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 tr67ofld|
function z = zernfun(n,m,r,theta,nflag) MlcoOi!
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. {KQ-Ce-6
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N &&QDEDszp
% and angular frequency M, evaluated at positions (R,THETA) on the 7=M'n;!Mh
% unit circle. N is a vector of positive integers (including 0), and RE*S7[ge
% M is a vector with the same number of elements as N. Each element _`Yvfz3
% k of M must be a positive integer, with possible values M(k) = -N(k) _c7
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, H&>>]DD
% and THETA is a vector of angles. R and THETA must have the same 3
v,ae7$U&
% length. The output Z is a matrix with one column for every (N,M) *7D$;?"
% pair, and one row for every (R,THETA) pair. nH3b<k;S
% YQ[&h
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike bUg 2Bm!y
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), :N'[de
% with delta(m,0) the Kronecker delta, is chosen so that the integral 6[Pr<4J
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, S r#fyr
% and theta=0 to theta=2*pi) is unity. For the non-normalized `
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% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. MdTd$ 4J3
% }?ac<> u&
% The Zernike functions are an orthogonal basis on the unit circle. hcqmjqJ
% They are used in disciplines such as astronomy, optics, and `a1R "A
% optometry to describe functions on a circular domain. Dm`U|<o
% _$jJpy
% The following table lists the first 15 Zernike functions. 3E2.v5*
% NB6h/0*v
% n m Zernike function Normalization tZ{q\+h
% -------------------------------------------------- PFn[[~5V
% 0 0 1 1 }Us$y0W\
% 1 1 r * cos(theta) 2 5t1DB'K9$_
% 1 -1 r * sin(theta) 2 fm2M i~}0
% 2 -2 r^2 * cos(2*theta) sqrt(6) u C8T!z
% 2 0 (2*r^2 - 1) sqrt(3) _/w-gL{
% 2 2 r^2 * sin(2*theta) sqrt(6) x jUH<LFxy
% 3 -3 r^3 * cos(3*theta) sqrt(8) o4
OEA)k)=
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) LyPBFo[?
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) #d i_V"
% 3 3 r^3 * sin(3*theta) sqrt(8) ~X(xa
% 4 -4 r^4 * cos(4*theta) sqrt(10) a0W\?
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ke6cZV5w
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) l$~bkVNL
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Q1&dB{L
% 4 4 r^4 * sin(4*theta) sqrt(10) YGO 7lar
% -------------------------------------------------- 5$G??="K
% T|iF/p]F
% Example 1: JGNxJ S<]
% tS\NO@E_Jh
% % Display the Zernike function Z(n=5,m=1) umn~hb5O
% x = -1:0.01:1; qO3BQ]UF
% [X,Y] = meshgrid(x,x); 1kw4'#J8
% [theta,r] = cart2pol(X,Y); U$JIF/MO_
% idx = r<=1; ^{+:w:g
% z = nan(size(X)); >u#VHaB
% z(idx) = zernfun(5,1,r(idx),theta(idx)); Y/I6.K3
% figure DWxh{h">
% pcolor(x,x,z), shading interp |Ie`L("
% axis square, colorbar m-FDCiN>
% title('Zernike function Z_5^1(r,\theta)') 2}C>{*}yQ
% ->9xw
% Example 2: 1Moh`
% *xVAm7_v
% % Display the first 10 Zernike functions x{o5Ha{
% x = -1:0.01:1; SpiC0
% [X,Y] = meshgrid(x,x); cZT.vA#
% [theta,r] = cart2pol(X,Y); M@@O50~
% idx = r<=1; 1e| M6*
% z = nan(size(X)); 3NZFW{u
% n = [0 1 1 2 2 2 3 3 3 3]; xVX||rrh
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; Yf`.Cq_:
% Nplot = [4 10 12 16 18 20 22 24 26 28]; Afl'-
% y = zernfun(n,m,r(idx),theta(idx)); 9+Hb`
% figure('Units','normalized') _%%"Y}
% for k = 1:10 Z_WTMs:x!
