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 DY9]$h*y
function z = zernfun(n,m,r,theta,nflag) AYgXqmH~+
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. b>Y{,`E3
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N fGO\f;P
% and angular frequency M, evaluated at positions (R,THETA) on the wapSpSt
% unit circle. N is a vector of positive integers (including 0), and 7@
)
% M is a vector with the same number of elements as N. Each element wD=]U@t`,
% k of M must be a positive integer, with possible values M(k) = -N(k) Ml7
(<J
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, K)BQ0v.:[
% and THETA is a vector of angles. R and THETA must have the same *8WB($T}
% length. The output Z is a matrix with one column for every (N,M) u '7h(1@
% pair, and one row for every (R,THETA) pair. ?oFd%|I
% ATl?./T u
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike Y}1c>5{bE
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), xEp?|Q$
% with delta(m,0) the Kronecker delta, is chosen so that the integral fEX=csZ86
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, o87kF!x
% and theta=0 to theta=2*pi) is unity. For the non-normalized FO5a<6
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. aL( hWE
% -cM1]soT
% The Zernike functions are an orthogonal basis on the unit circle. p,goYF??
% They are used in disciplines such as astronomy, optics, and MDU#V
% optometry to describe functions on a circular domain. &CQO+Yr$l
% 0Gc@AG{
% The following table lists the first 15 Zernike functions. -}9^$}PR
% N,c!1:b
% n m Zernike function Normalization DK\XC%~m
% -------------------------------------------------- /\c'kMAW!
% 0 0 1 1 t/\
% 1 1 r * cos(theta) 2 H*'1bLzq
% 1 -1 r * sin(theta) 2 \3$!) z
% 2 -2 r^2 * cos(2*theta) sqrt(6) \&5V';
% 2 0 (2*r^2 - 1) sqrt(3) mK[Z#obc=
% 2 2 r^2 * sin(2*theta) sqrt(6) y
%Q. (
% 3 -3 r^3 * cos(3*theta) sqrt(8) ch8a
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) A^>@6d $2
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) MLu!8dgI
% 3 3 r^3 * sin(3*theta) sqrt(8) kFv*>>X`
% 4 -4 r^4 * cos(4*theta) sqrt(10) ('tXv"fT
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) k*\Bl4g
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) -GAF>
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 6
Rl[M+Q
% 4 4 r^4 * sin(4*theta) sqrt(10) C/!.VMl^
% -------------------------------------------------- <X:JMj+
% nt#9j',6Rn
% Example 1: ]>t~Bcnm
%
u]P|
% % Display the Zernike function Z(n=5,m=1) 9{*{Ba
% x = -1:0.01:1; #;]#NqFX
% [X,Y] = meshgrid(x,x); U!aM63F3
% [theta,r] = cart2pol(X,Y); GtVT^u_
% idx = r<=1; >
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% z = nan(size(X)); zj1~[$
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% z(idx) = zernfun(5,1,r(idx),theta(idx)); zuV%`n
% figure :\\NK/"
% pcolor(x,x,z), shading interp 0O9b
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% axis square, colorbar Vxh39eW
% title('Zernike function Z_5^1(r,\theta)') d:@+dS
% i6WH^IQ M
% Example 2: Y%XF64)6
% bj
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% % Display the first 10 Zernike functions tk&AZb,sP
% x = -1:0.01:1; zm"
% [X,Y] = meshgrid(x,x); {]k#=a4
% [theta,r] = cart2pol(X,Y); m/KaWrw/)
% idx = r<=1; Ghgn<YG
% z = nan(size(X)); IZ=Z=k{
% n = [0 1 1 2 2 2 3 3 3 3]; BJj'91B[d
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; ~_\Ra%
% Nplot = [4 10 12 16 18 20 22 24 26 28]; U.e!:f4{
% y = zernfun(n,m,r(idx),theta(idx)); ~"#0rPT
% figure('Units','normalized') hdPGqJE
% for k = 1:10 5/=$p:E>
% z(idx) = y(:,k); q)?%END
% subplot(4,7,Nplot(k)) uUI#^ A
% pcolor(x,x,z), shading interp k=]e7~!
% set(gca,'XTick',[],'YTick',[]) (Q*q#U
% axis square :_8K8Sa
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) &C9IR,&
% end B\J[O5},
% Kh]es,$D
% See also ZERNPOL, ZERNFUN2. ,:?ibE=
5 pCicwea#
% Paul Fricker 11/13/2006 -9b=-K.y
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% Check and prepare the inputs: $*C
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% ----------------------------- Kxsd@^E
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) gP%<<yl
error('zernfun:NMvectors','N and M must be vectors.') !j6k]BgZ
end TO6F
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if length(n)~=length(m) QVT0.GzR
error('zernfun:NMlength','N and M must be the same length.') '12m4quO
end q8{Bx03m6
xV>
.]
n = n(:); 1=5"j]0hY
m = m(:); K*@?BE
if any(mod(n-m,2)) 'V&g"Pb
error('zernfun:NMmultiplesof2', ... K)'[^V Xh
'All N and M must differ by multiples of 2 (including 0).') Y=XDN:
end 3r~8:F"g
8-;.Ejz!\A
if any(m>n) x6/u+Urn
error('zernfun:MlessthanN', ... $bE"3/uf
'Each M must be less than or equal to its corresponding N.') .x=abA$!9
end f7&ni#^Ztj
4@{;z4*`
if any( r>1 | r<0 ) {]IY;cL
error('zernfun:Rlessthan1','All R must be between 0 and 1.') mS%4
end AROHe
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) B&MDn']fV/
error('zernfun:RTHvector','R and THETA must be vectors.') WI1YP0V
end +Z"Wa0wA
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r = r(:); @1j*\gYz
theta = theta(:); \n}%RD-Ce
length_r = length(r); t]B`>SL3W
if length_r~=length(theta) [vr"FLM|9
error('zernfun:RTHlength', ... fHaF9o+/b
'The number of R- and THETA-values must be equal.') 3cJ'tRsp<
end |;J`~H"K
)a^&7
% Check normalization: Aw7N'0K9UN
% -------------------- KcT(/!
if nargin==5 && ischar(nflag) ;1~ n|IY
isnorm = strcmpi(nflag,'norm'); *L<EGFP
if ~isnorm E?]$Y[KJKs
error('zernfun:normalization','Unrecognized normalization flag.') Ea4zC|;
end +P))*0(c_
else pauO_'j_1p
isnorm = false; >FeCa
hFn
end HDhkg-QC
B}7j20:Z
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% );HhV,$n
% Compute the Zernike Polynomials 3=wcA/"!
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% EwBrOq`C
,L%]}8EL"
% Determine the required powers of r: whN<{AG
% ----------------------------------- bM'F8Fi
m_abs = abs(m); J[}j8x?r
rpowers = []; &tUX(
for j = 1:length(n) LTf)`SN %'
rpowers = [rpowers m_abs(j):2:n(j)]; ce$[H}rDB
end q>+!Ete1p
rpowers = unique(rpowers); y:E$n!
gR/?MJ(v
% Pre-compute the values of r raised to the required powers, kP5I+B
% and compile them in a matrix: [m! P(o
% ----------------------------- wKJ|;o4;L
if rpowers(1)==0 eS Z':p
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); XnYX@p
rpowern = cat(2,rpowern{:}); (e;/Smol
rpowern = [ones(length_r,1) rpowern]; oHfr
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else `j3 OFC{7E
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); QUkP&