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 11%<bmJ]Q3
function z = zernfun(n,m,r,theta,nflag) vM_UF{a$=
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. QU4/hS;Ux
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N .M3]\I u
% and angular frequency M, evaluated at positions (R,THETA) on the PQ6.1}
% unit circle. N is a vector of positive integers (including 0), and [)K?e!c8
% M is a vector with the same number of elements as N. Each element q)Qd+:a7{
% k of M must be a positive integer, with possible values M(k) = -N(k) V`F]L^m=L
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, T#ktC0W]h
% and THETA is a vector of angles. R and THETA must have the same Ce:2Tw
% length. The output Z is a matrix with one column for every (N,M) 6Fp}U
% pair, and one row for every (R,THETA) pair. QWqEe|}6
% 99GzhX_
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike T(Q(7
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), mmE!!J`B
% with delta(m,0) the Kronecker delta, is chosen so that the integral Q-scL>IkCb
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, Lye^G%{
% and theta=0 to theta=2*pi) is unity. For the non-normalized [sx J<
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 1}O&q6\"J
% in>Os@e#
% The Zernike functions are an orthogonal basis on the unit circle. r]GG9si
% They are used in disciplines such as astronomy, optics, and 1y\-Iz^
% optometry to describe functions on a circular domain. {51<EvyE*
% ^T(v4'7
% The following table lists the first 15 Zernike functions. xqP DL9\
% Ancka
% n m Zernike function Normalization ii< /!B(
% -------------------------------------------------- -&L(0?*qo
% 0 0 1 1 {#-I;I:
% 1 1 r * cos(theta) 2 *@2+$fgz
% 1 -1 r * sin(theta) 2 :Nry |
% 2 -2 r^2 * cos(2*theta) sqrt(6) a]JQZo1$
% 2 0 (2*r^2 - 1) sqrt(3) J |$(O$hYy
% 2 2 r^2 * sin(2*theta) sqrt(6) jv8diQ.
% 3 -3 r^3 * cos(3*theta) sqrt(8) d A[MjOd3
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) l1<]pdLTR
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) \FE
% 3 3 r^3 * sin(3*theta) sqrt(8) W3AtO
% 4 -4 r^4 * cos(4*theta) sqrt(10) Sbf+;:D
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) w;e42.\
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) S,Y\ox-
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Qyh_o
% 4 4 r^4 * sin(4*theta) sqrt(10) l"T{!Oq
% -------------------------------------------------- m%?+;V
% 3Ryae/Nk
% Example 1: ymNL`GYN[
% vdhwFp~Y
% % Display the Zernike function Z(n=5,m=1) 8`I/\8;H'p
% x = -1:0.01:1; E~[v.3`
% [X,Y] = meshgrid(x,x); 0uw3[,I
% [theta,r] = cart2pol(X,Y); "Uk "
% idx = r<=1; erhxZ|."P
% z = nan(size(X)); \Z?.Po`!j
% z(idx) = zernfun(5,1,r(idx),theta(idx)); F<W`zQ46
% figure Mk:k0,z
% pcolor(x,x,z), shading interp >q+q];=(
% axis square, colorbar ")gd)_FOS
% title('Zernike function Z_5^1(r,\theta)') ,McwPHEMB
% ZxvqLu
% Example 2: E%+ aqA)f
% $e99[y@
% % Display the first 10 Zernike functions [ X7LV
% x = -1:0.01:1; do-mkvk
% [X,Y] = meshgrid(x,x); l(o;O.dLt
% [theta,r] = cart2pol(X,Y); GnCO{"n
% idx = r<=1; 8 !{;yz
% z = nan(size(X)); kdr?I9kwW
% n = [0 1 1 2 2 2 3 3 3 3]; != @U~X|cu
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; =|Q7k +b
% Nplot = [4 10 12 16 18 20 22 24 26 28]; l.Psh7B2
% y = zernfun(n,m,r(idx),theta(idx)); k+D32]b@
% figure('Units','normalized') |FR'?y1
% for k = 1:10 dn? #}^,"
% z(idx) = y(:,k); G$P|F6
% subplot(4,7,Nplot(k)) sKIpL(_I$
% pcolor(x,x,z), shading interp -pF3q2zb
% set(gca,'XTick',[],'YTick',[]) |=\w b^l+
% axis square U\<8}+x
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) )P)Zds@F
% end W-72&\7
% q#=}T~4j
% See also ZERNPOL, ZERNFUN2. #iZ%CY\
Q?1'
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% Paul Fricker 11/13/2006 }@+{;"
JQ[~N-
xs'vd:l.Pp
% Check and prepare the inputs: \W;+@w|c
% ----------------------------- MO1t0My c
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ljS~>&
error('zernfun:NMvectors','N and M must be vectors.') dxz.%a@PW
end {I]X-+D|_
tB,1+I=
if length(n)~=length(m) )|d]0/<
error('zernfun:NMlength','N and M must be the same length.') Sz"rp9x+
end Ah|,`0dw
f{[ ]m(X;
n = n(:); fw[Z7`\Q5
m = m(:); 8M"0o}wx
if any(mod(n-m,2)) xM#+jI
error('zernfun:NMmultiplesof2', ... Lwy9QZL
'All N and M must differ by multiples of 2 (including 0).') *8a8Ng
end V~9s+>
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if any(m>n) _Di";fe?
error('zernfun:MlessthanN', ... @$e!|.{1q
'Each M must be less than or equal to its corresponding N.') )`*=P}D
end ++Z,U
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