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 h( QYxI,|
function z = zernfun(n,m,r,theta,nflag) =dP{ Gh
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. @MR?6 n*k
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 6qvp*35Cx
% and angular frequency M, evaluated at positions (R,THETA) on the O!1TthI
% unit circle. N is a vector of positive integers (including 0), and (LAXM
x
% M is a vector with the same number of elements as N. Each element RH;:9_*F
% k of M must be a positive integer, with possible values M(k) = -N(k) 0pe3L
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, 0Sl]!PZR1
% and THETA is a vector of angles. R and THETA must have the same 1[nG}
% length. The output Z is a matrix with one column for every (N,M) }}{!u0N},V
% pair, and one row for every (R,THETA) pair. M<?Q4a'Q
% ;+ "f
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike woH)0v
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 5wt TP ;P
% with delta(m,0) the Kronecker delta, is chosen so that the integral Q'B6^%:<~
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, qd@&59zSh
% and theta=0 to theta=2*pi) is unity. For the non-normalized sPAg)6&M
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 5__+_hO
;3
% em@EDMvI
% The Zernike functions are an orthogonal basis on the unit circle. [BbutGvj
% They are used in disciplines such as astronomy, optics, and c2SC|s]
% optometry to describe functions on a circular domain. U4?(A@z9^
% Doze8pn
% The following table lists the first 15 Zernike functions. (AY9oei>
% fg%&N2/(.B
% n m Zernike function Normalization p 5u_1U0
% -------------------------------------------------- (3vHY`9
% 0 0 1 1 )YW<" $s
% 1 1 r * cos(theta) 2 6&v?)o
% 1 -1 r * sin(theta) 2 )(Iy<Y?#
% 2 -2 r^2 * cos(2*theta) sqrt(6) tY W>t9
% 2 0 (2*r^2 - 1) sqrt(3) o(A|)c4k
% 2 2 r^2 * sin(2*theta) sqrt(6) .?C%1a&_l
% 3 -3 r^3 * cos(3*theta) sqrt(8) G*[P<<je_
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) }b3/b
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) lw%?z/HDf
% 3 3 r^3 * sin(3*theta) sqrt(8) e>'H
IO
% 4 -4 r^4 * cos(4*theta) sqrt(10) >gtQw!
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) {kI#A?M
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) #PLEPB
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) H!e 3~+)
% 4 4 r^4 * sin(4*theta) sqrt(10) R_P}~l
% -------------------------------------------------- Tz&Y]#h_
% &6 -k#r
% Example 1: GDaN
% yWPIIWHx!
% % Display the Zernike function Z(n=5,m=1) k ^'f[|}
% x = -1:0.01:1; lB8il2&
% [X,Y] = meshgrid(x,x); UsVMoX^
% [theta,r] = cart2pol(X,Y); e`tLR- &
% idx = r<=1; !%mAh81{&/
% z = nan(size(X)); y2HxP_s?P?
% z(idx) = zernfun(5,1,r(idx),theta(idx)); |8_JY2
R
% figure jPvDFT^d/
% pcolor(x,x,z), shading interp $L4/I !Yf
% axis square, colorbar 6+rlXmd
% title('Zernike function Z_5^1(r,\theta)') u?ek|%Ok
% vZ7gS
% Example 2: ~Z/
^c,[:
% ".*x!l0y7
% % Display the first 10 Zernike functions V5}nOGV9
% x = -1:0.01:1; 7"X>?@
% [X,Y] = meshgrid(x,x); :S@1
% [theta,r] = cart2pol(X,Y); t5k!W7C
% idx = r<=1; 5`/@N{e
% z = nan(size(X)); <hnCUg1
% n = [0 1 1 2 2 2 3 3 3 3]; ]36sZ
*
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; cNpe_LvW
% Nplot = [4 10 12 16 18 20 22 24 26 28]; oj,lz?
