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 +!6aB|-
function z = zernfun(n,m,r,theta,nflag) i[/g&fx
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. N@lTn}U
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 9"O z-!Y4
% and angular frequency M, evaluated at positions (R,THETA) on the ?2zVWZ
% unit circle. N is a vector of positive integers (including 0), and x*Y&s<
% M is a vector with the same number of elements as N. Each element ZdJwy%
% k of M must be a positive integer, with possible values M(k) = -N(k) R5c
Ya
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, o?$kcI4
% and THETA is a vector of angles. R and THETA must have the same jFY6}WY)}7
% length. The output Z is a matrix with one column for every (N,M) (lq7 ct
% pair, and one row for every (R,THETA) pair. r63_|~JVB<
% '^)Ve:K-.
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike HgPRz C
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), YhYcqE8
% with delta(m,0) the Kronecker delta, is chosen so that the integral 1OJD!juL$
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, Fk@A;22N
% and theta=0 to theta=2*pi) is unity. For the non-normalized 8\+kfK
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. rxH*h`Xx@
% }CnqJ@>C5
% The Zernike functions are an orthogonal basis on the unit circle. P9=L?t.
% They are used in disciplines such as astronomy, optics, and U]tbV<m%
% optometry to describe functions on a circular domain. 2`hc0
IE
% ++d(}^C;
% The following table lists the first 15 Zernike functions. g+;)?N*j
% 7\m.xWX e
% n m Zernike function Normalization /fC@T
% -------------------------------------------------- ?muI8b
% 0 0 1 1 z/6/
% 1 1 r * cos(theta) 2 xP%`QTl\
% 1 -1 r * sin(theta) 2 J0CEZ
% 2 -2 r^2 * cos(2*theta) sqrt(6) l!CWE
% 2 0 (2*r^2 - 1) sqrt(3) B f33%I~
% 2 2 r^2 * sin(2*theta) sqrt(6) }_93}e
% 3 -3 r^3 * cos(3*theta) sqrt(8) 6REv( E]
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) F4'g}yOLd
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) =67dpQ'y
% 3 3 r^3 * sin(3*theta) sqrt(8) /cHd&i,>
% 4 -4 r^4 * cos(4*theta) sqrt(10) gdkl,z3N3
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) wv0d"PKTS
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) 5[l9`Cn&A
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) M:x?I_JG8
% 4 4 r^4 * sin(4*theta) sqrt(10) u=NpL^6s<
% -------------------------------------------------- RzCC>-
% I{Hl2?CnI,
% Example 1: ^*.S7.;2o
% c&r