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 P$w0.XZa
function z = zernfun(n,m,r,theta,nflag) Jzfzy0$
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. FK+jfr [
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N O </<
% and angular frequency M, evaluated at positions (R,THETA) on the ~ZL}j+L/
% unit circle. N is a vector of positive integers (including 0), and J *^|ojX
% M is a vector with the same number of elements as N. Each element {{giSW'
% k of M must be a positive integer, with possible values M(k) = -N(k) s8 3_Bd
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, r@iGMJx$
% and THETA is a vector of angles. R and THETA must have the same dNbN]gHC
% length. The output Z is a matrix with one column for every (N,M) .F> cZ,
% pair, and one row for every (R,THETA) pair. P 7gS
M
% HO$s&}t
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike $s?q>Z)
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), +#n[55d
% with delta(m,0) the Kronecker delta, is chosen so that the integral w^P4_Yr[T
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, [NH[n#
% and theta=0 to theta=2*pi) is unity. For the non-normalized _DH,$evS%
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. &9TG&~(+
% syV&Ds)
% The Zernike functions are an orthogonal basis on the unit circle. J6&;pCAi
% They are used in disciplines such as astronomy, optics, and o6oZk0
% optometry to describe functions on a circular domain. QT?fp
>'
% 1Te:&d
% The following table lists the first 15 Zernike functions. MW`q*J`Yo
% '7wWdq
% n m Zernike function Normalization -pcYhLIn
% -------------------------------------------------- Z7OWpujCvN
% 0 0 1 1
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% 1 1 r * cos(theta) 2 iB[%5i-
% 1 -1 r * sin(theta) 2 Wh 8fC(BE
% 2 -2 r^2 * cos(2*theta) sqrt(6) /sC$;l
% 2 0 (2*r^2 - 1) sqrt(3) F)
< f8F
% 2 2 r^2 * sin(2*theta) sqrt(6) { \r{$<s
% 3 -3 r^3 * cos(3*theta) sqrt(8) kG\+f>XQ
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) &Zq43~
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) k\1q Jr
% 3 3 r^3 * sin(3*theta) sqrt(8) n T\W|
% 4 -4 r^4 * cos(4*theta) sqrt(10) D4;V8(w=#
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) [;#}BlbN
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) PNc^)|4^Q
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) QT^W00h
% 4 4 r^4 * sin(4*theta) sqrt(10) ?%B%[u
% -------------------------------------------------- " c}pY ^(
% 3 uhwoE
% Example 1: YVqhX]/
% '$4o,GA8
% % Display the Zernike function Z(n=5,m=1) [C/h{WPC-
% x = -1:0.01:1; uppA`>
% [X,Y] = meshgrid(x,x); VA.:'yQtJ
% [theta,r] = cart2pol(X,Y); ~Ui<y=d
% idx = r<=1; 9cX
~
% z = nan(size(X)); >wz-p
nD
% z(idx) = zernfun(5,1,r(idx),theta(idx)); rhwY5FD?
% figure xHe<TwkI
% pcolor(x,x,z), shading interp `'.u$IBW
% axis square, colorbar Gl`Yyw@84
% title('Zernike function Z_5^1(r,\theta)') ;R 'OdQ$o
% d;V
% Example 2: cm]8 m_!
% P+,\x&Vr
% % Display the first 10 Zernike functions Y7]N.G3,]
% x = -1:0.01:1; Bk~WHg>@G
% [X,Y] = meshgrid(x,x); Ah)_mxK
% [theta,r] = cart2pol(X,Y); )m
\}ITf
% idx = r<=1; X=mzo\Aos
% z = nan(size(X)); xgnt)&7T
% n = [0 1 1 2 2 2 3 3 3 3]; Xn9TQ"[4
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; 8%>
Ls
% Nplot = [4 10 12 16 18 20 22 24 26 28]; _`*x}
% y = zernfun(n,m,r(idx),theta(idx)); ?VO*s-G:J
% figure('Units','normalized') wp$CJ09f*
% for k = 1:10 \0'o*nlJ
% z(idx) = y(:,k); #V*<G#B
% subplot(4,7,Nplot(k)) eHm!
% pcolor(x,x,z), shading interp j+w*Absh
% set(gca,'XTick',[],'YTick',[]) D />REC^
% axis square 3zGxe-
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) UYD(++
% end 1E=%:? d
% =-1d m+P
% See also ZERNPOL, ZERNFUN2. <s)+V6\E
M
E4MZt:>
% Paul Fricker 11/13/2006 Cd"O'<^Sb
-U'6fx) +
9)3ok#pQ/
% Check and prepare the inputs: G!L=W#{
% ----------------------------- DNq=|?qn]
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) /{\tkvv-Z
error('zernfun:NMvectors','N and M must be vectors.') bJmVq%>;
end w91{''sK
"ALR)s,1,
if length(n)~=length(m) 6 80i?=z
error('zernfun:NMlength','N and M must be the same length.')