% y = zernfun(n,m,r(idx),theta(idx)); <<A`aU^fX
% figure('Units','normalized') ^(}585b
% for k = 1:10 `L;eba
% z(idx) = y(:,k); O^>jdl!TZ
% subplot(4,7,Nplot(k)) %b.UPS@I
% pcolor(x,x,z), shading interp Gnm4gF!BI
% set(gca,'XTick',[],'YTick',[]) WnFG{S{s
% axis square $S*4r&8ZD
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) iJFs0?*
% end 07T70[G
% _;A $C(
% See also ZERNPOL, ZERNFUN2. 57{oh")
Dz=k7zRg"
% Paul Fricker 11/13/2006 a\uie$"cr]
hw_JDv+
r9 y.i(j
% Check and prepare the inputs: ;32#t[ib
% ----------------------------- u.pxz8
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 8 S`9dSc
error('zernfun:NMvectors','N and M must be vectors.') 9ILIEm:
end :^ i9]
O[17";P
if length(n)~=length(m) YO{GU7
error('zernfun:NMlength','N and M must be the same length.') ~wnOV#v
end I:(m aMc
$DFv30 f
n = n(:); bok.j
m = m(:); `D(
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if any(mod(n-m,2)) 7z6b@$,
error('zernfun:NMmultiplesof2', ... &MR/6"/s
'All N and M must differ by multiples of 2 (including 0).') G |*(8r()
end vqslirC
%HQ.|
if any(m>n) $ZPX]2D4B#
error('zernfun:MlessthanN', ... _fFU#k:MU
'Each M must be less than or equal to its corresponding N.') gV1[3dW
end {eJt,[Y *
wyx(FinIH
if any( r>1 | r<0 ) L(;WxHL
error('zernfun:Rlessthan1','All R must be between 0 and 1.') 1:C:?ZC#c
end _s,ao'/
%sh>;^58P
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Z!d7&T}
error('zernfun:RTHvector','R and THETA must be vectors.') ?B@;QjhjiJ
end q:>^ "P{
5 /",<1
r = r(:); e[u?_h
theta = theta(:); -!RtH |P
length_r = length(r); J;t 7&Zpe
if length_r~=length(theta) ivO/;)=t
error('zernfun:RTHlength', ... djQv[Vc{
'The number of R- and THETA-values must be equal.') =*BIB5
end JE5
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% Check normalization: H?1xjY9sl
% -------------------- v7
if nargin==5 && ischar(nflag) pD"vRbYF
isnorm = strcmpi(nflag,'norm'); i>L+gLW
if ~isnorm `Ycf]2.,$
error('zernfun:normalization','Unrecognized normalization flag.') h<<>3 A
end t9gfU5?
else qIUfPA=/_
isnorm = false; dhg~$CVO
end ?rVy2!
x} /,yaWZ
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |!|^ v
% Compute the Zernike Polynomials <^.=>Q0S\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Eh</? Qv\
2A`A\19t
% Determine the required powers of r: [sV"ws
% ----------------------------------- -W{DxN1
m_abs = abs(m); "|Fy+'5}
rpowers = []; v!3A9!.
for j = 1:length(n) 5[l8y,
rpowers = [rpowers m_abs(j):2:n(j)];
xp'_%n~K@
end oeSN9O
rpowers = unique(rpowers); ;DA8B'^>
~fl@ 2
% Pre-compute the values of r raised to the required powers, ^VW
PdH/Fe
% and compile them in a matrix: rVvR!"//yH
% ----------------------------- hDP/JN8y
if rpowers(1)==0 bUV >^d
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); U/ V
rpowern = cat(2,rpowern{:}); gXT9 r' k
rpowern = [ones(length_r,1) rpowern]; +:=(#Y
else m`#Od^vk
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); |@?%Ct
rpowern = cat(2,rpowern{:}); ( m\$hX
end _iKq~\v2
6%`&+Lq
% Compute the values of the polynomials: #
?1Sm/5k`
% -------------------------------------- Ng><n}
y = zeros(length_r,length(n)); @Q&3L~K"
for j = 1:length(n) =@Dwlze
s = 0:(n(j)-m_abs(j))/2; \}6;Kf}\
pows = n(j):-2:m_abs(j); Dih6mTP{
for k = length(s):-1:1 %+ 7p lM
p = (1-2*mod(s(k),2))* ... -m'j]1
prod(2:(n(j)-s(k)))/ ... G CRz<)1
prod(2:s(k))/ ... Vt^3iX{!
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... Sw^X2$h
prod(2:((n(j)+m_abs(j))/2-s(k))); ~AYN
idx = (pows(k)==rpowers); a8u9aEB
y(:,j) = y(:,j) + p*rpowern(:,idx); :.(;<b<\
end ?1L.:CS
eD$M<Eu
if isnorm )m6M9eC
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); QY/hI`
end tMj;s^P1
end i|
\6JpNA:
% END: Compute the Zernike Polynomials kP#e((f,
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% kdz=ltw
]&Z))H
% Compute the Zernike functions: f~E*Zz`;
% ------------------------------ R [H+qr
idx_pos = m>0; %6Q4yk
idx_neg = m<0; >56>*BHD
pZ`|iLNl-
z = y; bNT9 H`P
if any(idx_pos) ob+euCuJ
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); xw{-9k-~
end #T`t79*N
if any(idx_neg) 0CSv10Tg
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); y"]n:M:(
end Ehzo05/!
ntNI]~z&
% EOF zernfun