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 #I#_gjJkx
function z = zernfun(n,m,r,theta,nflag) H=9{|%iS
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. r|y\FL
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ;A0ZcgF
% and angular frequency M, evaluated at positions (R,THETA) on the -/_hO$|W
% unit circle. N is a vector of positive integers (including 0), and Yn>FSq^Wp-
% M is a vector with the same number of elements as N. Each element |}@teN^J*U
% k of M must be a positive integer, with possible values M(k) = -N(k) d}wE4(]b
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, _)6r@fZ.p
% and THETA is a vector of angles. R and THETA must have the same JY%l1:}G3
% length. The output Z is a matrix with one column for every (N,M) o;>qsn8
% pair, and one row for every (R,THETA) pair. G<Urj+3/Xo
% .e\PCf9v
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike WLH ;{
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 57EL&V%j
% with delta(m,0) the Kronecker delta, is chosen so that the integral MRzY<MD
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 'l3 DP
% and theta=0 to theta=2*pi) is unity. For the non-normalized /as+ TU`A
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. :0p$r
pJP
% y2nT)nL
% The Zernike functions are an orthogonal basis on the unit circle. =-avzuy#
% They are used in disciplines such as astronomy, optics, and oo1h"[
% optometry to describe functions on a circular domain. @*WrHoa2N
% ek d[|g
% The following table lists the first 15 Zernike functions. /< Dtu UM
% QiaBZAol
% n m Zernike function Normalization gFXz:!A
% -------------------------------------------------- A2.4#Qb'
% 0 0 1 1 vnqLcNB H
% 1 1 r * cos(theta) 2 TXqtE("BDl
% 1 -1 r * sin(theta) 2 0Y8Cz /$
% 2 -2 r^2 * cos(2*theta) sqrt(6) ~SI G0U8
% 2 0 (2*r^2 - 1) sqrt(3) 2B+qS'OT
% 2 2 r^2 * sin(2*theta) sqrt(6) P.djR)YI
% 3 -3 r^3 * cos(3*theta) sqrt(8) fFXnD
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) 7_J0[C!G
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) g|j15&x
% 3 3 r^3 * sin(3*theta) sqrt(8) )GOio+{H
% 4 -4 r^4 * cos(4*theta) sqrt(10) 0JW
=RW
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) PB~
r7O]
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) [4+I1UR`
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) !1l~'/r
% 4 4 r^4 * sin(4*theta) sqrt(10) v3wq-
% -------------------------------------------------- O"wo&5b_
% m14'u GC
% Example 1: CW FE{
% %0'7J@W
% % Display the Zernike function Z(n=5,m=1) Rpj{!Ia
% x = -1:0.01:1; Sx1OY0)s
% [X,Y] = meshgrid(x,x); z~ua#(z1S
% [theta,r] = cart2pol(X,Y); !Oi':OQG
% idx = r<=1; 1JV-X G6
% z = nan(size(X)); k&npC8oA
% z(idx) = zernfun(5,1,r(idx),theta(idx)); Lzx2An@R
% figure UYzNaw4/x
% pcolor(x,x,z), shading interp 9xeg,#1
% axis square, colorbar 8YQ7XB
% title('Zernike function Z_5^1(r,\theta)') 9)uJ\NMy
% GtKSA#oYZB
% Example 2: cI-@nV
% 5>hXqNjP2
% % Display the first 10 Zernike functions lBudC
% x = -1:0.01:1; onm"7JsO'
% [X,Y] = meshgrid(x,x); J|([(
% [theta,r] = cart2pol(X,Y); 7tne/Yz
% idx = r<=1; #$l:%
% z = nan(size(X)); E@-5L9eJ\
% n = [0 1 1 2 2 2 3 3 3 3]; xl9S=^`=
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; *d31fBCk%
% Nplot = [4 10 12 16 18 20 22 24 26 28]; 2SlI5+u
% y = zernfun(n,m,r(idx),theta(idx)); o
^ 08<
% figure('Units','normalized') un}!&*+
% for k = 1:10 4~2 9,
% z(idx) = y(:,k); w%(D4ldp
% subplot(4,7,Nplot(k)) bk]g}s
% pcolor(x,x,z), shading interp lHE \Z`
% set(gca,'XTick',[],'YTick',[]) # hw;aQ
% axis square +`!>lo{X
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}'])
j} w
% end YD0j&@.
