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 +aF}oA&X[
function z = zernfun(n,m,r,theta,nflag) O=SkAsim
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. %AOja+
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N MX4]Vpv
% and angular frequency M, evaluated at positions (R,THETA) on the PP:(EN1
% unit circle. N is a vector of positive integers (including 0), and r]3'74j:
% M is a vector with the same number of elements as N. Each element E*L iM5+I
% k of M must be a positive integer, with possible values M(k) = -N(k) N]KxAttt
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, _k8A$s<d
% and THETA is a vector of angles. R and THETA must have the same lEHzyh}2k
% length. The output Z is a matrix with one column for every (N,M) [7_56\G4
% pair, and one row for every (R,THETA) pair. yV_4?nh
% p!k7C&]E
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike lds-T
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 54
> -
% with delta(m,0) the Kronecker delta, is chosen so that the integral vad12WrG<
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, >.dWjb6t
% and theta=0 to theta=2*pi) is unity. For the non-normalized \J+*
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. "4vy lHIo
% *@d&5
% The Zernike functions are an orthogonal basis on the unit circle. 3~nnCR[R
% They are used in disciplines such as astronomy, optics, and *tm0R> ?!
% optometry to describe functions on a circular domain. Y0D}g3`
% PJ cwH6m
% The following table lists the first 15 Zernike functions. \(t@1]&jw
% %tG*C,l]
% n m Zernike function Normalization Gmf B
% -------------------------------------------------- el:9 wq
% 0 0 1 1 8]&i-VFof
% 1 1 r * cos(theta) 2 +}f9
% 1 -1 r * sin(theta) 2 r5!/[_l
% 2 -2 r^2 * cos(2*theta) sqrt(6) @+ atBmt
% 2 0 (2*r^2 - 1) sqrt(3) fN'HE#W1Xa
% 2 2 r^2 * sin(2*theta) sqrt(6) nLV9<M
Zm
% 3 -3 r^3 * cos(3*theta) sqrt(8) !Hys3AP
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) WVY\&|)$
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) R(n^)^?
% 3 3 r^3 * sin(3*theta) sqrt(8) Bz5-ITX
% 4 -4 r^4 * cos(4*theta) sqrt(10) i1S>yV^l
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 2h[85\4
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) [HCAmnb
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) keB&Bjd&
% 4 4 r^4 * sin(4*theta) sqrt(10) .BFYY13H
% -------------------------------------------------- K_K5'2dE
% e["2QIOe
% Example 1: /z BxJT0
% F<!)4>2@
% % Display the Zernike function Z(n=5,m=1) NJNJjdD>
% x = -1:0.01:1; -?(E_^ng
% [X,Y] = meshgrid(x,x); 61xs%kxb..
% [theta,r] = cart2pol(X,Y); bQ~j=\[r
% idx = r<=1; +[5.WC7J
% z = nan(size(X)); -eX5z
% z(idx) = zernfun(5,1,r(idx),theta(idx)); da (km+
% figure !qX_I db\
% pcolor(x,x,z), shading interp ~#kT_*sw)
% axis square, colorbar UKM2AZ0lb
% title('Zernike function Z_5^1(r,\theta)') uL[.ND2._&
% 5Kkdo!z
% Example 2: ve\X3"p#
% WJ_IuX51'
% % Display the first 10 Zernike functions _6wFba@>/n
% x = -1:0.01:1; w:
>5=mfk
% [X,Y] = meshgrid(x,x); -%L6#4m4o
% [theta,r] = cart2pol(X,Y); 1 5A*7|
% idx = r<=1; }!6\|;Qsz,
% z = nan(size(X)); a{[x4d,z
% n = [0 1 1 2 2 2 3 3 3 3]; g55`A`5%C
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; _cu:aktf2
% Nplot = [4 10 12 16 18 20 22 24 26 28]; TC<@e<-%Sq
% y = zernfun(n,m,r(idx),theta(idx)); 1AU#%wIEP
% figure('Units','normalized') R+Y4|
% for k = 1:10 `3:.??7N
% z(idx) = y(:,k); >Jp:O
7
% subplot(4,7,Nplot(k)) x:nKfY5
% pcolor(x,x,z), shading interp =9j8cC5y
% set(gca,'XTick',[],'YTick',[]) P{u0ftyX}
% axis square d9q(xZ5
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) v'e[GB0
% end EOm:!D\
% VO"("7L
% See also ZERNPOL, ZERNFUN2. C*`mM'#
w+N> h;j
% Paul Fricker 11/13/2006 3"O>&Q0c
