切换到宽版
  • 广告投放
  • 稿件投递
  • 繁體中文
    • 10438阅读
    • 9回复

    [求助]ansys分析后面型数据如何进行zernike多项式拟合? [复制链接]

    上一主题 下一主题
    离线niuhelen
     
    发帖
    19
    光币
    28
    光券
    0
    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 ~>HzAo9e  
    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! Rw|'LaW  
     
    分享到
    离线phoenixzqy
    发帖
    4352
    光币
    8426
    光券
    1
    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)  t/\   
    6e%@uB}$  
    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。 jYFJk&c  
    RqtBz3v  
    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
    发帖
    850
    光币
    833
    光券
    0
    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
    发帖
    59
    光币
    0
    光券
    0
    只看该作者 7楼 发表于: 2012-09-28
    我也正在找啊
    离线niuhelen
    发帖
    19
    光币
    28
    光券
    0
    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
    发帖
    19
    光币
    28
    光券
    0
    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) T/_u;My;  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. H9mNnZ_k  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of {821e&r  
    %   order N and frequency M, evaluated at R.  N is a vector of 'J,UKK\5  
    %   positive integers (including 0), and M is a vector with the g8<ODU0[g  
    %   same number of elements as N.  Each element k of M must be a cx\E40WD  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) )9YDNVo*-  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is g:o/^_  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix * Gg7(cnpw  
    %   with one column for every (N,M) pair, and one row for every OS(`H5D  
    %   element in R. y, l[v39  
    % AxH;psj  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- .xT?%xSi/  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is I-]G{  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to hX.cdt_?  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 uY]';Ot G  
    %   for all [n,m]. \p4*Q}t  
    % *k{Llq  
    %   The radial Zernike polynomials are the radial portion of the OrkcY39"~a  
    %   Zernike functions, which are an orthogonal basis on the unit h4hAzFQ.s  
    %   circle.  The series representation of the radial Zernike aTvyz r1  
    %   polynomials is )Te\6qM  
    % <Wn~s=  
    %          (n-m)/2 o?baiOkH  
    %            __ 7{#p'.nc5  
    %    m      \       s                                          n-2s 2{ F-@}=  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r xV> .]  
    %    n      s=0 [k7( t|Q{  
    % ;jzJ6~<  
    %   The following table shows the first 12 polynomials. iC#a+G*N_M  
    % La ?A@SD  
    %       n    m    Zernike polynomial    Normalization n!4}Hwz!  
    %       --------------------------------------------- o?a2wY^_  
    %       0    0    1                        sqrt(2) b] 5dBZ(  
    %       1    1    r                           2 -'&l!23a~  
    %       2    0    2*r^2 - 1                sqrt(6) {!I`EN]  
    %       2    2    r^2                      sqrt(6) $bE" 3/uf  
    %       3    1    3*r^3 - 2*r              sqrt(8) .x=abA$!9  
    %       3    3    r^3                      sqrt(8) f7&ni#^Ztj  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) 4@{;z4*`  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) {]IY; cL  
    %       4    4    r^4                      sqrt(10) mS%4  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) H k}P  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) Ftyxz&-4$p  
    %       5    5    r^5                      sqrt(12) -RP{viG WK  
    %       --------------------------------------------- Z\0wQ;}  
    % qsj$u-xhX  
    %   Example: K3zY-yIco  
    % @1j*\gYz  
    %       % Display three example Zernike radial polynomials @WazSL;N  
    %       r = 0:0.01:1; eEqcAUn  
    %       n = [3 2 5]; 9O- otAGM  
    %       m = [1 2 1]; 6nA9r5Ghv  
    %       z = zernpol(n,m,r); _N5pxe`  
    %       figure dw6ysOR@  
    %       plot(r,z) ,zjz "7'  
    %       grid on gbdzS6XW~  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') PcsYy]Q/  
    % u&*[   
    %   See also ZERNFUN, ZERNFUN2. R*6TS"aL  
    5%TSUU+<I  
    % A note on the algorithm. N1Y uLG:  
    % ------------------------ 5B~]%_gZr  
    % The radial Zernike polynomials are computed using the series nzbVI  
    % representation shown in the Help section above. For many special ^2 dQVV.  
