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 J?1 uKR
function z = zernfun(n,m,r,theta,nflag)
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%ZERNFUN Zernike functions of order N and frequency M on the unit circle. 1CD+B=pQG
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N LgU_LcoM*
% and angular frequency M, evaluated at positions (R,THETA) on the rQs)O<jl
% unit circle. N is a vector of positive integers (including 0), and dr}`H,X"3
% M is a vector with the same number of elements as N. Each element mHTXni<!
% k of M must be a positive integer, with possible values M(k) = -N(k) ZohCP
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, TDKki(o=~
% and THETA is a vector of angles. R and THETA must have the same l`{\"#4
% length. The output Z is a matrix with one column for every (N,M) }5[qo`M
% pair, and one row for every (R,THETA) pair. BwGfTua
% qvsd5P eCO
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike sN*N&XG
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), X1|njJGO1
% with delta(m,0) the Kronecker delta, is chosen so that the integral qp}Cqi
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, %QGC8Tz
% and theta=0 to theta=2*pi) is unity. For the non-normalized ,j{,h_Op
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. hGe/;@%
% J.b9F:&}
% The Zernike functions are an orthogonal basis on the unit circle. AaOuL,l
% They are used in disciplines such as astronomy, optics, and *uf'zQ<9
% optometry to describe functions on a circular domain. 0B/,/KX
% wLH>:yKUU
% The following table lists the first 15 Zernike functions. m|n%$$S&
% L|:`^M+^w
% n m Zernike function Normalization nI-w}NQ
% -------------------------------------------------- Nq[uoaT
% 0 0 1 1 <tNBxa$gS
% 1 1 r * cos(theta) 2 KIf dafRL
% 1 -1 r * sin(theta) 2 w^|*m/h|@u
% 2 -2 r^2 * cos(2*theta) sqrt(6) ?k&Vy
% 2 0 (2*r^2 - 1) sqrt(3) vn!3l1\+J
% 2 2 r^2 * sin(2*theta) sqrt(6) k 8[n+^
% 3 -3 r^3 * cos(3*theta) sqrt(8) R6 .hA_ih
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) '&tG?gb&
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) +H-6e P
% 3 3 r^3 * sin(3*theta) sqrt(8) 6+|do+0Icg
% 4 -4 r^4 * cos(4*theta) sqrt(10) @[<><uTH
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) :Zbg9`d*
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) ,{u
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% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Oi'5ytsES
% 4 4 r^4 * sin(4*theta) sqrt(10) y<|7z99L
% -------------------------------------------------- ]d0BN`*U.
% /<=u\e'rE
% Example 1: >V?eog%~
% Ys!82M$g
% % Display the Zernike function Z(n=5,m=1) Eqd<