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 [bAv|;
function z = zernfun(n,m,r,theta,nflag) U7OW)tUf
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. 8u>E(Vmpu
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N +m"iJW0
% and angular frequency M, evaluated at positions (R,THETA) on the RtSk;U1
% unit circle. N is a vector of positive integers (including 0), and 1iUy*p65:
% M is a vector with the same number of elements as N. Each element {pVD`#Tl[
% k of M must be a positive integer, with possible values M(k) = -N(k) _vad>-=D*U
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, E@?jsN7
% and THETA is a vector of angles. R and THETA must have the same DY1o!thz)
% length. The output Z is a matrix with one column for every (N,M) $Uzc
% pair, and one row for every (R,THETA) pair. X{)M}WO+r
% 46*?hA7@r(
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike %;gD_H4mm
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), TygRG+G-
% with delta(m,0) the Kronecker delta, is chosen so that the integral ^CX~>j\(
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 9khD7v
% and theta=0 to theta=2*pi) is unity. For the non-normalized ;yH/GN#O
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. X/?3ifP6I
% 2lQ'rnqS)
% The Zernike functions are an orthogonal basis on the unit circle. |XeuqZa
% They are used in disciplines such as astronomy, optics, and Q?vGg{>
% optometry to describe functions on a circular domain. x
ha!.&DO
% 67d0JQTu
% The following table lists the first 15 Zernike functions. mWtwp-
% hd\iW7
% n m Zernike function Normalization vQA: \!
% -------------------------------------------------- )4j#gHN\
% 0 0 1 1 mI}'8.
% 1 1 r * cos(theta) 2 WO]dWO6Mm
% 1 -1 r * sin(theta) 2 Hq=RtW2
% 2 -2 r^2 * cos(2*theta) sqrt(6) QQqWJq~
% 2 0 (2*r^2 - 1) sqrt(3) "}EydG"=
% 2 2 r^2 * sin(2*theta) sqrt(6) ++xEMP)
% 3 -3 r^3 * cos(3*theta) sqrt(8) &}rh+z
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) ^G15]Pyw
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) P\SE_*&
% 3 3 r^3 * sin(3*theta) sqrt(8) `6UW?1_Z5
% 4 -4 r^4 * cos(4*theta) sqrt(10) aVd{XVE
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 2OEOb,`
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) qW),)i
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) gg5`\}
% 4 4 r^4 * sin(4*theta) sqrt(10) X|X6^}
% -------------------------------------------------- HdLVXaD/
% <jfi"SJu
% Example 1: xEGI'lt
% [&6l=a
% % Display the Zernike function Z(n=5,m=1) .I[uXd
% x = -1:0.01:1; BH\qm
(X
% [X,Y] = meshgrid(x,x); aM~M@wS
% [theta,r] = cart2pol(X,Y); BB9Z?}
% idx = r<=1; !<@Zf4m
% z = nan(size(X)); G.1pg]P!
% z(idx) = zernfun(5,1,r(idx),theta(idx)); 9MVW~V
% figure l1a=r:WhH
% pcolor(x,x,z), shading interp A\gj\&B0"
% axis square, colorbar (m})V0/`
% title('Zernike function Z_5^1(r,\theta)') bc%7-%
% @r'8<6hVO
% Example 2: 8z\WyDz
% \3Ys8umKq
% % Display the first 10 Zernike functions B$aboL2
% x = -1:0.01:1; (V}DPA
% [X,Y] = meshgrid(x,x); |>Kf_b Y#
% [theta,r] = cart2pol(X,Y); &!a[rvtZ+
% idx = r<=1; 9w (QM-u
% z = nan(size(X)); b>?X8)f2e
% n = [0 1 1 2 2 2 3 3 3 3]; h$y1"!N(
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; FX 0^I 0
% Nplot = [4 10 12 16 18 20 22 24 26 28]; 7'd_]e-.