]8T!qS(UJd
;$z$@@WC
% Check and prepare the inputs: )HvnoUO0
% ----------------------------- "I
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if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) kcQ'$<Mz<
error('zernfun:NMvectors','N and M must be vectors.') O9r>E3-q
end 95z]9UL
{Lm~r+
U
if length(n)~=length(m) qM.bF&&Go
error('zernfun:NMlength','N and M must be the same length.') lv]hTH 4T
end <A#
l
35
3"P }n
n = n(:); ?2oHZ%G
m = m(:); .B\ 5OI,]
if any(mod(n-m,2)) P><o,s"v
error('zernfun:NMmultiplesof2', ... PTEHP
'All N and M must differ by multiples of 2 (including 0).') _vZ"4L+Iw+
end W16,Alf:
LU9A#
if any(m>n) l\s U
error('zernfun:MlessthanN', ... !=N"vD*
'Each M must be less than or equal to its corresponding N.') CjiVnWSz<
end u{*SX k
YJo["Q
if any( r>1 | r<0 ) phgm0D7
error('zernfun:Rlessthan1','All R must be between 0 and 1.') $ mI0Bk
end }oNhl^JC
2/0v B>
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) L>YU,I\o
error('zernfun:RTHvector','R and THETA must be vectors.') oIefw:FE,a
end rp0ZvEX
|gU(s
r = r(:); }6@pJG
theta = theta(:); K=,F#kn
length_r = length(r); IEzaK
if length_r~=length(theta) ,JEFGI{
error('zernfun:RTHlength', ... ;dzL}@we
'The number of R- and THETA-values must be equal.') sxt-Vs7+6
end ka3u&3"
u5Ftu?t
% Check normalization: r3\cp0P;s
% -------------------- sx`O8t
if nargin==5 && ischar(nflag) QI3Nc8t_2
isnorm = strcmpi(nflag,'norm'); |0%+wB
if ~isnorm P<f5*L#HD
error('zernfun:normalization','Unrecognized normalization flag.') ^/U|2'$'>E
end 1Y]TA3:
else Grk@dZI
isnorm = false; jb^N|zb
end \xS&v7b
48*Do}l]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% k0Uyf~p~
% Compute the Zernike Polynomials -
h9?1vc7
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% d{E}6)1=
7__Q1>o
% Determine the required powers of r: 7IjQi=#:
% ----------------------------------- 7Ddaf>
m_abs = abs(m); mljh|[
rpowers = []; 1Q.\s_2
for j = 1:length(n) P[k$vD
rpowers = [rpowers m_abs(j):2:n(j)]; uI DuGrt
end KFFSv{m[
rpowers = unique(rpowers); kVy\b E0o
H(&4[%;MP
% Pre-compute the values of r raised to the required powers, \}
^E`b
% and compile them in a matrix: :"!9_p(,,
% ----------------------------- >z.<u|r2
if rpowers(1)==0 /*c\qXA5
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 1M}&Z H
rpowern = cat(2,rpowern{:}); 1 %,a =,v
rpowern = [ones(length_r,1) rpowern]; txPIG/
else -P]sRl3O;
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); h@LHRMO
rpowern = cat(2,rpowern{:}); F<(i.o(
end *>+,(1Fz
= hN
!;7G
% Compute the values of the polynomials: Qx'`PNU9\
% -------------------------------------- /0eYMG+K=
y = zeros(length_r,length(n)); J:kmqk!
for j = 1:length(n) @, W vvh
s = 0:(n(j)-m_abs(j))/2; T0]*{k(FR
pows = n(j):-2:m_abs(j); w&x!,yd;
for k = length(s):-1:1 dS5a
p = (1-2*mod(s(k),2))* ... <!pvqNApg
prod(2:(n(j)-s(k)))/ ... HX6Ma{vBk
prod(2:s(k))/ ... Y}vr>\
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... gB4U*D0[e~
prod(2:((n(j)+m_abs(j))/2-s(k))); 4NdN<#Lr
idx = (pows(k)==rpowers); 5T:i9h
y(:,j) = y(:,j) + p*rpowern(:,idx); bHI<B)=`
end u@4V7;L
kWrp1`
if isnorm q]\g,a
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); c~v~2DM
end gc?#pP
end (k|_J42[
% END: Compute the Zernike Polynomials <Engi!
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% UAyC.$!
>(snII
% Compute the Zernike functions: &RTX6%'KY
% ------------------------------ 51QRM32Y
idx_pos = m>0;
$/7pYl\n
idx_neg = m<0; pm6>_Kz
A.5i"Ci[ie
z = y; 3ux0Jr2yT
if any(idx_pos) \{EpduwZ
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); "XT"|KF|D
end R+7oRXsu
if any(idx_neg) > z^#
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); {b@KYR9K
end {N#KkYH{"
A mwa)
% EOF zernfun