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 Y 6NoNc]h
function z = zernfun(n,m,r,theta,nflag) "-y2En
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. !b !C+ \v
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N NZu\ Ae
% and angular frequency M, evaluated at positions (R,THETA) on the ;-aF\}D@n
% unit circle. N is a vector of positive integers (including 0), and L9lN AiOH
% M is a vector with the same number of elements as N. Each element rLkUIG
% k of M must be a positive integer, with possible values M(k) = -N(k) S_Tv Ix/7&
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, 0XkLWl|k
% and THETA is a vector of angles. R and THETA must have the same TO(2n8'fdO
% length. The output Z is a matrix with one column for every (N,M) Lc&LF*
% pair, and one row for every (R,THETA) pair. 4$5d*7
% ?&ow:OH+
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike i8h(b2odQ
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), c
`[,>
% with delta(m,0) the Kronecker delta, is chosen so that the integral 7o+JQ&fF;
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, @ij8AGE:
% and theta=0 to theta=2*pi) is unity. For the non-normalized yN'<iTh
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. S!LLC{
% Sckt gp8
% The Zernike functions are an orthogonal basis on the unit circle. ;)6LX-
% They are used in disciplines such as astronomy, optics, and #NoY}*
% optometry to describe functions on a circular domain. 3SI~?&HU!/
% "mbjS(-eg
% The following table lists the first 15 Zernike functions. 5l(8{,NDt
% )2nx5"
% n m Zernike function Normalization $uPM.mPFE
% -------------------------------------------------- P#8+GN+bF
% 0 0 1 1 2qA"emUM
% 1 1 r * cos(theta) 2 ?{)s dJe
% 1 -1 r * sin(theta) 2 ;^[VqFpeS
% 2 -2 r^2 * cos(2*theta) sqrt(6) #5Q?Q~E@
% 2 0 (2*r^2 - 1) sqrt(3) P"Scs$NOU?
% 2 2 r^2 * sin(2*theta) sqrt(6) &Zzd6[G+
% 3 -3 r^3 * cos(3*theta) sqrt(8) &J]|pf3m
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) a/4!zT
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) vU4Gw4
% 3 3 r^3 * sin(3*theta) sqrt(8) \zdY$3z
% 4 -4 r^4 * cos(4*theta) sqrt(10) ~o<+tL
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ~BUzyc%
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) @Sik~Mm_h
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) mY)Y47iL
% 4 4 r^4 * sin(4*theta) sqrt(10) =6sA49~M
% -------------------------------------------------- M1Frn n
% n#US4&uT4A
% Example 1: b0PQ;?R#V
% l[,RA?i
{
% % Display the Zernike function Z(n=5,m=1) j O-H1@;
% x = -1:0.01:1; N!W# N$
% [X,Y] = meshgrid(x,x); L~Hl?bK
% [theta,r] = cart2pol(X,Y); x)]_]_vX
% idx = r<=1; tx+KxOt9Y
% z = nan(size(X)); M%3P@GRg
% z(idx) = zernfun(5,1,r(idx),theta(idx)); MV(Sb:RZ
% figure FX->_}kL=
% pcolor(x,x,z), shading interp Ej[:!L
% axis square, colorbar 9Kpzj43
% title('Zernike function Z_5^1(r,\theta)') 1"hd5a
% 7])cu>/
% Example 2: fQ[&
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% Vgj&hdbd
% % Display the first 10 Zernike functions b|rMmx8vA
% x = -1:0.01:1; MF41q%9p
% [X,Y] = meshgrid(x,x); 'XbrO|%
% [theta,r] = cart2pol(X,Y); !{WIN%O
% idx = r<=1; QE#Ar8tU
% z = nan(size(X)); I7S#vIMXR.
% n = [0 1 1 2 2 2 3 3 3 3]; 34Fc
oud);
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; *EB`~s
% Nplot = [4 10 12 16 18 20 22 24 26 28]; yF _@^V
% y = zernfun(n,m,r(idx),theta(idx)); %k"qpu
% figure('Units','normalized') pA%Sybw+
% for k = 1:10 &az
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% z(idx) = y(:,k); 5PRS|R7
% subplot(4,7,Nplot(k)) *l-f">?|
% pcolor(x,x,z), shading interp -|FSdzvg
% set(gca,'XTick',[],'YTick',[]) hoDE*>i
% axis square 4Y>J,c
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) )-u0n],
% end yu~o9
% NI%&Xhn!*>
% See also ZERNPOL, ZERNFUN2. H}p5qW.tH:
&Q>tV+*
% Paul Fricker 11/13/2006 $vR#<a,7>
zxo"
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FG6bKvEQm^
% Check and prepare the inputs: K<g<xW* X
% ----------------------------- P<OSm*;U:
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) c*@#0B
error('zernfun:NMvectors','N and M must be vectors.') r](%9Y
end P@xb
e -yL
if length(n)~=length(m) $*k9e ^{S
error('zernfun:NMlength','N and M must be the same length.') l$\OSG
end 45qSt2
sN_c4"\q
n = n(:); Hd8 O3_5
m = m(:); 89kxRH\IhG
if any(mod(n-m,2)) J?1U'/Wx2
error('zernfun:NMmultiplesof2', ... 2d:5~fEJp
'All N and M must differ by multiples of 2 (including 0).') 5j{jbo=!
