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 #Is/j =
function z = zernfun(n,m,r,theta,nflag) WN_i-A1G/h
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. 6pKb!JJ
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N PN +<C7/
% and angular frequency M, evaluated at positions (R,THETA) on the QIcg4\d%s
% unit circle. N is a vector of positive integers (including 0), and _kJ?mTk
% M is a vector with the same number of elements as N. Each element qXb{A*J
% k of M must be a positive integer, with possible values M(k) = -N(k) ckZZ)lW`*
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, 9AbSt&#
% and THETA is a vector of angles. R and THETA must have the same 3 E~d
% length. The output Z is a matrix with one column for every (N,M) )Q!3p={S*
% pair, and one row for every (R,THETA) pair. b')Lj]%;k
% H=f'nm]dQ
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike p{sbf;-x}
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 9qqzCMrI0e
% with delta(m,0) the Kronecker delta, is chosen so that the integral 7n_'2qY
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ub#>kCL9
% and theta=0 to theta=2*pi) is unity. For the non-normalized HLPnbI-+
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. IO(Y_7
% E@f2hW2
% The Zernike functions are an orthogonal basis on the unit circle. _;M46o%h
% They are used in disciplines such as astronomy, optics, and AIx,c1G]K
% optometry to describe functions on a circular domain. RCS91[
% Pdg %:aY
% The following table lists the first 15 Zernike functions. !JkH$~
% j.5;0b_L^
% n m Zernike function Normalization Fp`MX>F
% -------------------------------------------------- K)h\X~s
% 0 0 1 1 :*{>=BD
% 1 1 r * cos(theta) 2 #kuk3}&
% 1 -1 r * sin(theta) 2 0%m}tfQ5
% 2 -2 r^2 * cos(2*theta) sqrt(6) '+
8.nN
% 2 0 (2*r^2 - 1) sqrt(3) "DW; 6<m
% 2 2 r^2 * sin(2*theta) sqrt(6) ?^# h|aUp.
% 3 -3 r^3 * cos(3*theta) sqrt(8) !A 6l\_
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) e^Ds|}{V
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) {O"?_6',
% 3 3 r^3 * sin(3*theta) sqrt(8) V&'
:S{i
% 4 -4 r^4 * cos(4*theta) sqrt(10) zeXMi:X
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Hko(@z
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) _>/T<Db
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) V?k"BU
% 4 4 r^4 * sin(4*theta) sqrt(10) /eoS$q
% -------------------------------------------------- zW@OSKq4
% CD]2a@j{
% Example 1: d^&F%)AT
% e|L$e0
% % Display the Zernike function Z(n=5,m=1) )>! IY Q
% x = -1:0.01:1; I3 %P_oW'
% [X,Y] = meshgrid(x,x); W[dMf!(
% [theta,r] = cart2pol(X,Y); Dm3/i|Y
% idx = r<=1; is3nLm(
% z = nan(size(X)); Wgh4DhAW
% z(idx) = zernfun(5,1,r(idx),theta(idx)); <Wn"_Ud=
% figure yxECK&&P0#
% pcolor(x,x,z), shading interp +3c!.] o;
% axis square, colorbar wGqQR)a
% title('Zernike function Z_5^1(r,\theta)') K|H&x"t
% $ljgFmR_
% Example 2: U#B,Q6~
% I92c!`{
% % Display the first 10 Zernike functions ,sAN,?eG~
% x = -1:0.01:1; R|Oy/RGY$
% [X,Y] = meshgrid(x,x); S;o U'KOY
% [theta,r] = cart2pol(X,Y); %^L:K5V
% idx = r<=1; 8Ee bWs*1
% z = nan(size(X)); /12D >OK
% n = [0 1 1 2 2 2 3 3 3 3]; "CEy r0h
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; W~1/vJ.*l
% Nplot = [4 10 12 16 18 20 22 24 26 28]; ]~,V(K
% y = zernfun(n,m,r(idx),theta(idx)); xHml"Y1
% figure('Units','normalized') ~YIGOL"?
% for k = 1:10 N.J;/!%!
