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 >y06s{[
function z = zernfun(n,m,r,theta,nflag) EBL,E:_)
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. !Bd*
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% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N J%O4IcE
% and angular frequency M, evaluated at positions (R,THETA) on the LN3dp?;_{
% unit circle. N is a vector of positive integers (including 0), and NV:XPw/
% M is a vector with the same number of elements as N. Each element o YI=p3l
% k of M must be a positive integer, with possible values M(k) = -N(k) s*~jvL
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, Ag-?6v
% and THETA is a vector of angles. R and THETA must have the same @tv];t
% length. The output Z is a matrix with one column for every (N,M) + x;ML
% pair, and one row for every (R,THETA) pair. g7}z
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% vL=--#
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 8,H5G`
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), [|;Zxb:
% with delta(m,0) the Kronecker delta, is chosen so that the integral /&!d
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, RnBmy^l"
% and theta=0 to theta=2*pi) is unity. For the non-normalized F6GZZKj
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. e'?doP
% \F+o=
% The Zernike functions are an orthogonal basis on the unit circle. QVRokI`BF
% They are used in disciplines such as astronomy, optics, and Ccd7|L1
% optometry to describe functions on a circular domain. "KI,3g _V
% ://#
%SE
% The following table lists the first 15 Zernike functions. eN?P) ,
% J)yy}[Fx
% n m Zernike function Normalization :iNAXy
% -------------------------------------------------- U!I_i*:U
% 0 0 1 1 \gzwsT2&
% 1 1 r * cos(theta) 2 't%%hw-m}
% 1 -1 r * sin(theta) 2 w3bH|VnU8;
% 2 -2 r^2 * cos(2*theta) sqrt(6) pA,EUh|H
% 2 0 (2*r^2 - 1) sqrt(3) >0+|0ba
% 2 2 r^2 * sin(2*theta) sqrt(6) &'ETx"
% 3 -3 r^3 * cos(3*theta) sqrt(8) [oN> :
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) $\@ V4
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) Q]g 4gj
% 3 3 r^3 * sin(3*theta) sqrt(8) >]Yha}6h
% 4 -4 r^4 * cos(4*theta) sqrt(10) #IrP"j^
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) '%RK KA
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) gsR9M%mv
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) aE cg_es
% 4 4 r^4 * sin(4*theta) sqrt(10) AW;)_|xM
% -------------------------------------------------- sv6U%qV
% HXV73rDA
% Example 1: f]A6Mx6
% Y&