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 pQB."[n
function z = zernfun(n,m,r,theta,nflag) V0mn4sfs
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. JxU5 fe
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N VIf.q)_k
% and angular frequency M, evaluated at positions (R,THETA) on the ?S=mybp
% unit circle. N is a vector of positive integers (including 0), and X:{!n({r=
% M is a vector with the same number of elements as N. Each element %?/X=}sE
% k of M must be a positive integer, with possible values M(k) = -N(k) @=u3ZVD
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, \ Cj7k^
% and THETA is a vector of angles. R and THETA must have the same Ow,b^|
% length. The output Z is a matrix with one column for every (N,M) FS1z`wYP
% pair, and one row for every (R,THETA) pair. )4 ;`^]F
% 8u]2xB=K
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike YS_;OFsd
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), e*1_ 8I#2
% with delta(m,0) the Kronecker delta, is chosen so that the integral Vxt+]5X
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, Z;"vW!%d
% and theta=0 to theta=2*pi) is unity. For the non-normalized .=;
;
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. \~ wMfP8
% @C aG9]
% The Zernike functions are an orthogonal basis on the unit circle. #g!.T g'
% They are used in disciplines such as astronomy, optics, and F:DrX_O%
% optometry to describe functions on a circular domain. |y!A&d=xYn
% <~=Vg
% The following table lists the first 15 Zernike functions. q@2siI~W
% Znv,9-
% n m Zernike function Normalization -UT}/:a
% -------------------------------------------------- d/@,@8:
% 0 0 1 1 BJ(M2|VH
% 1 1 r * cos(theta) 2 `M6)f?|$.
% 1 -1 r * sin(theta) 2 /qw.p#
% 2 -2 r^2 * cos(2*theta) sqrt(6) #`s"WnP9'!
% 2 0 (2*r^2 - 1) sqrt(3) \73ch
% 2 2 r^2 * sin(2*theta) sqrt(6) }(u
ol
% 3 -3 r^3 * cos(3*theta) sqrt(8) >
Nr#O
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) )!T/3|C
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) x,V r=FB
% 3 3 r^3 * sin(3*theta) sqrt(8) [Vt\$
% 4 -4 r^4 * cos(4*theta) sqrt(10) +ck}l2
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) *8XEYZa
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) |Q>IrT
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) /a o5FL
% 4 4 r^4 * sin(4*theta) sqrt(10) :BTq!>s
% -------------------------------------------------- e>7i_4(C
% Z/J y'$x
% Example 1: &+R?_Ooibk
% rrv%~giU
% % Display the Zernike function Z(n=5,m=1) <9
;!3xG
% x = -1:0.01:1; HpnWoDM
% [X,Y] = meshgrid(x,x); KK &?gTa
% [theta,r] = cart2pol(X,Y); qIqM{#' ^
% idx = r<=1; 8\gjST*
% z = nan(size(X)); cN9t{.m
% z(idx) = zernfun(5,1,r(idx),theta(idx)); %~S&AE-
% figure ReeH@.74
% pcolor(x,x,z), shading interp ~PNub E
% axis square, colorbar ;A!BVq
% title('Zernike function Z_5^1(r,\theta)') @s^-.z
% |zE'd!7E
% Example 2: >&k-'`Nw
% pD]OT-8
% % Display the first 10 Zernike functions -Y;3I00(
% x = -1:0.01:1; L j$;:/G
% [X,Y] = meshgrid(x,x); `y* }lg T
% [theta,r] = cart2pol(X,Y); _wL BA^d^
% idx = r<=1; &jr3B;g!C
% z = nan(size(X)); Z EO WO
% n = [0 1 1 2 2 2 3 3 3 3]; dC4'{n|7
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; Ecx<OTo
% Nplot = [4 10 12 16 18 20 22 24 26 28]; >-{Hyx
% y = zernfun(n,m,r(idx),theta(idx)); >@AB<$A
% figure('Units','normalized') B?o7e<l[
% for k = 1:10 SK.: Q5:
% z(idx) = y(:,k); \5cpFj5%
% subplot(4,7,Nplot(k)) OK
gqT!
% pcolor(x,x,z), shading interp jlg(drTo
% set(gca,'XTick',[],'YTick',[]) (_{yB[z>`
% axis square 4nz 35BLr
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) @ur+;IK$
% end 6\S~P/PkE
% &Y eA:i?
