| niuhelen |
2011-03-12 23:00 |
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 4&`66\p; function z = zernfun(n,m,r,theta,nflag) {P = {) %ZERNFUN Zernike functions of order N and frequency M on the unit circle. <v5toyA % Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N [Q^kO; % and angular frequency M, evaluated at positions (R,THETA) on the ]JhtO{ % unit circle. N is a vector of positive integers (including 0), and U*6-Y%7 % M is a vector with the same number of elements as N. Each element );,#H`' % k of M must be a positive integer, with possible values M(k) = -N(k) 4)XN1r: % to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, jhg!K.A % and THETA is a vector of angles. R and THETA must have the same LO` (V % length. The output Z is a matrix with one column for every (N,M)
;$|+H"g| % pair, and one row for every (R,THETA) pair. ytAhhwN~ % ~zRW*pd % Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike qqkZbsN % functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), d628@~Ekn % with delta(m,0) the Kronecker delta, is chosen so that the integral R[_7ab]A % of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, oh:t ex< % and theta=0 to theta=2*pi) is unity. For the non-normalized 9V"^F.> % polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 4|XE
f, % @aj"12 % The Zernike functions are an orthogonal basis on the unit circle. 2;kab^iv' % They are used in disciplines such as astronomy, optics, and m6IZGl7% % optometry to describe functions on a circular domain. XeZv%` ? % F htf4 % The following table lists the first 15 Zernike functions. 7Y!^88,f. % <-lz_ % n m Zernike function Normalization <BO|.(ys % -------------------------------------------------- 'z!I#Y!Y % 0 0 1 1 u 6%56 %^f % 1 1 r * cos(theta) 2 *nH ?o* # % 1 -1 r * sin(theta) 2 _~_Hup % 2 -2 r^2 * cos(2*theta) sqrt(6) 8fDnDA.e % 2 0 (2*r^2 - 1) sqrt(3) S++}kR);
% 2 2 r^2 * sin(2*theta) sqrt(6) R'9TD=qEK % 3 -3 r^3 * cos(3*theta) sqrt(8) #z5'5|3 % 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) NtA}I)'SWU % 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) i\`[0dfY % 3 3 r^3 * sin(3*theta) sqrt(8) J@R+t6$3O % 4 -4 r^4 * cos(4*theta) sqrt(10) @&2T0UB % 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Kh5:+n_X % 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) Rf8|-G-}# % 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) DU[UGJg % 4 4 r^4 * sin(4*theta) sqrt(10)
-6 % -------------------------------------------------- 4}NFa;M1 %
o>W}1_ % Example 1: x^C,xP[#Y; % ]jy6C'Mp % % Display the Zernike function Z(n=5,m=1) #s]]\ % x = -1:0.01:1; k_y@vW3 % [X,Y] = meshgrid(x,x); =e ;\I/ % [theta,r] = cart2pol(X,Y); \!631FcQ % idx = r<=1; TuX#;!p6 % z = nan(size(X)); 6*]Kow? % z(idx) = zernfun(5,1,r(idx),theta(idx)); zlXkD~GV % figure t[^}/
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