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 1--Ka& H
function z = zernfun(n,m,r,theta,nflag) gfKv$~
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. $EL:Jx2<
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N mNsd&Rk'
% and angular frequency M, evaluated at positions (R,THETA) on the EeGTBVms
% unit circle. N is a vector of positive integers (including 0), and {B4.G8%Z
% M is a vector with the same number of elements as N. Each element L4ZB0PmN'
% k of M must be a positive integer, with possible values M(k) = -N(k) $&&+2?cx0
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, DPkH:X
% and THETA is a vector of angles. R and THETA must have the same ?I u=os>*
% length. The output Z is a matrix with one column for every (N,M) cdN =HM~I
% pair, and one row for every (R,THETA) pair. G=LK
irj(
% &A=c[pc
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike Ir=G\/A
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), _T_} k:&X
% with delta(m,0) the Kronecker delta, is chosen so that the integral /!N=@z)
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, F,V|In
% and theta=0 to theta=2*pi) is unity. For the non-normalized ]0g p.R
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. Ko)f:=Qo
% n(i/jW~0w
% The Zernike functions are an orthogonal basis on the unit circle. 13 %:3W(
% They are used in disciplines such as astronomy, optics, and ErgWs Aw-
% optometry to describe functions on a circular domain. p=\Q7<Z6d,
% }Syd*%BR[
% The following table lists the first 15 Zernike functions. 0\,!
% nTD4^'
% n m Zernike function Normalization YABi`;R]'
% -------------------------------------------------- =MvB9gx@r
% 0 0 1 1 qC5IV}9`
% 1 1 r * cos(theta) 2 x[u6_6=q9
% 1 -1 r * sin(theta) 2 C{+JrHV%h
% 2 -2 r^2 * cos(2*theta) sqrt(6) !,C8
% 2 0 (2*r^2 - 1) sqrt(3) lPrAx0m13%
% 2 2 r^2 * sin(2*theta) sqrt(6) /}
h"f5
% 3 -3 r^3 * cos(3*theta) sqrt(8) QKhGEW~G
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) 0M?zotv0#
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) T^{=cx9x9
% 3 3 r^3 * sin(3*theta) sqrt(8) 2H`>Kj
% 4 -4 r^4 * cos(4*theta) sqrt(10) xu{VU^'Y
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ,LC(Ax'.F
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) m/;fY>}3
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) itg"dGDk
% 4 4 r^4 * sin(4*theta) sqrt(10) !R@jbM
% -------------------------------------------------- ML0_Uc3en
% 8n:N#4Dh^
% Example 1: Q- w_@~
% suYbD!`(
% % Display the Zernike function Z(n=5,m=1) dk"@2%xJ2d
% x = -1:0.01:1; `sgW0Uf
% [X,Y] = meshgrid(x,x); IkG;j+=
% [theta,r] = cart2pol(X,Y); Az-!X!O*f
% idx = r<=1; ;/kmV~KG
% z = nan(size(X)); IM,4Si2
% z(idx) = zernfun(5,1,r(idx),theta(idx)); <;uM/vSi
% figure oX'@,(6)
% pcolor(x,x,z), shading interp +zXcTT[V
% axis square, colorbar ( uG;Q
% title('Zernike function Z_5^1(r,\theta)') (;H% r &
% TKiYEh
% Example 2: $*LBZcL
% &0H_W xKeB
% % Display the first 10 Zernike functions V-E 77u6{0
% x = -1:0.01:1; YK5(o KFN
% [X,Y] = meshgrid(x,x); ZE=
Yn~XM
% [theta,r] = cart2pol(X,Y); /5Vv5d/Z4!
% idx = r<=1; 5:#|Op N
% z = nan(size(X)); (_6JQn
% n = [0 1 1 2 2 2 3 3 3 3]; id" l"
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; ~Nf|,{[(5
% Nplot = [4 10 12 16 18 20 22 24 26 28]; JT=ax/%Mo
% y = zernfun(n,m,r(idx),theta(idx)); l ?YO!$
% figure('Units','normalized') NciIqF
% for k = 1:10 >yVp1Se
% z(idx) = y(:,k); 2m} bddS
% subplot(4,7,Nplot(k)) O%6D2d
% pcolor(x,x,z), shading interp ?RW1%+[
% set(gca,'XTick',[],'YTick',[]) h%NM%;"H/
% axis square ,yvS c
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) kbJ4CF}H
% end ~B?Wg!
