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 Ifj&S'():
function z = zernfun(n,m,r,theta,nflag) P3$eomX'
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. *Z5^WHwg
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N E$e7(D
% and angular frequency M, evaluated at positions (R,THETA) on the `@]s[1?f
% unit circle. N is a vector of positive integers (including 0), and [I $+wWW_
% M is a vector with the same number of elements as N. Each element k*.]*]
% k of M must be a positive integer, with possible values M(k) = -N(k) RU )35oEV|
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, ]b/]^1-(b
% and THETA is a vector of angles. R and THETA must have the same NR{wq|"
% length. The output Z is a matrix with one column for every (N,M) C$3*[
% pair, and one row for every (R,THETA) pair. *2'8d8>R%]
% @fL ^I&++
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ou|emAV
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), )?L=o0
% with delta(m,0) the Kronecker delta, is chosen so that the integral 0J)s2&H
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, H"Pb)t
% and theta=0 to theta=2*pi) is unity. For the non-normalized gg;r;3u
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. R$awg SE
% d"$8-_K
% The Zernike functions are an orthogonal basis on the unit circle. )xGAe#E~j
% They are used in disciplines such as astronomy, optics, and (
?V`|[+u
% optometry to describe functions on a circular domain. L+%"ew
% TOYK'|lwM
% The following table lists the first 15 Zernike functions. ]Z JoC!u
% P:qmg"i@3
% n m Zernike function Normalization LSo!_tY
% -------------------------------------------------- YdY-Jg Xm
% 0 0 1 1 h`fVQN.3
% 1 1 r * cos(theta) 2
CI|lJ
% 1 -1 r * sin(theta) 2 &c`-/8c
% 2 -2 r^2 * cos(2*theta) sqrt(6) p|4qkJK8
% 2 0 (2*r^2 - 1) sqrt(3) "q4tvcK.
% 2 2 r^2 * sin(2*theta) sqrt(6) BG{f)2F\
% 3 -3 r^3 * cos(3*theta) sqrt(8) g|=_@
pL
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) R#(0C(FI^
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) .=_p6_G
% 3 3 r^3 * sin(3*theta) sqrt(8) ]6&NIz`:,
% 4 -4 r^4 * cos(4*theta) sqrt(10) 7\u+%i;YZ
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) SGd]o"VF
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) d1/emwH
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) '[F:uA
% 4 4 r^4 * sin(4*theta) sqrt(10) .u`[|:K
% -------------------------------------------------- \/-c)
% ?I.9?cQXZ
% Example 1: fGgt[f[
% r;cDYg
% % Display the Zernike function Z(n=5,m=1) 0MQ= Rt
% x = -1:0.01:1; JuRoeq.
% [X,Y] = meshgrid(x,x); mp|pz%U
% [theta,r] = cart2pol(X,Y); kH!Z|Ps?R
% idx = r<=1; Pg4&}bX:I
% z = nan(size(X)); +bjy#=
% z(idx) = zernfun(5,1,r(idx),theta(idx)); fS]&?$q
% figure Ur j*V0^
% pcolor(x,x,z), shading interp x^eu[olN
% axis square, colorbar <QtZ6-;_f
% title('Zernike function Z_5^1(r,\theta)')
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% Ku;fZN[g
% Example 2: l=^A41L_
% wZrFu(_
% % Display the first 10 Zernike functions XLpP*VH3
% x = -1:0.01:1; wI#8|,]"z
% [X,Y] = meshgrid(x,x);
D+8d^-:
% [theta,r] = cart2pol(X,Y); ]{-.?W*$
% idx = r<=1; 4Nm >5*]
% z = nan(size(X)); ?0b-fL^^+l
% n = [0 1 1 2 2 2 3 3 3 3]; z,HhSW?&^
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; L
aTcBcI
% Nplot = [4 10 12 16 18 20 22 24 26 28]; c0Ug5Vr
% y = zernfun(n,m,r(idx),theta(idx)); owVvbC2<b(
% figure('Units','normalized') t2&kGf"
% for k = 1:10 K/4@2vF
% z(idx) = y(:,k); vwR_2u
% subplot(4,7,Nplot(k)) >WLPE6E
% pcolor(x,x,z), shading interp ?z
,!iK`
% set(gca,'XTick',[],'YTick',[]) &|SWy
2N
% axis square uL'f8Pqg
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) |5@Ra@0
% end h!"2Ux3!x
% A`c22Ls]
% See also ZERNPOL, ZERNFUN2. # @\3{;{R
