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 n\4sNoFI
function z = zernfun(n,m,r,theta,nflag) H[iR8<rhQ
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. j{NcDepLn
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N hzy#%FaB
% and angular frequency M, evaluated at positions (R,THETA) on the @yn1#E,
% unit circle. N is a vector of positive integers (including 0), and k Rp$[^ma
% M is a vector with the same number of elements as N. Each element h\OMWJ~
% k of M must be a positive integer, with possible values M(k) = -N(k) EYKV}`
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, ?xCWg.#l4V
% and THETA is a vector of angles. R and THETA must have the same <a%RKjQvT
% length. The output Z is a matrix with one column for every (N,M) NB)22 %
% pair, and one row for every (R,THETA) pair. ]AB4w+6!
% P?YcZAJT*
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike oei2$uu
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ,A!0:+
% with delta(m,0) the Kronecker delta, is chosen so that the integral USyOHHPW@
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, YZ^;xV
% and theta=0 to theta=2*pi) is unity. For the non-normalized ksli-Px
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. *Ag,/Cm]
% fnU;DS]W
% The Zernike functions are an orthogonal basis on the unit circle. 10e~Yc
% They are used in disciplines such as astronomy, optics, and Z[zRZ2'i5
% optometry to describe functions on a circular domain. ,CQg6-[
% &\M<>>IB
% The following table lists the first 15 Zernike functions. rW0-XLbL5H
% &qae+p?
% n m Zernike function Normalization 7,Q>>%/0P
% -------------------------------------------------- xEqr3(
% 0 0 1 1 0 5o
1
% 1 1 r * cos(theta) 2 lH1gWe
% 1 -1 r * sin(theta) 2 W v!%'IB
% 2 -2 r^2 * cos(2*theta) sqrt(6) j.7BoV
% 2 0 (2*r^2 - 1) sqrt(3) D1f}g
% 2 2 r^2 * sin(2*theta) sqrt(6) QNgfvy
% 3 -3 r^3 * cos(3*theta) sqrt(8) 5TS&NefM
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) L+2<J,
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) 7y'2
% 3 3 r^3 * sin(3*theta) sqrt(8) ?=0BU}
% 4 -4 r^4 * cos(4*theta) sqrt(10) NuC+iC$_/
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) [$%O-_x
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) QlK]2r9
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 2"!s8x1$
% 4 4 r^4 * sin(4*theta) sqrt(10) <]h?_)
% -------------------------------------------------- O
p,_d^
% <e@+w6Kp'7
% Example 1: (od9adSehV
% aLt2fB1 )
% % Display the Zernike function Z(n=5,m=1) XMw*4j2E
% x = -1:0.01:1; {E$smX
% [X,Y] = meshgrid(x,x); R*r;`x
% [theta,r] = cart2pol(X,Y); &-hXk!A
% idx = r<=1; fu $<*Sa2
% z = nan(size(X)); U/9_:
% z(idx) = zernfun(5,1,r(idx),theta(idx)); Q?]-/v
% figure J>p6')Y6~
% pcolor(x,x,z), shading interp S<UWv@`U"
% axis square, colorbar YzVhNJWpw
% title('Zernike function Z_5^1(r,\theta)') E]dmXH8A
% M.?[Xpa
% Example 2: VQwF9Iq]`
% VH7nyqEM
% % Display the first 10 Zernike functions , IDCbJ
% x = -1:0.01:1; 5V@c~1\
% [X,Y] = meshgrid(x,x); b ]u01T-
% [theta,r] = cart2pol(X,Y); fuF!3Q
% idx = r<=1; kBg8:bo~
% z = nan(size(X)); ,v$Q:n|
% n = [0 1 1 2 2 2 3 3 3 3]; VHqHG`}:
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; Gqs)E"h
% Nplot = [4 10 12 16 18 20 22 24 26 28]; dh
S7}n
% y = zernfun(n,m,r(idx),theta(idx)); ^c| _%/
% figure('Units','normalized') qPF`=#
% for k = 1:10 5)iOG#8qJ
% z(idx) = y(:,k); z1S
p'h$
% subplot(4,7,Nplot(k)) 2rPmu
% pcolor(x,x,z), shading interp ce:p*
% set(gca,'XTick',[],'YTick',[]) HvzXAd
% axis square x>$e*
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ]I'dnd3e
% end Cd2A&RB
% +o-jMvK9
% See also ZERNPOL, ZERNFUN2. i8->3uB
dTZ$92<
% Paul Fricker 11/13/2006 6W[~@~D=
