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 }`M53>C,gQ
function z = zernfun(n,m,r,theta,nflag) 3NRxf8
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. _):V7Zv
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N <8#Q5
% and angular frequency M, evaluated at positions (R,THETA) on the ]4f;%pE
% unit circle. N is a vector of positive integers (including 0), and +mP&B<=H)
% M is a vector with the same number of elements as N. Each element AY{#!RtV
% k of M must be a positive integer, with possible values M(k) = -N(k) dER#)bGj
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, ^~~&[wY
% and THETA is a vector of angles. R and THETA must have the same Khd"
% length. The output Z is a matrix with one column for every (N,M) #LRN@?P
% pair, and one row for every (R,THETA) pair. &<8Q/m]5
% 0\3mS{s
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ^wesuW@=
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), m>dZ n
% with delta(m,0) the Kronecker delta, is chosen so that the integral ?Ne@OMc
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, +%vBDcf
% and theta=0 to theta=2*pi) is unity. For the non-normalized YNV!(>\GE
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. xszGao'
% 7d&_5Tj:
% The Zernike functions are an orthogonal basis on the unit circle. {;.q?mj
% They are used in disciplines such as astronomy, optics, and h'Tn&2r6
% optometry to describe functions on a circular domain. 9$[I~I#z
% f+>l-6M+p
% The following table lists the first 15 Zernike functions. Fe8JsB-
% c 32IO&W4
% n m Zernike function Normalization
!]]QbB
% -------------------------------------------------- [KrWL;[1<
% 0 0 1 1 hT :+x3
% 1 1 r * cos(theta) 2 J[E_n;d1
% 1 -1 r * sin(theta) 2 0ox
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% 2 -2 r^2 * cos(2*theta) sqrt(6) ~3k& =3d]
% 2 0 (2*r^2 - 1) sqrt(3) W_k;jy_{9
% 2 2 r^2 * sin(2*theta) sqrt(6) JNhHQvi\
% 3 -3 r^3 * cos(3*theta) sqrt(8) "E`;8SZa
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) 9=,^^,q
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) /*g9drwaa
% 3 3 r^3 * sin(3*theta) sqrt(8) }6/L5j:+
% 4 -4 r^4 * cos(4*theta) sqrt(10) h{zE;!+)D
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 4R_Vi[i
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) jDI )iW`P
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Z4YQ5O5
% 4 4 r^4 * sin(4*theta) sqrt(10) '[u=q
-Lv
% -------------------------------------------------- sj;8[Xy's
% Q`$Q(/
% Example 1: aoNTRJc$
% VAkZ@
u3'~
% % Display the Zernike function Z(n=5,m=1) 3$Ecq|4J:
% x = -1:0.01:1; >r Nff!Ow
% [X,Y] = meshgrid(x,x); Be"Swz(n
% [theta,r] = cart2pol(X,Y); zqEMR>px
% idx = r<=1; P'o:Vhm_H
% z = nan(size(X)); cSdkhRAn
% z(idx) = zernfun(5,1,r(idx),theta(idx)); ejq2]^O4c
% figure +r EqE/QF
% pcolor(x,x,z), shading interp rNzsc|a:
% axis square, colorbar {"<6'2T3
% title('Zernike function Z_5^1(r,\theta)') c&zZsJ"~
% *2MM
% Example 2: _4E .
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% $lkd9r1
% % Display the first 10 Zernike functions [~&C6pR
% x = -1:0.01:1; ]W,K}~!
% [X,Y] = meshgrid(x,x); -ya0!D
% [theta,r] = cart2pol(X,Y); ;K[ G]8
% idx = r<=1; l!2hwRR
% z = nan(size(X)); q/w U7P\%
% n = [0 1 1 2 2 2 3 3 3 3]; BoZG^
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; 8J|pj4ce
% Nplot = [4 10 12 16 18 20 22 24 26 28]; 1FfdW>ay*
% y = zernfun(n,m,r(idx),theta(idx)); QusEWq)}<
% figure('Units','normalized') Qxds]5WB/
% for k = 1:10 aQax85
% z(idx) = y(:,k); Q;O\tl
% subplot(4,7,Nplot(k)) 6bL+q`3>
% pcolor(x,x,z), shading interp J"w!Q\_
% set(gca,'XTick',[],'YTick',[]) 4m++>q
% axis square .K![<eZ
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) XQEGMaZ
% end j7;v'eA`;7
% |_l\.
% See also ZERNPOL, ZERNFUN2. GD1=Fb"&)
G?-27Jk8
% Paul Fricker 11/13/2006 ?p{xt$<p
L2ePWctq}
j=v 1:E
% Check and prepare the inputs: %
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% ----------------------------- &s!"pEZWck
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) < 4DWH
error('zernfun:NMvectors','N and M must be vectors.') #8;|_RU
end .%+`e
oF/5mh__(K
if length(n)~=length(m) 4)=LOGW
error('zernfun:NMlength','N and M must be the same length.') pL$UI3VCP
end RVN"lDGA
@+",f]
n = n(:); )>LQ{X.
m = m(:); ?WWnt^
if any(mod(n-m,2))
?{#P.2
error('zernfun:NMmultiplesof2', ... gF%lwq
'All N and M must differ by multiples of 2 (including 0).') -B2>~#L
end lo:]r.lX{
bo&!oY#
if any(m>n) b?-%Uzp<
error('zernfun:MlessthanN', ... g#ZR,q
'Each M must be less than or equal to its corresponding N.') Z,o*M#}
end Ah)OyO6
{+f@7^/i.
if any( r>1 | r<0 ) LGT\1u
error('zernfun:Rlessthan1','All R must be between 0 and 1.') Tgp}k%R~
end XgKtg-,
5VWXUNe@_q
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) HZ=Dd4!
error('zernfun:RTHvector','R and THETA must be vectors.') M;W{A)0i1
end )8oI
s
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r = r(:); 7\^b+*
theta = theta(:); JnCY O^Qj
length_r = length(r); [ (tgoh/
if length_r~=length(theta) w5jH#ja
error('zernfun:RTHlength', ... UuxWP\~2
'The number of R- and THETA-values must be equal.') T3['6%
end ro37H2^Ty
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% Check normalization: 1W-t})!a
% -------------------- D0PP
if nargin==5 && ischar(nflag) ) 0$7{3
isnorm = strcmpi(nflag,'norm'); AW6]S*rh
if ~isnorm ^BjwPh4Z#
error('zernfun:normalization','Unrecognized normalization flag.') fl~k')s
end IDzP<u8v
else BW:&AP@B
isnorm = false; \~xsBPX+x
end xXZ$#z\Z,
5d|*E_yu
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {a_=4a
% Compute the Zernike Polynomials mT@UQCG
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ezlp~z"_k
5<4njo?k
% Determine the required powers of r: PiI ):B>
% ----------------------------------- 'O]_A57
m_abs = abs(m); e`R*6^e
rpowers = []; >;o^qi_$
for j = 1:length(n) Pf)<