    % functions, direct evaluation using the series representation can  D?Beg F  
    % produce poor numerical results (floating point errors), because P*k n}:  
    % the summation often involves computing small differences between e\}@w1  
    % large successive terms in the series. (In such cases, the functions kiF}+,z"  
    % are often evaluated using alternative methods such as recurrence O C;~ H{  
    % relations: see the Legendre functions, for example). For the Zernike OTYkJEC8\N  
    % polynomials, however, this problem does not arise, because the 5]G%MB/|$  
    % polynomials are evaluated over the finite domain r = (0,1), and tO&n$$  
    % because the coefficients for a given polynomial are generally all d\-*Fmp(S  
    % of similar magnitude. 6 (7 56  
    % %Ja0:e  
    % ZERNPOL has been written using a vectorized implementation: multiple  c{kpg N  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] blomB2vQ  
    % values can be passed as inputs) for a vector of points R.  To achieve p63fpnH  
    % this vectorization most efficiently, the algorithm in ZERNPOL $JOtUB{  
    % involves pre-determining all the powers p of R that are required to {JdXn  
    % compute the outputs, and then compiling the {R^p} into a single $$$[Vn_H<  
    % matrix.  This avoids any redundant computation of the R^p, and dOaOWMrfdf  
    % minimizes the sizes of certain intermediate variables. |7 K>`  
    % `j {q  
    %   Paul Fricker 11/13/2006 *QN,w BQ  
    xsU%?"r  
    +6:  
    % Check and prepare the inputs: a,fcKe&B  
    % ----------------------------- J<0sT=/2$  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) d v4~CW%Td  
        error('zernpol:NMvectors','N and M must be vectors.') D<70rBf2  
    end C\{ KB@C\*  
    H{*rV>%  
    if length(n)~=length(m) jcC"vr'u|  
        error('zernpol:NMlength','N and M must be the same length.') SP<(24zdd  
    end {.U:Ce  
    X6}W]  
    n = n(:); o]I8Ghk>/z  
    m = m(:); I@qGDKz;  
    length_n = length(n); qQf NT.  
    JS03B Itt  
    if any(mod(n-m,2)) O=LW[h!  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') le_a IbB"P  
    end l_;6xkv4  
    u[SqZftmO  
    if any(m<0) ;wJe%Nw?  
        error('zernpol:Mpositive','All M must be positive.') -F(luRBS(W  
    end 7'At_oG  
    /)RH-_63  
    if any(m>n) e1b?TF@lz  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') 0i5S=L`j  
    end u)zv`m  
    `'3&tAy  
    if any( r>1 | r<0 ) xVYa-I[Z  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') !ni 1 qM  
    end GwA\>qXw  
    #I MaN%  
    if ~any(size(r)==1) : &nF>  
        error('zernpol:Rvector','R must be a vector.') |Ch ,C  
    end amExZ/  
    3_9CREZCl  
    r = r(:); HNc/p4z  
    length_r = length(r); O46v  
    _PGd\>Ve  
    if nargin==4 UlNiH  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); J8@.qC'!  
