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 9nR\7!_
function z = zernfun(n,m,r,theta,nflag) ;wwc;wQ'
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. /p
!A:8
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N PYCN3s#Gi
% and angular frequency M, evaluated at positions (R,THETA) on the #8Bh5L!SJ1
% unit circle. N is a vector of positive integers (including 0), and 2>o[
% M is a vector with the same number of elements as N. Each element | N/d}
% k of M must be a positive integer, with possible values M(k) = -N(k) >V6t
L;+
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, OjGI
!
% and THETA is a vector of angles. R and THETA must have the same -Q20af-
% length. The output Z is a matrix with one column for every (N,M) G^.N$wcv
% pair, and one row for every (R,THETA) pair. D0Q9A]bD;
% ^8
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% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike jrp>Y:
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), u''Ce`N
% with delta(m,0) the Kronecker delta, is chosen so that the integral =v:?rY}
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, T]tP!a;K
% and theta=0 to theta=2*pi) is unity. For the non-normalized Cx TAd[az
% polynomials, max(Znm(r=1,theta))=1 for all [n,m].
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% rI>x'0Go*
% The Zernike functions are an orthogonal basis on the unit circle. $yx\2
% They are used in disciplines such as astronomy, optics, and eIvZhi
% optometry to describe functions on a circular domain. ` @ QZK0Ox
% :;_
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% The following table lists the first 15 Zernike functions. zM0}(5$m
% i(.e=
% n m Zernike function Normalization ei5YxV6I
% -------------------------------------------------- F{x+1hct0
% 0 0 1 1 8 W
% 1 1 r * cos(theta) 2 qPPe)IM'Sc
% 1 -1 r * sin(theta) 2 Wk[a|>
% 2 -2 r^2 * cos(2*theta) sqrt(6) FI^Wh7J
% 2 0 (2*r^2 - 1) sqrt(3) cKVFykwM
% 2 2 r^2 * sin(2*theta) sqrt(6) M/ 64`lcb
% 3 -3 r^3 * cos(3*theta) sqrt(8) VlV
X
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) -DkD*64wu
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8)
y-hTTd"{
% 3 3 r^3 * sin(3*theta) sqrt(8) 'C5id7O&
% 4 -4 r^4 * cos(4*theta) sqrt(10) ':n`0+Eh
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) |S!RQ-CF
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) o898pg
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) j:%,lcF
% 4 4 r^4 * sin(4*theta) sqrt(10) ~uhW~bT
% -------------------------------------------------- `jeATxWv
% xeF>"6\
% Example 1: YYT;a$GTo
% 'APx
% % Display the Zernike function Z(n=5,m=1) Pxl, "
% x = -1:0.01:1; Z:{|
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% [X,Y] = meshgrid(x,x); `Abd=1nH
% [theta,r] = cart2pol(X,Y); ,SIS3A>s
% idx = r<=1; "}3sL#|z
% z = nan(size(X)); pN^g.
% z(idx) = zernfun(5,1,r(idx),theta(idx)); m.+h@
% figure hNXZL>6
% pcolor(x,x,z), shading interp JRB6T _U
% axis square, colorbar sTd@/>S?p
% title('Zernike function Z_5^1(r,\theta)') L -Q8iFW'
% ?-j/X6(\(
% Example 2: tl_3 %$s
% DzR,ou
% % Display the first 10 Zernike functions e(s0mbJE
% x = -1:0.01:1; $z_yx
`5
% [X,Y] = meshgrid(x,x); atZe`0
% [theta,r] = cart2pol(X,Y); 5M=U*BI
% idx = r<=1; N}5
% z = nan(size(X)); JL,Y9G*]s
% n = [0 1 1 2 2 2 3 3 3 3]; S})f`X9_}
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; 6)1PDlB
% Nplot = [4 10 12 16 18 20 22 24 26 28]; }F]Z1('
% y = zernfun(n,m,r(idx),theta(idx)); U$5x#{AFp
% figure('Units','normalized') fnX[R2KZ
% for k = 1:10 8oE`>Y
% z(idx) = y(:,k); {H/%2
% subplot(4,7,Nplot(k)) {|oWU8.l
% pcolor(x,x,z), shading interp u4hn9**a1
% set(gca,'XTick',[],'YTick',[]) suQTi'K1
% axis square DCp8rvUI
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) _S) K+C|@
% end N5K(yY_T
% brTNwRze
% See also ZERNPOL, ZERNFUN2. a]Pi2:S
6c!F%xU}
% Paul Fricker 11/13/2006 }aOqoi7w
F`4W5~`
eZ:iW#YF
% Check and prepare the inputs: )<HvIr(xr
% ----------------------------- `!cdxKLR
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) d*|RFU
error('zernfun:NMvectors','N and M must be vectors.') y CHOg
end 4Wgzp51Aq!
