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 C+'-TLeu
function z = zernfun(n,m,r,theta,nflag) (7qlp*8.s
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. !H\;X`W|~D
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N l>;hQ h
% and angular frequency M, evaluated at positions (R,THETA) on the J$6WU z:?
% unit circle. N is a vector of positive integers (including 0), and ,P9F*;Dj
% M is a vector with the same number of elements as N. Each element 4 bk`i*-O
% k of M must be a positive integer, with possible values M(k) = -N(k) *)RKU),3nL
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, [)V~U?
% and THETA is a vector of angles. R and THETA must have the same NFTv4$5d
% length. The output Z is a matrix with one column for every (N,M) #QFz /6
% pair, and one row for every (R,THETA) pair. gnH{_
% ,ciX *F"
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike iZG-ca
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), JtO}i{A
% with delta(m,0) the Kronecker delta, is chosen so that the integral )B]s.w
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, bD{tsxm[9
% and theta=0 to theta=2*pi) is unity. For the non-normalized s4|tWfZ
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. __b4dv
% s?HK2b^;D
% The Zernike functions are an orthogonal basis on the unit circle. PE5*]+lW.
% They are used in disciplines such as astronomy, optics, and '1D$ ;
% optometry to describe functions on a circular domain. P%:?"t+J`;
% lG-B)
F
% The following table lists the first 15 Zernike functions. *OA(v^@tx7
% kSV(T'#x
% n m Zernike function Normalization )n)AmNpq
% -------------------------------------------------- wn@~80)$
% 0 0 1 1 (kR
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% 1 1 r * cos(theta) 2 ,marNG
% 1 -1 r * sin(theta) 2 ,<
g%}P/
% 2 -2 r^2 * cos(2*theta) sqrt(6) E2M<I;:EA
% 2 0 (2*r^2 - 1) sqrt(3) XMS:F]HN
% 2 2 r^2 * sin(2*theta) sqrt(6) |fKT@2(
% 3 -3 r^3 * cos(3*theta) sqrt(8) 4^r6RS@z
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) /Pextj<
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) "m {i`<,
% 3 3 r^3 * sin(3*theta) sqrt(8) ,Vq$>T@z
% 4 -4 r^4 * cos(4*theta) sqrt(10) ]){ZL
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ;=n}61
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) 'ge$}L}4
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) A5j?Yts
% 4 4 r^4 * sin(4*theta) sqrt(10) <n,QSy#
% -------------------------------------------------- 6hj[/O)E
% CJk"yW[,|
% Example 1: (-$5YKm
% B>1,I'/$.
% % Display the Zernike function Z(n=5,m=1) JOG-i
% x = -1:0.01:1; Pd+*syOM
% [X,Y] = meshgrid(x,x); SZT n=\
% [theta,r] = cart2pol(X,Y); VWzQXo
% idx = r<=1; R ?s;L
r
% z = nan(size(X)); X'b3CS4
% z(idx) = zernfun(5,1,r(idx),theta(idx)); NxF:s,a6
% figure 3Iqvc v
% pcolor(x,x,z), shading interp r^6@Zwox]
% axis square, colorbar 3ibQbk
% title('Zernike function Z_5^1(r,\theta)') E
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% G%k&|
% Example 2: gHc1_G]
% 1Du5Z9AM
% % Display the first 10 Zernike functions 8?8V;
% x = -1:0.01:1; ;`/a. /bc
% [X,Y] = meshgrid(x,x); %Mj,\J!
% [theta,r] = cart2pol(X,Y); <.Zh{"$qo
% idx = r<=1; i#4+l$q
% z = nan(size(X)); T%oJmp?0
% n = [0 1 1 2 2 2 3 3 3 3]; Sed8Q-m
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; /RJ]MQ\*O
% Nplot = [4 10 12 16 18 20 22 24 26 28]; U\Y0v.11
% y = zernfun(n,m,r(idx),theta(idx)); }J6:D]Q
% figure('Units','normalized') ?{aC-3VAT
% for k = 1:10 ~]?sA{
% z(idx) = y(:,k); [>mH
% subplot(4,7,Nplot(k)) )C"ixZ>2xQ
% pcolor(x,x,z), shading interp j^#p#`m
% set(gca,'XTick',[],'YTick',[]) UF^[?M =
% axis square U]}F A2
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 2FaCrc/
% end
ilQ}{p6I
% S`YT"|~
% See also ZERNPOL, ZERNFUN2. qpFxl
31c*^ZE.
