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 "TS
function z = zernfun(n,m,r,theta,nflag) +jP~s
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. IQH[Q9%
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 7~1IO|4t
% and angular frequency M, evaluated at positions (R,THETA) on the ~9\zWRh
% unit circle. N is a vector of positive integers (including 0), and 89~ =eY
% M is a vector with the same number of elements as N. Each element Ysi
g T
% k of M must be a positive integer, with possible values M(k) = -N(k) a%vrt)Gx
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, en*d/>OVJ
% and THETA is a vector of angles. R and THETA must have the same E?)656F[
% length. The output Z is a matrix with one column for every (N,M) sJG5/w
% pair, and one row for every (R,THETA) pair. 58V[mlW)O0
% 9`Q<Yy"du
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike Ts$@s^S]
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), >[10H8~bI/
% with delta(m,0) the Kronecker delta, is chosen so that the integral MXD4|r(
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ,*I@
% and theta=0 to theta=2*pi) is unity. For the non-normalized 3oy~=
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. w5=tlb
% %A/_5;PZ/
% The Zernike functions are an orthogonal basis on the unit circle. Q{g;J`Z)p
% They are used in disciplines such as astronomy, optics, and h"+ `13
% optometry to describe functions on a circular domain. tBATZ0nK`Q
% I=DxRgt
% The following table lists the first 15 Zernike functions. zj{r^D$
% XT>.`, sv
% n m Zernike function Normalization qJ4T]FVN
% -------------------------------------------------- Zw1U@5}A
% 0 0 1 1 rN)V[5R#M
% 1 1 r * cos(theta) 2 J% H;%ROx
% 1 -1 r * sin(theta) 2 [K/m
% 2 -2 r^2 * cos(2*theta) sqrt(6) _~u2: yl(
% 2 0 (2*r^2 - 1) sqrt(3) IiBD?}
% 2 2 r^2 * sin(2*theta) sqrt(6) PxFWJ?=
% 3 -3 r^3 * cos(3*theta) sqrt(8) bi4f]^hQz
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) aGZi9O7G}
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) b</9Ai=
% 3 3 r^3 * sin(3*theta) sqrt(8) Vr[czfROz'
% 4 -4 r^4 * cos(4*theta) sqrt(10) cvd\/pG)
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) -_C#wtC
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) 1BHG'y
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) `Z"Q^
% 4 4 r^4 * sin(4*theta) sqrt(10) :#~U<C@o
% -------------------------------------------------- <
0M:"^f
% "XgmuSQ!
% Example 1: !~]<$WZV
% 5q9s,r_
% % Display the Zernike function Z(n=5,m=1) 7Z ;?b0W
% x = -1:0.01:1; Fs_]RfG
% [X,Y] = meshgrid(x,x); %UUH"
% [theta,r] = cart2pol(X,Y); z!;1i[|x
% idx = r<=1; 8mT M$#\
% z = nan(size(X)); 66x?A0P
% z(idx) = zernfun(5,1,r(idx),theta(idx)); ",aTWQgN
% figure mrIh0B:`
% pcolor(x,x,z), shading interp dovZ#D@Q
% axis square, colorbar x<Vm5j
% title('Zernike function Z_5^1(r,\theta)') M-)RQ-h
% <@wj7\pQ
% Example 2: L( T12s
% =OIw*L8C"I
% % Display the first 10 Zernike functions ui q^|5Z
% x = -1:0.01:1; m7EcnQf
% [X,Y] = meshgrid(x,x); ;Gx)Noo/>
% [theta,r] = cart2pol(X,Y); /sM~Uq?
% idx = r<=1; xx{!3 F
% z = nan(size(X)); J^R=dT!
% n = [0 1 1 2 2 2 3 3 3 3]; 0Wa}<]:^
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; o<IAeH {+
% Nplot = [4 10 12 16 18 20 22 24 26 28]; )-*5v
D
% y = zernfun(n,m,r(idx),theta(idx)); cdqB,]"
% figure('Units','normalized') dL7E<?l
% for k = 1:10 bVP"(H]
% z(idx) = y(:,k); !Z
VU,b>
% subplot(4,7,Nplot(k)) xGTP;NT_H
% pcolor(x,x,z), shading interp kmzH'wktt
% set(gca,'XTick',[],'YTick',[]) lj+u@Z<xA
% axis square V%$/#sza
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) pym!U@$t
% end 4DZ-bt'
% 0TpK#OlI|c
% See also ZERNPOL, ZERNFUN2. Z{&cuo.@<]
D}8EER b
% Paul Fricker 11/13/2006 Eu"_MgD
|5Xq0nvCe
>pUtwIP
% Check and prepare the inputs: ODZ|bN0>
% ----------------------------- {( r6e
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) UAoh`6vFF8
error('zernfun:NMvectors','N and M must be vectors.') &0f5:M{P
end ;WR,eI..
