| 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 2o\GU function z = zernfun(n,m,r,theta,nflag) uQYBq)p| %ZERNFUN Zernike functions of order N and frequency M on the unit circle. `"#0\Wh % Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N Bp.z6x4 % and angular frequency M, evaluated at positions (R,THETA) on the <"8<< % unit circle. N is a vector of positive integers (including 0), and m$U rY(6d
% M is a vector with the same number of elements as N. Each element #SR"Q`P % k of M must be a positive integer, with possible values M(k) = -N(k) \i +=tGY % to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, R[/]iK+!& % and THETA is a vector of angles. R and THETA must have the same k\~A\UIYo % length. The output Z is a matrix with one column for every (N,M) &M6cCT]&M % pair, and one row for every (R,THETA) pair. :6
\?{xD % -H;%1y$A- % Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike -nvK*rn>} % functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), qUMM}ls % with delta(m,0) the Kronecker delta, is chosen so that the integral oV7A"8L^a % of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, )t/[z3rn % and theta=0 to theta=2*pi) is unity. For the non-normalized | gou#zi % polynomials, max(Znm(r=1,theta))=1 for all [n,m]. P!Mz5QZ+ % B 3Yj % The Zernike functions are an orthogonal basis on the unit circle. CLU[')H0 % They are used in disciplines such as astronomy, optics, and !{L6
4qI % optometry to describe functions on a circular domain. lYz$~/sd % NyJ=^=F# % The following table lists the first 15 Zernike functions. >;ucwLi % j+p=ik % n m Zernike function Normalization XP$ 1CWI % -------------------------------------------------- lk5}bnd5 % 0 0 1 1 &;)6G1X1 % 1 1 r * cos(theta) 2 wF`9}9q % 1 -1 r * sin(theta) 2 _DAAD,'<a % 2 -2 r^2 * cos(2*theta) sqrt(6) [P*w$Hn % 2 0 (2*r^2 - 1) sqrt(3) 6
s+ Z % 2 2 r^2 * sin(2*theta) sqrt(6) L'>t:^QTh % 3 -3 r^3 * cos(3*theta) sqrt(8) `B^HW8 % 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) 54A ndyeA % 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) 8")1, % 3 3 r^3 * sin(3*theta) sqrt(8) L%7?o: % 4 -4 r^4 * cos(4*theta) sqrt(10) h.\9a3B:r % 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) mST/u>' % 4 0 6*r^4 - 6*r^2 + 1 sqrt(5)
igV4nL % 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) #hBDOXHPf % 4 4 r^4 * sin(4*theta) sqrt(10) ^c9~~m16+ % -------------------------------------------------- \\qw"w9 % sf|[oD % Example 1: "~f=7
% tcg sXB/t % % Display the Zernike function Z(n=5,m=1) D [#1~M % x = -1:0.01:1; =;1MpD % [X,Y] = meshgrid(x,x); XZaei\rUn) % [theta,r] = cart2pol(X,Y); JvHGu&Nr! % idx = r<=1; 4Qr16,Us % z = nan(size(X)); =9oN#4mWK % z(idx) = zernfun(5,1,r(idx),theta(idx)); $=j}JX}z % figure C?Sy90f % pcolor(x,x,z), shading interp j}=$2|}8{ % axis square, colorbar N[~"X**x % title('Zernike function Z_5^1(r,\theta)') h5bQ % |zV-a2K%J % Example 2: K4vl#*qn % lW,rzJ1 % % Display the first 10 Zernike functions Y%UfwbX!g % x = -1:0.01:1; eeuTf % [X,Y] = meshgrid(x,x); H\f.a R= % [theta,r] = cart2pol(X,Y); ]F@XGJN % idx = r<=1; \advFKN % z = nan(size(X)); Y9TaU]7] % n = [0 1 1 2 2 2 3 3 3 3]; Z[baQO % m = [0 -1 1 -2 0 2 -3 -1 1 3]; ;[-dth % Nplot = [4 10 12 16 18 20 22 24 26 28]; FuNc#n> % y = zernfun(n,m,r(idx),theta(idx)); m~fA=#l
l % figure('Units','normalized') _u^ S[ % for k = 1:10 1{oq8LB % z(idx) = y(:,k); Y5~_y?BX % subplot(4,7,Nplot(k)) s|U=_,. % pcolor(x,x,z), shading interp +2kJuoj: % set(gca,'XTick',[],'YTick',[]) o;XzJ#P % axis square 4VjP:>*p % title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) b!Q|0X.? % end IYq)p
/ % ZJ9J*5!C % See also ZERNPOL, ZERNFUN2. ]q0mo1-EZ! V`V
Z[ % Paul Fricker 11/13/2006 sXm/+I^ 6@-VLO))O Y" & | |