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 %S;AM\o4
function z = zernfun(n,m,r,theta,nflag) rhbz|Uq
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. 0>Snps3*Z
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N > v%.q]E6n
% and angular frequency M, evaluated at positions (R,THETA) on the kEnGr6e
% unit circle. N is a vector of positive integers (including 0), and dEtjcId
% M is a vector with the same number of elements as N. Each element H?];8wq$G
% k of M must be a positive integer, with possible values M(k) = -N(k) jeWv~JA%L|
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, (T#$0RFq
% and THETA is a vector of angles. R and THETA must have the same Cjr]l!
% length. The output Z is a matrix with one column for every (N,M) ;,[0 bmL
% pair, and one row for every (R,THETA) pair. {WrEe7dLy
% [w'Q9\,p
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike p?y2j
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), c%/b*nQ(=
% with delta(m,0) the Kronecker delta, is chosen so that the integral ?'TK~,dG/
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, g~y0,0'j1\
% and theta=0 to theta=2*pi) is unity. For the non-normalized ]5"k%v|
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. "u7[[.P)
% ;yomaAr
% The Zernike functions are an orthogonal basis on the unit circle. &~P4yI;,
% They are used in disciplines such as astronomy, optics, and N9_* {HOy
% optometry to describe functions on a circular domain. j+gxn_E
% XYzaSp=bb
% The following table lists the first 15 Zernike functions. \uOM,98xS
% bwXeEA@{
% n m Zernike function Normalization V'j+)!w5
% -------------------------------------------------- |ZH(Z}m
% 0 0 1 1 t|>zke!'
% 1 1 r * cos(theta) 2 f"FFgQMkv
% 1 -1 r * sin(theta) 2 h5'hP>b#
% 2 -2 r^2 * cos(2*theta) sqrt(6) >n09K8
A
% 2 0 (2*r^2 - 1) sqrt(3) Y 3ApW vS
% 2 2 r^2 * sin(2*theta) sqrt(6) *yRsFC{,
% 3 -3 r^3 * cos(3*theta) sqrt(8) [
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% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) g4h{dFb|_
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) i7.8H*z'
% 3 3 r^3 * sin(3*theta) sqrt(8) ":udo VS!
% 4 -4 r^4 * cos(4*theta) sqrt(10) :>fT=$i@
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ;bB#Pg
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) 9O3 #d
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) o4kLgY !Q
% 4 4 r^4 * sin(4*theta) sqrt(10)
=Pl@+RgK+
% -------------------------------------------------- [j0[c9.p[
%
[Jt}^
% Example 1: T%eBgseS
% 8D )nM|
% % Display the Zernike function Z(n=5,m=1) *,$5EN
% x = -1:0.01:1; 1X2j%qI&
% [X,Y] = meshgrid(x,x); (lM,'
% [theta,r] = cart2pol(X,Y); <}RI<96
% idx = r<=1; ~9+01UU^
% z = nan(size(X)); O% T?+1E
% z(idx) = zernfun(5,1,r(idx),theta(idx)); w[-)c6J yE
% figure <t"T'\3
% pcolor(x,x,z), shading interp ;0 B1P|7zK
% axis square, colorbar z,TH}s6
% title('Zernike function Z_5^1(r,\theta)') Qfm$q~`D^W
% <==uK>pET
% Example 2: g3$'Ghf
% Czjb.c:a.Y
% % Display the first 10 Zernike functions ,' |J
% x = -1:0.01:1; MV" n{1B
% [X,Y] = meshgrid(x,x); s?EQ
% [theta,r] = cart2pol(X,Y); `MI;.t
% idx = r<=1; $t;:"i>
% z = nan(size(X)); S1|u@d'
% n = [0 1 1 2 2 2 3 3 3 3]; K<J,n!zc
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; ~b~Tq
% Nplot = [4 10 12 16 18 20 22 24 26 28]; ^+P.f[
% y = zernfun(n,m,r(idx),theta(idx)); WEj{2+
% figure('Units','normalized') G]ek-[-
% for k = 1:10 A2 +%
% z(idx) = y(:,k); {1SsHir>
% subplot(4,7,Nplot(k)) S oeoUI]m
% pcolor(x,x,z), shading interp .2E/(VM
% set(gca,'XTick',[],'YTick',[]) n|{K_! f
% axis square Fe0M2%e;|
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) VP#KoX85
% end ;mU;+~YE
% ' 4FH9J
% See also ZERNPOL, ZERNFUN2. GI]\
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% Paul Fricker 11/13/2006 KHAc!4lA
1cK'B<5">]
n2mO-ZXud
% Check and prepare the inputs: aoey
5hts
% ----------------------------- n&:ohOH%
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) sjyr9AF
error('zernfun:NMvectors','N and M must be vectors.') V~dhTdQ5}
end x:FZEyalG
]<^2B?}
if length(n)~=length(m) r E m/Q!
