Mx  
    % ----------------------------- b^vQpiz  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) tw)mepwB  
        error('zernpol:NMvectors','N and M must be vectors.') }3WxZv]I}  
    end 2=!RQv~%  
    Xne1gms  
    if length(n)~=length(m) =~LJ3sIX  
        error('zernpol:NMlength','N and M must be the same length.')  6(R<{{  
    end +D*Z_Yh6  
    !^G\9"4A  
    n = n(:); l,aay-E  
    m = m(:); w7&A0M  
    length_n = length(n); zX i 'kB  
    #}5uno  
    if any(mod(n-m,2)) (A.C]hD  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') (+hK%}K>  
    end ~v6D#@%A  
    j3ls3H&  
    if any(m<0) X1_5KH  
        error('zernpol:Mpositive','All M must be positive.') :7;@ZEe  
    end lr&a;aZp  
    lPAQ3t!,  
    if any(m>n) w_VP J  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') Z0r'S]fe  
    end buHJB*?9  
    86a\+Kz%%L  
    if any( r>1 | r<0 ) ba9?(+i$h  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') es0hm2HT3  
    end kD"{g#c  
    $<[79al#  
    if ~any(size(r)==1) }c:M^Ff  
        error('zernpol:Rvector','R must be a vector.') _DEjF)S  
    end ~pky@O#b  
    <(!:$  
    r = r(:); YuwI&)l  
    length_r = length(r); %J-GKpo/S  
    1G`Pmh@  
    if nargin==4 tfWS)y7  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); :[d9tm  
        if ~isnorm 4?01s-Y  
            error('zernpol:normalization','Unrecognized normalization flag.') 8H`[*|{'  
        end llDkJ)\  
    else `XDl_E+>l  
        isnorm = false; uhq8   
    end akTk(  
    M D#jj3y  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% F((4U"   
    % Compute the Zernike Polynomials ;5AcFB  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% :Llb< MY2  
    /dIzY0<aO  
    % Determine the required powers of r: HjwE+:w  
    % ----------------------------------- B`sAk %  
    rpowers = []; 'Z]w^<  
    for j = 1:length(n) pQQH)`J|t  
        rpowers = [rpowers m(j):2:n(j)]; /g.U&oI]D  
    end asqV~n  
    rpowers = unique(rpowers); iU:cW=W|M\  
    y|jq?M<A  
    % Pre-compute the values of r raised to the required powers, z{r}~{{E  
    % and compile them in a matrix: yIE!j %u  
    % ----------------------------- )LCHy^'  
    if rpowers(1)==0 V]?R>qhgu  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 0tJ Z4(0  
        rpowern = cat(2,rpowern{:}); ?&uu[y  
        rpowern = [ones(length_r,1) rpowern]; -F3-{E  
    else vw@S>G lGg  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); qcRs$-J  
        rpowern = cat(2,rpowern{:}); :~SyL!  
    end uEx-]F  
    UGatWj  
    % Compute the values of the polynomials: 3iU=c&P  
    % -------------------------------------- hCo|HB  
    z = zeros(length_r,length_n); 4I(Xy]wm  
    for j = 1:length_n H6gSO(U  
        s = 0:(n(j)-m(j))/2; Kf-JcBsrT  
        pows = n(j):-2:m(j); |V7*l1  
        for k = length(s):-1:1 7PF%76TO  
            p = (1-2*mod(s(k),2))* ... VS|2|n1<6  
                       prod(2:(n(j)-s(k)))/          ... ,]/X\t5]D  
                       prod(2:s(k))/                 ... /Gfw8g\}  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... :MDKC /mC  
                       prod(2:((n(j)+m(j))/2-s(k))); 'O-"\J\  
            idx = (pows(k)==rpowers); >5 BJ3Hf  
            z(:,j) = z(:,j) + p*rpowern(:,idx); bQ5\ ]5M  
        end 4`=m u}Y2  
         G]aOHJ:.  