% )heHERbJ
% See also ZERNPOL, ZERNFUN2. qJ<l$Ig
n)z:C{
% Paul Fricker 11/13/2006 b'z\|jY
SLUQFoz}
/Ahh6=qQY
% Check and prepare the inputs: p )]x,F
% ----------------------------- Hl'AnxE
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) N R4\TU
error('zernfun:NMvectors','N and M must be vectors.') 7$t['2j3
end ]0[ot$Da6
Oamz>Hplu
if length(n)~=length(m) [7g-M/jvY
error('zernfun:NMlength','N and M must be the same length.') *OIBMx#qxn
end L6;'V5Mg72
[hk/Rp7{
n = n(:); TJ_6:;4,|_
m = m(:); {`T^&bk
if any(mod(n-m,2)) [tElt4uG
error('zernfun:NMmultiplesof2', ... ,A)Z.OWOq
'All N and M must differ by multiples of 2 (including 0).') 5tzO=gO[
end i[ws%GfEv
8OO[Le]1
if any(m>n) fO
.=i1
E}
error('zernfun:MlessthanN', ... m6]6!_
'Each M must be less than or equal to its corresponding N.') ll- KK`Ka
end }$r]\v
8xG"hJR
if any( r>1 | r<0 ) x5Fo?E
error('zernfun:Rlessthan1','All R must be between 0 and 1.') K5\l
(BB
end 4x3 _8/=
N:S2X+}(
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) N
7Y X
error('zernfun:RTHvector','R and THETA must be vectors.') G007[|
end xU}J6 Tv
(/!@
-]1
r = r(:); qDz[=6BF
theta = theta(:); E*fa&G~s )
length_r = length(r); 7^mQfQv
if length_r~=length(theta) " vc4QH$
error('zernfun:RTHlength', ... 1oQbV`P
'The number of R- and THETA-values must be equal.') Zk>m!F>,p
end DUH_LnHw)
0>]&9'cn
% Check normalization: moh,a B#
% -------------------- {XUSw8W'
if nargin==5 && ischar(nflag) C>mFylN
isnorm = strcmpi(nflag,'norm'); W- nS{v(
if ~isnorm mFxt +\
error('zernfun:normalization','Unrecognized normalization flag.') Msfxce
end :}/\hz
,
else e"XolM0IM
isnorm = false; 1$6
u
end >!{8)ti
Ggsts
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% TXS`ey
% Compute the Zernike Polynomials 8Gy*BpmJn
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% }d iE'
0Zo><=
% Determine the required powers of r: s{V&vRr
% ----------------------------------- .;.Zbhm
m_abs = abs(m); ~Fl\c-
rpowers = []; \u(Gj]B#"
for j = 1:length(n) oIIi_yc
rpowers = [rpowers m_abs(j):2:n(j)]; `T ^0&#
end Gm=&[?}
rpowers = unique(rpowers); ggYi 7Wzsd
|TkicgeS
% Pre-compute the values of r raised to the required powers, kM=&Tfpj
% and compile them in a matrix: Yl?s^]SFU
% ----------------------------- aG4 ^xOD
if rpowers(1)==0 61OlnmvE
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); E<0Mluk
rpowern = cat(2,rpowern{:}); Cw kQhj?
rpowern = [ones(length_r,1) rpowern]; qe(C>qjMbG
else hNgT/y8
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); x_?K6[G&}
rpowern = cat(2,rpowern{:}); b&1hj[`)
end X.,R%>O}`P
_v,Wl/YAp
% Compute the values of the polynomials: ,HmGp
% -------------------------------------- R[fQ$` M
y = zeros(length_r,length(n)); },Grg~l
for j = 1:length(n) AeN:wOm
s = 0:(n(j)-m_abs(j))/2; nmE H/a
pows = n(j):-2:m_abs(j); T2)CiR-b
for k = length(s):-1:1 t7xJ"
p = (1-2*mod(s(k),2))* ... {)!ua7GF0H
prod(2:(n(j)-s(k)))/ ... d7zZ~n
prod(2:s(k))/ ... tx`^'%GMA
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... \_(0V"
prod(2:((n(j)+m_abs(j))/2-s(k))); xXmlHo<D
idx = (pows(k)==rpowers); o8%o68py
y(:,j) = y(:,j) + p*rpowern(:,idx); a\Gd;C ^`
end "[7'i<,AI
;'Z"CbS+
if isnorm \9od*y
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); <@+L^Ps~z
end oY,{9H37b
end OPqhdqo
% END: Compute the Zernike Polynomials ",,.xLI7
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ; 4/ n~
/~fu,2=7
% Compute the Zernike functions: ,nP nH1vb
% ------------------------------ FB>P39u
idx_pos = m>0; -O/[c
idx_neg = m<0; )-}<}< oO
UCFFF%
z = y; ,+gtr.
if any(idx_pos) bu\(KR$s
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); HL>l.IG?
end een62-`
if any(idx_neg) <veypLi"R
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); HxLuJ
end ,lFzL3'_0x
mY XL
% EOF zernfun