% z(idx) = y(:,k); @17hB h
% subplot(4,7,Nplot(k)) AUloP?24
% pcolor(x,x,z), shading interp CqXD z
% set(gca,'XTick',[],'YTick',[]) 67I6]3[Z
% axis square u_aln[oIv
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) Y$^x.^dT,
% end 7]_lSYwrb
% ZCQ7xQD
% See also ZERNPOL, ZERNFUN2. 7'[C+/:
HQ%-e5Q
% Paul Fricker 11/13/2006 $*| :A
(D'Z4Y
TQ?D*&
% Check and prepare the inputs: )Oq N\
% ----------------------------- 4#5w^
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) i<g|+}I
error('zernfun:NMvectors','N and M must be vectors.') `_]Z#X&&h
end WUid5e2
U*ZP>Vv
if length(n)~=length(m) p[(VhbN
error('zernfun:NMlength','N and M must be the same length.') 8#%p[TLj
end ,L+tm>I
#@,39!;,:O
n = n(:); v>3)^l:=Y*
m = m(:); Sti)YCXH
if any(mod(n-m,2)) Q6y883>9
error('zernfun:NMmultiplesof2', ... W{Cc wq
'All N and M must differ by multiples of 2 (including 0).') ;lST@>
end %$j)?e
.>0e?A4,5?
if any(m>n) =2#a@D6Bl
error('zernfun:MlessthanN', ... O)MKEMuA
'Each M must be less than or equal to its corresponding N.') \?[#>L4
end _=Y]ZX`j
6h
N~<
if any( r>1 | r<0 ) $Yt29AQ
error('zernfun:Rlessthan1','All R must be between 0 and 1.') #Zpp*S55
end 2}u hPW+
zCD?5*7
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) a z
7Vy-
error('zernfun:RTHvector','R and THETA must be vectors.') p6[a"~y
end 5y!
4ny_
w>T1D
r = r(:); rt%.IQdY
theta = theta(:); r)<]W@Pr
length_r = length(r); 05:`(vl
if length_r~=length(theta) b
r)o Sw
error('zernfun:RTHlength', ... . m_y5J
'The number of R- and THETA-values must be equal.') 8NJ(l
end U">D_ 8
h0NM5
% Check normalization: OpY2Z7_
% -------------------- [~bfM6Jw
if nargin==5 && ischar(nflag) @.fuR#
isnorm = strcmpi(nflag,'norm'); 4KE"r F
if ~isnorm 1)u
3
error('zernfun:normalization','Unrecognized normalization flag.') 2O {@W +Mt
end KyW6[WA9
else FG7}MUu
isnorm = false; ?eT^gWX
end /-<S F T`
fGJPZe
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% #NVtZs!V/
% Compute the Zernike Polynomials M#on-[
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \_FX}1Wc2.
cu|gM[
% Determine the required powers of r: < pI2}
% ----------------------------------- #M6@{R2_
m_abs = abs(m); cj-P&D[Ny[
rpowers = []; <CJua1l\
for j = 1:length(n) ,+P!R0PNH
rpowers = [rpowers m_abs(j):2:n(j)]; I,vy__sZ
end )JE;#m0q
rpowers = unique(rpowers); .Vux~A
Lm\N`
% Pre-compute the values of r raised to the required powers, Z{`;Ys:zk
% and compile them in a matrix: ;rpjXP
% ----------------------------- T%K(opISc(
if rpowers(1)==0 VO>A+vx3M
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); #EAP<h
rpowern = cat(2,rpowern{:}); L5""
rpowern = [ones(length_r,1) rpowern]; 8Cz_LyL
else }pj>BK>
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); Z}.N4 /
rpowern = cat(2,rpowern{:}); *. l,_68
end DDn@M|*$
KDgJ~T
% Compute the values of the polynomials: /j./
% -------------------------------------- Gvv~P3Dm
y = zeros(length_r,length(n)); aM?Xi6
U5
for j = 1:length(n)
bLGgu#
s = 0:(n(j)-m_abs(j))/2; [=9-AG~}
pows = n(j):-2:m_abs(j); vmL%%7
for k = length(s):-1:1 >|!F.W
p = (1-2*mod(s(k),2))* ... KgX~PP>
prod(2:(n(j)-s(k)))/ ... M~w
=ZJ@
prod(2:s(k))/ ... 2}>jq8Y47
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ,xB&{J
prod(2:((n(j)+m_abs(j))/2-s(k))); t>f<4~%MJ
idx = (pows(k)==rpowers); ,rc5r3
y(:,j) = y(:,j) + p*rpowern(:,idx); uQWJ7Xm
end lz@fXaZM
C_=! ( @`8
if isnorm EP&