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 t3.;W/0_
function z = zernfun(n,m,r,theta,nflag) _
a|zvH
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. CfA^Xp@vc
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N R g7 O
% and angular frequency M, evaluated at positions (R,THETA) on the {i)k# `
% unit circle. N is a vector of positive integers (including 0), and hTZaI *
% M is a vector with the same number of elements as N. Each element y_:i'Ri.
% k of M must be a positive integer, with possible values M(k) = -N(k) vlAYKtl3]
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, VQO6!ToKY
% and THETA is a vector of angles. R and THETA must have the same #`rvL6W q}
% length. The output Z is a matrix with one column for every (N,M) b/='M`D}#G
% pair, and one row for every (R,THETA) pair. x8xSA*@k
% E=.4(J7K
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike qr[H0f]
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), z^to"j
% with delta(m,0) the Kronecker delta, is chosen so that the integral pmR6(/B#
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, \e64Us>"x
% and theta=0 to theta=2*pi) is unity. For the non-normalized o/bmS57
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ;H5PiSq;z
% Q<.847 )
% The Zernike functions are an orthogonal basis on the unit circle. UGK4uK+I`
% They are used in disciplines such as astronomy, optics, and V8w!yc
% optometry to describe functions on a circular domain. 5"=qVmT)
% /(Se:jH$>
% The following table lists the first 15 Zernike functions. pJ7M.C!
% 7KOM,FWKe
% n m Zernike function Normalization e$M \HPc
% -------------------------------------------------- u/3 4E=
% 0 0 1 1 &)@|WLW
% 1 1 r * cos(theta) 2 o;+$AU1f
% 1 -1 r * sin(theta) 2 hiWfVz{~
% 2 -2 r^2 * cos(2*theta) sqrt(6) E(F<shT#
% 2 0 (2*r^2 - 1) sqrt(3) V)CS,w
% 2 2 r^2 * sin(2*theta) sqrt(6) :!a'N3o>
% 3 -3 r^3 * cos(3*theta) sqrt(8) C~IsYdln
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) Zb<IZ)i# 1
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) vs-%J6}G
% 3 3 r^3 * sin(3*theta) sqrt(8) ,C%fA>?UF8
% 4 -4 r^4 * cos(4*theta) sqrt(10) <RfPd+</
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) #;59THdtPk
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) pBV_'A}ioh
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) c|8[$_2
% 4 4 r^4 * sin(4*theta) sqrt(10) AvF:$kG
% -------------------------------------------------- M8oCh
% dYdZt<6W<(
% Example 1: `,XCD-R^
% d?G~k[C!a
% % Display the Zernike function Z(n=5,m=1) .}W#YN$
% x = -1:0.01:1; m%Ah]x;
% [X,Y] = meshgrid(x,x); 2JNO@
% [theta,r] = cart2pol(X,Y); 9~ 8 A>
% idx = r<=1; z DDvXz
% z = nan(size(X)); Gzxq] Mg
% z(idx) = zernfun(5,1,r(idx),theta(idx)); bjvpYZC\5
% figure vovc,4}
% pcolor(x,x,z), shading interp Uf#.b2]
% axis square, colorbar R4+Gmx1
% title('Zernike function Z_5^1(r,\theta)') 0F$;]zg
% 8zv=@`4@G
% Example 2: cNX,%
% Ve,h]/G
% % Display the first 10 Zernike functions >\=~2>FCD
% x = -1:0.01:1; !;'#fxW[
% [X,Y] = meshgrid(x,x); =WFn+#&^
% [theta,r] = cart2pol(X,Y); q3a`Y)aVB
% idx = r<=1; HAa2q=
% z = nan(size(X)); _&!%yW@
% n = [0 1 1 2 2 2 3 3 3 3]; 6[g~p< 8n}
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; 5ve4 u
% Nplot = [4 10 12 16 18 20 22 24 26 28]; 6( 1xU\x
% y = zernfun(n,m,r(idx),theta(idx)); f>$Ld1
% figure('Units','normalized') [C)JI; \
% for k = 1:10 ^MJT lRUb
% z(idx) = y(:,k); u2=gG.
