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 &aht K}u
function z = zernfun(n,m,r,theta,nflag) G u I sM
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. H.S|njn:r
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ba1QFzN
% and angular frequency M, evaluated at positions (R,THETA) on the rG%_O$_dO
% unit circle. N is a vector of positive integers (including 0), and 2&f=4b`Z
% M is a vector with the same number of elements as N. Each element V1V4 <Zj
% k of M must be a positive integer, with possible values M(k) = -N(k) 6Kc7@oO~
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, U`4Zj1y
% and THETA is a vector of angles. R and THETA must have the same !Yi<h/:
% length. The output Z is a matrix with one column for every (N,M) 5DBd
[u3
% pair, and one row for every (R,THETA) pair. _4#psxl[M
% |,~A9
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike t`3T_t Y
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), )8>f
% with delta(m,0) the Kronecker delta, is chosen so that the integral vPq\reKe
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, /9#jv]C:
% and theta=0 to theta=2*pi) is unity. For the non-normalized _C#()#
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. KT?s\w
% @G{DOxE*
% The Zernike functions are an orthogonal basis on the unit circle. .Bn2;nO
% They are used in disciplines such as astronomy, optics, and +~AI(h
% optometry to describe functions on a circular domain. qUg4-Z4
% *\+'tFT6
% The following table lists the first 15 Zernike functions. AUpC HG7
% VDN]P3
% n m Zernike function Normalization 3CRBu:)m
% -------------------------------------------------- tzN;;h4C
% 0 0 1 1 e;3 (,
% 1 1 r * cos(theta) 2 s*WfRY*=V
% 1 -1 r * sin(theta) 2 |*a>6y
% 2 -2 r^2 * cos(2*theta) sqrt(6) P
&._-[
% 2 0 (2*r^2 - 1) sqrt(3) e-meUf9
% 2 2 r^2 * sin(2*theta) sqrt(6) u^[v{hv'H
% 3 -3 r^3 * cos(3*theta) sqrt(8) |0 %UM}
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) mMWNUkDq
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) ~PAn
_]Z
% 3 3 r^3 * sin(3*theta) sqrt(8) Kf5 p*AI
% 4 -4 r^4 * cos(4*theta) sqrt(10) d)sl)qt}0
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) VX%\_@
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) j!H?dnE||
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 5X-(@GwN
% 4 4 r^4 * sin(4*theta) sqrt(10) oOz6Er[KO
% -------------------------------------------------- e.H"!X!0#H
% (#Aq*2Z.
% Example 1: U.x.gZRo[
% /_(Dq8^g@
% % Display the Zernike function Z(n=5,m=1) V>SA3
% x = -1:0.01:1; |7fBiVo
% [X,Y] = meshgrid(x,x); o(qmI/h
% [theta,r] = cart2pol(X,Y); SQk!o{
% idx = r<=1; t,6=EK*3T
% z = nan(size(X)); S_6;e|
% z(idx) = zernfun(5,1,r(idx),theta(idx)); VG^-aR_F
% figure _m-r}9au
% pcolor(x,x,z), shading interp n-_w0Y
% axis square, colorbar \_'pUp22
% title('Zernike function Z_5^1(r,\theta)') ']D( ({%g
% lU&IS?^?
% Example 2: jL1UPN
% p}uw-$O
% % Display the first 10 Zernike functions `#bcoK5
% x = -1:0.01:1; J-c7ZcTt
% [X,Y] = meshgrid(x,x); hT#mM*`
% [theta,r] = cart2pol(X,Y); Q0-~&e_'
% idx = r<=1; zYsGI<4
% z = nan(size(X)); 7h~M&\M
% n = [0 1 1 2 2 2 3 3 3 3]; hSH-Ck@Qy
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; 'r
CR8>k
% Nplot = [4 10 12 16 18 20 22 24 26 28]; q?\D9aT9
% y = zernfun(n,m,r(idx),theta(idx)); yAe}O#dy
% figure('Units','normalized') ER+[gT1CQ
% for k = 1:10 \ZH=$c*W
% z(idx) = y(:,k); na)_8r~
% subplot(4,7,Nplot(k)) [u:_Jqf-
% pcolor(x,x,z), shading interp fM{Vy])J
% set(gca,'XTick',[],'YTick',[]) =t2epIr5
% axis square zx*f*L,6F
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) hZy*E [i
% end k6\c^%x
% LTHS&3%2
% See also ZERNPOL, ZERNFUN2. i%2K%5{)$D
COafVlJ,l
