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 6Z#$(oC
function z = zernfun(n,m,r,theta,nflag) !O 0{ .k
%ZERNFUN Zernike functions of order N and frequency M on the unit circle.
9)*218.
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N )#l&BV5
% and angular frequency M, evaluated at positions (R,THETA) on the tjg?zlj
% unit circle. N is a vector of positive integers (including 0), and M(U<H;Csk
% M is a vector with the same number of elements as N. Each element @j<Q2z^
% k of M must be a positive integer, with possible values M(k) = -N(k) QAzwNXE+
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, z 4qEC
% and THETA is a vector of angles. R and THETA must have the same hw({>cH\
% length. The output Z is a matrix with one column for every (N,M) v\2-%
% pair, and one row for every (R,THETA) pair. QV[#^1
% $d*PY_
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike *X /i<
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), <nU8.?\?~
% with delta(m,0) the Kronecker delta, is chosen so that the integral ?0tm{qP
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, :MihVL F
% and theta=0 to theta=2*pi) is unity. For the non-normalized RxE.t[
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ?*^HZ~O1
% t{-*@8Ke
% The Zernike functions are an orthogonal basis on the unit circle. |OiM(E(
% They are used in disciplines such as astronomy, optics, and x~QZVL=:
% optometry to describe functions on a circular domain. jG`,k*eUrJ
% a0&L,7mu<'
% The following table lists the first 15 Zernike functions. $ftxid8
% _BoYyJQH
% n m Zernike function Normalization w0X})&,{`m
% -------------------------------------------------- H X{K5 +
% 0 0 1 1 F~sUfqiJ'
% 1 1 r * cos(theta) 2 #T=e p0
% 1 -1 r * sin(theta) 2 q 7-ZPX
% 2 -2 r^2 * cos(2*theta) sqrt(6) ;}H*|"z;!
% 2 0 (2*r^2 - 1) sqrt(3) VG_xNM
% 2 2 r^2 * sin(2*theta) sqrt(6) 4_-L1WH
% 3 -3 r^3 * cos(3*theta) sqrt(8) q"i]&dMr
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) /@64xrvIl=
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) ~t1?oJ
% 3 3 r^3 * sin(3*theta) sqrt(8) 9-Z?
% 4 -4 r^4 * cos(4*theta) sqrt(10) Vn65:" O
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) NJCSo(O
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) v7/k0D .
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ;+1ooeU
% 4 4 r^4 * sin(4*theta) sqrt(10) wf_ $#.;m
% -------------------------------------------------- A=sz8?K+`
% NiYT%K%
% Example 1: y&V%xE/
% <v!jS=T
% % Display the Zernike function Z(n=5,m=1) pVM1%n:#
% x = -1:0.01:1; :F_>`{
% [X,Y] = meshgrid(x,x); ZnBGNr
% [theta,r] = cart2pol(X,Y); i|rC Ga0}
% idx = r<=1; V4&a+MJ@
% z = nan(size(X)); ibn\&}1
% z(idx) = zernfun(5,1,r(idx),theta(idx)); \5-Dp9vG
% figure Aho-\9/x%
% pcolor(x,x,z), shading interp w"O{@2B3:H
% axis square, colorbar LLL;SNY
% title('Zernike function Z_5^1(r,\theta)') D&x.io
% M8IU[Pz4
% Example 2: a ?\:,5=
% 6~l+wu<$
% % Display the first 10 Zernike functions 6tGF
% x = -1:0.01:1; 22*~CIh~x
% [X,Y] = meshgrid(x,x); .Fx3WryF
% [theta,r] = cart2pol(X,Y); >2v<;.
% idx = r<=1; d@tf+_Ih
% z = nan(size(X)); Y$#6%`*#>n
% n = [0 1 1 2 2 2 3 3 3 3]; Tb!FO"o
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; $b[Ha{9(v
% Nplot = [4 10 12 16 18 20 22 24 26 28]; uPC(|U%
% y = zernfun(n,m,r(idx),theta(idx)); 5jv*C]z
% figure('Units','normalized') Fkg%_v$
% for k = 1:10 9fWR8iV
% z(idx) = y(:,k); RXo 6y(^
% subplot(4,7,Nplot(k)) uqD|j:~ =k
% pcolor(x,x,z), shading interp QQ=Kj%R
% set(gca,'XTick',[],'YTick',[]) 1,7
}ah_
% axis square I%b5a`7
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 2.^CIJc
% end x |gYxZ
% 2PSkLS&IM
% See also ZERNPOL, ZERNFUN2. O`I}Lg]~q
~pHuh#>
% Paul Fricker 11/13/2006 f\r"7j
G .$KP
O0s,)8+z5D
% Check and prepare the inputs: }=JSd@`_
% ----------------------------- o+L[o_er
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) S;u.Ds&
error('zernfun:NMvectors','N and M must be vectors.') $$SJLV
end J*_^~t
\6bvk _
if length(n)~=length(m) +_25E.>ml
error('zernfun:NMlength','N and M must be the same length.') JDW/Mc1bh
end ^/cqE[V~,
?B@3A)a
n = n(:); t1~k+
m = m(:); v
V;]?
