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 ^{yb4yQ
0
function z = zernfun(n,m,r,theta,nflag) "e\73?P
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. wMF1HT<*
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N Brg0: 5H
% and angular frequency M, evaluated at positions (R,THETA) on the < :eKXH2
% unit circle. N is a vector of positive integers (including 0), and aAoAjV NkK
% M is a vector with the same number of elements as N. Each element Gg6cjc =dC
% k of M must be a positive integer, with possible values M(k) = -N(k) 2mj>,kS?c
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, gDfM} 2]/
% and THETA is a vector of angles. R and THETA must have the same 6"?#s/fk
% length. The output Z is a matrix with one column for every (N,M) #9"lL1
% pair, and one row for every (R,THETA) pair.
KYcc jX
% @AG=Eq9<o
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ) tV]h#4
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), O{]}{Ss
% with delta(m,0) the Kronecker delta, is chosen so that the integral 0~<t :q!
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, (#je0ES
% and theta=0 to theta=2*pi) is unity. For the non-normalized 'uUa|J1mu
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ioTqT:.
% k3OnvnJb
% The Zernike functions are an orthogonal basis on the unit circle. E.VEW;=
% They are used in disciplines such as astronomy, optics, and &#q%#M:
% optometry to describe functions on a circular domain. /$vX1T
% )Knsy
% The following table lists the first 15 Zernike functions. g5Hsz,x
% OZObx
% n m Zernike function Normalization d9
8pv%
% -------------------------------------------------- &:+_{nc,
% 0 0 1 1 T?__
% 1 1 r * cos(theta) 2 =g@hh)3wP
% 1 -1 r * sin(theta) 2 A]V<K[9:b
% 2 -2 r^2 * cos(2*theta) sqrt(6) AQ.q?'vE)
% 2 0 (2*r^2 - 1) sqrt(3) 4f0dc\$
% 2 2 r^2 * sin(2*theta) sqrt(6) f'Xz4;
% 3 -3 r^3 * cos(3*theta) sqrt(8) DUm/0q&
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) 1^;&?E
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) \v9<L'NP)
% 3 3 r^3 * sin(3*theta) sqrt(8) ~>$(5s2
% 4 -4 r^4 * cos(4*theta) sqrt(10) v#sx9$K T
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10)
93`
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) ?~Vev D
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) -H_7GVSnl
% 4 4 r^4 * sin(4*theta) sqrt(10) K&