        if isnorm a09<!0Rp  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); 3 8`<:{^Y  
        end Xlt|nX~#;  
    end XB5DPx  
    9o!Bzy+_  
    % EOF zernpol
    离线niuhelen
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) [opGZ`>)j"  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. ku M$UYTTX  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated o[D9I hs  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive 3HK\BS  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, ] @fk] ]R  
    %   and THETA is a vector of angles.  R and THETA must have the same )Xyn q(  
    %   length.  The output Z is a matrix with one column for every P-value, I1&aM}y{G  
    %   and one row for every (R,THETA) pair. }SCM I4\  
    % Y\'}a+:@Ph  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike Y`wSv NU  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) Jj%K=sw  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) g< .qUBPKX  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 `5Zz5V  
    %   for all p. jZr q{Z<  
    % Eu04e N  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 eh#(eua0/  
    %   Zernike functions (order N<=7).  In some disciplines it is [z9Z5sLO  
    %   traditional to label the first 36 functions using a single mode 0+b1vhQ  
    %   number P instead of separate numbers for the order N and azimuthal Yc*; /T}  
    %   frequency M. lsNd_7k  
    % C$)onk  
    %   Example: Pj% |\kbNs  
    % ^sWT:BDh  
    %       % Display the first 16 Zernike functions _v]MsT-q  
    %       x = -1:0.01:1; x ]ot 2  
    %       [X,Y] = meshgrid(x,x); A&jlizN7  
    %       [theta,r] = cart2pol(X,Y); R ViuJ;  
    %       idx = r<=1; U :_^#\p  
    %       p = 0:15; 0_t!T'jr7  
    %       z = nan(size(X)); uY'HT|@:{  
    %       y = zernfun2(p,r(idx),theta(idx)); 7. ;3e@s  
    %       figure('Units','normalized') D. XvG_  
    %       for k = 1:length(p) BIL Lq8)  
    %           z(idx) = y(:,k); ;sFF+^~L  
    %           subplot(4,4,k) J5jvouR  
    %           pcolor(x,x,z), shading interp l1Fc>:o{  
    %           set(gca,'XTick',[],'YTick',[]) .#pU=v#/[  
    %           axis square Thit  
    %           title(['Z_{' num2str(p(k)) '}']) v|2T%y_ u  
    %       end <Q?F?.^e  
    % du^J2m{f  
    %   See also ZERNPOL, ZERNFUN. *c+ (-  
    45>?o  
    %   Paul Fricker 11/13/2006 <2qr}K{'A  
    |ZBI *  
    lHX72s|V  
    % Check and prepare the inputs: i~J'%a<Qp  
    % ----------------------------- 1&Zj  
    if min(size(p))~=1 ]z9=}=If  
        error('zernfun2:Pvector','Input P must be vector.') cExS7~*  
    end Th%Sjgsn  
    \)|hogI|f  
    if any(p)>35 4{`{WI{  
        error('zernfun2:P36', ... ekCC5P!  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... MFk5K  
               '(P = 0 to 35).']) @;RXLq/8  
    end M/K5#8Arj  
    DR<9#RRD  
    % Get the order and frequency corresonding to the function number: vRO _Q?  
    % ---------------------------------------------------------------- }pu27F)&  
    p = p(:); @MCg%Afw  
    n = ceil((-3+sqrt(9+8*p))/2); `W*U4?M  
    m = 2*p - n.*(n+2); D}X\Ca"h  
    S^\Vgi(  
    % Pass the inputs to the function ZERNFUN:  kPLxEwl  
    % ---------------------------------------- <e</m)j  
    switch nargin pIX`MlBdF  
        case 3 p.?rey<%  
            z = zernfun(n,m,r,theta); 3/n5#&c\4  
        case 4 }9fTF:P  
            z = zernfun(n,m,r,theta,nflag); e**qF=HCw  
        otherwise "LTad`]<Ro  
            error('zernfun2:nargin','Incorrect number of inputs.') <W$mj04@  
    end Npy :!  
    W:L AP R  
    % EOF zernfun2
    离线niuhelen
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    只看该作者 3楼 发表于: 2011-03-12
    回 2楼(phility) 的帖子
    非常感谢啊,我手上也有zernike多项式的拟合的源程序,也不知道对不对,不怎么会有 Z}Ft:7   
    function z = zernfun(n,m,r,theta,nflag) VS8Rx.?  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. Fy-t T]Q9  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N }Ud*TOo`  
    %   and angular frequency M, evaluated at positions (R,THETA) on the L0WN\|D  
    %   unit circle.  N is a vector of positive integers (including 0), and |4 0`B% Z  
    %   M is a vector with the same number of elements as N.  Each element b2&0Hx  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) Gu\q%'I  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, bAtSVu  
    %   and THETA is a vector of angles.  R and THETA must have the same `&ckZiq  
    %   length.  The output Z is a matrix with one column for every (N,M) U#WF ;q0L  
    %   pair, and one row for every (R,THETA) pair. zue~ce73J  
    % %aVq+kC h  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike i6Emhji  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), \n|EM@=eE  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral 5uj?#)N  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, H ]Z$OpI  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized ={@6{-tl  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. JO6)-U$7UG  
    % N~zdWnSZ@G  
    %   The Zernike functions are an orthogonal basis on the unit circle. U>}w2bZ*  
    %   They are used in disciplines such as astronomy, optics, and ?QdWrE_  
    %   optometry to describe functions on a circular domain. _5Ct]vy  
    % .;`AAH'k  
    %   The following table lists the first 15 Zernike functions. a'yK~;+_9  
    % Wf>R&o6tr  
    %       n    m    Zernike function           Normalization h^(* Tv-!  
