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 1IRlFC
function z = zernfun(n,m,r,theta,nflag) #A '|O\RGP
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. bijE]:<AE7
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N !$i*u-%4
% and angular frequency M, evaluated at positions (R,THETA) on the |nFg"W
% unit circle. N is a vector of positive integers (including 0), and P:gN"f6
% M is a vector with the same number of elements as N. Each element R|Lr@k{6+r
% k of M must be a positive integer, with possible values M(k) = -N(k) DL0i
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, M{ mdh\
% and THETA is a vector of angles. R and THETA must have the same =6sL}$
% length. The output Z is a matrix with one column for every (N,M) ,>rr|O
% pair, and one row for every (R,THETA) pair. c{dge/2yb
% MWxv\o
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike "=S< xT+
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), X <<hb
% with delta(m,0) the Kronecker delta, is chosen so that the integral l"#}g%E
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, :7w^2/ZGo
% and theta=0 to theta=2*pi) is unity. For the non-normalized }Ra'`;D$
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. &(]@L\A
% dMnJ)R
% The Zernike functions are an orthogonal basis on the unit circle. C Ahkv0?8
% They are used in disciplines such as astronomy, optics, and cJnAwIs_e`
% optometry to describe functions on a circular domain. K2u$1OKv
% .%pbKi
`
% The following table lists the first 15 Zernike functions.
1UHStR
% Vg0$5@
% n m Zernike function Normalization EN =oA P
% -------------------------------------------------- El}."}l&
% 0 0 1 1 IU8/B+hM~
% 1 1 r * cos(theta) 2 Ie[8Iot?bn
% 1 -1 r * sin(theta) 2 4\.1phe$a
% 2 -2 r^2 * cos(2*theta) sqrt(6) d&dp#)._8
% 2 0 (2*r^2 - 1) sqrt(3) B|~tW21
% 2 2 r^2 * sin(2*theta) sqrt(6) /=5YHq>
% 3 -3 r^3 * cos(3*theta) sqrt(8) lAxbF
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) )Bl0
W
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) UKBVCAK
% 3 3 r^3 * sin(3*theta) sqrt(8) L@"1d.k_
% 4 -4 r^4 * cos(4*theta) sqrt(10) +$hqwNh@Z@
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) :jol
Nl|a
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) cBl
F
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) '8Q:}{
% 4 4 r^4 * sin(4*theta) sqrt(10) |6%B2I&c
% -------------------------------------------------- FY^[?lj
% (QPfrR=J4
% Example 1: Ye '=F
% TV~<1vj
% % Display the Zernike function Z(n=5,m=1) (8(7:aE$
% x = -1:0.01:1; 'w?*4H
% [X,Y] = meshgrid(x,x); lzQmD/i*
% [theta,r] = cart2pol(X,Y); TTS.wBpR,
% idx = r<=1; Mpfdl65
% z = nan(size(X)); |mSF a8G@
% z(idx) = zernfun(5,1,r(idx),theta(idx)); tSr.0'CE
% figure nN=o/z d
% pcolor(x,x,z), shading interp Ue>;h9^
% axis square, colorbar R6^U9fDG
% title('Zernike function Z_5^1(r,\theta)') ionFPc].
% tgy= .o]
% Example 2: GOT@
% p)5j~Nl
% % Display the first 10 Zernike functions "f/Su(6{0
% x = -1:0.01:1; VJK?"mX
% [X,Y] = meshgrid(x,x); #J1vN]g
% [theta,r] = cart2pol(X,Y); N$8do?
% idx = r<=1; HLL[r0P`F
% z = nan(size(X)); ea"!:cL(g
% n = [0 1 1 2 2 2 3 3 3 3]; njbEw4nX
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; G~SgI>Q
% Nplot = [4 10 12 16 18 20 22 24 26 28]; %\5wHT+)
% y = zernfun(n,m,r(idx),theta(idx)); \7W4)>At-
% figure('Units','normalized') Mw=sW5Z
% for k = 1:10 "|{3V:e>a
% z(idx) = y(:,k); f}jo18z%
% subplot(4,7,Nplot(k)) |'w_5?|4
% pcolor(x,x,z), shading interp LaI(
% set(gca,'XTick',[],'YTick',[]) b)7uz>I
% axis square WD wW`
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) h/I'9&J>*
% end .~)[>
% 9.<d S
% See also ZERNPOL, ZERNFUN2. t`PA85.|d
W<J".2D
% Paul Fricker 11/13/2006 ?\_N*NEtK
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% Check and prepare the inputs: 2h<{~;
% ----------------------------- ',?9\xEB
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) a&>Tk%
error('zernfun:NMvectors','N and M must be vectors.') ^P5+ _P
end ]<9=%m
5k0r{^#M
if length(n)~=length(m) Au+SCj
error('zernfun:NMlength','N and M must be the same length.') R5`"~qP-
end fz%I'+!
