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 f_c\uN@f
function z = zernfun(n,m,r,theta,nflag) T?8BAxC?K
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. ]#o;`5'
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N KuR]X``2
% and angular frequency M, evaluated at positions (R,THETA) on the 9Yt|Wj
% unit circle. N is a vector of positive integers (including 0), and kV'zAF
v
% M is a vector with the same number of elements as N. Each element /YJo"\7
% k of M must be a positive integer, with possible values M(k) = -N(k) !>48`o^
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, <cTX;&0=
% and THETA is a vector of angles. R and THETA must have the same $kUB%\`
% length. The output Z is a matrix with one column for every (N,M) q{w|`vIb
% pair, and one row for every (R,THETA) pair. !tq]kKJ3:
% <B6md
i'R
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike LUQ.=:mBR
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 8"h;+;
% with delta(m,0) the Kronecker delta, is chosen so that the integral V( ELrjB0
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, Cy-p1s
% and theta=0 to theta=2*pi) is unity. For the non-normalized SeHrj&5U
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. L^qCE-[
% 13?:a[~=Y
% The Zernike functions are an orthogonal basis on the unit circle. qiz(k:\o
% They are used in disciplines such as astronomy, optics, and mV}bQ^*?Z
% optometry to describe functions on a circular domain. y[7M(K
% GCl
*x:
% The following table lists the first 15 Zernike functions. wDvu2iC=
% bF _]j/
% n m Zernike function Normalization {
j_-iF
% -------------------------------------------------- 8F[];LF>
% 0 0 1 1 aE0R{yup Z
% 1 1 r * cos(theta) 2 \GEz.Vb
% 1 -1 r * sin(theta) 2 'Xik2PaO
% 2 -2 r^2 * cos(2*theta) sqrt(6) [{Wo:c9Qq1
% 2 0 (2*r^2 - 1) sqrt(3) Ta[2uv>
% 2 2 r^2 * sin(2*theta) sqrt(6) 0moA mfc
% 3 -3 r^3 * cos(3*theta) sqrt(8) jf)cDj2
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) EjfQF C
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) uO4
LD}A
% 3 3 r^3 * sin(3*theta) sqrt(8) 2TGND-(j
% 4 -4 r^4 * cos(4*theta) sqrt(10) 2/3yW.C
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 7rD 8
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) A;8kC}
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 5WI
bnV@
% 4 4 r^4 * sin(4*theta) sqrt(10) /Xi21W/
% -------------------------------------------------- / =9Y(v
% p&I>xu8fl
% Example 1: q{h,}[U=
% 3$"V,_TBZ
% % Display the Zernike function Z(n=5,m=1) :2j`NyLI.
% x = -1:0.01:1; 6aB]&WO1@
% [X,Y] = meshgrid(x,x); / /NV_^$y
% [theta,r] = cart2pol(X,Y); (rFkXK4^J
% idx = r<=1; d'(n/9K
% z = nan(size(X)); /T6bc^nOW
% z(idx) = zernfun(5,1,r(idx),theta(idx)); H!Gw@u]E
% figure pj_W^,*/
% pcolor(x,x,z), shading interp vyS>3(NZ
% axis square, colorbar #~p;s>
% title('Zernike function Z_5^1(r,\theta)') +mjwX?yF
% $'l<2h>4
% Example 2: B-g-T>8
% @95p [
% % Display the first 10 Zernike functions @7}XBg[pI
% x = -1:0.01:1; ou0TKE9
_
% [X,Y] = meshgrid(x,x); TDw~sxtv&
% [theta,r] = cart2pol(X,Y); >V8!OaY5n
% idx = r<=1; A$p&<#
% z = nan(size(X)); }Bv1fbD4U
% n = [0 1 1 2 2 2 3 3 3 3]; OGcdv{,P
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; -`8@
% Nplot = [4 10 12 16 18 20 22 24 26 28]; ft7M9<#v
% y = zernfun(n,m,r(idx),theta(idx)); g5U,
% figure('Units','normalized') 8^EWD3N`
% for k = 1:10 y9mV6.r
% z(idx) = y(:,k); AyQ5jkIE^{
% subplot(4,7,Nplot(k)) u$tst_y-
% pcolor(x,x,z), shading interp OybmyGHY
% set(gca,'XTick',[],'YTick',[]) P,ZQ*Ju
% axis square uPl7u1c
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) Q"s6HZ"YI
% end Ak3^en
% G\tN(%.f
% See also ZERNPOL, ZERNFUN2. iJdJP)!tz6
.WSn Y71
% Paul Fricker 11/13/2006 W/A@q o"
<
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<mP_K^9c
% Check and prepare the inputs: _3W .:
% ----------------------------- r;b `@
.