2mEvoWnJ
G4]( !f!Kv
% Check and prepare the inputs: B-UsMO
% ----------------------------- y~n1S~5cI
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) .bY
R
error('zernfun:NMvectors','N and M must be vectors.') Ake@krh>$
end YpI|=mv
5XoM)
if length(n)~=length(m) 2"6bz^>}
error('zernfun:NMlength','N and M must be the same length.') `br$kB
end yQ0:M/r;0
$Da?)Hz'F
n = n(:); *}) W>
m = m(:); <.".,Na(J0
if any(mod(n-m,2)) C?j:+
error('zernfun:NMmultiplesof2', ... qWM+!f
'All N and M must differ by multiples of 2 (including 0).') f0&%
end F.),|t$\
rXP~k]tC
if any(m>n) }Xvm(
;
error('zernfun:MlessthanN', ... {B-*w%}HU
'Each M must be less than or equal to its corresponding N.') i&YWutG
end Swr4De_5
-}_1f[b
if any( r>1 | r<0 ) JED\"(d(
error('zernfun:Rlessthan1','All R must be between 0 and 1.') }i{A4f`
end k(he<-GF\
3$ wK*xK
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) )
RdBIbm
error('zernfun:RTHvector','R and THETA must be vectors.') J6Vx7
end }O Y/0p-Z
pX+4B=*
r = r(:); UmR4zGM}
theta = theta(:); 0SDnMij&bf
length_r = length(r); 5] LfJh+"n
if length_r~=length(theta) 5th?m>
error('zernfun:RTHlength', ... ``%yVVg}
'The number of R- and THETA-values must be equal.') !2h ZtX
end k.z(.uc=
,u>[cRqw
% Check normalization: eR0$CTSw
% -------------------- u*/+cT
if nargin==5 && ischar(nflag) V';l H2
isnorm = strcmpi(nflag,'norm'); "([/G?QAG
if ~isnorm |nE4tN#J<
error('zernfun:normalization','Unrecognized normalization flag.') stUUez>
end @{W"mc+
else [Q+k2J_h
isnorm = false; oKb"Ky@s
end cPv(VjS1;
tva=DS
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% f7y.##W G
% Compute the Zernike Polynomials qV6WT&)T
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% . P+Qu
,J8n}7aI
% Determine the required powers of r: C!|LGzs0
% ----------------------------------- "Kdn`zN{
m_abs = abs(m); K8R>O *~
rpowers = []; &gPP#D6A
for j = 1:length(n) BlQX$s]
rpowers = [rpowers m_abs(j):2:n(j)]; kRc+OsY9
end r!
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rpowers = unique(rpowers); aL%E#
fbU3-L?
% Pre-compute the values of r raised to the required powers, N#2ldY *
% and compile them in a matrix: 1[T7;i$
% ----------------------------- *= ?|n
if rpowers(1)==0 vENf3;o0
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); r0 )ne|&Hp
rpowern = cat(2,rpowern{:}); P:t .Nr"
rpowern = [ones(length_r,1) rpowern]; l<BV{Gl
else -58q6yA
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 4e Y?#8
rpowern = cat(2,rpowern{:}); |?ssHW
end ?*%_:fB
bi^?SH\
% Compute the values of the polynomials: w7o`BR
% -------------------------------------- ,T`,OZm
y = zeros(length_r,length(n)); #K6cBfqI
for j = 1:length(n) P/dnH
s = 0:(n(j)-m_abs(j))/2; 8'HS$J;C
pows = n(j):-2:m_abs(j); F,{mF2U*$
for k = length(s):-1:1 [IQ|c?DxpL
p = (1-2*mod(s(k),2))* ... 0'fswa)
prod(2:(n(j)-s(k)))/ ... bD{k=jum
prod(2:s(k))/ ... ~y2zl
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... -X~mW
prod(2:((n(j)+m_abs(j))/2-s(k))); YT
Zi[/
idx = (pows(k)==rpowers); )muNfs m
y(:,j) = y(:,j) + p*rpowern(:,idx); !~Uj 'w
end M{Z
;7n'
_BmObXOp.
if isnorm NOuG# P
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); pX
^^0
end S k~"-HL|
end `om+p?j
% END: Compute the Zernike Polynomials C=/B\G/.9
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% XS [L-NHG
. \"k49M`
% Compute the Zernike functions: Zn'tNt/
% ------------------------------ sfj+-se(K.
idx_pos = m>0; 67YC;J]n=z
idx_neg = m<0; )&Oc7\J,
r8Mx+r
z = y; IB/3=4n^|
if any(idx_pos) t82'K@sq
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); o)/Pr7Qn
end NEIkG>\7q
if any(idx_neg) &(rWl`eTY`
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); La;G S
end BVNW1<_:
rtRbr_
% EOF zernfun