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 d>aZpJ[.
function z = zernfun(n,m,r,theta,nflag) T2Vj&EA@
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. 4+W}TKw
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N U^,ld`
% and angular frequency M, evaluated at positions (R,THETA) on the {#;6$dU;(
% unit circle. N is a vector of positive integers (including 0), and SOUA,4
% M is a vector with the same number of elements as N. Each element J*;t{M5
% k of M must be a positive integer, with possible values M(k) = -N(k) jAJkCCG
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, R]r~TJ o
% and THETA is a vector of angles. R and THETA must have the same 2N]y)S_<V
% length. The output Z is a matrix with one column for every (N,M) =_UPZ]
% pair, and one row for every (R,THETA) pair. -~aVt~{k/
% #A))#sT'R
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike M9N|Ql
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 2+^#<Uok
% with delta(m,0) the Kronecker delta, is chosen so that the integral |4'E&(BU-
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, tl4;2m3w
% and theta=0 to theta=2*pi) is unity. For the non-normalized z^oi15D|{
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. LD6fi
% Z@h]dU5%a
% The Zernike functions are an orthogonal basis on the unit circle. 4s"HO/
% They are used in disciplines such as astronomy, optics, and QHQj6]
% optometry to describe functions on a circular domain. g=%W"v
% '2
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% The following table lists the first 15 Zernike functions. 62zlO{ >rJ
% 3oIoQj+D
% n m Zernike function Normalization b"zq3$6*
% -------------------------------------------------- J
L Z
% 0 0 1 1 .58AXg
% 1 1 r * cos(theta) 2 Mdy H/.Te
% 1 -1 r * sin(theta) 2 pkT
a^I
% 2 -2 r^2 * cos(2*theta) sqrt(6) Y# lE
% 2 0 (2*r^2 - 1) sqrt(3) oFsMQ Py
% 2 2 r^2 * sin(2*theta) sqrt(6) U "}Kth
% 3 -3 r^3 * cos(3*theta) sqrt(8) 6F<L4*4U
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) z;3}GxE-si
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) ~pw_*AN
% 3 3 r^3 * sin(3*theta) sqrt(8) ,fNiZ
% 4 -4 r^4 * cos(4*theta) sqrt(10) lz>5bR'
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) G)putk@
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) ^6`R:SV4Gx
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) x7/2e{p
uu
% 4 4 r^4 * sin(4*theta) sqrt(10) #._!.P
% -------------------------------------------------- dk.da&P
% 9XoKOR(
% Example 1: [&39Yv.k,7
% 8"4`W~ 3
% % Display the Zernike function Z(n=5,m=1) ``NjNd
% x = -1:0.01:1; PEBQ|k8g&
% [X,Y] = meshgrid(x,x); CZuxH
% [theta,r] = cart2pol(X,Y); $Qm;F%
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% idx = r<=1; ^*0;Z<_
% z = nan(size(X)); aE;!mod
% z(idx) = zernfun(5,1,r(idx),theta(idx)); m\VJ=
% figure w
S;(u[W
% pcolor(x,x,z), shading interp qS7*.E~j|]
% axis square, colorbar sX=!o})0
% title('Zernike function Z_5^1(r,\theta)') crmnh4-
% q| j;dI&
% Example 2: `t8e2?GH
% Pjx9@i
% % Display the first 10 Zernike functions m t*v@'l.
% x = -1:0.01:1; 0W>O,%z&P#
% [X,Y] = meshgrid(x,x); GY4yZa
% [theta,r] = cart2pol(X,Y); 7kb`o
y;(^
% idx = r<=1; Onk~1ks:
% z = nan(size(X)); U}
g%`<
% n = [0 1 1 2 2 2 3 3 3 3]; rKjQEO$yi
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; J=Jw"? f
% Nplot = [4 10 12 16 18 20 22 24 26 28]; F:H76O` 8
% y = zernfun(n,m,r(idx),theta(idx)); |Rl|Th
% figure('Units','normalized') 7'<4'BGzl]
% for k = 1:10 Mr&]RTEE
% z(idx) = y(:,k); /wK7l-S
% subplot(4,7,Nplot(k)) V*/))n?