9 k)?-
end !!%vs
6
\[%[`m
n = n(:); 6Z\[{S];
m = m(:); 4%aODr8
if any(mod(n-m,2)) #]q<fhJhr$
error('zernfun:NMmultiplesof2', ... 7-nwfp&|$
'All N and M must differ by multiples of 2 (including 0).') 593D/^}D
end @{j'Pf'
d_-{-@
if any(m>n) ?9i
7w1`
error('zernfun:MlessthanN', ... {ckA
'Each M must be less than or equal to its corresponding N.') #K yb9Qg
end w*e O9k
k?o(j/
if any( r>1 | r<0 ) g0 \c
error('zernfun:Rlessthan1','All R must be between 0 and 1.') ZUVk~X3
end APsd^J
/ 9/=]
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 5YQ4]/h
error('zernfun:RTHvector','R and THETA must be vectors.') N^Xb_jg;J
end S6*3."Sk
=iB0ak
r = r(:); o;7!$v>uK
theta = theta(:); RM|<(kq
length_r = length(r); XwOj`N{!H
if length_r~=length(theta) N0,.cd]y`
error('zernfun:RTHlength', ... Mmq{]q~At
'The number of R- and THETA-values must be equal.') !ANv XPp
end SuMK=^>%
6!
\a8q'z
% Check normalization: L0/0<d(K
% -------------------- ?dVF@
if nargin==5 && ischar(nflag) WJ9Jj69
isnorm = strcmpi(nflag,'norm'); x\)0+c~\}x
if ~isnorm Q|rrbx b
error('zernfun:normalization','Unrecognized normalization flag.') H5j~<@STC
end rQC{"hS1
else hub1rY|No
isnorm = false; ]d&6 ?7 !>
end cxFfAk\,en
/>S=Y"a/7
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ~Y<x-)R
% Compute the Zernike Polynomials Q+*o-
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 9He>F7J:p'
a.L ?J
% Determine the required powers of r: Edj}\e*-J
% ----------------------------------- U{gJn#e/.
m_abs = abs(m); w8:~LX.n
rpowers = []; dW
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for j = 1:length(n) M$d DExd~
rpowers = [rpowers m_abs(j):2:n(j)]; ( ?3 )l
end 'KMyaEh.u
rpowers = unique(rpowers); ~v$gk
i|0H {q
% Pre-compute the values of r raised to the required powers, m*tmmP4R
% and compile them in a matrix: s
de|t
% ----------------------------- @[D-2s
if rpowers(1)==0 ~rN~Ql%S
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); LGGC=;{}
rpowern = cat(2,rpowern{:}); &uI`Xq.
rpowern = [ones(length_r,1) rpowern]; dWwh?{n
else id8a#&t]
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); yf(VwU,
x
rpowern = cat(2,rpowern{:}); ZP61T*n
end NdZv*
*D!$gfa
% Compute the values of the polynomials: tbrjTeC
% -------------------------------------- % zHsh
y = zeros(length_r,length(n)); ?u{y[pI6
for j = 1:length(n) fn>MOD!l
s = 0:(n(j)-m_abs(j))/2; zFmoo4P/
pows = n(j):-2:m_abs(j); /xj^TyWM
for k = length(s):-1:1 l 3bo
p = (1-2*mod(s(k),2))* ... T B~C4H K=
prod(2:(n(j)-s(k)))/ ... )"s <hR,
prod(2:s(k))/ ... U@x5cw:
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... Xs$k6C3
prod(2:((n(j)+m_abs(j))/2-s(k))); s|.V:%9e
idx = (pows(k)==rpowers); H@GiHej
y(:,j) = y(:,j) + p*rpowern(:,idx); q|0Lu
end k;/U6,LQ*
P#]%C
if isnorm z)I.^
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); U@yn%k9
end Vi[* a
end PB*mD7"
% END: Compute the Zernike Polynomials `?{i dg
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% }a6tG
DS0c0lsx
% Compute the Zernike functions: eS{lr4-]
% ------------------------------ |pqc(B u
idx_pos = m>0; *}DCxv
idx_neg = m<0; //S/pCqED
c L}}^
z = y; 8%q:lI
if any(idx_pos) i;>Yx#
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 6 Ty;m>j
end H5j6$y|I|N
if any(idx_neg) E-\Wo3
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); D&KRJQ/
end *o <S{
]JF>a_2wG
% EOF zernfun