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 C{U?0!^
function z = zernfun(n,m,r,theta,nflag) .yz}ROmN^
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. >CHrg]9
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N <g$~1fa
% and angular frequency M, evaluated at positions (R,THETA) on the #d6)#:uss
% unit circle. N is a vector of positive integers (including 0), and 8X[:j&@
% M is a vector with the same number of elements as N. Each element 1`=nWy='
% k of M must be a positive integer, with possible values M(k) = -N(k) E|iQc8gr&
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, qm/)ku0
% and THETA is a vector of angles. R and THETA must have the same N sXHO
% length. The output Z is a matrix with one column for every (N,M) Q+[n91ey**
% pair, and one row for every (R,THETA) pair. RoPRQCE
% jIJ~QpNE
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike o~`/_+
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), yD zc<p\`
% with delta(m,0) the Kronecker delta, is chosen so that the integral EV]1ml k$
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 4h|c<-`>t
% and theta=0 to theta=2*pi) is unity. For the non-normalized 0Tx6zO
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. F1*>y
% uXn1
'K<'2
% The Zernike functions are an orthogonal basis on the unit circle. Mk"^?%PxT
% They are used in disciplines such as astronomy, optics, and `dq,>HdW
% optometry to describe functions on a circular domain. %)1y AdG
8
% ~%<X0s|
% The following table lists the first 15 Zernike functions. 8\+uec]k
% G<65H+)M\
% n m Zernike function Normalization (A9Fhun
% -------------------------------------------------- | )K8N<n
% 0 0 1 1 xF!,IKlBBp
% 1 1 r * cos(theta) 2 Z^3rLCa
% 1 -1 r * sin(theta) 2 >g1~CEMN#
% 2 -2 r^2 * cos(2*theta) sqrt(6) :CG`t?N9M
% 2 0 (2*r^2 - 1) sqrt(3) +$ 'Zf0U
% 2 2 r^2 * sin(2*theta) sqrt(6) hOjk3
k
% 3 -3 r^3 * cos(3*theta) sqrt(8) ZMQZs~;~d
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) u^^[Q2LDU}
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) M?1Y,5
% 3 3 r^3 * sin(3*theta) sqrt(8) ,wQ5.U,
% 4 -4 r^4 * cos(4*theta) sqrt(10) DX#Nf""Pw
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Ag-(5:
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) p|U?86t
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) +}Dw3;W}m
% 4 4 r^4 * sin(4*theta) sqrt(10) *#,7d"6W5
% -------------------------------------------------- R@1 xt@?
% <FV1Wz
% Example 1: .s?L^Z^
% _>&X\`D
% % Display the Zernike function Z(n=5,m=1) n\mO6aJ
% x = -1:0.01:1; /6)<}#
% [X,Y] = meshgrid(x,x); f\|w'
% [theta,r] = cart2pol(X,Y); o_izl\
% idx = r<=1; D+rxT:
d
% z = nan(size(X)); G|bT9f$
% z(idx) = zernfun(5,1,r(idx),theta(idx)); *7uH-u"5d
% figure rD*jp6Cl
% pcolor(x,x,z), shading interp h0g8*HY+}
% axis square, colorbar Wf+cDpK
% title('Zernike function Z_5^1(r,\theta)') y6(Z`lx
% d[iQ`YW5
% Example 2: h79}qU
% =9H7N]*h
% % Display the first 10 Zernike functions uy>q7C
% x = -1:0.01:1; k
=>oO9`
% [X,Y] = meshgrid(x,x); 7`*h2 mgY
% [theta,r] = cart2pol(X,Y); ; 5*&xz
% idx = r<=1; Zu*F#s!tUI
% z = nan(size(X)); j*|VctM
% n = [0 1 1 2 2 2 3 3 3 3]; yuh *
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; zYH&i6nj
% Nplot = [4 10 12 16 18 20 22 24 26 28]; L^1NY3=$
% y = zernfun(n,m,r(idx),theta(idx)); (d(CT;
% figure('Units','normalized') ]%;:7?5l
% for k = 1:10 )v'WWwXY>
% z(idx) = y(:,k); 6fkRrD
% subplot(4,7,Nplot(k)) U7?;UCmX
% pcolor(x,x,z), shading interp g_;\iqxL
% set(gca,'XTick',[],'YTick',[]) Z%gh3
% axis square 'NWfBJm
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) /p/]t,-j2
% end ]vAz
% Sj3+l7S?