% See also ZERNPOL, ZERNFUN2. W+1^4::+
*4_Bd=5(U
% Paul Fricker 11/13/2006 /|#fejPh
f:P}*^
Gw
8e"gW >f
% Check and prepare the inputs: 0Fr?^3h
% ----------------------------- IdxzE_@
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) BDZ?Ez\Sg
error('zernfun:NMvectors','N and M must be vectors.') 9JKEw
end ymcLFRu,
EDs\,f}
if length(n)~=length(m) :T(|&F[(
error('zernfun:NMlength','N and M must be the same length.') 5QO9Q]I#_\
end b\+`e b8_
##4HYQ%E
n = n(:); ROZF)|l
m = m(:); w"&n?L
if any(mod(n-m,2)) fa2kG&, _
error('zernfun:NMmultiplesof2', ... 9k[9P;"F:
'All N and M must differ by multiples of 2 (including 0).') !_Z&a
end 5.J.RE"M
vEz"xz1j!]
if any(m>n) 2T[9f;jM'
error('zernfun:MlessthanN', ... R,=fv
'Each M must be less than or equal to its corresponding N.') yJe>JK~)
end 26x[X.C:
QnX(V[
if any( r>1 | r<0 ) T37XBg H
error('zernfun:Rlessthan1','All R must be between 0 and 1.') Yk Qd
end wJY'
j^2j&Ta
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) z,%$+)K
error('zernfun:RTHvector','R and THETA must be vectors.') IRqy%@)
end PRE|+=w$
&~U ] ~;@
r = r(:); G'aDb/
theta = theta(:); 1D!<'`)AY
length_r = length(r); ^\,E&=/}M
if length_r~=length(theta) 2Q:+_v
error('zernfun:RTHlength', ... -!]ZMi9
'The number of R- and THETA-values must be equal.') l0i^uMS
end @>H75
F`]2O:[
% Check normalization: `&6dnSC},P
% -------------------- .y:U&Rw4
if nargin==5 && ischar(nflag) jdJ>9O0A,
isnorm = strcmpi(nflag,'norm'); OprkR
if ~isnorm G[q$QB+
error('zernfun:normalization','Unrecognized normalization flag.') 5bpEYW+
end BsYa3d=}
else
ls)%c
isnorm = false; c6]D-YNFG
end 2*#|Nj=^
UU0,!?o4
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% "AGLVp.zT
% Compute the Zernike Polynomials ]
{HI?V
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Y,zxbXZv'5
cDH^\-z
% Determine the required powers of r: s.NGA.]$
% ----------------------------------- QGmn#]w\\
m_abs = abs(m); n^6j9FQ7
rpowers = []; 'Ne@e)s9
for j = 1:length(n) N_[*H
rpowers = [rpowers m_abs(j):2:n(j)]; !f&g-V
end ^eYVWQ'
rpowers = unique(rpowers); k7A-J\
P3 ^Y"Pv?
% Pre-compute the values of r raised to the required powers, !ff&W1@
% and compile them in a matrix: Czu\RXJR
% ----------------------------- "o}+Ciul
if rpowers(1)==0 N7R!C)!IL
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); {fn!'
rpowern = cat(2,rpowern{:}); M?uC%x+S$_
rpowern = [ones(length_r,1) rpowern]; "vE4E|
else ]yPqLJ
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); H\tUpan6fy
rpowern = cat(2,rpowern{:}); (n9gkO&8"
end r#]WI|
6 3,H{
% Compute the values of the polynomials: !^Y(^RS@
% -------------------------------------- =h73s0]
y = zeros(length_r,length(n)); tS8u
for j = 1:length(n) B%+T2=&$7
s = 0:(n(j)-m_abs(j))/2; ax5<#3__
pows = n(j):-2:m_abs(j); ?R.j^S^
for k = length(s):-1:1 E#t>Qn
p = (1-2*mod(s(k),2))* ... v^iL5y!
prod(2:(n(j)-s(k)))/ ... A#'8X w|
prod(2:s(k))/ ... ,>+p-M8ZL
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ':m,)G5&
prod(2:((n(j)+m_abs(j))/2-s(k))); *w0%d1
idx = (pows(k)==rpowers); PQ$%H>{
y(:,j) = y(:,j) + p*rpowern(:,idx); *CTlOy
end a8Nh=^Py
EV@X*| w
if isnorm N `F~n%N
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); |02gup qqi
end GKc`xIQ
end dP]\Jo=Yh
% END: Compute the Zernike Polynomials =CVB BuVy
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \%N!5>cZ{
g:Xhw$x9
% Compute the Zernike functions: t$#jL5
% ------------------------------ R)ITy!z
idx_pos = m>0; =`s!;
idx_neg = m<0; 74k dsgQf
VYImI>.t{
z = y; 6 EC*
if any(idx_pos) JKmIvZ)8
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); G`BU=Fi
end lHe{\N[C
if any(idx_neg) ly_HWuFJ3
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); HqD^B[jS
end ZO$m["|
@x'"~"%7b
% EOF zernfun