% Paul Fricker 11/13/2006 F?tWx+N<{
$)@D(m,ybd
p*5_+u
% Check and prepare the inputs: pYzop4
% ----------------------------- CyLwCS{V\
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) iS)-25M'
error('zernfun:NMvectors','N and M must be vectors.') 4Cu\|"5)
end 'm`}XGUBS
7w2$?k',-
if length(n)~=length(m) VqvjOeCbH
error('zernfun:NMlength','N and M must be the same length.') L7{}`O/g7
end ~tWh6-:|{J
),vDn}>
n = n(:); q
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m = m(:); [uGsF0#e
if any(mod(n-m,2)) ~C^:SND7
error('zernfun:NMmultiplesof2', ... Z8Ig,
'All N and M must differ by multiples of 2 (including 0).') NA2={RB;
end .-iW
T4Dn
7'esJ)2
if any(m>n) T0dD:s N
error('zernfun:MlessthanN', ... /d}"s.3p
'Each M must be less than or equal to its corresponding N.') RHBQgD$
end O'IU1sU
(_4DZMf
if any( r>1 | r<0 ) _p4]\LA
error('zernfun:Rlessthan1','All R must be between 0 and 1.') Lu6g`O:['
end {|>Wwa2e
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) `XH0S`B
error('zernfun:RTHvector','R and THETA must be vectors.') b
MD|
end "P#1=
>w<w*pC
r = r(:); v=iiS}s
theta = theta(:); :,JjN&
length_r = length(r); v[|W\y@H/3
if length_r~=length(theta) ^wWbW&<Tg
error('zernfun:RTHlength', ... Q;VuoHj!
'The number of R- and THETA-values must be equal.') Z6${nUX
end C`t@tgT
(eU 4{X7
% Check normalization: 'I/_vqp@
% -------------------- }NyQ<,+mq&
if nargin==5 && ischar(nflag) h_#=f(.'j
isnorm = strcmpi(nflag,'norm'); WtZI1`\qe
if ~isnorm 8u~
error('zernfun:normalization','Unrecognized normalization flag.') -O\i^?lD;
end HdxP:s.T
else F^bY]\-5
isnorm = false; % Q6
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end Y&yfm/R u
ciODTq?
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% D {Ol8:
% Compute the Zernike Polynomials 2lsUCQI;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% J6s]vV q"
R]X 0D.
% Determine the required powers of r: Sj ~SG
% ----------------------------------- "."(<c/3
m_abs = abs(m); rWL;pM<
rpowers = []; o5a=>|?p>
for j = 1:length(n) q 7%p3
rpowers = [rpowers m_abs(j):2:n(j)]; L>~Tc
end :K^J bQ
rpowers = unique(rpowers); T#-;>@a}
kd^H}k
% Pre-compute the values of r raised to the required powers, o:Kw<z,$H
% and compile them in a matrix: Cty#|6k
% ----------------------------- Oq.ss!/z
if rpowers(1)==0 A_9^S!
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); $!>.h*np
rpowern = cat(2,rpowern{:}); 3U >-~-DS
rpowern = [ones(length_r,1) rpowern]; { V6pC
else To>,8E+GAb
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); RX>P-vp
rpowern = cat(2,rpowern{:}); iv$YUM+
end 2.z-&lFBZ
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% Compute the values of the polynomials: %nY\"
% -------------------------------------- L_!ShE
y = zeros(length_r,length(n)); CfU|]<
for j = 1:length(n) pc*)^S
s = 0:(n(j)-m_abs(j))/2; RA[j=RxK
pows = n(j):-2:m_abs(j); #3qeRl
for k = length(s):-1:1 j-ej7
p = (1-2*mod(s(k),2))* ... 7tcadXk0
prod(2:(n(j)-s(k)))/ ... %BGg?&
prod(2:s(k))/ ... AChz}N$C
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ;_(f(8BO
prod(2:((n(j)+m_abs(j))/2-s(k))); [oTe8^@[
idx = (pows(k)==rpowers); g&FTX>wX
y(:,j) = y(:,j) + p*rpowern(:,idx); 12n:)yQy
end u)0I$Tc"
C")genMH
if isnorm #; ?3kuq(
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); } jj)
end ?+d`_/IB
end d5m-f/
% END: Compute the Zernike Polynomials 3^y(@XFt
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% "OjAhKfG
!B3TLeh
% Compute the Zernike functions: )SmnLvL
% ------------------------------ <