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 iJ' xh n
function z = zernfun(n,m,r,theta,nflag) /walu+]h
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. Cxod[$8
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ;Vik5)D2D
% and angular frequency M, evaluated at positions (R,THETA) on the @+F4YJmB?l
% unit circle. N is a vector of positive integers (including 0), and klgy;jSEr
% M is a vector with the same number of elements as N. Each element &N~ZI*^
% k of M must be a positive integer, with possible values M(k) = -N(k) fb~=Y$|
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, 'J&f%kx"
% and THETA is a vector of angles. R and THETA must have the same BBG3OAyg_
% length. The output Z is a matrix with one column for every (N,M) |2\{z{?
% pair, and one row for every (R,THETA) pair. `LAR@a5i
% r_e7a6
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike C98]9
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 'bld,Do6
% with delta(m,0) the Kronecker delta, is chosen so that the integral I+>%uShm
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 6
5y+Z
% and theta=0 to theta=2*pi) is unity. For the non-normalized mbnV[
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. )irRO 8
% rqPFU6
% The Zernike functions are an orthogonal basis on the unit circle. 'TH15r@
% They are used in disciplines such as astronomy, optics, and a22Mufl
% optometry to describe functions on a circular domain. \I"Z2N>^z
% *_E|@y
% The following table lists the first 15 Zernike functions. "YdEE\
% Hqnxq
% n m Zernike function Normalization ?Kvl!F!`
% -------------------------------------------------- [.RO'>2z
% 0 0 1 1 7\*FEjRM]
% 1 1 r * cos(theta) 2 %AOja+
% 1 -1 r * sin(theta) 2 E0%~!b
% 2 -2 r^2 * cos(2*theta) sqrt(6) pwwH<0[
% 2 0 (2*r^2 - 1) sqrt(3) jM-)BP6f4
% 2 2 r^2 * sin(2*theta) sqrt(6) h~{aGo
% 3 -3 r^3 * cos(3*theta) sqrt(8) 7eWk7&Xul
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) DvvT?K
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) ) ri}nL.
% 3 3 r^3 * sin(3*theta) sqrt(8) ?4H i-
% 4 -4 r^4 * cos(4*theta) sqrt(10) 2I*;A5$N1
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10)
Bs?7:kN(
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) /Q~gU<
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) &Tl
0Pf
% 4 4 r^4 * sin(4*theta) sqrt(10) zIP6\u
% -------------------------------------------------- ` PYJ^I0
% WTImRXK4
% Example 1: ,`ZYvF^%
% Hwo$tVa:=
% % Display the Zernike function Z(n=5,m=1) ~QvqG{bFB
% x = -1:0.01:1; []a[v%PkG
% [X,Y] = meshgrid(x,x); aK`@6F,]j
% [theta,r] = cart2pol(X,Y); Y&/]O$<
% idx = r<=1; 1hcjSO
% z = nan(size(X)); u,}{I}x_
% z(idx) = zernfun(5,1,r(idx),theta(idx)); vjjSKP6B
% figure u%~igt@x
% pcolor(x,x,z), shading interp LM&y@"wfm
% axis square, colorbar CHV*vU<N
% title('Zernike function Z_5^1(r,\theta)') $Of0n` e
% !"8fdSfg
w
% Example 2: p~*UpU8u
% ,t\* ZTt$
% % Display the first 10 Zernike functions R(n^)^?
% x = -1:0.01:1; Bz5-ITX
% [X,Y] = meshgrid(x,x); i1S>yV^l
% [theta,r] = cart2pol(X,Y); 2h[85\4
% idx = r<=1; |&Ym@Jyj
% z = nan(size(X)); 0ez(A
% n = [0 1 1 2 2 2 3 3 3 3]; TDd{.8qf
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; rj6#1kt
% Nplot = [4 10 12 16 18 20 22 24 26 28]; oh$Q6G
% y = zernfun(n,m,r(idx),theta(idx)); Ur*6Gi6
% figure('Units','normalized') wm+/e#'&
% for k = 1:10 u]vQ>Uu
% z(idx) = y(:,k); 'uq#ai[5I
% subplot(4,7,Nplot(k)) eds26(
% pcolor(x,x,z), shading interp )T k1 QHU
% set(gca,'XTick',[],'YTick',[]) #!)n
{h+
% axis square tU_y6
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) C+|b1/N-
% end ?JL:CBvCp
% ,\qs4&
% See also ZERNPOL, ZERNFUN2. _x!7}O#k
jg?x&'u\)
% Paul Fricker 11/13/2006 5Kkdo!z
ve\X3"p#
H@ t'~ZO
% Check and prepare the inputs: W"Gkq!3u{
% ----------------------------- `X3^fg
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ~i `>adJ:
error('zernfun:NMvectors','N and M must be vectors.') =2@B&
end n5{Xj:}
6 ~>FYX
if length(n)~=length(m) Br`IW
error('zernfun:NMlength','N and M must be the same length.') }fKSqB]T-
end /{|fyKo\?
