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 db{NKwpj'
function z = zernfun(n,m,r,theta,nflag) `Mo~EHso.
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. ~Y1"k]J
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N tfi2y]{A
% and angular frequency M, evaluated at positions (R,THETA) on the wlm3~B\64
% unit circle. N is a vector of positive integers (including 0), and j)6@q@P/
% M is a vector with the same number of elements as N. Each element Q.j-C}a
% k of M must be a positive integer, with possible values M(k) = -N(k) M3hy5j(b
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, sL!;hKK
% and THETA is a vector of angles. R and THETA must have the same &@mvw=d
% length. The output Z is a matrix with one column for every (N,M) ^JYF1
% pair, and one row for every (R,THETA) pair. >g5T;NgH9
% 0-8ELX[#
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike |usnY
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ~0VwF
% with delta(m,0) the Kronecker delta, is chosen so that the integral +W V@o'
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 0!3!?E <
% and theta=0 to theta=2*pi) is unity. For the non-normalized ==jkp
U*=
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. Jm{As*W>
% F!z! :yp
% The Zernike functions are an orthogonal basis on the unit circle. V/QTYy1
% They are used in disciplines such as astronomy, optics, and ,gAr|x7_
% optometry to describe functions on a circular domain. OGSEvfW
% eLHa9R{)B
% The following table lists the first 15 Zernike functions. o`<h=+a\
% J,dG4.ht
% n m Zernike function Normalization ')5jllxv
% -------------------------------------------------- v:'P"uU;4
% 0 0 1 1 +^^S'mP8
% 1 1 r * cos(theta) 2 >m)2ox_B
% 1 -1 r * sin(theta) 2 [8V(N2
% 2 -2 r^2 * cos(2*theta) sqrt(6) S*~Na]nS0
% 2 0 (2*r^2 - 1) sqrt(3) LM'*OtpDG
% 2 2 r^2 * sin(2*theta) sqrt(6) pl1EJ <
% 3 -3 r^3 * cos(3*theta) sqrt(8) Li?{e+ g
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) S>/I?(J
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) (P]^8qc
% 3 3 r^3 * sin(3*theta) sqrt(8) : L6-{9$
% 4 -4 r^4 * cos(4*theta) sqrt(10) )_x8?:lv
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) A-AN6.
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) sT;=7L<TA
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ^)eessZ
% 4 4 r^4 * sin(4*theta) sqrt(10) Gaw,1Ow!`2
% -------------------------------------------------- -r6(=A
% a9mr-`<
% Example 1: .@x"JI>;
% 2vW,.]95M
% % Display the Zernike function Z(n=5,m=1) @=aq&gb
% x = -1:0.01:1; +e{djp@m
% [X,Y] = meshgrid(x,x); `9G$p|6
% [theta,r] = cart2pol(X,Y); OTy4"%
% idx = r<=1; K>DnD0
% z = nan(size(X)); ^{6UAT~!R
% z(idx) = zernfun(5,1,r(idx),theta(idx)); &CPe$'FYI
% figure ]R2Z -2
% pcolor(x,x,z), shading interp #!<+:y'S?
% axis square, colorbar g-T X;(
% title('Zernike function Z_5^1(r,\theta)') 5
\.TZMB
% j*3sjOoC
% Example 2: lHj7O&+
% U_zpLpm^
% % Display the first 10 Zernike functions c,[qjr#\>
% x = -1:0.01:1; $[^ KCNB
% [X,Y] = meshgrid(x,x); q4IjCu+
% [theta,r] = cart2pol(X,Y); LcQ\?]w`]
% idx = r<=1; _UbR8
% z = nan(size(X)); !O%f)v?
% n = [0 1 1 2 2 2 3 3 3 3]; TF([yZO'
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; EC\rh](d
1
% Nplot = [4 10 12 16 18 20 22 24 26 28]; X\^3,k."
% y = zernfun(n,m,r(idx),theta(idx)); \:f}X?:
% figure('Units','normalized') w4&v( m
% for k = 1:10 ,2:L{8_L
% z(idx) = y(:,k); XTn{1[.O
% subplot(4,7,Nplot(k)) ,_X,V!
% pcolor(x,x,z), shading interp jy)9EU=
% set(gca,'XTick',[],'YTick',[]) =tvm=
% axis square ^PCL^]W
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) @_tA"E
% end , K"2tb
% enfu%"(K)
% See also ZERNPOL, ZERNFUN2. A_4\$NZ^
*rMN,B@
% Paul Fricker 11/13/2006 s-YV_
N[?4yV2s
g275{2G9
% Check and prepare the inputs: fPuQ,J2=
% ----------------------------- -QHzf&D?
