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 Ip>^O/}$1
function z = zernfun(n,m,r,theta,nflag) DeA @0HOxh
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. :OHSxb>[
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N DWuRJ
% and angular frequency M, evaluated at positions (R,THETA) on the ]a)IMIh;
% unit circle. N is a vector of positive integers (including 0), and 0HjJaML
% M is a vector with the same number of elements as N. Each element ,MRvuw0P
% k of M must be a positive integer, with possible values M(k) = -N(k) @|^jq
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, ]yo_wGiwY
% and THETA is a vector of angles. R and THETA must have the same (%i!%{!]
% length. The output Z is a matrix with one column for every (N,M) B9wp*:.
% pair, and one row for every (R,THETA) pair. fzl=d_
% K~USK?Q%
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike NzAQ@E2d:
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), P!5Z]+B#
% with delta(m,0) the Kronecker delta, is chosen so that the integral %Hh3u$Y,
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 1sD~7KPg?
% and theta=0 to theta=2*pi) is unity. For the non-normalized 8AryIgy>@
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. j?( c}!}
% Bgf=\7;5
% The Zernike functions are an orthogonal basis on the unit circle. C+`xx('N9
% They are used in disciplines such as astronomy, optics, and Y7-*2"!
% optometry to describe functions on a circular domain. T\jAk+$Jo
% C}9Kx }q
% The following table lists the first 15 Zernike functions. @2u#93Y
% }0Y`|H\v
% n m Zernike function Normalization 2@fa
rx:
% -------------------------------------------------- _y>}#6B
% 0 0 1 1 =w6}\ 'X
% 1 1 r * cos(theta) 2 q=njKC
% 1 -1 r * sin(theta) 2 au}s=ua~i
% 2 -2 r^2 * cos(2*theta) sqrt(6) Ym'7vW#~
% 2 0 (2*r^2 - 1) sqrt(3) +uELTHH=
% 2 2 r^2 * sin(2*theta) sqrt(6) xLZ bU4
% 3 -3 r^3 * cos(3*theta) sqrt(8) w m19T7*L
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) =C#*!N73
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) ":V%(c
% 3 3 r^3 * sin(3*theta) sqrt(8) ?;_H{/)m
% 4 -4 r^4 * cos(4*theta) sqrt(10) |<1M&\oaQ'
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) e^=NL>V6p
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) X CzXS.
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) bGu([VB
% 4 4 r^4 * sin(4*theta) sqrt(10) 5E`JD
% -------------------------------------------------- /,X7.t_-
% SAy{YOLtl
% Example 1: o ~;M"
% 4j^bpfb,
% % Display the Zernike function Z(n=5,m=1) N2T&,&,t
% x = -1:0.01:1; J]dW1boT@
% [X,Y] = meshgrid(x,x); /w0w*nH
% [theta,r] = cart2pol(X,Y); .D!WO
% idx = r<=1; <}cZi4l'
% z = nan(size(X)); -8/ JP
% z(idx) = zernfun(5,1,r(idx),theta(idx)); FJ}gUs{m
% figure \ZsP]};*
% pcolor(x,x,z), shading interp eKyqU9
% axis square, colorbar
^iuo^2+
% title('Zernike function Z_5^1(r,\theta)') 7C?E z%a@
% *y?[<2"$
% Example 2: t| _{;!^
% mVt3WZa
% % Display the first 10 Zernike functions ?;_O
9
% x = -1:0.01:1; K_Re}\D
% [X,Y] = meshgrid(x,x); <