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 D.f=!rT7E7
function z = zernfun(n,m,r,theta,nflag) 0^^i=iE-u
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. &Gl&m@-j
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N XCoOs<O:@
% and angular frequency M, evaluated at positions (R,THETA) on the @x4Dt&:"
% unit circle. N is a vector of positive integers (including 0), and |+''d
% M is a vector with the same number of elements as N. Each element {F[Xe_=#"
% k of M must be a positive integer, with possible values M(k) = -N(k) N<%,3W_-_
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, 2e=Hjf
)
% and THETA is a vector of angles. R and THETA must have the same \x}UjHYIc&
% length. The output Z is a matrix with one column for every (N,M) XjNu|H/
% pair, and one row for every (R,THETA) pair. +UtK2<^:o
% m+ YgfR
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike zq&lxySa
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), *WG}K?"/
% with delta(m,0) the Kronecker delta, is chosen so that the integral ~E~J*R Ze
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, UQ?8dw:E~
% and theta=0 to theta=2*pi) is unity. For the non-normalized zKr(Gt8
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. l|{<!7a
% b iD7(AK
% The Zernike functions are an orthogonal basis on the unit circle. &$f?XdZ7
% They are used in disciplines such as astronomy, optics, and N0f}q1S<-A
% optometry to describe functions on a circular domain. NM ]/OKs'H
% 2}-W@R
% The following table lists the first 15 Zernike functions. =\.|'
% m` cG&Ar5
% n m Zernike function Normalization 2)YLs5>W%
% -------------------------------------------------- ai RNd~\
% 0 0 1 1 Pe.D[]S
% 1 1 r * cos(theta) 2 0Og =H79<
% 1 -1 r * sin(theta) 2 `1gsrHi4N
% 2 -2 r^2 * cos(2*theta) sqrt(6) U$}]zaB
% 2 0 (2*r^2 - 1) sqrt(3) sBMHf9u
% 2 2 r^2 * sin(2*theta) sqrt(6) t~Ax#H
% 3 -3 r^3 * cos(3*theta) sqrt(8) dmne+ufB
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) Nx__zC^r
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) 8*X8U:.0o
% 3 3 r^3 * sin(3*theta) sqrt(8) iuEdm:pW
% 4 -4 r^4 * cos(4*theta) sqrt(10) 6gXc-}dp
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) )C[8#Q-:
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) wpdT "
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) `4MPXfoBL
% 4 4 r^4 * sin(4*theta) sqrt(10) RD^o&