% pcolor(x,x,z), shading interp Mc\lzq8\ 1
% set(gca,'XTick',[],'YTick',[]) ]f-e/8$`@
% axis square CBvBBt*
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) "=RB
#
% end {=(4
% }x8fXdd
% See also ZERNPOL, ZERNFUN2. z=u4&x|xA
=CJs&Qa2
% Paul Fricker 11/13/2006 ;1y\!f3#V~
q`{.2yV
)XNcy"
% Check and prepare the inputs: $iB(N ZV
% ----------------------------- }M1<a4~
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 9R E;50h
error('zernfun:NMvectors','N and M must be vectors.') Oc8+an1m
end 3b_#xr-
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if length(n)~=length(m) 1n5&PNu
error('zernfun:NMlength','N and M must be the same length.') jALo;PDJ
end &v`kyc
: Z.mM5
n = n(:); y"]> Rr
m = m(:); n^A=ar.
if any(mod(n-m,2)) Pgo5&SQb
error('zernfun:NMmultiplesof2', ... kBT cND|
'All N and M must differ by multiples of 2 (including 0).') H11Wb(6Wu
end LRmO6>y
jG/kT5S
if any(m>n) Rp|:$5&nE
error('zernfun:MlessthanN', ... vuK 5DG4
'Each M must be less than or equal to its corresponding N.') PK~okz4b
end X(1.Hjh
SrKF\h%/+
if any( r>1 | r<0 ) 5-g0 2g
error('zernfun:Rlessthan1','All R must be between 0 and 1.') k{;:KW|
end j9,X.?Xvx
Zaj<*?\
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Fb*;5VNU.
error('zernfun:RTHvector','R and THETA must be vectors.') [;b9'7j'
end 'R$~U?i8
/)G9w]|T
r = r(:); J d`NS3;*p
theta = theta(:); c9&
8kq5
length_r = length(r); >s>5k
O
if length_r~=length(theta) }%}eyLm(
error('zernfun:RTHlength', ... HsXFglQ
'The number of R- and THETA-values must be equal.') ="4jk=on
end }Jc^p
6yR7RF}
% Check normalization: Oll\T GXP!
% -------------------- v14[G@V~\
if nargin==5 && ischar(nflag) bv] ZUF0
isnorm = strcmpi(nflag,'norm'); cEN^H
if ~isnorm I7TMv.
error('zernfun:normalization','Unrecognized normalization flag.') Rbl(oj#
end 9*x9sfCv9
else 1k7E[G~G|
isnorm = false; \ pq]q
end }skXh_Vu4
UOwj"#
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% EEaFi8
% Compute the Zernike Polynomials B>'\g
O\2
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ]l\J"*"aB
+uH1rF_&@
% Determine the required powers of r: lyT~>.?{
% ----------------------------------- 8Ej2JMc
m_abs = abs(m); -V+fQGZe
rpowers = []; [~;9Mi.XL
for j = 1:length(n) rN*4Y
rpowers = [rpowers m_abs(j):2:n(j)]; yb]a p
end
[g/g(RL
rpowers = unique(rpowers); mT9TSW}
c1Hv^*Y
% Pre-compute the values of r raised to the required powers, +Gjy%JFp
% and compile them in a matrix: 5=$D~>-#
% ----------------------------- 4RK^efnp
if rpowers(1)==0 \;sUJr"$
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); xOt|j4
rpowern = cat(2,rpowern{:}); m/{rmtA4
rpowern = [ones(length_r,1) rpowern]; |5W u0T
else c~Ha68
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); Lkb?,j5
rpowern = cat(2,rpowern{:}); `yf#(YP
end *AJW8tIP
)D@
NX/}
% Compute the values of the polynomials: YS/DIH{9e
% -------------------------------------- 2#rF/!`^
y = zeros(length_r,length(n)); VMNihx0FJ
for j = 1:length(n) 7N:,F9V<
s = 0:(n(j)-m_abs(j))/2; 7y60-6r
pows = n(j):-2:m_abs(j); -yC},tK
for k = length(s):-1:1 hxv/285B
p = (1-2*mod(s(k),2))* ... .NPai4V'
prod(2:(n(j)-s(k)))/ ... jKtbGVZ7r
prod(2:s(k))/ ... 9\dC8
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ;MO
%))
prod(2:((n(j)+m_abs(j))/2-s(k))); Vdjca:`
idx = (pows(k)==rpowers); *l5/q\D
y(:,j) = y(:,j) + p*rpowern(:,idx); 8J@REP4
end jfI|( P
FkRrW^?5G
if isnorm tewC *%3V
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); )Z]8SED
end :*\JJ w
end 1_F2{n:yp
% END: Compute the Zernike Polynomials yDHH05Yl
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% l.&6|
"d{ |_Cf
% Compute the Zernike functions: U/TF,JUI
% ------------------------------ QYg2'`(
idx_pos = m>0; O* 7"Q&
idx_neg = m<0; O8M;q!)y
D VC};
z = y; a*o=,!
if any(idx_pos) QupCr/Hs
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); $L3UDX+F
end G"C'/
if any(idx_neg) &L;0%
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); -l^ u1z
end ]r|X[9
_57i[U r
% EOF zernfun