% Paul Fricker 11/13/2006 Tj:F Qnx
2~ a4ib
JI(|sAH
% Check and prepare the inputs: )uP= o
% ----------------------------- "( xu
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 3@PVUJ0B|
error('zernfun:NMvectors','N and M must be vectors.') {Bx\Z0+'&
end 2S3F]fG0
|u[gI+TUE
if length(n)~=length(m) ^.Q),{%Xo
error('zernfun:NMlength','N and M must be the same length.') .:}\Z27-c
end nYY U
M=%p$\x
n = n(:); ,bJx|
K
m = m(:); 2Xosj(H
if any(mod(n-m,2)) b,wO^07-3^
error('zernfun:NMmultiplesof2', ... u CXd%
CzE
'All N and M must differ by multiples of 2 (including 0).') xS'So7: h
end _19k@a
'J}lnt[V
if any(m>n) p%BO:%v
error('zernfun:MlessthanN', ... f
36rU
'Each M must be less than or equal to its corresponding N.') P+xZaf
H
end TocqoYX{{
RN0Rk 8AC
if any( r>1 | r<0 ) {ib`mC^
error('zernfun:Rlessthan1','All R must be between 0 and 1.') !?96P|G
end 8eNGPuoL)
Kmtr.]Nj
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) =g ]C9'I3
error('zernfun:RTHvector','R and THETA must be vectors.') B(~D*H2T[
end I`|>'$E[r
.*,ZcO
r = r(:); r*Mm5QozA
theta = theta(:); +x1sV *S
length_r = length(r); O3Uu{'=0
if length_r~=length(theta) GC~::m~
error('zernfun:RTHlength', ... F]&9Lp}
"
'The number of R- and THETA-values must be equal.') j2z$kw%
end |Z<adOg
xnArYm
% Check normalization: Z7 @#0;g{
% -------------------- ; -3M
if nargin==5 && ischar(nflag) aaBBI S
isnorm = strcmpi(nflag,'norm'); 0o#lB^e;l
if ~isnorm \l`;]cA
error('zernfun:normalization','Unrecognized normalization flag.') nv={.H
end W{%M+a[#l
else 8/=2N
isnorm = false; =LC5o2bLy
end '{|87kI
?h5Y^}8Qg
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ."2V:;;
% Compute the Zernike Polynomials 4#o` -vcW
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *]rV,\z:
!"wIb.j}0
% Determine the required powers of r: rkD(KG9E
% ----------------------------------- te`4*t
m_abs = abs(m); )_BteLo-
rpowers = []; :r\<DVj
for j = 1:length(n) uaS?y1:c
rpowers = [rpowers m_abs(j):2:n(j)]; SXhJz=h
end vt1!|2{
h
rpowers = unique(rpowers); Fax73vl|^a
!({[^[!
% Pre-compute the values of r raised to the required powers, 3KqylC&.
% and compile them in a matrix: m~}nM |m%
% ----------------------------- GK)hK-
if rpowers(1)==0 G#csN&|,
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); g,.iM8
rpowern = cat(2,rpowern{:}); jWm<!<~
rpowern = [ones(length_r,1) rpowern]; p4/D%*G^`
else /rquI y^
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); J[^-k!9M
rpowern = cat(2,rpowern{:}); CkOd>Kn
end dfNNCPu]+
CzwnmSv{.
% Compute the values of the polynomials: $+Xohtt
% -------------------------------------- ?&[`=ZVn
y = zeros(length_r,length(n)); Ts.61Rx
for j = 1:length(n) H#f
FU
s = 0:(n(j)-m_abs(j))/2; LEY$St
pows = n(j):-2:m_abs(j); bkV_ ^8
for k = length(s):-1:1 ^JH 4:
h
p = (1-2*mod(s(k),2))* ... }^=J]
prod(2:(n(j)-s(k)))/ ... s8R.?mhH=
prod(2:s(k))/ ... m~2PpO
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... WXJ%bH
prod(2:((n(j)+m_abs(j))/2-s(k))); W &*0F~
idx = (pows(k)==rpowers); z+;+c$X
y(:,j) = y(:,j) + p*rpowern(:,idx); /:B!hvpw
end /W fpA\4S
tY VmB:l
if isnorm 1B 2>8N
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); m'Ran3rp
end O
Qd,.m
end 6L8wsz CW
% END: Compute the Zernike Polynomials P#iBwmwN+.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% v&|o5om
aCQAh[T
% Compute the Zernike functions: {>90d(j
% ------------------------------ j2V^1
idx_pos = m>0; 3~Ah8,
idx_neg = m<0; +dlN^P647
<&B)i\j8=b
z = y; Zhf+u
r
if any(idx_pos) ^`ny]3JA
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 3b~k)t4R
end m#ID%[hg$
if any(idx_neg) ?nE<Aig
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ?3[as<GZ8
end 67^?v)|
"OkJPu2!W
% EOF zernfun