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 '~RP+
function z = zernfun(n,m,r,theta,nflag) umrfA
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. [z$th
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N m72r6Yq2@
% and angular frequency M, evaluated at positions (R,THETA) on the xJ>U_Gd
% unit circle. N is a vector of positive integers (including 0), and q"OvuHBSOn
% M is a vector with the same number of elements as N. Each element S6gg(nNe
% k of M must be a positive integer, with possible values M(k) = -N(k) H:
Rd4dl,
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, 'Z#8]YP`
% and THETA is a vector of angles. R and THETA must have the same hjywYd]8
% length. The output Z is a matrix with one column for every (N,M) T+7O+X#
% pair, and one row for every (R,THETA) pair. &*\wr}a!
% _p/
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% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike tW;1
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), gT$`a
% with delta(m,0) the Kronecker delta, is chosen so that the integral Q?KWiFA}'
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, bD[W`yW0
% and theta=0 to theta=2*pi) is unity. For the non-normalized (K"U# Zn
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. mt$0p|B8
% 4(>|f_$
% The Zernike functions are an orthogonal basis on the unit circle. 6m_
fEkS[
% They are used in disciplines such as astronomy, optics, and Y.&nxT95=
% optometry to describe functions on a circular domain. A L|F
Bd
% +CL`]'~;E-
% The following table lists the first 15 Zernike functions. =n>&Bl-Bl
% r9<OB`)3+
% n m Zernike function Normalization [F4]pR(
% -------------------------------------------------- /1ZRjf^
% 0 0 1 1 L=4%MyZ.e
% 1 1 r * cos(theta) 2 3B#qQ#
% 1 -1 r * sin(theta) 2 f0+)%gO{
% 2 -2 r^2 * cos(2*theta) sqrt(6) !t/I
j ~o
% 2 0 (2*r^2 - 1) sqrt(3) Eb66GXF[
% 2 2 r^2 * sin(2*theta) sqrt(6) Q$!dPwDg
% 3 -3 r^3 * cos(3*theta) sqrt(8) t'Zq>y;yg
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) lt\.
)Y>4
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8)
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% 3 3 r^3 * sin(3*theta) sqrt(8) Y=+pz^/"
% 4 -4 r^4 * cos(4*theta) sqrt(10) $'#hCs
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Nv!If$d
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) gE$D#PZa
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ha+)ZF
% 4 4 r^4 * sin(4*theta) sqrt(10) *@`Sx'5!
% -------------------------------------------------- k I
% KITC,@xE_O
% Example 1: yClX!OL
% &`+tWL6L
% % Display the Zernike function Z(n=5,m=1) W]b>k lp;
% x = -1:0.01:1; PhTMXv<cE
% [X,Y] = meshgrid(x,x); J:g4ES-/
% [theta,r] = cart2pol(X,Y); r'!L}^n
% idx = r<=1; o9I=zAGjy
% z = nan(size(X)); ~n9x
,
% z(idx) = zernfun(5,1,r(idx),theta(idx)); a=n*}.
% figure ,*_=w^;Rr
% pcolor(x,x,z), shading interp SB"Uu2)wZ
% axis square, colorbar ZBYFQTEE
% title('Zernike function Z_5^1(r,\theta)') DJ)Q,l*|N9
% [t#xX59
% Example 2: />^ sGB
% g
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% % Display the first 10 Zernike functions fCC^hB]'
% x = -1:0.01:1; =^a Ngq
% [X,Y] = meshgrid(x,x); EjxzX1:
% [theta,r] = cart2pol(X,Y); ?r
P'PUB
% idx = r<=1; CshYUr -
% z = nan(size(X)); 44@yQ?
