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 (#LV*&K%IC
function z = zernfun(n,m,r,theta,nflag) }T4"#'`
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. H:y.7
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N f>-OwL($P
% and angular frequency M, evaluated at positions (R,THETA) on the Fgt/A#`fz
% unit circle. N is a vector of positive integers (including 0), and 2/qfK+a
% M is a vector with the same number of elements as N. Each element )#IiHBF
% k of M must be a positive integer, with possible values M(k) = -N(k) J3y5R1?EP
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, m0XK?;\V
% and THETA is a vector of angles. R and THETA must have the same mi%d([)%<
% length. The output Z is a matrix with one column for every (N,M) '1^\^)&q
% pair, and one row for every (R,THETA) pair. C03ehjT<
% 8WfF: R;
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike :}e*3={4
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), m:II<tv
% with delta(m,0) the Kronecker delta, is chosen so that the integral .2[>SI
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, OUnt?[U\
% and theta=0 to theta=2*pi) is unity. For the non-normalized >L?/Ph %d
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. {$oZR"MP
% %+Mi~k*A'
% The Zernike functions are an orthogonal basis on the unit circle. BLuILE:$
% They are used in disciplines such as astronomy, optics, and m"X0Owx
% optometry to describe functions on a circular domain. +cQ4u4
% {cq; SH
% The following table lists the first 15 Zernike functions. i2)rDek3]T
% WTSY:kvcCY
% n m Zernike function Normalization n]6xrsE
% -------------------------------------------------- }!lLA4XRr
% 0 0 1 1 tJ bOn$]2"
% 1 1 r * cos(theta) 2 9I+;waLlB
% 1 -1 r * sin(theta) 2 !`)-seTm
% 2 -2 r^2 * cos(2*theta) sqrt(6) l4|bpR Cp
% 2 0 (2*r^2 - 1) sqrt(3) Yg<o 9x$
% 2 2 r^2 * sin(2*theta) sqrt(6) N[){yaj
% 3 -3 r^3 * cos(3*theta) sqrt(8) W>bhSKV%
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) 9k& lq$
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) Xr6lYO _R
% 3 3 r^3 * sin(3*theta) sqrt(8) 3yZtyXRPn
% 4 -4 r^4 * cos(4*theta) sqrt(10) Y}(v[QGV
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) p_!Y:\a5
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) z,I7 PY& G
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 573wK~9oMh
% 4 4 r^4 * sin(4*theta) sqrt(10) &