F:x [
if length(n)~=length(m) dOa%9[
error('zernfun:NMlength','N and M must be the same length.') :
]C~gc
end k)EX(T\
4apL4E"r
n = n(:); jLg9H/w{
m = m(:); ]_N|L|]M
if any(mod(n-m,2)) cnTaJ/o
error('zernfun:NMmultiplesof2', ... oudxm[/U
'All N and M must differ by multiples of 2 (including 0).') )GHq/:1W
end M4as
w@,zFV
if any(m>n) E>l~-PaZY
error('zernfun:MlessthanN', ... \rv<$d@L
'Each M must be less than or equal to its corresponding N.') H;RwO@v
end @S|XGf
|i++0BU
if any( r>1 | r<0 ) iLSr*`
o
error('zernfun:Rlessthan1','All R must be between 0 and 1.') m *JaXa
end 4?B\O`sy.
|\pbir
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) %c4Hse#Y
error('zernfun:RTHvector','R and THETA must be vectors.') 82l~G;.n3
end ` V##Y
O%bEB g
r = r(:); gEjdN.
theta = theta(:); d3xmtG {i
length_r = length(r); J{Q|mD=
if length_r~=length(theta) #\=F O>
error('zernfun:RTHlength', ... ^0Mt*e{q
'The number of R- and THETA-values must be equal.') `nu''B
H
end u?C#4
E>K!Vrh-L
% Check normalization: ov, hI>0!D
% -------------------- A}l3cP;
`#
if nargin==5 && ischar(nflag) wpN=,&!
isnorm = strcmpi(nflag,'norm'); >7 ="8
if ~isnorm %^jMj2
error('zernfun:normalization','Unrecognized normalization flag.') LGn:c;
end 6Yln,rC
else RCpR3iC2
isnorm = false; ]9^sa-8
end %KLpig
PpzP 7
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {tWf
% Compute the Zernike Polynomials D A\2rLs
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% m^zUmrj[
K|epPGRr
% Determine the required powers of r: `x*Pof!Io
% ----------------------------------- YuO.yh_
m_abs = abs(m); z:wutqru
rpowers = []; wfH^<jY)E
for j = 1:length(n) iUN Ib
rpowers = [rpowers m_abs(j):2:n(j)]; F'21jy&
end ,0!}7;j_c
rpowers = unique(rpowers); lNYt`xp
)?anOD[
% Pre-compute the values of r raised to the required powers, ;>Ib^ov
% and compile them in a matrix: xA$XT[D
% ----------------------------- 2fL;-\!y(
if rpowers(1)==0 dl.p\t(1
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ,
K~}\CR
rpowern = cat(2,rpowern{:}); 5j?3a1l0
rpowern = [ones(length_r,1) rpowern]; _z|65H
else >G25m'&,7
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); gi1^3R[
rpowern = cat(2,rpowern{:}); gtppv6<Mj4
end ;@oN s-
ZbdZrE$
% Compute the values of the polynomials: m+]K;}.}R
% -------------------------------------- NXrJfp
y = zeros(length_r,length(n)); 3EPv"f^V
for j = 1:length(n) ?Lk)gO^C
s = 0:(n(j)-m_abs(j))/2; a.k.n<
pows = n(j):-2:m_abs(j); :74y!
for k = length(s):-1:1 u 7>],<
p = (1-2*mod(s(k),2))* ... ig/xv
prod(2:(n(j)-s(k)))/ ... m;GCc8
prod(2:s(k))/ ... zHM(!\8K
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... #Lh;CSS
prod(2:((n(j)+m_abs(j))/2-s(k))); 8}O lL,fP
idx = (pows(k)==rpowers); +nFu|qM}
y(:,j) = y(:,j) + p*rpowern(:,idx); nksLWfpG?B
end v dc\R?
hcsP2
0s
if isnorm rlOAo`hd
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); B|C2lu
end _@
qjV~%Sy
end iP ->S\
% END: Compute the Zernike Polynomials w;4<h8Wn5
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% <L8'! q}
*k.G5>@
% Compute the Zernike functions: ;n*.W|Uph
% ------------------------------ S%Uutj\/W
idx_pos = m>0; aC8} d
idx_neg = m<0; -=)H{
f<d`B]$(
z = y; 2DrP"iGq5
if any(idx_pos) p>v$FiV2N
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); $9_xGfx}
end *av<E
if any(idx_neg) B9jC?I |`
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); h+g_rvIG*
end @=}0`bE
BYL)nCc
% EOF zernfun