        if ~isnorm [zq2h3r  
            error('zernpol:normalization','Unrecognized normalization flag.') = [: E  
        end kVCWyZh4  
    else _Wk*h}x  
        isnorm = false; -ON-0L  
    end FSz<R*2  
    ;"#yHP`  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% #Muh|P]%\  
    % Compute the Zernike Polynomials RO3q!+a$/  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ZI4dD.B  
    raSga'uT;  
    % Determine the required powers of r: 0;)Q  
    % ----------------------------------- .x] pJ9  
    rpowers = []; W2v'2qAs  
    for j = 1:length(n) x)Zm5&"Gg  
        rpowers = [rpowers m(j):2:n(j)]; F?jD5M08t/  
    end jAcKSx$}y"  
    rpowers = unique(rpowers); R i,_x  
    KJ S-{ed  
    % Pre-compute the values of r raised to the required powers, x![.C,O  
    % and compile them in a matrix: N^wHO<IO 1  
    % ----------------------------- 9@IL547V  
    if rpowers(1)==0 %CnNu  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); z Fj|E  
        rpowern = cat(2,rpowern{:}); }CZw'fhVWO  
        rpowern = [ones(length_r,1) rpowern]; A)j!Wgs^z  
    else ;/pI@C k  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); cX4]ViXSr  
        rpowern = cat(2,rpowern{:}); Z]tQmV8e  
    end ]9 _}S  
    ?*xH HI/  
    % Compute the values of the polynomials: Y-st2r[,  
    % -------------------------------------- 5}w   
    z = zeros(length_r,length_n); 3` oOoKX  
    for j = 1:length_n _Yp~Oj  
        s = 0:(n(j)-m(j))/2; ]xoG{%vgb  
        pows = n(j):-2:m(j); XjP;O,x  
        for k = length(s):-1:1  f}*:wj  
            p = (1-2*mod(s(k),2))* ... SsZSR.tD  
                       prod(2:(n(j)-s(k)))/          ... '3sySsD&O  
                       prod(2:s(k))/                 ... .m\0<8C  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... KZ#\ >  
                       prod(2:((n(j)+m(j))/2-s(k))); dS <*DP  
            idx = (pows(k)==rpowers); b5Q>e%i#  
            z(:,j) = z(:,j) + p*rpowern(:,idx); k.c.7%|~;  
        end d:^B2~j  
         Z^'\()3t  
        if isnorm TXyiCS3  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); W[j, QU  
        end GP %hf{  
    end gJ9"$fIPc  
    v4'kV:;&  
    % EOF zernpol
    离线niuhelen
    发帖
    19
    光币
    28
    光券
    0
    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) dkCU U  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. eMFxdtH  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated @@{5]Y  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive J>nBTY,_<  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, ,5ZQPICF  
    %   and THETA is a vector of angles.  R and THETA must have the same q-_!&kDK"  
    %   length.  The output Z is a matrix with one column for every P-value, NV9JMB{q  
    %   and one row for every (R,THETA) pair. Z(Y:  
    % h4F%lGot  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike b l+g7g;  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) y35~bz^2  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) 7[u>#8  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 ^i!6z2/  
    %   for all p. u-4@[*^T$  
    % !3mt<i]a"  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 Myiv#rQ)  
    %   Zernike functions (order N<=7).  In some disciplines it is A%$~  
    %   traditional to label the first 36 functions using a single mode S >CKm:7  
    %   number P instead of separate numbers for the order N and azimuthal w( XZSE  
    %   frequency M. +0UBP7kn  
    % G\;6n  
    %   Example: ^IIy>  
    % >-<iY4|[d  
    %       % Display the first 16 Zernike functions 1TGRIe)  
    %       x = -1:0.01:1;  <9yh:1"X  
    %       [X,Y] = meshgrid(x,x); 1,bE[_  
    %       [theta,r] = cart2pol(X,Y); [?KGLUmTAI  
    %       idx = r<=1; "UNFB3  
    %       p = 0:15; pb)8?1O|s  
    %       z = nan(size(X)); SZHgXl3:  
    %       y = zernfun2(p,r(idx),theta(idx)); b"N!#&O]  
    %       figure('Units','normalized') S**eI<QFSk  
    %       for k = 1:length(p) *tEqu%N1'  
    %           z(idx) = y(:,k); ^ W?cuJ8  
    %           subplot(4,4,k) "z3rH~q72  
    %           pcolor(x,x,z), shading interp qa )BbK^i  
    %           set(gca,'XTick',[],'YTick',[]) _(kaaWJ  
    %           axis square PzNPwd  
    %           title(['Z_{' num2str(p(k)) '}']) NE8W--Cg|  
    %       end Ihf :k_;  
    % Jut&J]{h  
    %   See also ZERNPOL, ZERNFUN. E8}evi  
    (A6~mi r!  
