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 ckDWY<@v
function z = zernfun(n,m,r,theta,nflag) >|j8j:S[
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. PB[Y^q
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N iO$Z?Dyg9
% and angular frequency M, evaluated at positions (R,THETA) on the Bs?B\k=
% unit circle. N is a vector of positive integers (including 0), and 3m;*gOLk6
% M is a vector with the same number of elements as N. Each element 3[_zz;Y*d
% k of M must be a positive integer, with possible values M(k) = -N(k) Hs9; &C
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, || p>O
% and THETA is a vector of angles. R and THETA must have the same MS Qz,nn
% length. The output Z is a matrix with one column for every (N,M) {H F,F=W
% pair, and one row for every (R,THETA) pair. lftT55Tki
% O@9<7@h+Nl
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 76IjM4&a
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), tJ3Hg8;
% with delta(m,0) the Kronecker delta, is chosen so that the integral Al93x
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, mFk6a{+YX
% and theta=0 to theta=2*pi) is unity. For the non-normalized b=87k
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. G~.bi<(v
% c]Z@L~WW
% The Zernike functions are an orthogonal basis on the unit circle. @#u'z~a)
% They are used in disciplines such as astronomy, optics, and ,ma4bqRMc
% optometry to describe functions on a circular domain. gdj,e ^
% +cXdF
% The following table lists the first 15 Zernike functions. TyGsSc
% r
&.gOC
% n m Zernike function Normalization [D$%LR X
% -------------------------------------------------- w^EUBRI-
% 0 0 1 1 PR+L6DT_
% 1 1 r * cos(theta) 2 pw,
<0UhV
% 1 -1 r * sin(theta) 2 [}*xxy
% 2 -2 r^2 * cos(2*theta) sqrt(6) .\rJ|HpZ1J
% 2 0 (2*r^2 - 1) sqrt(3) S\jIs [Dz
% 2 2 r^2 * sin(2*theta) sqrt(6) |'+ [ '
% 3 -3 r^3 * cos(3*theta) sqrt(8) R?Ys%~5
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) (_ TKDx_
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) "e ;wN3/bF
% 3 3 r^3 * sin(3*theta) sqrt(8) WHk rd8
% 4 -4 r^4 * cos(4*theta) sqrt(10) C@F3iwTtp
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Bk
yW
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) h.t2 ;O, b
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ^630%YO
% 4 4 r^4 * sin(4*theta) sqrt(10) B[IqLD'6
% -------------------------------------------------- be+]kp
% Y I?4e7Z+
% Example 1: SbYsa
% - ]Mbe2;
% % Display the Zernike function Z(n=5,m=1) K0 6 E:
% x = -1:0.01:1; +Rq7m]
% [X,Y] = meshgrid(x,x); lm[LDtc
% [theta,r] = cart2pol(X,Y); * .P3fVlZ
% idx = r<=1; \L5h&