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) +s_a{iMVP
error('zernfun:NMvectors','N and M must be vectors.') I!Dx)>E&
end G8]{pbX
XR8`,qH>
if length(n)~=length(m) De3;}]wC
error('zernfun:NMlength','N and M must be the same length.') Q~"Lyy8
end X*#\JF4$i
5M> p%/
n = n(:); zEQQ4)mA
m = m(:); auIW>0?}
if any(mod(n-m,2)) _"F=4`lJ
error('zernfun:NMmultiplesof2', ... ~i?Jg/qcxN
'All N and M must differ by multiples of 2 (including 0).') t{UWb~"
end A'![*O
[qxpu{
if any(m>n) Q,9KLi3
error('zernfun:MlessthanN', ... Uf_mwEE
'Each M must be less than or equal to its corresponding N.') qm#?DSLap
end pqv l,G5
sAO/yG
if any( r>1 | r<0 ) U(+QrC:
error('zernfun:Rlessthan1','All R must be between 0 and 1.') us5Zi# }
end &
:W6O)uY
Te!eM{_$T
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) StR)O))I
error('zernfun:RTHvector','R and THETA must be vectors.') S&=@Hj-
end 08@4u
L
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r = r(:); i`6utOq
theta = theta(:); r_ m|?U
%
length_r = length(r); h+dk2|a
if length_r~=length(theta) ,]qc#KDq-1
error('zernfun:RTHlength', ... ZJ)>gV
'The number of R- and THETA-values must be equal.') #mioT",bm=
end ;=%cA#}_0
i< imE#
% Check normalization: *XDe:A
% -------------------- `{yD\qDyX
if nargin==5 && ischar(nflag) @w%kOX
isnorm = strcmpi(nflag,'norm'); }#g &l*P
if ~isnorm kVeY} 8
error('zernfun:normalization','Unrecognized normalization flag.') ?TDmW8G}J
end Ozulp(8*
else Ir` l*:j$
isnorm = false; OvC@E]/+
end _MQh<,Z8
.GYdC'
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% )abH//Pps.
% Compute the Zernike Polynomials b!QRD'31'j
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% N>s3tGh
p&xj7qwp@F
% Determine the required powers of r: :hB6-CZkqN
% ----------------------------------- qbD_
m_abs = abs(m); ,o
`tRh<
rpowers = []; *!NW!,R
for j = 1:length(n) J| 46i
rpowers = [rpowers m_abs(j):2:n(j)]; D!)h92CIDm
end ( t"|XSF
rpowers = unique(rpowers); _+~jZ]o
N
Z0~,cO8~
% Pre-compute the values of r raised to the required powers, 8SiWAOQAL
% and compile them in a matrix: 2*-qEUl1
% ----------------------------- ;8]Hw a1!
if rpowers(1)==0 >FFp"%%
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); W"~"R
rpowern = cat(2,rpowern{:}); Cb|R
rpowern = [ones(length_r,1) rpowern]; hR>`I0|p&
else -&y&b-
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); <qoPBm])
rpowern = cat(2,rpowern{:}); 1JGww]JZo
end Me+)2S 9
EL
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% Compute the values of the polynomials: *z'Rl'j9[
% -------------------------------------- #\}xyPS
y = zeros(length_r,length(n)); +b dnTV6
for j = 1:length(n) M7gqoJM'Q
s = 0:(n(j)-m_abs(j))/2; ]#rmk!VT?
pows = n(j):-2:m_abs(j); O4W2X@
for k = length(s):-1:1 57N<OQWf
p = (1-2*mod(s(k),2))* ... *;
6LX
prod(2:(n(j)-s(k)))/ ... fb!>@@9Z
prod(2:s(k))/ ... w (,x{Bg\
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... UXS+GAWU
prod(2:((n(j)+m_abs(j))/2-s(k))); cPl$N5/5
idx = (pows(k)==rpowers); wD<W'K
y(:,j) = y(:,j) + p*rpowern(:,idx); oFu( J
end $O9#4A;
i:^
8zW
if isnorm Js,.$t
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); a3_pF~Qx
end pmDFmES
end }Do$oyAV$G
% END: Compute the Zernike Polynomials E-#}.}i5
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ,Ct1)%
Znh<r[p<
% Compute the Zernike functions: PkdL] !:
% ------------------------------ ,NU`aG-
idx_pos = m>0; u,Cf4H*xS
idx_neg = m<0; Z1+1>|-iW
L q;=UE
z = y; Czd)AVK
if any(idx_pos) {X&H
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); I6 Q{ Axy
end 5q.)K
f+
if any(idx_neg) Ivc/g,
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); k!wEPi]
end $)M5@KT
RZ:=';
% EOF zernfun