    %   Paul Fricker 11/13/2006 /kkUEo+  
    Z CPUNtOl  
    Q zaD\^OF  
    % Check and prepare the inputs: ^9q#,6  
    % ----------------------------- '=fk;AiQ  
    if min(size(p))~=1 Op ?"G  
        error('zernfun2:Pvector','Input P must be vector.') B<m0YD?>~>  
    end BrwC9:  
    x}?<9(nE c  
    if any(p)>35 3+>n!8x ;A  
        error('zernfun2:P36', ... ;t\h"K<,|  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... p,$N-22a  
               '(P = 0 to 35).']) &.*UVc2+Y  
    end tMiIlf!>p  
    K=v:qY4Z  
    % Get the order and frequency corresonding to the function number: !!^z6jpvn  
    % ---------------------------------------------------------------- =ZIT!B?4  
    p = p(:); AT~,  
    n = ceil((-3+sqrt(9+8*p))/2); &o;0%QgF  
    m = 2*p - n.*(n+2); !ou#g5Q@z  
    _2hLc\#  
    % Pass the inputs to the function ZERNFUN: @>(KEjQTz  
    % ---------------------------------------- bhpku=ov  
    switch nargin $?0ch15/  
        case 3 'YNdrvz  
            z = zernfun(n,m,r,theta); +ZOiL[rS  
        case 4 Jd>~gA}l  
            z = zernfun(n,m,r,theta,nflag); w#9Kt W,tt  
        otherwise PWpt\g  
            error('zernfun2:nargin','Incorrect number of inputs.') <GNLDpj  
    end ^}d]O(  
    2(xC|  
    % EOF zernfun2
    离线niuhelen
    发帖
    19
    光币
    28
    光券
    0
    只看该作者 3楼 发表于: 2011-03-12
    回 2楼(phility) 的帖子
    非常感谢啊,我手上也有zernike多项式的拟合的源程序,也不知道对不对,不怎么会有 p?:5 U[KM  
    function z = zernfun(n,m,r,theta,nflag) )S)L9('IxT  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. =#b@7Yw:  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N F3qK6Ah.  
    %   and angular frequency M, evaluated at positions (R,THETA) on the \WCQ>c?~  
    %   unit circle.  N is a vector of positive integers (including 0), and WVbrbs4  
    %   M is a vector with the same number of elements as N.  Each element L8QWEFB|  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) @Iv;y*y  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, G21o @38e  
    %   and THETA is a vector of angles.  R and THETA must have the same  .w9LJ  
    %   length.  The output Z is a matrix with one column for every (N,M) HgF;[rq3Q  
    %   pair, and one row for every (R,THETA) pair. K T}  
    % JQ&t"`\k  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike +$,Re.WnP  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), %t9C  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral LwH#|8F  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 'x!\pE-  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized m|@H`=`d  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. Z'JS@dV  
    % @p+;iS1}  
    %   The Zernike functions are an orthogonal basis on the unit circle. ~7P)$[  
    %   They are used in disciplines such as astronomy, optics, and ?['!0PF  
    %   optometry to describe functions on a circular domain. Tu#< {'1$  
    % ):D"L C  
    %   The following table lists the first 15 Zernike functions. =Ph8&l7~sp  
    % Sjpx G@k  
    %       n    m    Zernike function           Normalization p T(M>LP83  