OBGA~E;%
n = n(:); =@#[@Ia
m = m(:); @"M%ZnFu
if any(mod(n-m,2)) l djypEa}
error('zernfun:NMmultiplesof2', ... Qr`WPTQr"
'All N and M must differ by multiples of 2 (including 0).') Z]$RO
end # 2As-9
.#"O VI]#
if any(m>n) =bJj;bc'5
error('zernfun:MlessthanN', ... yNY *Fl!
'Each M must be less than or equal to its corresponding N.') 3"28=)o
end +\SNaq~&
x+j5vzhG)
if any( r>1 | r<0 ) xkv2#"*v
error('zernfun:Rlessthan1','All R must be between 0 and 1.') W~15[r0
end
:e-&,K
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) SvUC8y
error('zernfun:RTHvector','R and THETA must be vectors.') 3y!yz3E
end zz ^2/l
B_FfXFQm<
r = r(:); 5#~ARk*?a
theta = theta(:); 3L24|-GxH
length_r = length(r); 0sjw`<ic
if length_r~=length(theta) ?!H<V@a
error('zernfun:RTHlength', ... i2or/(u`
'The number of R- and THETA-values must be equal.') x)6yWr[ri%
end r>+Hwj0>
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% Check normalization: H-%)r&"vn
% -------------------- iE}jilU
if nargin==5 && ischar(nflag) vt`hY4
isnorm = strcmpi(nflag,'norm'); (>m3WI$d
if ~isnorm E.v~<[g
error('zernfun:normalization','Unrecognized normalization flag.') O@U[S.IK
end J~z;sTR
else tN|sHgs
isnorm = false; Gy36{*
end %
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ZOl
=zn
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 6T 2jVNg
% Compute the Zernike Polynomials +O23@G?x
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% fRo_rj _
)L#C1DP#
% Determine the required powers of r: {t: ZMUV
% ----------------------------------- w5"C<5^
m_abs = abs(m); 2Mx9Kd'a
r
rpowers = []; W(9fCDO;
for j = 1:length(n) a
pqzf
rpowers = [rpowers m_abs(j):2:n(j)]; {HeIY2
end Y>-|`2Z
rpowers = unique(rpowers); 4%O*2JAw
jh.W$.Oq
% Pre-compute the values of r raised to the required powers, tx;DMxN!W
% and compile them in a matrix: hd1H
% ----------------------------- l)E
\mo
8
if rpowers(1)==0 T{u!4Yu
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); sqHvrI
rpowern = cat(2,rpowern{:}); ~*D)L'`2M
rpowern = [ones(length_r,1) rpowern]; v=?U{{xQ
else +]Of f^s
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); pRmnS;*z&
rpowern = cat(2,rpowern{:}); pmXx2T#=
end UwY <3ul
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% Compute the values of the polynomials: E!'H,#"P
% -------------------------------------- cH6ie?KvAo
y = zeros(length_r,length(n)); >x)YdgJ*
for j = 1:length(n) \/4ipU.
s = 0:(n(j)-m_abs(j))/2; i](,s.
pows = n(j):-2:m_abs(j); 9"2.2li5$
for k = length(s):-1:1 8Q^yh6z
p = (1-2*mod(s(k),2))* ... e;pVoRI
prod(2:(n(j)-s(k)))/ ... EDvK9J
prod(2:s(k))/ ... i^sK+v
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... =25qY"Mf
prod(2:((n(j)+m_abs(j))/2-s(k))); VE^NSkOa&
idx = (pows(k)==rpowers); C1P{4 U
y(:,j) = y(:,j) + p*rpowern(:,idx); 'P}"ZHW
end ,5'LbO-
dN;kYWRK
if isnorm c&)H
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); :w(J=0Lt
end JU:!lyd
end 8-cG[/|0
% END: Compute the Zernike Polynomials k);z}`7
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% i9k7rEW^
O/gok+K
% Compute the Zernike functions: &d`Umm]
% ------------------------------ rui}a=rs
idx_pos = m>0; |K'{R'A
idx_neg = m<0; # j*$ `W;
x+|Fw d
z = y; xC`Hm?kM
if any(idx_pos) YS?P A#
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); {b^naE
end K%qunjv
if any(idx_neg) lZ0+:DaP2
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); yt>Pf<AI
end ,.]e~O4R
Jl Q%+$
% EOF zernfun