% n = [0 1 1 2 2 2 3 3 3 3]; :(7icHa
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; <5).(MTa
% Nplot = [4 10 12 16 18 20 22 24 26 28]; tZ|0wPp
% y = zernfun(n,m,r(idx),theta(idx)); L>xecep
% figure('Units','normalized') ;W"=s79
% for k = 1:10 +%E)]*Ym
% z(idx) = y(:,k); Klr+\R@(n
% subplot(4,7,Nplot(k)) i`k{}!F
% pcolor(x,x,z), shading interp #Y|t,x;
% set(gca,'XTick',[],'YTick',[]) oUSv)G.zb
% axis square M/<>'%sj
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ":igYh
% end ::<v; `l
% @J~hi\&`
% See also ZERNPOL, ZERNFUN2. o/dj1a~U
*z?Vy<u G
% Paul Fricker 11/13/2006 \tCxz(vKz
y6#AL<W@=
.|?UqZ(,
% Check and prepare the inputs: *I)F5M
% ----------------------------- pUV4oyGV
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 1s\
error('zernfun:NMvectors','N and M must be vectors.') =[_=y=G
end $X\deJ1Hi
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if length(n)~=length(m) rf\/Y"D
error('zernfun:NMlength','N and M must be the same length.') 8cKP_Ec
end OV>JmYe1{/
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n = n(:); gPJZpaS
m = m(:); 8?l/x
if any(mod(n-m,2)) j'IZ etT
error('zernfun:NMmultiplesof2', ... !_i;6UVG
'All N and M must differ by multiples of 2 (including 0).') PS:"mP7n
end eVRPjVzQ'Q
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if any(m>n) /^M|$JRI
error('zernfun:MlessthanN', ... yiO/0n Mp
'Each M must be less than or equal to its corresponding N.') ?GT,Y5
end ;ElwF&"!X
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if any( r>1 | r<0 ) fk5$z0 /
error('zernfun:Rlessthan1','All R must be between 0 and 1.') Fo.p}j+>
end (qyT,K8
oVAY}q|wU
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Oaj$Z-
f
error('zernfun:RTHvector','R and THETA must be vectors.') 3'jH,17lWV
end OAiSE`
v\ <4y P
r = r(:); ]8o[&50y
theta = theta(:); N+nv#]{
length_r = length(r); wAA9M4
if length_r~=length(theta) 8M6wc394
error('zernfun:RTHlength', ... Sv>bU4LHf
'The number of R- and THETA-values must be equal.') )RCva3Ul
end @3v[L<S{
h anS8
% Check normalization: QLLMSa+! \
% -------------------- 1e)5D& njS
if nargin==5 && ischar(nflag) s`dwE*~
isnorm = strcmpi(nflag,'norm');
=yCz!vc
if ~isnorm 0
zn }l6OS
error('zernfun:normalization','Unrecognized normalization flag.') qBDhCE
end jccSjGX@w
else D:=Q)Uh0I
isnorm = false; )YY8`\F>1
end ~{00moN"m
w:3CWF4q]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ?'/#Gt`
% Compute the Zernike Polynomials `gE_u
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% w|[{xn^R
L7"B`oa(p
% Determine the required powers of r: u5I#5
% ----------------------------------- cMZ-
m_abs = abs(m); ]yV,lp
rpowers = []; rp_Aw
for j = 1:length(n) @!KG;d:l
rpowers = [rpowers m_abs(j):2:n(j)]; h=o%\F4
end iPK:gK3Q
rpowers = unique(rpowers); B!AJ*
VK[`e[.C
% Pre-compute the values of r raised to the required powers, Aq,&p,m03
% and compile them in a matrix: :TRhk.
% ----------------------------- i~ITRi@
if rpowers(1)==0 fl+dL#]
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); E5Zxp3 N
rpowern = cat(2,rpowern{:}); _)a!g-Do7
rpowern = [ones(length_r,1) rpowern]; N?l
else &pFP=|Pq
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); &'"dYZj{
rpowern = cat(2,rpowern{:}); Z_(P^/
end JWVn@)s
7*(K%e"U
% Compute the values of the polynomials: z|v/hUrD
% -------------------------------------- zOn%\
y = zeros(length_r,length(n)); xY@<