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 )}!'VIe^!
function z = zernfun(n,m,r,theta,nflag) uek3Y[n
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. \[EWxu
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N n #I}!x>2
% and angular frequency M, evaluated at positions (R,THETA) on the JrTBe73.]j
% unit circle. N is a vector of positive integers (including 0), and l)s +"C#
% M is a vector with the same number of elements as N. Each element *,*qv^
% k of M must be a positive integer, with possible values M(k) = -N(k) 4/WCs$
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, Dys"|,F
% and THETA is a vector of angles. R and THETA must have the same X)OP316yx
% length. The output Z is a matrix with one column for every (N,M) hp4(f W
% pair, and one row for every (R,THETA) pair. pH%c7X/[3L
% -%l,Zd9
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike oJT@'{;*z
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), )`7+o9&
% with delta(m,0) the Kronecker delta, is chosen so that the integral 63Yu05'
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, %iC63)(M
% and theta=0 to theta=2*pi) is unity. For the non-normalized _ n4ma
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. {~g
% \#,#_
% The Zernike functions are an orthogonal basis on the unit circle. {VG[m@
% They are used in disciplines such as astronomy, optics, and 2z# @:Q
% optometry to describe functions on a circular domain. L.[uMuUa
% r.^X>?
% The following table lists the first 15 Zernike functions. [#'_@zZz
% )#~fS28j
% n m Zernike function Normalization m(:qZW
% -------------------------------------------------- K0=E4>z,`q
% 0 0 1 1 wLe&y4
% 1 1 r * cos(theta) 2 \<x_96jt!\
% 1 -1 r * sin(theta) 2 R6mJFE*6T9
% 2 -2 r^2 * cos(2*theta) sqrt(6) C*e[CP@u
% 2 0 (2*r^2 - 1) sqrt(3) `f+g A
% 2 2 r^2 * sin(2*theta) sqrt(6) nY-9
1q?Y
% 3 -3 r^3 * cos(3*theta) sqrt(8) ,ri--<
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) z2V8NUn
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) pY>-N
% 3 3 r^3 * sin(3*theta) sqrt(8) A;a(n\Sy
% 4 -4 r^4 * cos(4*theta) sqrt(10) c.A/{a
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) G$9|aaf`1#
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) ,<r 3Z$G
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) n12c075
% 4 4 r^4 * sin(4*theta) sqrt(10) S&]<;N_B
% -------------------------------------------------- ={@ @`yP^$
% qgsE7 ]
% Example 1: V?dK *8s
% ]J=)pDrk
% % Display the Zernike function Z(n=5,m=1) gs8@b5 RSb
% x = -1:0.01:1; U]EuDNkO{
% [X,Y] = meshgrid(x,x); `4$Qv'X*
% [theta,r] = cart2pol(X,Y); A<CXd t+t
% idx = r<=1; 0QH3,Ps1C
% z = nan(size(X)); )u/
^aK53^
% z(idx) = zernfun(5,1,r(idx),theta(idx)); `Mp7})
% figure D4 ]B>
% pcolor(x,x,z), shading interp JK]tcP
% axis square, colorbar m&~Dj#%(w
% title('Zernike function Z_5^1(r,\theta)') }\L!;6oy
% a{Hb7&
% Example 2: cPaWJ+c
% (Cd{#j<
% % Display the first 10 Zernike functions 9`n)"r
% x = -1:0.01:1; G$|;~'E
% [X,Y] = meshgrid(x,x); xXxh3 k\
% [theta,r] = cart2pol(X,Y); /A))"D
% idx = r<=1; t:sq*d
% z = nan(size(X)); =*:_swd
% n = [0 1 1 2 2 2 3 3 3 3]; bKMR7&e.Ep
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; v;}`?@G
% Nplot = [4 10 12 16 18 20 22 24 26 28]; C9Z\G 3
% y = zernfun(n,m,r(idx),theta(idx)); pH l2!{z
% figure('Units','normalized') KPd C9H
% for k = 1:10 p vQK6r
% z(idx) = y(:,k); hd
;S>K/C
% subplot(4,7,Nplot(k)) j484b2uj1
% pcolor(x,x,z), shading interp Ar7mH4M
% set(gca,'XTick',[],'YTick',[]) $EGRaps{j>
% axis square e=jT]i *cU
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) [H:GKhPC`
% end /<7C[^h{-
% DEQE7.]3 q
% See also ZERNPOL, ZERNFUN2. m_Ac/ctf
O:^LQ
% Paul Fricker 11/13/2006 3JZWhxkf[$
Xz.Y-5)
$7DcQ b9
% Check and prepare the inputs: K7xWE,y
% ----------------------------- [kuVQ$)
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) d:<H?~
error('zernfun:NMvectors','N and M must be vectors.') "(\)
&G
end !MJe+.