    %       -------------------------------------------------- 5(Q%XQV*P  
    %       0    0    1                                 1 ,uhb~N<  
    %       1    1    r * cos(theta)                    2 '$]97b7G  
    %       1   -1    r * sin(theta)                    2 O)n~](sC\  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) j/c&xv 7=  
    %       2    0    (2*r^2 - 1)                    sqrt(3) eF-."1  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) uo%)1NS!  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) o~y;j75{.*  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) =wV<hg)C  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) Pw`8Wj  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) R=2FNP  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) ,G?WAOy,  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) E,x+JeKV  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) YWO)HsjP  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 0)e\`Bv  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) Zaf:fsj>  
    %       -------------------------------------------------- .2Elr(&*h  
    % ?ri?GmI|  
    %   Example 1: LxSpctiNx  
    % ,Np0wg0  
    %       % Display the Zernike function Z(n=5,m=1) l'E*=Rn  
    %       x = -1:0.01:1; x}I+Iggi  
    %       [X,Y] = meshgrid(x,x); ~1AgD-:Jz  
    %       [theta,r] = cart2pol(X,Y); \aUC(K~o\;  
    %       idx = r<=1; By",rD- r  
    %       z = nan(size(X)); \\H}`0m:  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); k:#!zK}  
    %       figure 6@F9G 4<Z  
    %       pcolor(x,x,z), shading interp ;) z:fToh  
    %       axis square, colorbar Nv}=L : E  
    %       title('Zernike function Z_5^1(r,\theta)') ' ;FnIZ  
    % DGn;m\B  
    %   Example 2: Eib5  
    % a;qryUyG  
    %       % Display the first 10 Zernike functions ~#[yJNYQ  
    %       x = -1:0.01:1; i0kak`x0  
    %       [X,Y] = meshgrid(x,x); .*S#aq4S  
    %       [theta,r] = cart2pol(X,Y); ub#a`  
    %       idx = r<=1; P90yI  
    %       z = nan(size(X)); 'ud{m[|  
    %       n = [0  1  1  2  2  2  3  3  3  3]; li'YDtMKCY  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; $)ijN^hV  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; o!Ieb  
    %       y = zernfun(n,m,r(idx),theta(idx)); 6"5A%{ J  
    %       figure('Units','normalized') gpvYb7Of0  
    %       for k = 1:10 *-=(Q`3  
    %           z(idx) = y(:,k); Ls$D$/:q?  
    %           subplot(4,7,Nplot(k)) y@:h4u"3  
    %           pcolor(x,x,z), shading interp ^?7-r6  
    %           set(gca,'XTick',[],'YTick',[]) CR`Q#Yi  
    %           axis square ):68%,  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) Q4!_>YZ  
    %       end n&;85IF1  
    % 0$)>D==  
    %   See also ZERNPOL, ZERNFUN2. 6azGhxh  
    i$:*Pb3mV  
    %   Paul Fricker 11/13/2006 'qb E=  
    r?lf($ D*  
    ~hnQUS`A  
    % Check and prepare the inputs: JPc+rfF  
    % ----------------------------- 0y" $MC v  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) FxtQXu-g  
        error('zernfun:NMvectors','N and M must be vectors.') r6MMCJ|G  
    end G%AbC"  
    Yz/md1T$  
    if length(n)~=length(m) 5j<mbt}  
        error('zernfun:NMlength','N and M must be the same length.') vMi;+6'n>  
    end `iAF3:  
    6ryak!|[  
    n = n(:); dGYn4i2k?  
