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 om]4BRe
function z = zernfun(n,m,r,theta,nflag) Y_]De3:V0B
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. T}4/0yR2
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N A0fFv+RN3
% and angular frequency M, evaluated at positions (R,THETA) on the JqMDqPIQ
% unit circle. N is a vector of positive integers (including 0), and D`xHD#j h
% M is a vector with the same number of elements as N. Each element cKn`/\.H
% k of M must be a positive integer, with possible values M(k) = -N(k) ]ix!tb.Q
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, me'd6!O9-
% and THETA is a vector of angles. R and THETA must have the same
zcva-ze:;
% length. The output Z is a matrix with one column for every (N,M) g7&9"
% pair, and one row for every (R,THETA) pair. YSj+\Z$(
% 8X I?
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike &m[Qn!>i6
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 3;8!rNN
% with delta(m,0) the Kronecker delta, is chosen so that the integral Dc+'<"
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 9JWa$iBH@
% and theta=0 to theta=2*pi) is unity. For the non-normalized )x]3Zq
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 8Th` ]tI
% tqy@iEz+
% The Zernike functions are an orthogonal basis on the unit circle. in(U:04
% They are used in disciplines such as astronomy, optics, and EZYBeqv
% optometry to describe functions on a circular domain. Q6XRsFc
% bcAvM;
% The following table lists the first 15 Zernike functions.
!xwG%{_
% .cR
-V`
% n m Zernike function Normalization ki{3IEOr}
% -------------------------------------------------- JKX_q&bUw
% 0 0 1 1 /[9t`
% 1 1 r * cos(theta) 2 %eJGte-
% 1 -1 r * sin(theta) 2 0jzbG]pc:E
% 2 -2 r^2 * cos(2*theta) sqrt(6) Raw)9tUt
% 2 0 (2*r^2 - 1) sqrt(3) -_<}$9lz
% 2 2 r^2 * sin(2*theta) sqrt(6) vAX|hwn;
% 3 -3 r^3 * cos(3*theta) sqrt(8) 9W8]8sUeG
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) &EM\CjKv"
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) 7c;9$j
% 3 3 r^3 * sin(3*theta) sqrt(8) ,&d@O>$E:
% 4 -4 r^4 * cos(4*theta) sqrt(10) [y0O{,lI
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ~l$3uN[g
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) XTd3|Pm
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) @G:V
% 4 4 r^4 * sin(4*theta) sqrt(10) h1(j2S`:
% -------------------------------------------------- (708H_
% DaH?@Q
% Example 1: NWd<+-pC6
% XUF\r]B,9
% % Display the Zernike function Z(n=5,m=1) 0[F:'_
% x = -1:0.01:1; @A+RVg*=
% [X,Y] = meshgrid(x,x); KE1ao9H8wR
% [theta,r] = cart2pol(X,Y); } %'bullT
% idx = r<=1; )I\=BPo|B
% z = nan(size(X)); vm'5s]kdh
% z(idx) = zernfun(5,1,r(idx),theta(idx)); m{7^EF
% figure qClHP)<
% pcolor(x,x,z), shading interp unJ R=~E
% axis square, colorbar S2>c#BQ
% title('Zernike function Z_5^1(r,\theta)') @VN&t:/ l
% L.T?}o
% Example 2: 4G@nZn
% ?DH"V7bs
% % Display the first 10 Zernike functions O}[PJfvBHo
% x = -1:0.01:1; w0ZLcND{
% [X,Y] = meshgrid(x,x); b7/AnSR~Jt
% [theta,r] = cart2pol(X,Y); xBFJ} v
% idx = r<=1; p7ir*r/2
% z = nan(size(X)); m'-|{c
% n = [0 1 1 2 2 2 3 3 3 3]; F3oQ^;xB
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; @R m-CWa
% Nplot = [4 10 12 16 18 20 22 24 26 28]; \*\R1_+
% y = zernfun(n,m,r(idx),theta(idx)); -B$~`2-
% figure('Units','normalized') @h?shW=^
% for k = 1:10 3M0+"l(X
% z(idx) = y(:,k); ~Z ~v
% subplot(4,7,Nplot(k)) j$da8] !
