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 9R8q+2
function z = zernfun(n,m,r,theta,nflag) 5@>hjXi"Y
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. &2u
|7U.
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ]'Eg2(wy
% and angular frequency M, evaluated at positions (R,THETA) on the <J+Oh\8tad
% unit circle. N is a vector of positive integers (including 0), and xK_UkB-$i
% M is a vector with the same number of elements as N. Each element V WZpEi
% k of M must be a positive integer, with possible values M(k) = -N(k) ^AU-hVj
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, [ K/l;Zd
% and THETA is a vector of angles. R and THETA must have the same T2Z$*;,>T
% length. The output Z is a matrix with one column for every (N,M) ,V
52Fj
% pair, and one row for every (R,THETA) pair. N}U+K
% VC/n}7p
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike GYQ:G=
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), of*T,MUI
% with delta(m,0) the Kronecker delta, is chosen so that the integral 5B:"$vC{=
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, #sCR}
% and theta=0 to theta=2*pi) is unity. For the non-normalized K
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% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. DlCN
% 1W>/4l
% The Zernike functions are an orthogonal basis on the unit circle. K>.}>)0
% They are used in disciplines such as astronomy, optics, and 9~Sa7P
% optometry to describe functions on a circular domain. el5Pe{j'
% V.`hk^V,
% The following table lists the first 15 Zernike functions. Q +l{> sL
% j7&#R+f
% n m Zernike function Normalization )x\%*ewY
% -------------------------------------------------- m,+PYq
% 0 0 1 1 E8kD#tL
% 1 1 r * cos(theta) 2 9{fP.ifdv7
% 1 -1 r * sin(theta) 2 m33&obSP
% 2 -2 r^2 * cos(2*theta) sqrt(6) iSf%N>y'K
% 2 0 (2*r^2 - 1) sqrt(3) W gyRK2#!
% 2 2 r^2 * sin(2*theta) sqrt(6) d>F7i~W
% 3 -3 r^3 * cos(3*theta) sqrt(8) mr}o0@5av
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) KB~[nZs7
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) -'miM ~kG[
% 3 3 r^3 * sin(3*theta) sqrt(8) kXhd]7ru
% 4 -4 r^4 * cos(4*theta) sqrt(10) Y_n/rD>
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ^jL)<y4`
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) [\Wl~
a l
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ~\-=q^/!
% 4 4 r^4 * sin(4*theta) sqrt(10) Ynf "g#(
% -------------------------------------------------- fsOlg9
% 51eZf JB
% Example 1: am/}V%^
% +arh/pd_I
% % Display the Zernike function Z(n=5,m=1) 3"Oipt+
% x = -1:0.01:1; e^q^AP+*
% [X,Y] = meshgrid(x,x); _hV34:1F
% [theta,r] = cart2pol(X,Y); L>/$l(
% idx = r<=1; C&0f8PnD
% z = nan(size(X)); ^3sv2wh^|8
% z(idx) = zernfun(5,1,r(idx),theta(idx)); qdk!.A{
% figure 2d|^$$#`
% pcolor(x,x,z), shading interp FDuA5At
% axis square, colorbar 4IZAJqw(*
% title('Zernike function Z_5^1(r,\theta)') h/C{
% [MAPa
% Example 2: Iw[zN[oz
% %6fnL~A
% % Display the first 10 Zernike functions ]EF"QLNN(
% x = -1:0.01:1; $Xo_8SX,
% [X,Y] = meshgrid(x,x); -{*3<2rFK
% [theta,r] = cart2pol(X,Y); ;ja~Q .}4
% idx = r<=1; 4mW$+lzn
% z = nan(size(X)); iDDq<a.A
% n = [0 1 1 2 2 2 3 3 3 3]; V+sZ;$
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; ;/Y#ph[
% Nplot = [4 10 12 16 18 20 22 24 26 28]; }L Q%%
% y = zernfun(n,m,r(idx),theta(idx)); 2IHS)kkT|
% figure('Units','normalized') _\dC<K *>
% for k = 1:10 F6CuY$0m=
% z(idx) = y(:,k); V
7~ 9z\lW
% subplot(4,7,Nplot(k)) ]Y$Wv9S6
% pcolor(x,x,z), shading interp 'Sd+CXS
% set(gca,'XTick',[],'YTick',[]) D3g5#.$,}>
% axis square jm&[8ApW
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 76[qFz
% end ok ,O/|E}?
% ByoI+n* U
% See also ZERNPOL, ZERNFUN2. -|#/KKF
\s8h.xjU
% Paul Fricker 11/13/2006 kQ\l7xd
cJm},
B;Z _'.i,d
% Check and prepare the inputs: +{6:]
% ----------------------------- e"EGqn&!
