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 wUSWB{y
function z = zernfun(n,m,r,theta,nflag) 7>4t{aRf_8
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. WAQv4&xGM
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 7eq;dNB@gq
% and angular frequency M, evaluated at positions (R,THETA) on the A+dY~@*a
% unit circle. N is a vector of positive integers (including 0), and \mycn/e
% M is a vector with the same number of elements as N. Each element C=Zuy^
% k of M must be a positive integer, with possible values M(k) = -N(k) &v`kyc
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, : Z.mM5
% and THETA is a vector of angles. R and THETA must have the same y"]> Rr
% length. The output Z is a matrix with one column for every (N,M) n^A=ar.
% pair, and one row for every (R,THETA) pair. Pgo5&SQb
% kBT cND|
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike :_^YEm+A
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), jG/kT5S
% with delta(m,0) the Kronecker delta, is chosen so that the integral `W/6xm(X5;
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, '|+_~ZO*d
% and theta=0 to theta=2*pi) is unity. For the non-normalized vXf#gX!Y
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 6tzn% ?
% {!="PnB
% The Zernike functions are an orthogonal basis on the unit circle. dxd}:L~z
% They are used in disciplines such as astronomy, optics, and %:/;R_
% optometry to describe functions on a circular domain. FJD*A`a
% fY `A
% The following table lists the first 15 Zernike functions. #2dmki"~(
% E>[~"~x"pV
% n m Zernike function Normalization oNdO@i%.q4
% -------------------------------------------------- 'R$~U?i8
% 0 0 1 1 /)G9w]|T
% 1 1 r * cos(theta) 2 J d`NS3;*p
% 1 -1 r * sin(theta) 2 c9&
8kq5
% 2 -2 r^2 * cos(2*theta) sqrt(6) >s>5k
O
% 2 0 (2*r^2 - 1) sqrt(3) }%}eyLm(
% 2 2 r^2 * sin(2*theta) sqrt(6) HsXFglQ
% 3 -3 r^3 * cos(3*theta) sqrt(8) ="4jk=on
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) z4<h)hh"k6
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) U_J|{*4S.!
% 3 3 r^3 * sin(3*theta) sqrt(8) c=K M[s.
% 4 -4 r^4 * cos(4*theta) sqrt(10) :r6
bw
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ^=@%@mR/[C
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) wg[*]_,a
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) K>q,?x b
% 4 4 r^4 * sin(4*theta) sqrt(10) (2{1m#o
% -------------------------------------------------- ?LW1D+
% 63~i6
% Example 1: FkS{Z s
% )Y:CV,`
% % Display the Zernike function Z(n=5,m=1) q80?C.,`
% x = -1:0.01:1; \0:l9;^4
% [X,Y] = meshgrid(x,x); g"!B
|
% [theta,r] = cart2pol(X,Y); yf$7<gwX
% idx = r<=1; MdPwuXI
% z = nan(size(X)); bySw#h_
% z(idx) = zernfun(5,1,r(idx),theta(idx)); Sz._XY^
% figure 3sL#_@+yz
% pcolor(x,x,z), shading interp
vu1:8j
% axis square, colorbar CR_A{(
% title('Zernike function Z_5^1(r,\theta)') `,|7X]%b
% @Wv*`
% Example 2: n.T
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% Io:xG6yG
% % Display the first 10 Zernike functions D]0#A|nF
% x = -1:0.01:1; [`:\(( 8
% [X,Y] = meshgrid(x,x); ;TR.UUT
% [theta,r] = cart2pol(X,Y); .z9JoQ
% idx = r<=1; g6~uf4;
% z = nan(size(X)); y@(U6ZOyx
% n = [0 1 1 2 2 2 3 3 3 3]; Lkb?,j5
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; `yf#(YP
% Nplot = [4 10 12 16 18 20 22 24 26 28]; *AJW8tIP
% y = zernfun(n,m,r(idx),theta(idx)); %8v?dB;>x`
% figure('Units','normalized') +XQS
-=
% for k = 1:10 zi5;>Iv0}
% z(idx) = y(:,k); .IgCC_C9
% subplot(4,7,Nplot(k)) L-Hl.UV
% pcolor(x,x,z), shading interp Z)ObFJMG5
% set(gca,'XTick',[],'YTick',[]) wvgX5P>
% axis square )UxF lp;\
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ul:jn]S*
% end ;Z8K3p
% !]"T`^5,Y
% See also ZERNPOL, ZERNFUN2. uYs+xX_
g.veHh|;_
% Paul Fricker 11/13/2006 Mbi)mybM
JU~l
Xf.SJ8G
% Check and prepare the inputs: $V@IRBm
% ----------------------------- PB`94W
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) VbZZ=q=Kd
error('zernfun:NMvectors','N and M must be vectors.') gXF.e.uU
end H_jMl$f)j
1c\$ziB
if length(n)~=length(m) khyVuWN
error('zernfun:NMlength','N and M must be the same length.') -ERDW Y
end tW 9vo-{+
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n = n(:); |{Oe&j3|
m = m(:); OpiN,>;
if any(mod(n-m,2)) mH;\z;lyK
error('zernfun:NMmultiplesof2', ... +H+OYQ>^
'All N and M must differ by multiples of 2 (including 0).') i5rAb<q`
end V a<L[8
&OsJnkY<<
if any(m>n) o8Tt|Lxb$8
error('zernfun:MlessthanN', ... RU@`+6j+
'Each M must be less than or equal to its corresponding N.') oo<,hOv
end SkS
vu}
yQ h":"$k
if any( r>1 | r<0 ) k|&@xEbS
error('zernfun:Rlessthan1','All R must be between 0 and 1.') 0*+i~g,Kl@
end B1\}'g8%f
%\CsP!