% See also ZERNPOL, ZERNFUN2. z0d.J1VW
= }~hWL
% Paul Fricker 11/13/2006 #$.;'#u'so
%Tfbsyf%f
" s,1%Ltt
% Check and prepare the inputs: ?e%ZOI
% ----------------------------- oh4E7yN
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 8'[~2/
error('zernfun:NMvectors','N and M must be vectors.') i}cRi&2[
end 8=!D$t\3
Lc}LGq!
if length(n)~=length(m) n'"/KS+_
error('zernfun:NMlength','N and M must be the same length.') &5>Kl}7
end W~)}xy
N"Z{5A
n = n(:); ,<.V7(|t)
m = m(:); `~cqAs}6]Q
if any(mod(n-m,2))
,>:U2%
error('zernfun:NMmultiplesof2', ... |NlO7aQ>2H
'All N and M must differ by multiples of 2 (including 0).') :@yEQ#nFp
end [|v][Hwv
(|2t#'m
if any(m>n) z[N`s$;
error('zernfun:MlessthanN', ... }H53~@WP>
'Each M must be less than or equal to its corresponding N.') r-,%2y?
end <3nMx^
Usvl}{L[
if any( r>1 | r<0 ) %O;:af"Ja8
error('zernfun:Rlessthan1','All R must be between 0 and 1.') T9=I$@/
end &0d#Y]D4`
`Gs9Xmc|
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) z 'Hw
error('zernfun:RTHvector','R and THETA must be vectors.') ?d* z8w
end juJklSD
e1yt9@k,
r = r(:); Y/F6\oh
theta = theta(:); [+Iz@0q
length_r = length(r); Q3'llOx
if length_r~=length(theta) @NR>{Eg
error('zernfun:RTHlength', ... y
RqL9t
'The number of R- and THETA-values must be equal.') #<fRE"v:Q
end [NTzcSN.
q])K,)
% Check normalization: Xg6Jh``
% -------------------- 1er
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if nargin==5 && ischar(nflag) 1C+13LE$U
isnorm = strcmpi(nflag,'norm'); {p2!|A&a
if ~isnorm $c!p&
error('zernfun:normalization','Unrecognized normalization flag.') v&\Q8!r_
end <sbu;dQ`
else 70d 1ReQ
isnorm = false; Z-%\
<zT
end =IZT(8
2k~l$p>CN!
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% #~]zhHI
% Compute the Zernike Polynomials Fe*R
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% !)f\%lb
`7E;VL^Y1
% Determine the required powers of r: ,>a&"V^k
% ----------------------------------- [(i
m_abs = abs(m); ]h`&&B qt
rpowers = []; 6q\bB
for j = 1:length(n) (TtkFo'!U
rpowers = [rpowers m_abs(j):2:n(j)]; l:~/<`o
end ;fTKfa
rpowers = unique(rpowers); tAd%#:K
LVM%"sd?
% Pre-compute the values of r raised to the required powers, Y(ykng
% and compile them in a matrix: >b}o~F^J
% ----------------------------- mthA4sz
if rpowers(1)==0 g{)dP!}
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); :LQYo'@yB
rpowern = cat(2,rpowern{:}); tU5zF.%
rpowern = [ones(length_r,1) rpowern]; UW={[h{.|@
else =ZznFVJ`={
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); /KaZHR.
rpowern = cat(2,rpowern{:}); :`#d:.@]o@
end y-b%T|p9
u/0h$l
% Compute the values of the polynomials: g}oi!f$|
% -------------------------------------- C3f' {}
y = zeros(length_r,length(n)); "S]0
for j = 1:length(n) `g?Negt\v
s = 0:(n(j)-m_abs(j))/2; M]
%?>G
pows = n(j):-2:m_abs(j); [85spub&}
for k = length(s):-1:1 8NJqV+jn)t
p = (1-2*mod(s(k),2))* ... }"H,h)T
prod(2:(n(j)-s(k)))/ ... qBQ?HLK-
prod(2:s(k))/ ... iq8<ov
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... &m7]v,&
prod(2:((n(j)+m_abs(j))/2-s(k))); i^&~?2
idx = (pows(k)==rpowers); <$$yw=ef
y(:,j) = y(:,j) + p*rpowern(:,idx); H2 {+)
end 2 a)xTA#
wWP}C D
if isnorm +) om^e@.
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); m9WDT
end !-x$L>1$
end RLXL&
% END: Compute the Zernike Polynomials fw~Bza\e
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% >2)OiQ`zg
UgSB>V<?
% Compute the Zernike functions: bH9kj/q\b
% ------------------------------ jOunWv|
idx_pos = m>0; 8'[7
)I=
idx_neg = m<0; ua$GNm
mDABH@R
z = y; 2]jn '4
if any(idx_pos) f*% D$Mqg
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); X7MM2V
end 4he GnMD
if any(idx_neg) ek\ xx
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 4[r0G+
end xrz,\eTb
2;`1h[,-^
% EOF zernfun