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) _{if"
error('zernfun:NMvectors','N and M must be vectors.') -k>k<bDAI
end 4Z{R36 {
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if length(n)~=length(m) xZq, kP^
error('zernfun:NMlength','N and M must be the same length.') &> .QDO
end c;29GHs2
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n = n(:); Y -BZV |
m = m(:); o^uh3,.
if any(mod(n-m,2)) GdScYAC
error('zernfun:NMmultiplesof2', ... t,w/L*r+w
'All N and M must differ by multiples of 2 (including 0).') c-7Zk!LfD
end pIm ]WNX(
b+AxTe("
if any(m>n) N-}OmcO]e
error('zernfun:MlessthanN', ... 9-A@2&J1
'Each M must be less than or equal to its corresponding N.') rmX5-k
end g-x;a0MQx
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if any( r>1 | r<0 ) ga1b%5]v.
error('zernfun:Rlessthan1','All R must be between 0 and 1.') o+^e+ptc
end <VN< ~sz
HF&dHD2f
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) <T'fJcR
error('zernfun:RTHvector','R and THETA must be vectors.') 0^2e^qf
end Zia6m[ ^Q
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r = r(:); `x L@%
theta = theta(:); NXOvC!<
length_r = length(r); LBpAR|
if length_r~=length(theta) F')E)tV
error('zernfun:RTHlength', ... I2z7}*<u
'The number of R- and THETA-values must be equal.') Vhm^<I-d
end u91
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% Check normalization: ]/R>nT
% -------------------- >NpW$P{'
if nargin==5 && ischar(nflag) `Sj8IxO
isnorm = strcmpi(nflag,'norm'); @X/-p3729
if ~isnorm &t@ $]m(
error('zernfun:normalization','Unrecognized normalization flag.') hL;??h,!_
end A}"uEk(R
else ?K.!^G
isnorm = false; </fTn_{2s8
end G<*h,'B
b]8\%=d
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ws]d,]
% Compute the Zernike Polynomials Y(R],9h8
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% DEv,!8
b DF_
% Determine the required powers of r: wp/x|AV
% ----------------------------------- !\&4,l(
m_abs = abs(m); +hT9V1'-D
rpowers = []; xJvalb
for j = 1:length(n) P_@ty~u
rpowers = [rpowers m_abs(j):2:n(j)]; @6b;sv1W
end 8,m:
rpowers = unique(rpowers); ?H!X
p
Ga*
% Pre-compute the values of r raised to the required powers, LGCeYXic
% and compile them in a matrix: NL ceBok
% ----------------------------- cja-MljD
if rpowers(1)==0 Rn whkb&&
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); Y7yzM1?t
rpowern = cat(2,rpowern{:}); m)<N:|
rpowern = [ones(length_r,1) rpowern]; tkix@Q!;\
else qSVg.<+
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); <DdzDbgax
rpowern = cat(2,rpowern{:}); E~Y%x/oX
end z<<aT
ewinG-hX_
% Compute the values of the polynomials: o\yqf:V8
% -------------------------------------- rmnnV[@o
y = zeros(length_r,length(n)); =u&NdMy
for j = 1:length(n) }% ?WS
s = 0:(n(j)-m_abs(j))/2; 23UXOY0BW
pows = n(j):-2:m_abs(j); `VOLw*Ci
for k = length(s):-1:1 Z~7}
p = (1-2*mod(s(k),2))* ... e}"k8 ./
prod(2:(n(j)-s(k)))/ ... m9m~ 2
prod(2:s(k))/ ... ^m^,:]I0P
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... "}UYsXg
prod(2:((n(j)+m_abs(j))/2-s(k))); \cq.M/p
idx = (pows(k)==rpowers); %u$dN9cw
y(:,j) = y(:,j) + p*rpowern(:,idx); O[')[uo8s
end }pPt- k
i>rsq[l
if isnorm [k6,!e[/uG
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); C)s*1@af
end ["?WVXCF8|
end j(=zc6m
% END: Compute the Zernike Polynomials u]Y NF[]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% N_8L8ds5
: ]JsUb{YK
% Compute the Zernike functions: C}mWX7<Z.
% ------------------------------ 9!6yo
idx_pos = m>0; K,GX5c5
idx_neg = m<0; 1HNX6
vro5G')
z = y; }\\6"90g*
if any(idx_pos) r;aP`MVO<
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); _b 8XF&O
end ?GGh )";y
if any(idx_neg) zn{[]J
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); g7W\
&
end EC|b7
j~Xn\~*n
% EOF zernfun