    m = m(:); :0j?oY~e  
    if any(mod(n-m,2)) J!v3i*j\  
        error('zernfun:NMmultiplesof2', ... hk(ZM#Bh  
              'All N and M must differ by multiples of 2 (including 0).') Pmr5S4Ka  
    end @uqd.Q  
    I {S;L  
    if any(m>n) nzuX&bSw  
        error('zernfun:MlessthanN', ... 1MP~dRZ$  
              'Each M must be less than or equal to its corresponding N.') iZ3IdiZ  
    end hYT0l$Ng  
    nA-.mWD_C  
    if any( r>1 | r<0 ) H1pO!>M  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') QuF:p  
    end \}u Y'F  
    c)TPM/>(p  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) F# ,90F'  
        error('zernfun:RTHvector','R and THETA must be vectors.') BOb">6C  
    end DkY4MH?  
    q1$N>;&  
    r = r(:); c?f4Q,%|  
    theta = theta(:); ';w#w<yaI  
    length_r = length(r); ;v)JnbsH}  
    if length_r~=length(theta) (Y.k8";)`  
        error('zernfun:RTHlength', ... (^8Y|:Tz  
              'The number of R- and THETA-values must be equal.') :j9l"5"  
    end n71r_S*  
    Xk~D$~4<  
    % Check normalization:  4C6YO  
    % -------------------- NR 5gj-B[  
    if nargin==5 && ischar(nflag) a?I= !js  
        isnorm = strcmpi(nflag,'norm'); ?/wm(uL  
        if ~isnorm Uu10)/.LC  
            error('zernfun:normalization','Unrecognized normalization flag.') \+oQd=K@  
        end EA@ .,7F  
    else ?Ny9'g>?  
        isnorm = false; GfxZ'VIn  
    end $-OA'QwB]  
    >a!/QMh  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Thp[+KP>  
    % Compute the Zernike Polynomials aD<A.Lhy  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |sJ[0z  
    :)-Sk$  
    % Determine the required powers of r: 9tU]`f  
    % ----------------------------------- d\&U*=  
    m_abs = abs(m); n$MO4s8)  
    rpowers = []; `&r+F/Ap2  
    for j = 1:length(n) SB;&GHq"n  
        rpowers = [rpowers m_abs(j):2:n(j)]; YiXk5B0Uh  
    end 7Kr*P<-G  
    rpowers = unique(rpowers); j"t(0 m  
    |{z:IQLv  
    % Pre-compute the values of r raised to the required powers, a5dLQx b  
    % and compile them in a matrix: 4qb/da E:Z  
    % ----------------------------- (+w*[qHe  
    if rpowers(1)==0 & TCkpS  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 1jmjg~W  
        rpowern = cat(2,rpowern{:}); lZd(emH@  
        rpowern = [ones(length_r,1) rpowern]; 9a[9i}_  
    else yJ[0WY8<kC  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); A]_7}<<N  
        rpowern = cat(2,rpowern{:}); 2 ~dE<}  
    end b <tNk]7  
    /7(W?xOe  
    % Compute the values of the polynomials: !4ocZmj\  
    % -------------------------------------- ]}>2D,;  
    y = zeros(length_r,length(n)); z 4e7PW|  
    for j = 1:length(n) vz@A;t  
        s = 0:(n(j)-m_abs(j))/2; <v"R.<  
        pows = n(j):-2:m_abs(j); nQF(vTDN  
        for k = length(s):-1:1 J@/kIrx  
            p = (1-2*mod(s(k),2))* ... ")1:F>  
                       prod(2:(n(j)-s(k)))/              ... vSGH[nyCY  
                       prod(2:s(k))/                     ... @JiLgIe `  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... H9Gh>u]}  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); PF0_8,@U  
            idx = (pows(k)==rpowers); [CTnXb  
            y(:,j) = y(:,j) + p*rpowern(:,idx); F;Spi  
        end :;v~%e{k  
         ^7`BP%6  
        if isnorm xBj 9y u  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); dUD[e,?  
        end 4V"E8rUL(  
    end lwR<(u31e  
    % END: Compute the Zernike Polynomials 7RQR)DG  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Ky`qskvu  
    ;_XFo&@  
    % Compute the Zernike functions: 1;* cq  
    % ------------------------------ a)!o @  
    idx_pos = m>0; av(6wht8  
    idx_neg = m<0; j\ZXG=j  
    f'F?MINJP  
    z = y; +Z,;,5'5G  
    if any(idx_pos) x o;QCOH  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); *j|~$e}C  
    end 9v#CE!  
    if any(idx_neg) H[T?\Lq  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); t"sBPLU\  
    end Q1lyj7c#x  
    XjBW9a  
    % 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