% $"va8,
% See also ZERNPOL, ZERNFUN2. <YrsS-9
<v?2p{U%
% Paul Fricker 11/13/2006 <4CqG4}Y
/v.<h*hxWy
%g69kizoWi
% Check and prepare the inputs: Pfd%[C/vdm
% ----------------------------- X]dN1/_
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) #}Bv/`t
error('zernfun:NMvectors','N and M must be vectors.') gLlA'`!
end L]l?_#*x
QHd|cg
if length(n)~=length(m) '@5x=>
error('zernfun:NMlength','N and M must be the same length.') 1B$8<NCQ=?
end 7/K'nA
EJNHZ<
n = n(:); l-5O5|C
m = m(:); Vddod
if any(mod(n-m,2)) g[;&_gL
error('zernfun:NMmultiplesof2', ... L@J$kqWY
'All N and M must differ by multiples of 2 (including 0).') rS+ >oP}
end X^i3(N
<SdOb#2
if any(m>n) M0hR]4T
error('zernfun:MlessthanN', ... :*-O;Yw?S@
'Each M must be less than or equal to its corresponding N.') >fD%lq;
end }H/94]~tH
=6N=5JePB
if any( r>1 | r<0 ) "B9zQ,[Q
error('zernfun:Rlessthan1','All R must be between 0 and 1.') rddn"~lm1
end ?"kU+tCxg
Jg$ NYs.xZ
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) D0Ls~qr
error('zernfun:RTHvector','R and THETA must be vectors.') [C!m,4
end y^ D3}ds
q?8#D
r = r(:); J ayax]u7J
theta = theta(:); T0 cm+|S
length_r = length(r); "9Br)3
if length_r~=length(theta) p*JP='p
error('zernfun:RTHlength', ... }:*?w>=
'The number of R- and THETA-values must be equal.') VeH%E.:
end B5_QH8kt7
Np;tpq~
% Check normalization: a, `B.I
% -------------------- `:2np{
if nargin==5 && ischar(nflag) mXu";?2
isnorm = strcmpi(nflag,'norm'); 5nK|0vv%2
if ~isnorm ncpA\E;ff^
error('zernfun:normalization','Unrecognized normalization flag.') @@}muW>;T
end -*2b/=$u
else k"cKxzB
isnorm = false; TLg 9`UA
end tq*{Hil>P`
i6i;{\tc
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% $GVf;M2*
% Compute the Zernike Polynomials `g{eWY1l
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% }(WUZ^L
nVGOhYn
% Determine the required powers of r: u%Z4 8wr
% ----------------------------------- Rb
<{o8
m_abs = abs(m); Z#K0a'
rpowers = []; @s\}ER3
for j = 1:length(n) VD{_6
rpowers = [rpowers m_abs(j):2:n(j)]; g}vU*g
;
end ul"Z%
1]
rpowers = unique(rpowers); Ge24Lp;Y6
s3~6[T?8
% Pre-compute the values of r raised to the required powers, Y1BxRd?D
% and compile them in a matrix: 5'xZ9K
% ----------------------------- " j:15m5
if rpowers(1)==0 \d w ["k
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); x/ P\qI
rpowern = cat(2,rpowern{:}); 1z3I^gI*i
rpowern = [ones(length_r,1) rpowern]; prxmDI
else QFhQfn
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 8)J,jh9q
rpowern = cat(2,rpowern{:}); eT8h:+k
end |mz0
]
X<H+Z2d
% Compute the values of the polynomials: S_Vquw(+
% -------------------------------------- \BSPv]d
y = zeros(length_r,length(n)); w_q=mKu
for j = 1:length(n) ?\a';@h
s = 0:(n(j)-m_abs(j))/2; `y.i(~^1
pows = n(j):-2:m_abs(j); QSOJHRl=C
for k = length(s):-1:1 @2 SL$0!QA
p = (1-2*mod(s(k),2))* ... ~ o5h}OU"
prod(2:(n(j)-s(k)))/ ... Q\$cBSJC1
prod(2:s(k))/ ... lpefOnO[
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... hPUYq7B
prod(2:((n(j)+m_abs(j))/2-s(k))); ,q
Bu5t
idx = (pows(k)==rpowers); cp+eh
y(:,j) = y(:,j) + p*rpowern(:,idx); n\YWWW[wf
end xCm`g{
uC1v^!D
if isnorm 0y#TGM|0D
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); j<i:rk|
end ` ln=D$
end /A`Lyp#
% END: Compute the Zernike Polynomials ' :,p6
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Eyy^pb
O[&G6+
% Compute the Zernike functions: 82z<Q*YP
% ------------------------------ BP@Lhii
idx_pos = m>0; =[^_x+x
hE
idx_neg = m<0; fkr;
a`<W
LtBm }0
z = y; &v_b7h
if any(idx_pos) dp>Lh TLc
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); Jm
G)=$,
end +JL"Z4b@R}
if any(idx_neg) t8b,@J`R
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ,vUMy&AV
end %g%#=a;]q
Yy8%vDdJO
% EOF zernfun