    %       -------------------------------------------------- HGDrH   
    %       0    0    1                                 1 e#(Ck{e  
    %       1    1    r * cos(theta)                    2 >Jz9wo`  
    %       1   -1    r * sin(theta)                    2 :HkBP90o  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) 7RAB"T;?Q  
    %       2    0    (2*r^2 - 1)                    sqrt(3) 5'~_d@M  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) dj0; tQ=C  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) kmI0V[Y  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) 7F^d-  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) (yOkf-e2y  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) <uH8Fivb  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) }]?Si6_ZZ  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Ifj&S'():  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) kSQ8kU_w+  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) <B"sp r&1  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) [VCC+_  
    %       -------------------------------------------------- rH+OXGoB  
    % c7Z4u|G  
    %   Example 1: _FLEz|%~  
    % hRcb}>pr  
    %       % Display the Zernike function Z(n=5,m=1) o`?rj!\  
    %       x = -1:0.01:1; S&op|Z)1  
    %       [X,Y] = meshgrid(x,x); l\HdB"nT  
    %       [theta,r] = cart2pol(X,Y); _"DS?`z6  
    %       idx = r<=1; I5$P9UE+^9  
    %       z = nan(size(X)); Nk`UQ~g$  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); uy'ghF  
    %       figure `2~>$Tr  
    %       pcolor(x,x,z), shading interp W .7rHa  
    %       axis square, colorbar kX 1}/l  
    %       title('Zernike function Z_5^1(r,\theta)') 5\-uo&#  
    % OW:*qY c;:  
    %   Example 2: f& 4_:'-,  
    % !liV Y]  
    %       % Display the first 10 Zernike functions PxHFH pL  
    %       x = -1:0.01:1; vh9* >[i  
    %       [X,Y] = meshgrid(x,x); W L$^B@gXQ  
    %       [theta,r] = cart2pol(X,Y); XC4Z,,ah"  
    %       idx = r<=1; K~x,so  
    %       z = nan(size(X)); G1"iu8 9d  
    %       n = [0  1  1  2  2  2  3  3  3  3]; ,b+NhxdZ  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; VSj!Gm0LB  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; B~Q-V&@o  
    %       y = zernfun(n,m,r(idx),theta(idx)); 5sD,gZ7  
    %       figure('Units','normalized') "(koR Q  
    %       for k = 1:10 Y4T")  
    %           z(idx) = y(:,k); 'm%{Rz>j  
    %           subplot(4,7,Nplot(k)) WA{igj@\  
    %           pcolor(x,x,z), shading interp GN.O a$  
    %           set(gca,'XTick',[],'YTick',[]) A]1Nm3@  
    %           axis square $ |4C]Me (  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) zd?@xno  
    %       end ZS Med(//b  
    % D)_ C@*q  
    %   See also ZERNPOL, ZERNFUN2. ;`9f<d#\  
    ,!ZuH?Z  
    %   Paul Fricker 11/13/2006 rCyb3,W  
    R+sT &d  
    ajbe7#}  
    % Check and prepare the inputs: HDyf]2N*N  
    % ----------------------------- od;-D~  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) :fRXLe1=  
        error('zernfun:NMvectors','N and M must be vectors.') fSh5u/F!  