Zfyo-Wk
n = n(:); QcgfBsv96
m = m(:); .w]GWL
if any(mod(n-m,2)) < P`u}
error('zernfun:NMmultiplesof2', ... )KP5WudX
'All N and M must differ by multiples of 2 (including 0).') F+@5C:<?
end '3?\K3S4i
:H c0b=
if any(m>n) !%c'$f/
error('zernfun:MlessthanN', ... VO"("7L
'Each M must be less than or equal to its corresponding N.') C*`mM'#
end 8cA~R-
s M +WkN}{
if any( r>1 | r<0 ) Aj0Tfdxy
error('zernfun:Rlessthan1','All R must be between 0 and 1.') zD<or&6
end f4BnX(1u
VqS#waNrx
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) AZmb!}m+d
error('zernfun:RTHvector','R and THETA must be vectors.') 9D4NX<_
end HQB(*
D&S26jrZ
r = r(:); ;g~TWy^o
theta = theta(:); 6,9o>zT%H
length_r = length(r); N&M~0iw
if length_r~=length(theta) &-mX ,
error('zernfun:RTHlength', ... !tp1:'KG
'The number of R- and THETA-values must be equal.') 8KRba4[
end Jej` ;I
J.8IwN1E
% Check normalization: L@gWzC~?Q
% -------------------- C?2'+K
if nargin==5 && ischar(nflag) #b~JDO(
isnorm = strcmpi(nflag,'norm'); 46 PoM
if ~isnorm ,13Lq-
error('zernfun:normalization','Unrecognized normalization flag.') /FIE:Io
end W]nSR RWco
else A$w4PVS
isnorm = false; PnoPbk[<
end |M+<m">E
)LyojwY_g
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% APO>y
% Compute the Zernike Polynomials rSJ9v:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% WH= EPOR,
%wSj%>&-R
% Determine the required powers of r: p1|f<SF')
% ----------------------------------- (x3.poSt
m_abs = abs(m); WoBo9aR
rpowers = []; MzL1Bh!M
for j = 1:length(n) D)d~3`=#
rpowers = [rpowers m_abs(j):2:n(j)]; 'UYR5Y>
end V,G|k!!
rpowers = unique(rpowers); +9")KQT
r3\cp0P;s
% Pre-compute the values of r raised to the required powers, PZ*pQ=`
% and compile them in a matrix: !Uq^7Mw
% ----------------------------- W]5USFan
if rpowers(1)==0 $t6e2=7
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); iySRY^
rpowern = cat(2,rpowern{:}); ?G-e](]^<
rpowern = [ones(length_r,1) rpowern]; UNkCL4N
else `YIf_a{
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); g2T -TG'd
rpowern = cat(2,rpowern{:}); %y%j*B!%
end YE9,KVV;$n
oD$J0{K6
% Compute the values of the polynomials: x*Y@Q?`>5W
% -------------------------------------- 4'LB7}WG
y = zeros(length_r,length(n)); &Y^WP?HS
for j = 1:length(n) yn/rW$
s = 0:(n(j)-m_abs(j))/2; 1Q.\s_2
pows = n(j):-2:m_abs(j); E,f>1meN=
for k = length(s):-1:1 iX4Iu3
p = (1-2*mod(s(k),2))* ... ~PHB_cyth
prod(2:(n(j)-s(k)))/ ... Y14W?|KOB
prod(2:s(k))/ ...
3dRr/Ilc
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... gw}Mw
prod(2:((n(j)+m_abs(j))/2-s(k)));
Yl.0aS
idx = (pows(k)==rpowers); hc'-Dh
y(:,j) = y(:,j) + p*rpowern(:,idx); Ed
,D8ND
end C,.Ee3T
_z1(y}u}
if isnorm Z%n(O(^L
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 2[r^M'J
end jWYV#ifs2
end xQp|;oW;z
% END: Compute the Zernike Polynomials 8{Fsm;UsY
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% HO''&hz
C(K; zo*S(
% Compute the Zernike functions: xQ'2BAEa
% ------------------------------ oI#a_/w
idx_pos = m>0; vVgg0Y2
idx_neg = m<0; zD?K>I =
//4Xq8y
z = y; /mK?E5H'r1
if any(idx_pos) Y}vr>\
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); gB4U*D0[e~
end h)Ff2tX
if any(idx_neg) NmSo4Dg`U
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); j8sH#b7Z
end Rv/Bh<t
+(+Itmx2&
% EOF zernfun