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 EE!}$qOR
function z = zernfun(n,m,r,theta,nflag) K}R+~<bIY
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. i
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% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N Fr8GGN~/
% and angular frequency M, evaluated at positions (R,THETA) on the b]+F/@h~]
% unit circle. N is a vector of positive integers (including 0), and F`nQS&y
% M is a vector with the same number of elements as N. Each element Mn.,?IF`K
% k of M must be a positive integer, with possible values M(k) = -N(k) ~-BF7f6C
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, a[O6xA%
% and THETA is a vector of angles. R and THETA must have the same neDXzMxF
% length. The output Z is a matrix with one column for every (N,M) ~t`s&t'c|
% pair, and one row for every (R,THETA) pair. _6ZjF>f
% 7w/IHM L
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ;hX( /T
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), H,!xTy"Wh
% with delta(m,0) the Kronecker delta, is chosen so that the integral 7z{wYCw
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, <=q}
Nd\
% and theta=0 to theta=2*pi) is unity. For the non-normalized _95296
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. g\JJkXjD#
% `a& kD|Yh
% The Zernike functions are an orthogonal basis on the unit circle. zN4OrG0
% They are used in disciplines such as astronomy, optics, and Zh<;r;2
% optometry to describe functions on a circular domain. @^W`Yg)C
%
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% The following table lists the first 15 Zernike functions. Z',!LK!
% 0CrsZt X
% n m Zernike function Normalization _/s"VYFZ
% -------------------------------------------------- /~[Lr
% 0 0 1 1 TC\+>LXiZ
% 1 1 r * cos(theta) 2 bmfM_oz
% 1 -1 r * sin(theta) 2 IU%|K~_n
% 2 -2 r^2 * cos(2*theta) sqrt(6) SiratkP9n7
% 2 0 (2*r^2 - 1) sqrt(3) yw3"jdcl
% 2 2 r^2 * sin(2*theta) sqrt(6) y/lF1{}5
% 3 -3 r^3 * cos(3*theta) sqrt(8) ,fVD`RR(W?
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) 11[lc2
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) :S+K\
% 3 3 r^3 * sin(3*theta) sqrt(8) \*xB<mq
% 4 -4 r^4 * cos(4*theta) sqrt(10) o\IMYT
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) x *Lt]]A
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) -u2i"I730
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) B`5<sW
% 4 4 r^4 * sin(4*theta) sqrt(10) jVj5 ; }
% -------------------------------------------------- `Lf'/q
% !'7fOP-J]
% Example 1: ;;Q^/rkC
% {4 Of.
% % Display the Zernike function Z(n=5,m=1) {meX2Z4
% x = -1:0.01:1; 0B
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% [X,Y] = meshgrid(x,x); NO"PO
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% [theta,r] = cart2pol(X,Y); *eVq(R9?T
% idx = r<=1; EdJL&*
% z = nan(size(X)); *z=_sD?1
% z(idx) = zernfun(5,1,r(idx),theta(idx)); p\6cpf
% figure ^PksXfk
% pcolor(x,x,z), shading interp ;?-AFd\i
% axis square, colorbar XpQ Ol
% title('Zernike function Z_5^1(r,\theta)') tT$OnZu&
% ]n22+]D
% Example 2: }I"C4'(a
% (C2 XFg_
% % Display the first 10 Zernike functions <AHpk5Sn{
% x = -1:0.01:1; W?
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% [X,Y] = meshgrid(x,x); i=8iK#2 h
% [theta,r] = cart2pol(X,Y); v<qh;2
% idx = r<=1; sGvbL-S-f:
% z = nan(size(X)); pJpapA2l*6
% n = [0 1 1 2 2 2 3 3 3 3]; Zo9<96I&
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; !liV Y]
% Nplot = [4 10 12 16 18 20 22 24 26 28]; PxHFH pL
% y = zernfun(n,m,r(idx),theta(idx)); vh9* >[i
% figure('Units','normalized') WL$^B@gXQ
% for k = 1:10 XC4Z ,,ah"
% z(idx) = y(:,k); K~x,so
% subplot(4,7,Nplot(k)) \u3\ TJ
% pcolor(x,x,z), shading interp )&DAbB!O
% set(gca,'XTick',[],'YTick',[]) l.[pnL D
% axis square KaGUpHw
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) /'O8RUjN
% end XX;4A
% Tt[zSlIMx
% See also ZERNPOL, ZERNFUN2. h$>F}n
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% Paul Fricker 11/13/2006 ?mQ^"9^XS
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ZO}Og&%
% Check and prepare the inputs: _`$LdqgE
% ----------------------------- e5=d
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if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) uFd$*`jS
error('zernfun:NMvectors','N and M must be vectors.') ^6 6!f 5^W
end
gwZ<$6
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if length(n)~=length(m) x^f<G
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error('zernfun:NMlength','N and M must be the same length.') ajbe7#}
end HDyf]2N*N
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n = n(:); K,f:X g!:
m = m(:); mgxIxusR
if any(mod(n-m,2)) w7nt $L5
error('zernfun:NMmultiplesof2', ... `h}eP[jA
'All N and M must differ by multiples of 2 (including 0).') ?@V R%z
end $o6/dEKQ
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if any(m>n) ^}3^|jF
error('zernfun:MlessthanN', ... ,m=F
H?5
'Each M must be less than or equal to its corresponding N.') *2X6;~
end TyxIlI4"
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if any( r>1 | r<0 ) }ze+ tf
error('zernfun:Rlessthan1','All R must be between 0 and 1.') 61\u{@o$
end 1I Yip\:lS
#GsOE#*>T
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) l,wlxh$}(
error('zernfun:RTHvector','R and THETA must be vectors.') p<R:[rz
end Hg+<GML
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r = r(:); ~ KK9aV{
theta = theta(:); V>$( N/1
length_r = length(r); [f6uwp
if length_r~=length(theta) <+8'H:wz
error('zernfun:RTHlength', ... ,'NasL8?We
'The number of R- and THETA-values must be equal.') >DL
end 2:+8]b 3i
|@ mz@
% Check normalization: npP C;KD
% -------------------- *0WVrM06?
if nargin==5 && ischar(nflag) Z:b?^u4.
isnorm = strcmpi(nflag,'norm'); OhF55,[
if ~isnorm 3CUQQ_
error('zernfun:normalization','Unrecognized normalization flag.') Z[vx0[av&
end M,Gy.ivz
else #J5BHY~
isnorm = false; -15e
end \u=d`}E
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% r}U6LE?>
% Compute the Zernike Polynomials %wD#[<BGn>
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% D(cD8fn,J
?y>N&\pt2
% Determine the required powers of r: 68%aDs
% -----------------------------------
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m_abs = abs(m); c
'|*{%<e2
rpowers = []; _h%Jf{nu
for j = 1:length(n) .Xg.,kW
rpowers = [rpowers m_abs(j):2:n(j)]; HC0juT OiO
end (qcFGM22U
rpowers = unique(rpowers); zI88IM7/
wG3L+[,
% Pre-compute the values of r raised to the required powers, E4#{&sRT
% and compile them in a matrix: aRd~T6I
% ----------------------------- bC&A@.g{
if rpowers(1)==0 b[%@3 }E
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); T2{e1 =Z7
rpowern = cat(2,rpowern{:}); FT).$h~+4
rpowern = [ones(length_r,1) rpowern]; S)CsH1Q
else "+DA)K
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); B=Hd:P|
rpowern = cat(2,rpowern{:}); h*%p%t<