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 9}Z;(,6/.\
function z = zernfun(n,m,r,theta,nflag) !_~/Y/M
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. r:E4Wi{\
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N UIg?3J}R
% and angular frequency M, evaluated at positions (R,THETA) on the ~-uf%=
% unit circle. N is a vector of positive integers (including 0), and c#1kg@q@
% M is a vector with the same number of elements as N. Each element 11Qi
_T\
% k of M must be a positive integer, with possible values M(k) = -N(k) Gm9
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, 7#oq|5
% and THETA is a vector of angles. R and THETA must have the same .O(9\3q\
% length. The output Z is a matrix with one column for every (N,M) Tp.]{*
% pair, and one row for every (R,THETA) pair. +Wy `X5v
% #Ufb
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 9^`cVjD5
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), {D :WXvI
% with delta(m,0) the Kronecker delta, is chosen so that the integral kdx06'4o
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, `kv7Rr}Q
% and theta=0 to theta=2*pi) is unity. For the non-normalized l{ql'm
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. C!9mygI
% b`j9}tZ
% The Zernike functions are an orthogonal basis on the unit circle. f\Hw Y)^>
% They are used in disciplines such as astronomy, optics, and Nh/i'q/
% optometry to describe functions on a circular domain. Kng=v~)N'
% 8;c\}D
% The following table lists the first 15 Zernike functions. O@W/s!&lFa
% 6#K.n&=*
% n m Zernike function Normalization P>)J:.tr0
% -------------------------------------------------- VAUd^6Xdwx
% 0 0 1 1 &2[Xu4*
% 1 1 r * cos(theta) 2 #R31VQwK5
% 1 -1 r * sin(theta) 2 T /IX(b'<
% 2 -2 r^2 * cos(2*theta) sqrt(6) 2EN}"Du]mj
% 2 0 (2*r^2 - 1) sqrt(3) {hN<Ot
% 2 2 r^2 * sin(2*theta) sqrt(6) &y|Ps eH"
% 3 -3 r^3 * cos(3*theta) sqrt(8) ycki0&n3
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) 8'bZR]
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) z8%qCq
% 3 3 r^3 * sin(3*theta) sqrt(8) bi+g=cS
% 4 -4 r^4 * cos(4*theta) sqrt(10) Eyk:pnKJb
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) BD}%RTeWKq
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) ? u".*!%
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) >SziRm>Y7
% 4 4 r^4 * sin(4*theta) sqrt(10) ZGI<L
% -------------------------------------------------- ) R5j?6}xF
% \-{$IC-L
% Example 1: &`vThs[x
% .f;@OqU
% % Display the Zernike function Z(n=5,m=1) :pz@'J
% x = -1:0.01:1; HkhZB^_V
% [X,Y] = meshgrid(x,x); #902x*Z'c"
% [theta,r] = cart2pol(X,Y); L]"$dF
% idx = r<=1; 9%3+\[s1
% z = nan(size(X)); V*(x@pF
% z(idx) = zernfun(5,1,r(idx),theta(idx)); "AKr;|m
% figure YRf$?xa
% pcolor(x,x,z), shading interp @OUBo;/
% axis square, colorbar }lhk;#r
% title('Zernike function Z_5^1(r,\theta)') PO0Od z
% >hq{:m
% Example 2: q@XJ,e1A
% *icaKy3
% % Display the first 10 Zernike functions _5(p=Zc
% x = -1:0.01:1; h"Wpb}FT
% [X,Y] = meshgrid(x,x); `'3 De(
% [theta,r] = cart2pol(X,Y); 5WxNH}{
% idx = r<=1; w2/3[VZ}l
% z = nan(size(X)); fO^s4gWTg
% n = [0 1 1 2 2 2 3 3 3 3]; /38I(0
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; YPq:z"`-y4
% Nplot = [4 10 12 16 18 20 22 24 26 28]; $3&XM
% y = zernfun(n,m,r(idx),theta(idx)); 'NfsAE
% figure('Units','normalized') tSoF!@6
% for k = 1:10 6@FhDj2X
% z(idx) = y(:,k); }aXS MxCd
% subplot(4,7,Nplot(k)) 4MW oGV9
% pcolor(x,x,z), shading interp kRb JK
% set(gca,'XTick',[],'YTick',[]) J&JZYuuf
% axis square "*l{ m2"
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) *3@8,~_tp
% end B1E:P`t
% I!u=.[5zdC
% See also ZERNPOL, ZERNFUN2. WS.g`%
n<> ^cD
% Paul Fricker 11/13/2006 \pTC[Ry1
WJa7
B~qo^ppVU
% Check and prepare the inputs: C\Yf]J
% ----------------------------- p W5D!z
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) T;{:a-8
error('zernfun:NMvectors','N and M must be vectors.') n6Uf>5
end _nxu8g]
N`fFYO
if length(n)~=length(m) v.TgB)
error('zernfun:NMlength','N and M must be the same length.') Y9vi&G?Jl
end ,g*3u
O<,\tZ'N
n = n(:); gV\Y>y4v
m = m(:); o]qwN:8^
if any(mod(n-m,2)) &OXx\}>MW
error('zernfun:NMmultiplesof2', ... c{?SFwgd
'All N and M must differ by multiples of 2 (including 0).') `Je1$)%
end W7_m,{q
}''0N1,/
if any(m>n) 0CXXCa7!