if any(mod(n-m,2)) $Ld-lQsL
error('zernfun:NMmultiplesof2', ... k2fJ
'All N and M must differ by multiples of 2 (including 0).') "a(e2H2&T4
end }{kn/m/
FS!9 j8
if any(m>n) &g>MZ"Z|
error('zernfun:MlessthanN', ... ';}:*nZ//_
'Each M must be less than or equal to its corresponding N.') vE1:;%Q
end B.KK@
Spu;
if any( r>1 | r<0 ) 0 d+b<J,
error('zernfun:Rlessthan1','All R must be between 0 and 1.') #DARZh U)
end \t`Vq JLyu
4E_u.tJ
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) t~)4f.F:
error('zernfun:RTHvector','R and THETA must be vectors.') n*i&o;5
end `t44.=%
i[MBO`FF
r = r(:); ,1cpV|mAr
theta = theta(:); -0BxZ AW=
length_r = length(r); !VXy67
if length_r~=length(theta) JG&E"j#q
error('zernfun:RTHlength', ... kM@e_YtpY
'The number of R- and THETA-values must be equal.') *M$mAy<
end N"tX K
I2pE}6q
% Check normalization: Dx=RLiU9
% -------------------- 0M)\([W9&
if nargin==5 && ischar(nflag) 2pvby`P4
isnorm = strcmpi(nflag,'norm'); ,7Ejb++/M,
if ~isnorm Yakrsi/jV}
error('zernfun:normalization','Unrecognized normalization flag.') 1<m.Q*
end t:P7ah
else }'86hnW
isnorm = false; Jr%F#/
end h?h)i>
}}u`*&,g
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% mkPqxzxbrL
% Compute the Zernike Polynomials >e(@!\ x
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% O_GHvLO=
gwsOw [;k
% Determine the required powers of r: L>&{<M_
% ----------------------------------- k {vd1,HZ
m_abs = abs(m); IP-M)_I
rpowers = []; -e?n4YO*\
for j = 1:length(n) [6
"5
rpowers = [rpowers m_abs(j):2:n(j)]; N})vrB;1
end @HnahD
rpowers = unique(rpowers); x^i97dZS^"
&U CtyCz
% Pre-compute the values of r raised to the required powers, ~|"uuA1/#O
% and compile them in a matrix: qsN_EMgbdn
% ----------------------------- m6H+4@Z-;(
if rpowers(1)==0 :8hX kQ
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); R?,v:S&i7;
rpowern = cat(2,rpowern{:}); gNZ"Kr o6
rpowern = [ones(length_r,1) rpowern]; O'xp" e,
else wuxOFlrg
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); %KN2iNq
rpowern = cat(2,rpowern{:}); ]/3!t=La
end f_;tFP
B
m*h O@M
% Compute the values of the polynomials: ^vv1cft
% -------------------------------------- PI9aKNt
y = zeros(length_r,length(n)); cVarvueS
for j = 1:length(n) (lq%4h
s = 0:(n(j)-m_abs(j))/2; tNOOaj9mw
pows = n(j):-2:m_abs(j); 7:=(yBG
for k = length(s):-1:1 7L6^IK
p = (1-2*mod(s(k),2))* ... MSp)Jc
prod(2:(n(j)-s(k)))/ ... 7|bBC+;(
prod(2:s(k))/ ... u[4h|*'"|
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... NXz/1ut%
prod(2:((n(j)+m_abs(j))/2-s(k))); "(~fl<;
idx = (pows(k)==rpowers); 8/y8tMm]
y(:,j) = y(:,j) + p*rpowern(:,idx); :uqEGnEut
end G9#3
|B-?
M\Wg|gpy
if isnorm teLZplC=f
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); E0aFHC[
end { i4`-w
end : Q2=t!
% END: Compute the Zernike Polynomials [Z;H=`
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 3RD+;^}q3
Nr"GxezU+A
% Compute the Zernike functions: (y\.uPu!
% ------------------------------ )(1tDQ`L>
idx_pos = m>0; *_Ih@f H
idx_neg = m<0; vfVF^
WOd
\q^dhY>)
z = y; <h<_''+
if any(idx_pos) [iyhrc:@
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); =%u=ma;
end B{S^t\T$
if any(idx_neg) 31%3&B:Ts
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); onS4ZE3B
end }XRfHQk
:;LaV
% EOF zernfun