s"(RdJ-,
% Paul Fricker 11/13/2006 #ydold{F
7KT*p&xm
~z[`G#dU
% Check and prepare the inputs: {1GJ,['qL
% ----------------------------- $Dg-;I
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) r}U6LE?>
error('zernfun:NMvectors','N and M must be vectors.') %wD#[<BGn>
end 6(|mdk`i
?y>N&\pt2
if length(n)~=length(m) HKN|pO3v
error('zernfun:NMlength','N and M must be the same length.') _S!^=9bJ
end }"Y<<e<z:
_h%Jf{nu
n = n(:); .Xg.,kW
m = m(:); 0Q3U\cDr
if any(mod(n-m,2)) o$_0Qs$
error('zernfun:NMmultiplesof2', ... UhB+c
'All N and M must differ by multiples of 2 (including 0).') :4AQhn^;"
end w+$$uz
D#jwI,n}x
if any(m>n) b3NIFKw
error('zernfun:MlessthanN', ... 1nVQYqT_
'Each M must be less than or equal to its corresponding N.') ]l7W5$26 @
end +]l?JKV
lH fZw})d
if any( r>1 | r<0 ) +Z#=z,.^
error('zernfun:Rlessthan1','All R must be between 0 and 1.') VO#rJ1J
end ?o<vmIge
(6,:X
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 8)B{x[?|
error('zernfun:RTHvector','R and THETA must be vectors.') X)g
X9DA
end A#>wbHjWF
]+lT*6P*
r = r(:); D@=]mh6vl
theta = theta(:); VPCI5mS_
length_r = length(r); ODJ"3 J
if length_r~=length(theta) 4+olyBht
error('zernfun:RTHlength', ... :kZ]Swi 5
'The number of R- and THETA-values must be equal.') &+9 ;
end bLT3:q#s
v[CR$@Y
% Check normalization: 88Pt"[{1
% -------------------- j/V_h'}
if nargin==5 && ischar(nflag) D5zc{) /
isnorm = strcmpi(nflag,'norm'); k-$Acv(
if ~isnorm e\)%<G5
error('zernfun:normalization','Unrecognized normalization flag.') Aez2n(yac
end [*%lm9 x
else T!
}G51
isnorm = false; <Qq
{&,Le
end )Rr6@o
#rHMf%0
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% <5Vf3KoC&
% Compute the Zernike Polynomials kV-<[5AWW
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% DksYKv
g5BL"Dn
% Determine the required powers of r: j;$f[@0o
% ----------------------------------- }0~$^J
m_abs = abs(m); (o>N*?,}
rpowers = []; |:)Bo<8
for j = 1:length(n) iBE|6+g~Cj
rpowers = [rpowers m_abs(j):2:n(j)]; 'O%*:'5k
end k_g@4x1y*
rpowers = unique(rpowers); osc8;B/
I!zoo[/)%
% Pre-compute the values of r raised to the required powers, +;,{`*W+N
% and compile them in a matrix: 74_?@Z(
% ----------------------------- c<13 r=+
if rpowers(1)==0 Yjg$o:M
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); besc7!S
rpowern = cat(2,rpowern{:}); n'rq
rpowern = [ones(length_r,1) rpowern]; e`Yj}i*bx]
else 8YSvBy
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ^;II@n
i
rpowern = cat(2,rpowern{:}); j,rc9
end ~HY)$Yp;
Zl_sbIY
% Compute the values of the polynomials: 5|0}bv O
% -------------------------------------- n%r>W^2j
y = zeros(length_r,length(n)); '[r: pwE
for j = 1:length(n) _d!sSyk`
s = 0:(n(j)-m_abs(j))/2;
,]wab6sY
pows = n(j):-2:m_abs(j); Vc\g"1x
for k = length(s):-1:1 2<7pe@c98
p = (1-2*mod(s(k),2))* ... <v ub
Q4
prod(2:(n(j)-s(k)))/ ... $y;w@^
prod(2:s(k))/ ... ]xf89[;0
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... /@"mQx~[q
prod(2:((n(j)+m_abs(j))/2-s(k))); y0O(n/
idx = (pows(k)==rpowers); hLfWDf*T|
y(:,j) = y(:,j) + p*rpowern(:,idx); 5ncW
s)
end s?<FS@k
%g*nd#wG
if isnorm LBio$67F
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); $%U}k=-
end /A5=L<T6F
end 3SM'vV0[
% END: Compute the Zernike Polynomials %n]jsdE^|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ]:ca=&>
Gb2|e.z
% Compute the Zernike functions: aT/2rMKPF
% ------------------------------ zt2#K
idx_pos = m>0; Qc33CA
idx_neg = m<0; M5[#YG'FlQ
ks$5$,^T2o
z = y; '>[ZfT
if any(idx_pos) Z4z|B&
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); .}E<,T
end `-nSH)GBM
if any(idx_neg) #WA7}tHb
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 0gyvRM@ x[
end Zy Df@(z`
6n:X
p_yO
% EOF zernfun