% subplot(4,7,Nplot(k)) . C_\xb
% pcolor(x,x,z), shading interp NHKIZx8sR
% set(gca,'XTick',[],'YTick',[]) 7O6VnKl
% axis square b'\a
4
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) sU>!sxW
% end cR.[4rG'
% TG ,T>'
% See also ZERNPOL, ZERNFUN2. |BrD:+
e_3KNQ`kA
% Paul Fricker 11/13/2006 r?Y+TtF\e
NPjh2 AJm
&^WJ:BvA|^
% Check and prepare the inputs: |)'gQvDM
% ----------------------------- ZZ 1s}TG
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 2w>lnJ-
error('zernfun:NMvectors','N and M must be vectors.') " jefB6k9h
end xi5/Wc6
6n9;t\'Gt
if length(n)~=length(m) P$4h_dw
error('zernfun:NMlength','N and M must be the same length.') pyPS5vWG
end qkX}pQkG)h
OE,uw2uaT
n = n(:); V&)lS Qw
m = m(:); XAN{uD^3\%
if any(mod(n-m,2)) v/% q*6@
error('zernfun:NMmultiplesof2', ... E8]PV,#xY
'All N and M must differ by multiples of 2 (including 0).') UPtWj8h
end y?BzZ16\bL
Jz(!eTVs
if any(m>n) Mv9q-SIc[
error('zernfun:MlessthanN', ... `V N $
S
'Each M must be less than or equal to its corresponding N.') (GnwK1f
end 7 ky$9+~
rx^vh%/
Q!
if any( r>1 | r<0 ) IEb"tsel
error('zernfun:Rlessthan1','All R must be between 0 and 1.') }Ip"j]h
end **I9Nw!IH
fneg[K
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) XxT7YCi
error('zernfun:RTHvector','R and THETA must be vectors.') '8g/^Y@
end .;gK*`G2W)
^Pc>/lY$Q%
r = r(:); .f'iod-
theta = theta(:); !6:q#B*
length_r = length(r); %\=oy=f
if length_r~=length(theta) p_hljgOV
error('zernfun:RTHlength', ... [oOA@
'The number of R- and THETA-values must be equal.') 5u ED
end ^/+0L[R
>-0b@ +j
% Check normalization: 3HsjF5?W
% -------------------- phIEz3Fu/
if nargin==5 && ischar(nflag) f3h&K}x
isnorm = strcmpi(nflag,'norm'); ns.[PJ"8
if ~isnorm A:"J&TbBx
error('zernfun:normalization','Unrecognized normalization flag.') )r
O`K
end )N
QtjB$
else a7G0
isnorm = false; dvUBuY^[
end l6.#s3I['
]Y[8|HJ8
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [0GM!3YJ7
% Compute the Zernike Polynomials _q([k_4h
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% )=\W
sQ
rN|c0N
% Determine the required powers of r: EXz5Rue
LV
% ----------------------------------- tK&.0)*=
m_abs = abs(m); LX<c(i
rpowers = []; 0D1yG(ck
for j = 1:length(n) Xq&x<td
rpowers = [rpowers m_abs(j):2:n(j)]; t;+6>sTu
end NEQcEUd?
rpowers = unique(rpowers); K[LTw_oE
5* 1wQlL
% Pre-compute the values of r raised to the required powers, .rj FhSr$
% and compile them in a matrix: H[ %Fo
% ----------------------------- 6l#1E#]|
if rpowers(1)==0 @]f"X>
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ]?F05!$ *
rpowern = cat(2,rpowern{:}); "r0z(j
rpowern = [ones(length_r,1) rpowern]; ~B%EvG7:n
else |7Z,z0 ?V
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ma LJ M\C
rpowern = cat(2,rpowern{:}); i L1.R+
end {+[~;ISL
=$5[uI2
% Compute the values of the polynomials: uPe4Rr
% -------------------------------------- 96F:%|yG
y = zeros(length_r,length(n)); o}5:vi]
for j = 1:length(n) 4 'rWy~`
V
s = 0:(n(j)-m_abs(j))/2; yy?|q0
pows = n(j):-2:m_abs(j); 1Qf21oN{
for k = length(s):-1:1 K@VXFV
p = (1-2*mod(s(k),2))* ... @M4~,O6-
prod(2:(n(j)-s(k)))/ ... s<qSelj
prod(2:s(k))/ ... CGg:e:4
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... K G~](4JE(
prod(2:((n(j)+m_abs(j))/2-s(k))); h~elF1dG
idx = (pows(k)==rpowers); $X5~9s1Wl
y(:,j) = y(:,j) + p*rpowern(:,idx); L}A R{
end 0c1}?$f[?%
kETA3(h'
if isnorm SPsq][5eR
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); .]Z M2
end (?R
end n:he`7.6O
% END: Compute the Zernike Polynomials UA,&0.7
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 5S7`gN.
iyOd&|.
% Compute the Zernike functions: 'KQ]7
% ------------------------------ *6*#"#D
idx_pos = m>0; Wnl8XHPn
idx_neg = m<0; 6u7(}K
!N,Z3p>Q
z = y; U^Z[6u
if any(idx_pos) N(&FATZUW
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); /db?ltb
end D4'?
V
Iz
if any(idx_neg) ao{>.b
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 8)rv.'A((E
end t@.gmUUA
E yNI]XEj
% EOF zernfun