,WB_C\.#XN
if length(n)~=length(m) 9kX=99kf[
error('zernfun:NMlength','N and M must be the same length.') D@\;@(
|
end %V(N U_o
~l*?D7[o
n = n(:); H&=n:'k^
m = m(:); r -q3+c^+
if any(mod(n-m,2)) 6(J4IzZ
error('zernfun:NMmultiplesof2', ... (YYj3#|
'All N and M must differ by multiples of 2 (including 0).') G]mWaA
end ,s><kHJ
M9s43XL(&
if any(m>n) pgd8`$(Q
error('zernfun:MlessthanN', ... qQxA@kdd
'Each M must be less than or equal to its corresponding N.') S2
"=B&,}
end 3 IWLBc
Yb%-tv:
if any( r>1 | r<0 ) bKuj
po6
error('zernfun:Rlessthan1','All R must be between 0 and 1.') p>:ef<.i
end K4k~r!&OU
y5/'!L)g
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) *|k;a]HT
error('zernfun:RTHvector','R and THETA must be vectors.') &(1H!
end ! FR%QGn1
{;&B^uz
]
r = r(:); 1 O7]3&L@
theta = theta(:); %h" qMs S
length_r = length(r); R>d@tr
if length_r~=length(theta) C1T=O
error('zernfun:RTHlength', ... ,]Ro',A&
'The number of R- and THETA-values must be equal.') )>y
k-
end Q'|0?nBOY
^}o7*
% Check normalization: &@%
b?~
% -------------------- _^ q\XPS
if nargin==5 && ischar(nflag) @GG(7r\/B
isnorm = strcmpi(nflag,'norm'); -Aa]aDAz68
if ~isnorm fimb]C I|x
error('zernfun:normalization','Unrecognized normalization flag.') ^Ue0mC7m
end o#xgrMB
else wy{ \/?~c
isnorm = false; w{3Q( =&
end ,{?q^"
I( ]BMMj
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -=$% {
% Compute the Zernike Polynomials 20UqJM8Ot
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% #M5_em4kN
$s-9|Lbs`
% Determine the required powers of r: <t{?7_ 8
% ----------------------------------- k7^R,.c@
m_abs = abs(m); M%$DT
rpowers = []; LY-lTr@A^
for j = 1:length(n) M[aT2A
rpowers = [rpowers m_abs(j):2:n(j)]; v@8S5KJ
end B(j02<-
rpowers = unique(rpowers); )Fqy%uR8
5M%,N-P^
% Pre-compute the values of r raised to the required powers, >dpbCPJ9[
% and compile them in a matrix: |l|_dn
% ----------------------------- ^
<qrM
if rpowers(1)==0 ! FNf>z+
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); GS\%mPZ
rpowern = cat(2,rpowern{:}); 1GtOA3,~;-
rpowern = [ones(length_r,1) rpowern]; `gBD_0<T7
else of9q"h
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ,>;!%Ui/p
rpowern = cat(2,rpowern{:}); 2B7h9P.N B
end GR,J0LT
fNkuX-om
% Compute the values of the polynomials: XQ]`&w(
% -------------------------------------- >']+OrQH
y = zeros(length_r,length(n)); BlXX:aZv
for j = 1:length(n) a{h%DpG
s = 0:(n(j)-m_abs(j))/2; I-xwJi9?,
pows = n(j):-2:m_abs(j); cDCJ]iDs
for k = length(s):-1:1 ]}Pl%.
p = (1-2*mod(s(k),2))* ... P7z:3o.
prod(2:(n(j)-s(k)))/ ... VS?dvZ1cC
prod(2:s(k))/ ...
jm[}M
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... BBcj=]"_
prod(2:((n(j)+m_abs(j))/2-s(k))); Bn\l'T
idx = (pows(k)==rpowers); $^t<9"t
y(:,j) = y(:,j) + p*rpowern(:,idx); ?^|QiuU:n
end < CDA"
TWUUvj`.
if isnorm <}d/v_+pnh
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 9Uk(0A
end sltk@
end \M9h&I\7
% END: Compute the Zernike Polynomials B={/nC}G~
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% uJgI<l'|e3
pA<eTlH
% Compute the Zernike functions: Q uB+vL
% ------------------------------ h1#S+k
idx_pos = m>0; Gz?2b#7v
idx_neg = m<0; RU6KIg{H
[g#s&bF
z = y; [OzzL\)3l
if any(idx_pos) U
IfH*6X
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 2}w#3K
end <
kz[:n:
if any(idx_neg) +P|2m"UA
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); <;%0T
xK|U
end iNQ0p:<k
W2`.RF^
% EOF zernfun