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) a!6OE"?QQ
error('zernfun:NMvectors','N and M must be vectors.') neMe<jr
end >Gu>T\jpe.
e715)_HD
if length(n)~=length(m) a0v1LT6
error('zernfun:NMlength','N and M must be the same length.') *IfIRR>3l(
end ]a@v)aa-
$@
#G+QQ_
n = n(:); E(K$|k_>
m = m(:); <a/ZOuBzZ
if any(mod(n-m,2)) Y&!McM!Jw
error('zernfun:NMmultiplesof2', ... c=c.p
i"s
'All N and M must differ by multiples of 2 (including 0).') I]S(tx!
end 0BU:(o&
qi5>GX^t]b
if any(m>n) $EHn;~w T
error('zernfun:MlessthanN', ... '&L
'Each M must be less than or equal to its corresponding N.') j2&OYg
end I>(z)"1
sC*E;7gT,
if any( r>1 | r<0 ) cH8H)55F
error('zernfun:Rlessthan1','All R must be between 0 and 1.') |Z)/
end X]qp~:4G
L bK1CGyA
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) KgkB)1s@n
error('zernfun:RTHvector','R and THETA must be vectors.') S>zKD
end T)?@E/VaS
<zWQ[^
r = r(:); Z-r0
D
theta = theta(:); !QzMeN;D
length_r = length(r); }t{^*(
if length_r~=length(theta) ViC76aJ
error('zernfun:RTHlength', ... :zk.^q
'The number of R- and THETA-values must be equal.') 6(;[ov1
end Q0cf]
6Yi,%#
% Check normalization: sg~/RSJ3
% -------------------- Sf8Xj|u
if nargin==5 && ischar(nflag) ,PtR^" Mf4
isnorm = strcmpi(nflag,'norm'); YH6K-}
if ~isnorm cyn]>1ZM
error('zernfun:normalization','Unrecognized normalization flag.') $7ME a"a
end =$`")3y3
else &5CeRx7%
isnorm = false; w@D@,q'x
end :=KGQ3V~eK
t5[JN:an
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% u(Q(UuI
% Compute the Zernike Polynomials "e?#c<p7
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ku8Z;ONeH
}LVE^6zyk
% Determine the required powers of r: KuAGy*:4T
% ----------------------------------- ~wV98u-N
m_abs = abs(m); 2+rao2
rpowers = []; (W6\%H2u
for j = 1:length(n) 5_T>HHR6
rpowers = [rpowers m_abs(j):2:n(j)]; HCCp<2D"C
end B,qZwc|
rpowers = unique(rpowers); S`PSFetC
8TM=AV
% Pre-compute the values of r raised to the required powers, MjosA R
% and compile them in a matrix: R1rfp;
% ----------------------------- R3=E?us!
if rpowers(1)==0 @MVZy
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false);
Rf$6}F
rpowern = cat(2,rpowern{:}); Kct +QO(
rpowern = [ones(length_r,1) rpowern]; }|,\?7,
else AZP>\Dq
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); w6Ny>(T/
rpowern = cat(2,rpowern{:}); k0=y_7
=(5
end "s^@PzQpN
*/qc%!YV9
% Compute the values of the polynomials: y(g
Otg
% -------------------------------------- Y'":OW#oN
y = zeros(length_r,length(n)); c_=zd6 b$S
for j = 1:length(n) X'p%$HsMG
s = 0:(n(j)-m_abs(j))/2; M0\[hps~X
pows = n(j):-2:m_abs(j); ;qQzF
for k = length(s):-1:1 %}MM+1eu
p = (1-2*mod(s(k),2))* ... N>iCb:_
T;
prod(2:(n(j)-s(k)))/ ... ~d8o,.n`1
prod(2:s(k))/ ... !KW)*
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... Vi~+C@96
prod(2:((n(j)+m_abs(j))/2-s(k))); tG&B D\
idx = (pows(k)==rpowers); -B! TA0=oJ
y(:,j) = y(:,j) + p*rpowern(:,idx); EnAw8Gm*
end p#NZ\qJ
cSWVHr
if isnorm l$@lk?dc
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); T0C'$1T
end `2+52q<FO
end "lAS
<dq
% END: Compute the Zernike Polynomials 8zv6Mx
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 1EzA@3:{
?NeB_<dLa`
% Compute the Zernike functions: QR8Q10
% ------------------------------ N_}Im>;!
idx_pos = m>0; z<XS"4l?W
idx_neg = m<0; *]u/,wCB
LZ$!=vg4
z = y; 8`<GplO
if any(idx_pos) J?DyTs3Z
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); )^3655mb
end A>S2BL#=
if any(idx_neg) b&&'b)
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); $RO=r90o
end )]Rr:i9n
.v!e=i}.
% EOF zernfun