qeM DC#N
if length(n)~=length(m) [.>=>KJ_
error('zernfun:NMlength','N and M must be the same length.') 80Y%C-Y:
end =+_nVO*
.iV=ybMT
n = n(:); P
DY :?/
m = m(:); fYuSfB+<
if any(mod(n-m,2)) Do(G;D`h+_
error('zernfun:NMmultiplesof2', ... !%$[p'
'All N and M must differ by multiples of 2 (including 0).') Y*@7/2,
end sq=EL+=j
CE
M4E
if any(m>n) A o*IshVh
error('zernfun:MlessthanN', ... [NE!
'Each M must be less than or equal to its corresponding N.') d_(>:|oh
end c9c]1XJ
@Nb/n
if any( r>1 | r<0 ) hRXnig{;3
error('zernfun:Rlessthan1','All R must be between 0 and 1.') J t.<Z&
end I._ A
/
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) egs P\ '
error('zernfun:RTHvector','R and THETA must be vectors.') /
^)3V}
end (P?|Bk[
:sw5@JdJ
r = r(:); *i*\dl
theta = theta(:); *JImP9SE
length_r = length(r); 3]1 !g6
if length_r~=length(theta) +E9G"Z65iP
error('zernfun:RTHlength', ... V^tD@N
'The number of R- and THETA-values must be equal.') |};d:LwX
end .pZYPKMaE
$MvKwQ/
% Check normalization: W\j'8^kI9
% -------------------- Q^<amM!
if nargin==5 && ischar(nflag) f'ld6jt|%
isnorm = strcmpi(nflag,'norm'); VEa"^{,w
if ~isnorm &(<>}
r
error('zernfun:normalization','Unrecognized normalization flag.') +h-% {
end t$qIJt$
else \roJf&O }
isnorm = false; jhz*Y}MX
end VS4Glx73
Ib{#dhV
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 3_G0eIE"u
% Compute the Zernike Polynomials $-^&AKc
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +D
@B eQu
sh,4n{+
% Determine the required powers of r: enxb
pq#
% ----------------------------------- V%[t'uh
m_abs = abs(m); >4bw4
Z1
rpowers = []; \a0{9Xx F
for j = 1:length(n) q8ZxeMqx%
rpowers = [rpowers m_abs(j):2:n(j)]; |5>A^a
end J|jvqt9C
rpowers = unique(rpowers); tHaHBx1P
+EA ")T<l
% Pre-compute the values of r raised to the required powers, 8=bn
TJf
% and compile them in a matrix: ?$)a[UnqX
% ----------------------------- cb'Ya_
if rpowers(1)==0 6VQQI9
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); F+VNrt-
rpowern = cat(2,rpowern{:}); 1
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rpowern = [ones(length_r,1) rpowern]; xxzUey
else QNE/SSL
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ;*K;)C
rpowern = cat(2,rpowern{:}); **-rPonM[
end =ZoNkj/^,
'H`:c+KDG`
% Compute the values of the polynomials: 5WHqD!7u
% -------------------------------------- KiMlbF.~V
y = zeros(length_r,length(n)); vS ( Y_6
for j = 1:length(n) +(`D'5EB(
s = 0:(n(j)-m_abs(j))/2; G \a`F'Oo
pows = n(j):-2:m_abs(j); HQF@@
for k = length(s):-1:1 B.?F^m@zS
p = (1-2*mod(s(k),2))* ... %qJgtu"8
prod(2:(n(j)-s(k)))/ ... KBi(Ns#+
prod(2:s(k))/ ... {B#w9>'b
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... N:'GNMu
prod(2:((n(j)+m_abs(j))/2-s(k))); DB|w&tygq
idx = (pows(k)==rpowers); LdOqV'&r
y(:,j) = y(:,j) + p*rpowern(:,idx); *Q2 oc:6
end Tw%1m
o=7e8l
if isnorm Dg~m}La
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 6 ym$8^
end hX,RuI
end #v$wjqK5
% END: Compute the Zernike Polynomials RI<smt.Ng
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% _8SB+s*
Qa2p34Z/
% Compute the Zernike functions: C_;nlG6
% ------------------------------ Y1AZ%{^0a
idx_pos = m>0; uZf
6W<a
idx_neg = m<0; m' j1
OP= oSfa
z = y; V"gKk$j7
if any(idx_pos) $M,Q"QL
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); n}9<7e~/
end ZJFF4($qN
if any(idx_neg) 8)s0$64Ra
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); $AZ=;iP-
end }"RVUYU
DIP%*b#l$\
% EOF zernfun