end xIlo@W6
H?a1XEY/
if any(m>n) h;lg^zlTb
error('zernfun:MlessthanN', ... d$?sS9"8(
'Each M must be less than or equal to its corresponding N.') &|
guPZ
end Z+%w|Sx
!4 =]@eFk
if any( r>1 | r<0 ) K8?]&.!
error('zernfun:Rlessthan1','All R must be between 0 and 1.') |u@/,x/t
end AY
B~{
fK?/o]vq
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) c(j|xQ\pE
error('zernfun:RTHvector','R and THETA must be vectors.') Af`qe+0E
end +5k^-
P2t{il
r = r(:); >%?kp[
theta = theta(:); h@H8oZ[
length_r = length(r); j]X$7
if length_r~=length(theta) zcrM3`Zh
error('zernfun:RTHlength', ... 6oA2"!u^w
'The number of R- and THETA-values must be equal.') ,'%wadOo
end 2Vwv#NAV k
(=eJceE!
% Check normalization: v{44`tR
% -------------------- ~B704i
if nargin==5 && ischar(nflag) mFa%d8Y
isnorm = strcmpi(nflag,'norm'); N0POyd/rL
if ~isnorm dR|*VT\
error('zernfun:normalization','Unrecognized normalization flag.') +WTO_J7
end TilCP"(6D
else ,|lDR@
isnorm = false; tSf$`4
end z,+LPr
/qwl;_Jcf
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% rQLl[a
% Compute the Zernike Polynomials O+w82!<:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% KM:k<pvi
+f"q^R IU
% Determine the required powers of r: rFLm!J]
% ----------------------------------- z^z,_?q;
m_abs = abs(m); pcC/$5FQ
rpowers = []; I8%Uyap{
for j = 1:length(n) O}Mu_edM
rpowers = [rpowers m_abs(j):2:n(j)]; A(84cmq!q
end Py^fWQ5I~%
rpowers = unique(rpowers); Ss$/Bh>hN
ON-zhT?v
% Pre-compute the values of r raised to the required powers, b
sM]5^
% and compile them in a matrix: 'jA>P\@8
% ----------------------------- *$ kpSph
if rpowers(1)==0 3k_bhK zI
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); <nk7vo?Ks
rpowern = cat(2,rpowern{:}); /3KPK4!m
rpowern = [ones(length_r,1) rpowern]; S(ky:
else YW7Pimks
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 6+LBs.vl}
rpowern = cat(2,rpowern{:}); 8:gUo8
end kD\7wz,ui
A V]7l}-
% Compute the values of the polynomials: H[ o > "@4
% -------------------------------------- U.A:'9K,
y = zeros(length_r,length(n)); es!>u{8)
for j = 1:length(n) pybE0]
s = 0:(n(j)-m_abs(j))/2; Z!foD^&R
pows = n(j):-2:m_abs(j); 8$~^-_>n/
for k = length(s):-1:1 ! lxq,Whr{
p = (1-2*mod(s(k),2))* ... p6AF16*f0
prod(2:(n(j)-s(k)))/ ... WvN{f*
prod(2:s(k))/ ... zXZXp~7)
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... }l<:^lX
prod(2:((n(j)+m_abs(j))/2-s(k))); ]WvV*FL9D3
idx = (pows(k)==rpowers); (,I9|
y(:,j) = y(:,j) + p*rpowern(:,idx); 8Xx4W^*_
end `_+%
E@/*eJ
if isnorm E2i'lO\P
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); ! z6T_;s
end F&u)wI'
end k{C03=xk
% END: Compute the Zernike Polynomials n%K^G4k^
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% L]Dq1q8`
e5$S2o~JF
% Compute the Zernike functions: ]Ei*I}
% ------------------------------ m"f3hd4D_q
idx_pos = m>0; ,!vI@>nhG
idx_neg = m<0; .r~M7 I
Px?zih!6
z = y; $nqVE{ksV
if any(idx_pos) :x3"Cj
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ,lDOo+eE%:
end gaWJzK
Yc_
if any(idx_neg) _V,bvHWlM
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); _^@ >I8ix
end 3W3)%[ 5
@MKf$O4K
% EOF zernfun