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 ?^M,Mt
function z = zernfun(n,m,r,theta,nflag) 0y6M;"&~E
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. Z %Ozzp/
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N uKd4+Km
% and angular frequency M, evaluated at positions (R,THETA) on the eZaSV>27
% unit circle. N is a vector of positive integers (including 0), and Fs].Fa
% M is a vector with the same number of elements as N. Each element AYgXqmH~+
% k of M must be a positive integer, with possible values M(k) = -N(k) #c5jCy}n
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, R(`:~@3\6
% and THETA is a vector of angles. R and THETA must have the same ^lAM /
% length. The output Z is a matrix with one column for every (N,M) :aK?Dt Z
% pair, and one row for every (R,THETA) pair. 8!rdqI
% !
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% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ZZ7qSyBs?
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), __2<v?\
% with delta(m,0) the Kronecker delta, is chosen so that the integral h%krA<G9
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, LP=j/qf|
% and theta=0 to theta=2*pi) is unity. For the non-normalized
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% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. W*t]
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% i=cST8!8N
% The Zernike functions are an orthogonal basis on the unit circle. X!p`|i
% They are used in disciplines such as astronomy, optics, and PO`p.("h
% optometry to describe functions on a circular domain. aPVzOBp
% -cM1]soT
% The following table lists the first 15 Zernike functions. p,goYF??
% MDU#V
% n m Zernike function Normalization &CQO+Yr$l
% -------------------------------------------------- V`1,s~"q
% 0 0 1 1 ;~EQS.Qp
% 1 1 r * cos(theta) 2 D]]wJQU2
% 1 -1 r * sin(theta) 2 @kqxN\DE
% 2 -2 r^2 * cos(2*theta) sqrt(6) !:^q_q4
% 2 0 (2*r^2 - 1) sqrt(3) L%T(H<