% z(idx) = y(:,k); zW`koRH@
% subplot(4,7,Nplot(k)) X[Gk!dr#
% pcolor(x,x,z), shading interp (uc)^lfX
% set(gca,'XTick',[],'YTick',[]) p+D6Z'B
% axis square /\I%)B47^9
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) BtApl)q#
% end Z*3}L
% ?^5*[H
% See also ZERNPOL, ZERNFUN2. ?G w89r
XB 7^Ka
% Paul Fricker 11/13/2006 y.<Y]m
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yyDBW`V((
% Check and prepare the inputs: Q8:ocEhR
% ----------------------------- ]arskmB]
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ,X6j$YLWp
error('zernfun:NMvectors','N and M must be vectors.') dph6aN(49
end _\;#a
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if length(n)~=length(m) DQnWLC"u
error('zernfun:NMlength','N and M must be the same length.') ;>Qd )'
end UH|.@7w
(.+n1)L?
n = n(:); E1g$WhXIS
m = m(:); Y\\nJuJo
if any(mod(n-m,2)) |:[vpJFK
error('zernfun:NMmultiplesof2', ... uelTsn
'All N and M must differ by multiples of 2 (including 0).') Ih"Ol(W
end [
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if any(m>n) dig76D_[e
error('zernfun:MlessthanN', ... 6LQ O>k
'Each M must be less than or equal to its corresponding N.') ?\r3
_
end r!iuwE@
/=} vPey
if any( r>1 | r<0 ) }dl(9H=4
error('zernfun:Rlessthan1','All R must be between 0 and 1.') X
jN.X
end zSCPp6
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Jl ?Q}SB
error('zernfun:RTHvector','R and THETA must be vectors.') "ukbqdKD
end fTgN2U
Ts6X:D4,
r = r(:); )>p6h]]a
theta = theta(:); (B#|3o
length_r = length(r); T,>e\
if length_r~=length(theta) sAlgp2-
error('zernfun:RTHlength', ... RoRVu,1
'The number of R- and THETA-values must be equal.') TD7ONa-,
end &r%3)Z8Et
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% Check normalization: T7'$A!c
% -------------------- ic#drpl,
if nargin==5 && ischar(nflag) q(W@=-uDK
isnorm = strcmpi(nflag,'norm'); -Ma"V
if ~isnorm
N\$wpDI~
error('zernfun:normalization','Unrecognized normalization flag.') q4=RE
end p6)UR~9Rs
else {%Sww:
isnorm = false; $n"Llw&)
end Efl+`6`J
}JsdgO&z
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Y~xZ{am
% Compute the Zernike Polynomials (C%'I
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% swrd
r~!lD9R~
% Determine the required powers of r: Ex3woT-
% ----------------------------------- OLwxGRYX
m_abs = abs(m); ewg WzB9c
rpowers = []; GZo4uwG@a
for j = 1:length(n) %*nZ,r
rpowers = [rpowers m_abs(j):2:n(j)]; .bGeZwvf:G
end !:5n
rpowers = unique(rpowers); 4KnDXQ%
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% Pre-compute the values of r raised to the required powers, rGWTpN
% and compile them in a matrix: /slML~$t<
% ----------------------------- 4Q5v8k=
if rpowers(1)==0 -,&Xp>u\
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 1F|+4
rpowern = cat(2,rpowern{:}); 3[rB:cE/
rpowern = [ones(length_r,1) rpowern]; wah`
else Qp,l>k
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); j^.P=;
rpowern = cat(2,rpowern{:}); O}Jb,?p
end C0RwW??t
o[6hUX0tN
% Compute the values of the polynomials: *)<tyIHd
% -------------------------------------- /L.a:Er$
y = zeros(length_r,length(n)); X#y l8k_
for j = 1:length(n) U VT8TN-T
s = 0:(n(j)-m_abs(j))/2; & \m\QI
pows = n(j):-2:m_abs(j); 0CROq}
for k = length(s):-1:1 u#\3T>o%@
p = (1-2*mod(s(k),2))* ... $gNCS:VG*
prod(2:(n(j)-s(k)))/ ... MYDSkW
prod(2:s(k))/ ... Tx5L
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... VA0TY/{
]
prod(2:((n(j)+m_abs(j))/2-s(k))); DKZ69^
idx = (pows(k)==rpowers); ;Yj}9[p;T
y(:,j) = y(:,j) + p*rpowern(:,idx); 7@F B^[H:y
end IjNm/${$
AZa3!e/1
if isnorm C N"c
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); >BX_Bou
end m"*:XfOL
end Ij+zR>P8=\
% END: Compute the Zernike Polynomials pqe**`z@y
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% pGIeW}2'9
,>$#e1!J
% Compute the Zernike functions: Hpt)(Nz:
% ------------------------------ !4E:IM63
idx_pos = m>0; NQAnvX;
idx_neg = m<0; $spf=t"nh
Cv| :.y
z = y; ( ;"ICk&
if any(idx_pos) K +~
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); %_
~[+~#
end >HFJm&lQ
if any(idx_neg) Q%7EC>V
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); TDoYp
end R/#*~tPi8
(\}IOCNS
% EOF zernfun