    end JFq wC=-  
    `h}eP[jA  
    if length(n)~=length(m) ? @V R%z  
        error('zernfun:NMlength','N and M must be the same length.') $o6/dEKQ  
    end Iw1Y?Qia  
    @WJ;T= L  
    n = n(:); BT_]=\zi  
    m = m(:); -F[8 ZiZ  
    if any(mod(n-m,2)) N&8TG  
        error('zernfun:NMmultiplesof2', ... KuNLu31%  
              'All N and M must differ by multiples of 2 (including 0).') r^9l/H~ $  
    end g14*6O:  
    t@RYJmW  
    if any(m>n) ,RP-)j"Wff  
        error('zernfun:MlessthanN', ... R^Rc!G}  
              'Each M must be less than or equal to its corresponding N.') c=\tf~}^Ms  
    end ^Fk;t  
    [ X*p [  
    if any( r>1 | r<0 ) 6*8Wtq  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') LvG.ocCG  
    end ,,3lH-C  
    9#LMK 1ge  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) sW^M  ]  
        error('zernfun:RTHvector','R and THETA must be vectors.') U}v`~' K  
    end 337.' |ZE  
    P -m_],  
    r = r(:); rK~-Wzwu  
    theta = theta(:); {+N< 9(O  
    length_r = length(r); (1/Sf&2i  
    if length_r~=length(theta) M 8^ID #  
        error('zernfun:RTHlength', ... 3 qYGEhxv  
              'The number of R- and THETA-values must be equal.') "EW8ll7r  
    end FOaA}D `]  
    ~cz}C("Z  
    % Check normalization: [hJ1]RW8  
    % -------------------- s8j |>R|k  
    if nargin==5 && ischar(nflag) `At.$3B  
        isnorm = strcmpi(nflag,'norm'); n}p G&&;q  
        if ~isnorm x"r0<RK  
            error('zernfun:normalization','Unrecognized normalization flag.') *MJm:  
        end b#2)"V(  
    else ",r v%i2 f  
        isnorm = false; F?L]Dff  
    end !0 7jr%-~  
    $ m`Dyu  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% G,8mFH  
    % Compute the Zernike Polynomials dg D-"-O  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Sn:>|y~  
    ;W|kc</R*  
    % Determine the required powers of r: [V}vd@*k  
    % ----------------------------------- T/$ gnn  
    m_abs = abs(m); 0 9H rn  
    rpowers = []; l;F"m+B!$  
    for j = 1:length(n) tldT(E6  
        rpowers = [rpowers m_abs(j):2:n(j)]; ]l7W5$26 @  
    end +]l?JKV  
    rpowers = unique(rpowers); YOxgpQ:i  
    q|5WHB  
    % Pre-compute the values of r raised to the required powers, SH*'<  
    % and compile them in a matrix: 9~ p;iiKGG  
    % ----------------------------- vE]ge  
    if rpowers(1)==0 ;H$ Cq' I  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ;R$G.5h  
        rpowern = cat(2,rpowern{:}); " <bjS  
        rpowern = [ones(length_r,1) rpowern]; h'ik3mLH  
    else +'H[4g`  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); H4i}gdR  
        rpowern = cat(2,rpowern{:}); Km2~nkQ  
    end N=mvr&arP  
    3=S |U,  
    % Compute the values of the polynomials: tpI/I bq  
    % -------------------------------------- ]dycesc'  
    y = zeros(length_r,length(n)); gx\V)8Zr  
    for j = 1:length(n) }OkzP)(  
        s = 0:(n(j)-m_abs(j))/2; YznL+TD  
        pows = n(j):-2:m_abs(j); 3mgvWR  
        for k = length(s):-1:1 -]%EX:bm  
            p = (1-2*mod(s(k),2))* ... -e_fn&2,Y  
                       prod(2:(n(j)-s(k)))/              ... u5CSx'h]  
                       prod(2:s(k))/                     ... O:]']' /  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... $G=^cNB|JB  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); z$<=8ox8e  
            idx = (pows(k)==rpowers); He_O+[sc  
            y(:,j) = y(:,j) + p*rpowern(:,idx); ]t[%.^5#  
        end mQj#\<*  
         #y\O+\4e  
        if isnorm QW..=}pL  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); ,7nu;fOT[  
        end >iyNZ]."\  
    end g}9 ,U&$]y  
    % END: Compute the Zernike Polynomials ~|u;z,\  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% W83d$4\d  
    4DIU7#GG  
    % Compute the Zernike functions: HoBx0N9\2  
    % ------------------------------ <?7CwW  
    idx_pos = m>0; tbQY&TO1  
    idx_neg = m<0; GEPWb[Oa  
    COi15( G2  
    z = y; F'~r?D  
    if any(idx_pos) kn#?+Q  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 3P_.SF  
    end s:<y\1Ay  
    if any(idx_neg) ?M90K)&g{  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 4Q+,_iP  
    end eKP >} `  
    za>%hZf\  
    % EOF zernfun
    离线phility
    发帖
    69
    光币
    11
    光券
    0
    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phility
    发帖
    69
    光币
    11
    光券
    0
    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