error('zernfun:MlessthanN', ... ! os@G
'Each M must be less than or equal to its corresponding N.') X !0 7QKs
end JTBt=u{6^
2DJg__("
if any( r>1 | r<0 ) KECW~e`
error('zernfun:Rlessthan1','All R must be between 0 and 1.') |#yT]0L%pA
end w{*V8S3h9
3#]II j`\
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) .j|uf[?h
error('zernfun:RTHvector','R and THETA must be vectors.') *HGhm04F{
end B|$o.$5
7j&EQm5\9
r = r(:); ;E.f%
theta = theta(:); s] ;P<
length_r = length(r); -m>3@"q
if length_r~=length(theta) U,6sR
error('zernfun:RTHlength', ... i^A=nsD`
'The number of R- and THETA-values must be equal.') '!?t+L%gO
end 5=<KA
41+WIa
L
% Check normalization: 1n7'\esC*
% -------------------- 5ZH3}B^L$
if nargin==5 && ischar(nflag) GJ2ZK=/
isnorm = strcmpi(nflag,'norm'); a;-%C{S9r
if ~isnorm % a.T@E
error('zernfun:normalization','Unrecognized normalization flag.') "zQ<)Q]U
end c$BH`" <*
else Y}t)!}p$r
isnorm = false; >BK/HuS
end P6ktA-Hv>
UHU ,zgM
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% N'aq4okoL
% Compute the Zernike Polynomials .7 LQ l?
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% c|aX4 =Z
WQiRbb X
% Determine the required powers of r: L+
XAbL)
% ----------------------------------- zks7wt]A
m_abs = abs(m); P?n4B \!
rpowers = []; xJU]py~o
for j = 1:length(n) bqA`oRb\
rpowers = [rpowers m_abs(j):2:n(j)]; [uHC
AP
end t?PqfVSq
rpowers = unique(rpowers); :&'jh/vRN
UQ7]hX9
% Pre-compute the values of r raised to the required powers, " Y^9g/
% and compile them in a matrix: YX)Rs
Vf
% ----------------------------- ElDeXLr'
if rpowers(1)==0 5kQ@]n:<k
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); .j^BWr
rpowern = cat(2,rpowern{:}); mD&I6F[s
rpowern = [ones(length_r,1) rpowern]; <-n^h~,4
else *mJ#|3I<
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); Y3(MKq
rpowern = cat(2,rpowern{:}); o~\.jQQxa
end SDE$ymPx
+Ss3Ph
% Compute the values of the polynomials: ~tRGw^<9
% -------------------------------------- "p|.[d
y = zeros(length_r,length(n)); |j9aTv[`
for j = 1:length(n) *V\.6,^v
s = 0:(n(j)-m_abs(j))/2; xLi3|^q
pows = n(j):-2:m_abs(j); 5p:BHw;%;
for k = length(s):-1:1 2fu<s^9dh
p = (1-2*mod(s(k),2))* ... HQ7g0:-^a>
prod(2:(n(j)-s(k)))/ ... !!V1#?0jw
prod(2:s(k))/ ... r<:d+5"
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... yTK3eK
prod(2:((n(j)+m_abs(j))/2-s(k))); pmWy:0 R
idx = (pows(k)==rpowers); gC iM\Qx
y(:,j) = y(:,j) + p*rpowern(:,idx); |o9`h 9i
end [+R_3'aK
qhcx\eD:?
if isnorm G/(,,T}eG
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi);
_(8#
end "M[&4'OM
end GQhy4ji'z
% END: Compute the Zernike Polynomials _xm<zy{`S
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% s2|.LmC3|B
_
A#lyp
% Compute the Zernike functions: 6S_mfWsi
% ------------------------------ Sa[lYMuB
idx_pos = m>0; 'v
CMf
idx_neg = m<0; U!uJ )mm
NQZ /E )f
z = y; u%yYLpaKf
if any(idx_pos) Eri007? D
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); P!IA;i
end K\fD';
if any(idx_neg) jN*wbqL
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); jXALL8[c
end -qaO$M^Q
]cS(2hP7
% EOF zernfun