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 F8-GnTxa
function z = zernfun(n,m,r,theta,nflag) q *&H
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. %eDSo9Y
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 7gf(5p5ZV
% and angular frequency M, evaluated at positions (R,THETA) on the 'fU #v`i
% unit circle. N is a vector of positive integers (including 0), and k37?NoT
% M is a vector with the same number of elements as N. Each element _D{A`z
% k of M must be a positive integer, with possible values M(k) = -N(k) Gkuqe3
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, >o1dc*
% and THETA is a vector of angles. R and THETA must have the same u.X]K:Yow
% length. The output Z is a matrix with one column for every (N,M) <?7qI8 5OT
% pair, and one row for every (R,THETA) pair. -z`FKej
% \[3~*eX6
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike v3Vve:}+
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), EO)JMV?6
% with delta(m,0) the Kronecker delta, is chosen so that the integral "D.<~!
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, +=E\sEe
% and theta=0 to theta=2*pi) is unity. For the non-normalized hO8xH +;
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. yk?bz
% HC$%"peN1b
% The Zernike functions are an orthogonal basis on the unit circle. aJ(/r.1G
% They are used in disciplines such as astronomy, optics, and C;m"W5+
% optometry to describe functions on a circular domain. r
1r@TG\
% qBBCnT
% The following table lists the first 15 Zernike functions. s oY\6mHio
% <7U~0@<Y
% n m Zernike function Normalization rk1,LsZVS
% -------------------------------------------------- b=lJ`|
% 0 0 1 1 .|[{$&B
% 1 1 r * cos(theta) 2 ]?=87w
% 1 -1 r * sin(theta) 2 NRtH?&7
% 2 -2 r^2 * cos(2*theta) sqrt(6) SDC|>e9i
% 2 0 (2*r^2 - 1) sqrt(3) *G.vY#h
% 2 2 r^2 * sin(2*theta) sqrt(6) J "I,]
% 3 -3 r^3 * cos(3*theta) sqrt(8) >b2!&dm
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) `r1}:`.m,
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) g0zzDv7~
% 3 3 r^3 * sin(3*theta) sqrt(8) n%F _3`
% 4 -4 r^4 * cos(4*theta) sqrt(10) h}SZ+G/L
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) iRrUIWx
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) gDU!dT
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) fVn4=d6X
% 4 4 r^4 * sin(4*theta) sqrt(10) Bz } nP9
% -------------------------------------------------- ~NK $rHwi%
% )&O2l
% Example 1: F&wAre<
% 9Q,>I6`l
% % Display the Zernike function Z(n=5,m=1) O`y3H lc
% x = -1:0.01:1; j_g(6uZhz3
% [X,Y] = meshgrid(x,x); %.=}v7&<z
% [theta,r] = cart2pol(X,Y); ~4~r
% idx = r<=1; D?_K5a&v,
% z = nan(size(X)); Ps@']]4>W
% z(idx) = zernfun(5,1,r(idx),theta(idx)); DehjV6t
% figure B%\&Q@X
% pcolor(x,x,z), shading interp bI
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% axis square, colorbar Cik1~5iF
% title('Zernike function Z_5^1(r,\theta)') i24k
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% _ VuWo
% Example 2: `r SOt*<
% f9K7^qwkiz
% % Display the first 10 Zernike functions .@)vJtH)
% x = -1:0.01:1; #[jS&rr(
% [X,Y] = meshgrid(x,x); VVSt,/SO
% [theta,r] = cart2pol(X,Y); GxzO|vFQ
% idx = r<=1; 4q] 6[/
% z = nan(size(X)); "e"#k}z9
% n = [0 1 1 2 2 2 3 3 3 3]; rNV3-#kU
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; C,+
% Nplot = [4 10 12 16 18 20 22 24 26 28]; X?t;uZI^
% y = zernfun(n,m,r(idx),theta(idx)); .4v?/t1
% figure('Units','normalized') q~> +x?30
% for k = 1:10 fhN\AjB6Td
% z(idx) = y(:,k); nRBS&&V
% subplot(4,7,Nplot(k)) OS#aYER~/
% pcolor(x,x,z), shading interp 3/]1m9x
% set(gca,'XTick',[],'YTick',[]) kZG=C6a
% axis square Sa<(F[p`
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 9jI muSZ
% end !nl-}P,
% A4f"v)vM
% See also ZERNPOL, ZERNFUN2. -OJ <Lf+"=
*>W<n1r@]
% Paul Fricker 11/13/2006 }T$BU>z33N
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r =x"E$
% Check and prepare the inputs: A2gFY}
% ----------------------------- < +*
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) WOj}+?/3 R
error('zernfun:NMvectors','N and M must be vectors.') iHNQxLkk{:
end +m./RlQ{
>s/_B//[
if length(n)~=length(m) ({rcH.:
error('zernfun:NMlength','N and M must be the same length.') j.] ]VA
end lU!_V%n
h.K"v5I*
n = n(:); -sA&1n"W&5
m = m(:); _<f%==
I'
if any(mod(n-m,2)) yJ!26
error('zernfun:NMmultiplesof2', ... !$l<'K$
'All N and M must differ by multiples of 2 (including 0).') @@*x/"GJG
end w` +,
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if any(m>n) \Ebh6SRp\
error('zernfun:MlessthanN', ... =aB+|E
'Each M must be less than or equal to its corresponding N.') a%c <3'
end % WDTnEm
<J%Z?3@T
if any( r>1 | r<0 ) #EUT"^:d
error('zernfun:Rlessthan1','All R must be between 0 and 1.') wA$?e}
end r4P%.YO+X
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) L@O>;zp;
error('zernfun:RTHvector','R and THETA must be vectors.') C<teZz8/w
end ]a/dvj}
i45.2,
r = r(:); S}rEQGGR{
theta = theta(:); TP#Ncqh
length_r = length(r); g8E5"jpXx3
if length_r~=length(theta) pBe1:
error('zernfun:RTHlength', ... NpGi3>5
'The number of R- and THETA-values must be equal.') %iNgHoH
end }^$#vJ(a7K
=XQGg`8<LB
% Check normalization: EoutB Vm
% -------------------- 873 bg|^hs
if nargin==5 && ischar(nflag) v\bWQs1
isnorm = strcmpi(nflag,'norm'); }JtcAuQt
if ~isnorm \2+ngq)
error('zernfun:normalization','Unrecognized normalization flag.') 8!35
K
end rNhS\1-
else l@SV!keQ
isnorm = false; Eg1TF oIWl
end vKW!;U9~P
^|oI^"IQ=
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% )6%*=-
% Compute the Zernike Polynomials #f(tzPD
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ;/V])4=
$hCS-9%&
% Determine the required powers of r: tt-ci,X+
% ----------------------------------- Kh4rl)L*+%
m_abs = abs(m); ,? <;zq
rpowers = []; <=_!8A
for j = 1:length(n) 6I(Y<LZ5
rpowers = [rpowers m_abs(j):2:n(j)]; h{"SV*Xpk/
end Z0H_l/g
rpowers = unique(rpowers); +pSo(e(
Q*Jb0f
% Pre-compute the values of r raised to the required powers, 0=
bXL!]
% and compile them in a matrix: 1E!.E=Y?M
% ----------------------------- .s"Og;g
if rpowers(1)==0 *MfH\X379
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); A-B>VX
rpowern = cat(2,rpowern{:}); cg^~P-i@*
rpowern = [ones(length_r,1) rpowern]; 4xT /8>v2|
else :mDOqlXW/
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); WYRC_U7
rpowern = cat(2,rpowern{:}); ?IQDk|<