error('zernfun:NMlength','N and M must be the same length.') b-<0\@`Z#
end _^BA;S@
Xq;|l?,O
n = n(:); 0>od1/`
m = m(:); &+#5gii1i
if any(mod(n-m,2)) -hXKCb4YU
error('zernfun:NMmultiplesof2', ... ^{uHph9ny
'All N and M must differ by multiples of 2 (including 0).') `D77CC]vU
end sE[`x^1'8
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if any(m>n) Lk !)G'42
error('zernfun:MlessthanN', ... J#$U<`j*G
'Each M must be less than or equal to its corresponding N.') 8%,#TMOg
end BY~Tc5
X84T F~2Y
if any( r>1 | r<0 ) Cy[G7A%
error('zernfun:Rlessthan1','All R must be between 0 and 1.') EHC7b^|3}
end "-=fi
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) tDRR 3=9pX
error('zernfun:RTHvector','R and THETA must be vectors.') )h}IZSm
end fbh,V%t7
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r = r(:); x-[ItJ% l
theta = theta(:); 9rMO=
length_r = length(r); v@=qVwX
if length_r~=length(theta) hoq2zDjD
error('zernfun:RTHlength', ... u#Ig!7iUu
'The number of R- and THETA-values must be equal.') Yj@Sy
end yJI~{VmU7
,ucRQ&P
% Check normalization: G[>NP#P
% -------------------- S~3|1Hw*tN
if nargin==5 && ischar(nflag) lEHx/#qt9
isnorm = strcmpi(nflag,'norm'); Z<;W*6J
if ~isnorm +Vk L?J
error('zernfun:normalization','Unrecognized normalization flag.') TaRPMKk
end 8%K{l g"
else
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isnorm = false; zP44
Xhz
end x@OBGKV
dx@dnWRT,
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% NA0nF8ek
% Compute the Zernike Polynomials 8'*x88+
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ;5ki$)v"
8{ZTHY-
% Determine the required powers of r: 86{>X5 +
% ----------------------------------- ,'0#q
m_abs = abs(m); 1b~21n
rpowers = []; ?b+Y])SJK
for j = 1:length(n) xq((]5P y
rpowers = [rpowers m_abs(j):2:n(j)]; !.'D"Me>
end D3C 7f'
rpowers = unique(rpowers); )h,yQ`.
T_S3_-|{==
% Pre-compute the values of r raised to the required powers, F%6wdM W
% and compile them in a matrix: 4 eLZ
% ----------------------------- 6Hnez @d
if rpowers(1)==0 ye.6tlW
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); <YW)8J
rpowern = cat(2,rpowern{:}); |#_p0yPy
rpowern = [ones(length_r,1) rpowern]; BaQyn 6B
else \x-2qlZ
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); gkd4)\9
rpowern = cat(2,rpowern{:}); ~3.*b%,
end RvAgv[8
A^,E~Z!x
% Compute the values of the polynomials: )jOa!E"
% -------------------------------------- `$/a-K}
y = zeros(length_r,length(n)); f- XUto
for j = 1:length(n) &b|RoPV
s = 0:(n(j)-m_abs(j))/2; Odo)h
pows = n(j):-2:m_abs(j); ?SNacN@r
for k = length(s):-1:1 <W#G)c0
p = (1-2*mod(s(k),2))* ... -*k2:i`
prod(2:(n(j)-s(k)))/ ... ~s+vJvWz
prod(2:s(k))/ ... bh@Ct nO
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... Yk|6?e{+)
prod(2:((n(j)+m_abs(j))/2-s(k))); i9L]h69r
idx = (pows(k)==rpowers); 1L*[!QT4
y(:,j) = y(:,j) + p*rpowern(:,idx); KyNu8s k
end _-C/sp^
xfeE D^?
if isnorm VZt%cq
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); mS'Ad<
end ^UKAD'_#%O
end C7dq=(p&
% END: Compute the Zernike Polynomials hV(^Y)f
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% C>Hdp_Lm
',R%Q0Q
% Compute the Zernike functions: &)OI!^ (
% ------------------------------ bN/8 ~!
idx_pos = m>0; IB&G#2M<
idx_neg = m<0; 7"'RE95
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z = y; :%h1Q>F
if any(idx_pos) :k_&Zd j,B
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); )pl5nu#<
end )vO"S
if any(idx_neg) \(pwHNSafk
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ];k!*lR)
end %nkP" Z#
* F%1~
% EOF zernfun