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 Z8#nu
function z = zernfun(n,m,r,theta,nflag) @M5+12FYt
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. c>_ti+
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ^ `y7JXI:
% and angular frequency M, evaluated at positions (R,THETA) on the EAGvP&~P
% unit circle. N is a vector of positive integers (including 0), and [a2]_]E%
% M is a vector with the same number of elements as N. Each element pCs3-&rI3
% k of M must be a positive integer, with possible values M(k) = -N(k) S4x9k{Xn
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, 9\_AB.Z:
% and THETA is a vector of angles. R and THETA must have the same "GO!^ZG]
% length. The output Z is a matrix with one column for every (N,M) G%
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% pair, and one row for every (R,THETA) pair. {EoYU\x
% /iU<\+ H
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike *#T:
_
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), dLiiJ6pl*
% with delta(m,0) the Kronecker delta, is chosen so that the integral '~D4%WKT
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, (p-q>@m
% and theta=0 to theta=2*pi) is unity. For the non-normalized xsZG(Tz
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. _QL|pLf-
% oMQ4q{&|
% The Zernike functions are an orthogonal basis on the unit circle. &B{zS K$N
% They are used in disciplines such as astronomy, optics, and "lh4Vg\7n
% optometry to describe functions on a circular domain. 4=L >
% msBoInhI
% The following table lists the first 15 Zernike functions. }?s-$@$R
% .G{cx=;
% n m Zernike function Normalization qVC+q8
% -------------------------------------------------- \f9WpAY
% 0 0 1 1 >dl5^
% 1 1 r * cos(theta) 2 v`A)GnNiN
% 1 -1 r * sin(theta) 2 7;EDU
% 2 -2 r^2 * cos(2*theta) sqrt(6) Nk7y2[
% 2 0 (2*r^2 - 1) sqrt(3) u#76w74
% 2 2 r^2 * sin(2*theta) sqrt(6) ~
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% 3 -3 r^3 * cos(3*theta) sqrt(8) k&$ov
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) Hr?lRaV
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) @+b$43^
% 3 3 r^3 * sin(3*theta) sqrt(8) COh#/-`\1
% 4 -4 r^4 * cos(4*theta) sqrt(10) 8^UF0>`'
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) LYDiqOrx
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) <_YdN)x
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) <?.eU<+O`S
% 4 4 r^4 * sin(4*theta) sqrt(10) d{S'6*`D
% -------------------------------------------------- }~
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% 1&bo