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) O,hT<
s "
error('zernfun:RTHvector','R and THETA must be vectors.') hg |DpP
end N5o jXX!l%
f BukrPsV
r = r(:); Z}WMpp^r
theta = theta(:); >NK*$r8
length_r = length(r); =%p0rz|b
if length_r~=length(theta) \y{C>!WX4
error('zernfun:RTHlength', ... s<aJ pi{n4
'The number of R- and THETA-values must be equal.') )]?sCNb
end r
5:DIA!
IL&Mf9m
% Check normalization: |&3[YZY
% -------------------- XZ}]H_, n
if nargin==5 && ischar(nflag) K&\xbT
isnorm = strcmpi(nflag,'norm'); ZI}7#K<9X
if ~isnorm 3u_[=a
error('zernfun:normalization','Unrecognized normalization flag.') AYfW}V"
end ,d$V-~2,
else >]s|'HTxF
isnorm = false; 3D(/k%;)
end )Z,O*u*
7gNJ}pLDx
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% B&VruOP0
% Compute the Zernike Polynomials (~#{{Ja
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 3* 1cCM42
2ntL7F<ow
% Determine the required powers of r: UBLr|e>dQE
% ----------------------------------- ^cn%]X#.
m_abs = abs(m); %`?IY <
rpowers = []; <Y9%oJn%
for j = 1:length(n) C%vR!Az
rpowers = [rpowers m_abs(j):2:n(j)]; /0A9d-Qd<
end scT,yNV
rpowers = unique(rpowers); xk7MMRb
&
=)HPzC
% Pre-compute the values of r raised to the required powers, \oA>%+]5
% and compile them in a matrix: 49W@?:b
% ----------------------------- \!x~FVA
if rpowers(1)==0 .nl!KzO6g
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); oc7&iL
rpowern = cat(2,rpowern{:}); "wy|gnQJ
rpowern = [ones(length_r,1) rpowern]; B<zoa=
else @y]ek/
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 8iA[w-Pv
rpowern = cat(2,rpowern{:}); G)t_;iNL|
end r$T\@oTL
<Nwqt[.
% Compute the values of the polynomials: 0n<>X&X
% -------------------------------------- >pdWR1ox
y = zeros(length_r,length(n)); y(^t &tgjS
for j = 1:length(n) @G,pM: t
s = 0:(n(j)-m_abs(j))/2; K2|2Ks_CS
pows = n(j):-2:m_abs(j); _Wg?H:\
for k = length(s):-1:1 :{BD/6
p = (1-2*mod(s(k),2))* ... A#k(0e!O
prod(2:(n(j)-s(k)))/ ... =p{55dR
prod(2:s(k))/ ... ]ie38tX$
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... wz`\RHL
prod(2:((n(j)+m_abs(j))/2-s(k))); :8h\x
idx = (pows(k)==rpowers); M~+}ss
y(:,j) = y(:,j) + p*rpowern(:,idx); 1K{u>T
end ( f]@lNmx
E.LD1Pm0
if isnorm KTtB!4by
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); Bm"-X:='
end ?TWve)U
end -+ylJo[D
% END: Compute the Zernike Polynomials fJ<I|ZZ
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% (w[#h9j
J,(@1R]KF:
% Compute the Zernike functions: 03pD<
% ------------------------------ N>
7sG(!'"
idx_pos = m>0; qtrN=c3x
idx_neg = m<0; %B}<5iO
NVnId p
z = y; {#` O'F>
if any(idx_pos) *Ri\7CqU"6
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); c~``)N
end I-Q@v`
if any(idx_neg)
}_mVXjF
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); .q90+9Ek=
end b!p]\B!
{{6D4M|s
% EOF zernfun