% y = zernfun(n,m,r(idx),theta(idx)); L3'o2@$
% figure('Units','normalized') a'rN&*P
% for k = 1:10 | \ C{R
% z(idx) = y(:,k); j?#S M!f
% subplot(4,7,Nplot(k)) &$|k<{j[<f
% pcolor(x,x,z), shading interp yD$rls:v<
% set(gca,'XTick',[],'YTick',[]) dyD=R
% axis square ~\(U&2t
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) =k'3rm*ld
% end 0;
M+8
% ?E=&LAI#
% See also ZERNPOL, ZERNFUN2. tNoo3&
w*OZ1|
% Paul Fricker 11/13/2006 3;@t{rIin
Wl?*AlFlk
+kmPQdO;*/
% Check and prepare the inputs: 32:q'
% ----------------------------- A{Jv`K
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) A7 E*w
error('zernfun:NMvectors','N and M must be vectors.') [_#9PH33
end M8Q-x-7
7?dB&m6W
if length(n)~=length(m) UXnd~DA
error('zernfun:NMlength','N and M must be the same length.') WEZ(4ah
end zsc8Lw
;spuBA)[X
n = n(:); A !x"*
m = m(:); eOE7A'X
if any(mod(n-m,2)) A!x_R {,yH
error('zernfun:NMmultiplesof2', ... %DbL|;z1
'All N and M must differ by multiples of 2 (including 0).') >x eKO2o
end #swzZyM$
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if any(m>n) ubju uha"
error('zernfun:MlessthanN', ... HJ:s)As
'Each M must be less than or equal to its corresponding N.') fr4#<6,
end EL;Ir tU
r*OSEzGUz
if any( r>1 | r<0 ) j|A *rzL8
error('zernfun:Rlessthan1','All R must be between 0 and 1.') b,cA mZ
end ;lB%N
t<,
b`usRoD{+
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) SL?
!
RQ
error('zernfun:RTHvector','R and THETA must be vectors.')
e%afK@c
end 1>[3(o3t
m1heU3BUWU
r = r(:); kS%FV;9>(
theta = theta(:); G!C2[:[g
length_r = length(r); u`xmF/jhQ
if length_r~=length(theta) !vHnMY~AG
error('zernfun:RTHlength', ... ?kI-o0@O.
'The number of R- and THETA-values must be equal.') 6@t4pML
end *!ZU"q}i
dP=1*
% Check normalization: @kenv3[Lc
% -------------------- /QZnN?k
if nargin==5 && ischar(nflag) nw+L _b
isnorm = strcmpi(nflag,'norm'); U}x2,`PI
if ~isnorm Ia=wf"JS)
error('zernfun:normalization','Unrecognized normalization flag.') 0m(/hK
end Xai ,
else z | Hl*T
isnorm = false; ; =ai]AYW
end L= O,OS+
v7&e,:r2E@
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% tKjPLi71
% Compute the Zernike Polynomials 3;zJ\a.+
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -v'7;L0K
mL?9AxO
% Determine the required powers of r: KJo[!|.
% ----------------------------------- bae .?+0[
m_abs = abs(m); EDcR:Dw3
rpowers = []; mT
<4@RrB
for j = 1:length(n) WO?EzQ ?
rpowers = [rpowers m_abs(j):2:n(j)]; ,B(UkPGT
end gbL99MZ@~
rpowers = unique(rpowers); (YVl5}V
\bw71( Q
% Pre-compute the values of r raised to the required powers, S7N3L."
% and compile them in a matrix: !@{_Qt1
% ----------------------------- T^B&GgW
if rpowers(1)==0 8 k9(iS
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); IAf,TKfe
rpowern = cat(2,rpowern{:}); ^hv
rpowern = [ones(length_r,1) rpowern]; DmEmv/N=
else Oh9wBV
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); .Qg!_C
rpowern = cat(2,rpowern{:}); z9}rT<hy
end Q.7Rv
XNw8
[yM{A<\L
% Compute the values of the polynomials: $v#Q'?jE
% -------------------------------------- O&.^67\|
y = zeros(length_r,length(n)); dd>|1'-]
for j = 1:length(n) Wp/!;
s = 0:(n(j)-m_abs(j))/2; )HNbWGu
pows = n(j):-2:m_abs(j); zNofI$U
for k = length(s):-1:1
LKieOgX
p = (1-2*mod(s(k),2))* ... dE!{=u(!i
prod(2:(n(j)-s(k)))/ ... RXh0hD
prod(2:s(k))/ ... 7Te`#"
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... M8X*fYn
prod(2:((n(j)+m_abs(j))/2-s(k))); K++pH~o
idx = (pows(k)==rpowers); -|B?pR
y(:,j) = y(:,j) + p*rpowern(:,idx); {:xINQ=}D
end )_"Cz".|9
s
Z(LT'}
if isnorm oe_l:Y%
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); M;OY+|uA
end x.qn$?3V]
end eUPG){"
% END: Compute the Zernike Polynomials 'uBXSP#
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -BfZ P5
FiMP_ y*S
% Compute the Zernike functions: e;~[PYeu
% ------------------------------ x^^;/%p
idx_pos = m>0; O|m-Uz"+
idx_neg = m<0; z=<x.F
wvvMesX<L
z = y; ';us;xR#
if any(idx_pos) >DVjO9Kf
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); pj;cL]L
end AX}l~
sv
if any(idx_neg) 9-[g/qrF
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ]^$&Ejpe#
end A1e| Y
H>AQlO+ J
% EOF zernfun