% pcolor(x,x,z), shading interp rtQHWRUn
% set(gca,'XTick',[],'YTick',[]) gq"k<C0
% axis square nJ?^?M'F%
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) dJ:MjQG`W
% end N4K8
u'f^
% WS2osBc
% See also ZERNPOL, ZERNFUN2. 7B3w\
NA$zd(
% Paul Fricker 11/13/2006 ,uz ]V1
}<jb vCeK
"&Qctk`<P
% Check and prepare the inputs: @mt0kV9
% ----------------------------- ZAuWx@}
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) )U:2z-X&e
error('zernfun:NMvectors','N and M must be vectors.') K~RoUE<3[
end }]UB;id'
L?Yoh<
if length(n)~=length(m) Q>qFM9Z
error('zernfun:NMlength','N and M must be the same length.') _)$PKOzbb
end 1\L[i];L8
pWE `x|J
n = n(:); |DF9cd^
m = m(:); -V %gVI[
if any(mod(n-m,2)) z5Qs@dG
error('zernfun:NMmultiplesof2', ... R)Mt(gFZT_
'All N and M must differ by multiples of 2 (including 0).') Oq(VvS/
end O)R(==P26P
wyxGe<1
if any(m>n) ;oH,~|K
error('zernfun:MlessthanN', ... iO1nwl !#
'Each M must be less than or equal to its corresponding N.') i;PL\Er:tX
end 4y}"Hy
MVCl.o
if any( r>1 | r<0 ) $mA5@O~C5\
error('zernfun:Rlessthan1','All R must be between 0 and 1.') %ICglF R
end 3UUGblg`~
7@%'wy&A
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) /"qcl7F
error('zernfun:RTHvector','R and THETA must be vectors.') $trAC@3O@
end |4Ck;gg!j
io4A>>W==/
r = r(:); o=fgin/E\
theta = theta(:); ~:sE:9$z
length_r = length(r); >x:EJV
if length_r~=length(theta) ^b?2N/m@
error('zernfun:RTHlength', ... +UWU|:
'The number of R- and THETA-values must be equal.') )wzV
$(~
end &217l2X
/
-dTLunv
% Check normalization: 9vGs;
% -------------------- 3mt%!}S
if nargin==5 && ischar(nflag) VFD%h
}
isnorm = strcmpi(nflag,'norm'); 10sK]XI
if ~isnorm \ SCy$,m
error('zernfun:normalization','Unrecognized normalization flag.') ~bA,GfSn0
end 0WxCSL$#I
else e5v`;(^M
isnorm = false; ek-!b!iI
end ^gro=Bp(
Ln#a<Rx.E7
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% GSVdb/+
% Compute the Zernike Polynomials rE!1wc>L
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% msTB'0
9 |:^k.
% Determine the required powers of r: [!*xO?yCJ
% ----------------------------------- M7y|EB))
m_abs = abs(m); {0jIY
rpowers = []; !DjT<dxf
for j = 1:length(n) cHvF* A
rpowers = [rpowers m_abs(j):2:n(j)]; \ a-CN>
end ddpl Pzm#
rpowers = unique(rpowers); Ns~&sE:
+QqH}=
M
% Pre-compute the values of r raised to the required powers, e 3@x*XI
% and compile them in a matrix: ]YD(`42 x
% ----------------------------- jD<pIHau
if rpowers(1)==0 E)'8U
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); wgd<3 X
rpowern = cat(2,rpowern{:}); } ~enEZ
rpowern = [ones(length_r,1) rpowern]; x6yW:tUG5
else R ZcH+?7
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); $-pbw@7
rpowern = cat(2,rpowern{:}); 0g(6r-2)7
end =&NOHT>
H*U`
% Compute the values of the polynomials: [QEwK|!L
% -------------------------------------- d?Y-;-|8Qh
y = zeros(length_r,length(n)); Sni=gZ K
for j = 1:length(n) {/UhUG
s = 0:(n(j)-m_abs(j))/2; ,w\ wQn>]K
pows = n(j):-2:m_abs(j); 03E3cp"
for k = length(s):-1:1 wL
eHQ]
p = (1-2*mod(s(k),2))* ... N~#D\X^t.
prod(2:(n(j)-s(k)))/ ... u(vw|nj`
prod(2:s(k))/ ... kV^?p
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... W8/(;K`/
prod(2:((n(j)+m_abs(j))/2-s(k))); lCFU1 GHH
idx = (pows(k)==rpowers); APHPN:v
y(:,j) = y(:,j) + p*rpowern(:,idx); Y1r,2 k
end 4]BJ0+|mT
lBiovT
if isnorm cF.mb*$K
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 8W{~wg`
end %h* 5xB]Tt
end EzP#Mnz^
% END: Compute the Zernike Polynomials NNX%Bq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ER<eX4oU
5#u.pu
% Compute the Zernike functions: >(tO
QeN
% ------------------------------ {E~l>Z88
idx_pos = m>0; u5 E/m
idx_neg = m<0; hDtKnF
3}4#I_<$F@
z = y; 1o#vhk/"+
if any(idx_pos) V4?Oc2mS
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); (5(fd.m+_
end C={mi#G[/
if any(idx_neg) C"No5r'K3
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Y(z}[`2
end zlMlMyG4
MgnE-6_c
% EOF zernfun