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 JI5?,
)-St
function z = zernfun(n,m,r,theta,nflag) waXA%u50
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. 3/o-\wWO
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N kc `Q-
N}
% and angular frequency M, evaluated at positions (R,THETA) on the nGGYKI
% unit circle. N is a vector of positive integers (including 0), and vWI9ocl`W
% M is a vector with the same number of elements as N. Each element XbYW,a@w2
% k of M must be a positive integer, with possible values M(k) = -N(k) &| el8;D
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, d$g-u8
% and THETA is a vector of angles. R and THETA must have the same %WHue
% length. The output Z is a matrix with one column for every (N,M) yL&F!+(/Ix
% pair, and one row for every (R,THETA) pair. 6Km@A M]
% $I!vQbi
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike u*Eb4
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), k2N[B(&4J
% with delta(m,0) the Kronecker delta, is chosen so that the integral 71nXROB
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, HgE^#qD?
% and theta=0 to theta=2*pi) is unity. For the non-normalized 9f;\fe
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. f&B&!&gZ
% +LV~%?W
% The Zernike functions are an orthogonal basis on the unit circle. ^3IO.`|
% They are used in disciplines such as astronomy, optics, and "#d}S)GlXM
% optometry to describe functions on a circular domain. fLAOA9
% PMjqcdBzm
% The following table lists the first 15 Zernike functions. 8 vK
Z;
% 95>(NwST4
% n m Zernike function Normalization &H;0N"Fn
% -------------------------------------------------- e?3 S0}
% 0 0 1 1 8.Wf^j$+{
% 1 1 r * cos(theta) 2 ZffK];D
% 1 -1 r * sin(theta) 2 t.c XrX`k
% 2 -2 r^2 * cos(2*theta) sqrt(6) #0"Fw$Pc
% 2 0 (2*r^2 - 1) sqrt(3) #A@*k}/+
% 2 2 r^2 * sin(2*theta) sqrt(6) Hn0,LH$/
% 3 -3 r^3 * cos(3*theta) sqrt(8) E"&fT!yi
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) " GkBX
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) G/\t<>O8o
% 3 3 r^3 * sin(3*theta) sqrt(8) qYZX,
x
% 4 -4 r^4 * cos(4*theta) sqrt(10) bcC;i~9
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 6;9SU+/
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) dGMBgj
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) *Ibl+
% 4 4 r^4 * sin(4*theta) sqrt(10) `omZ'n)
% -------------------------------------------------- DY'D]*'7$
% BZ<Q.:)
% Example 1: PYPs64kNC]
% ?SRG;G1
% % Display the Zernike function Z(n=5,m=1) w_q{C>-cR
% x = -1:0.01:1; >`Gys8T
% [X,Y] = meshgrid(x,x); }Zc.rk
% [theta,r] = cart2pol(X,Y); ]6Kx0mW
% idx = r<=1; a,x-akZWf
% z = nan(size(X)); J,%v`A ~N
% z(idx) = zernfun(5,1,r(idx),theta(idx)); z{Z'2 ,#
% figure *KN ' 0Z@W
% pcolor(x,x,z), shading interp |E&a3TQW
% axis square, colorbar .&=nP?ZPC6
% title('Zernike function Z_5^1(r,\theta)') x6\EU=,
% Zsc710_
% Example 2: 7RM$%'n\
% PsMoH/+"
% % Display the first 10 Zernike functions %WiDz0o
% x = -1:0.01:1; ^.aFns{wv
% [X,Y] = meshgrid(x,x); n.n;'p9t@
% [theta,r] = cart2pol(X,Y); e82SG8#]
% idx = r<=1; ({i}EC7{
% z = nan(size(X)); zMxHJNQ\D
% n = [0 1 1 2 2 2 3 3 3 3]; Pqli3(
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; 3#`_t :"A
% Nplot = [4 10 12 16 18 20 22 24 26 28]; ~cQ./G4
% y = zernfun(n,m,r(idx),theta(idx));
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% figure('Units','normalized') {Y6U%HG{{r
% for k = 1:10 d5T M_C
% z(idx) = y(:,k); XdjM/hB{fD
% subplot(4,7,Nplot(k)) .w/w]
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% pcolor(x,x,z), shading interp 3&:Us|}
% set(gca,'XTick',[],'YTick',[]) n*{aN}auJ
% axis square _>+!&_h
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) Fy37I/#)r&
% end GM=r{F
&
% s(jixAf
% See also ZERNPOL, ZERNFUN2. XFKe6:
w$8Su:g=
% Paul Fricker 11/13/2006 ?-%Q[W
jI%v[]V
}7&.FV"
% Check and prepare the inputs: k/o"E
% ----------------------------- Ndq/n21j
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) L"{qF<@V7&
error('zernfun:NMvectors','N and M must be vectors.') >uT,Z,7O
end WyciIO1
r0t4\d_&
if length(n)~=length(m)
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error('zernfun:NMlength','N and M must be the same length.') AGu#*,K
end $X<O\Kna
"`HkAW4GZa
n = n(:); Ey96XJV
m = m(:); j}O~6A>|
if any(mod(n-m,2)) MIma:N_c
error('zernfun:NMmultiplesof2', ... `Cq&;-u
'All N and M must differ by multiples of 2 (including 0).') /iURP-rl
end d1]CN6 7{G
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if any(m>n) HR'sMu3
error('zernfun:MlessthanN', ... 4FrP%|%E~
'Each M must be less than or equal to its corresponding N.') Nc;cb
end BV)oF2b:
0x BO5[w,Y
if any( r>1 | r<0 ) "i>?Tg^
error('zernfun:Rlessthan1','All R must be between 0 and 1.') S;@nPzhc
end `R[cM; c2
v2eLH:6
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) fk%W07x!
error('zernfun:RTHvector','R and THETA must be vectors.') Nl/^ga
end =T"R_3[NC
'kBg3E$y
r = r(:); ;IyA"C(i
theta = theta(:); wNc.z*+O"H
length_r = length(r); E$O-\)wY0
if length_r~=length(theta) h pf,44Kg
error('zernfun:RTHlength', ... @7S*
]
'The number of R- and THETA-values must be equal.') +/O3L=QyJ
end (|O9L s7N
\jA#RF.W
% Check normalization: I;xSd.-
% -------------------- #BtJo:
if nargin==5 && ischar(nflag) P=3mLz-
isnorm = strcmpi(nflag,'norm'); 9-:\ NH^;
if ~isnorm OHRkhwF.
error('zernfun:normalization','Unrecognized normalization flag.') hp|.hN(kS]
end '#<4oW\]
else Xz,fjKUnN
isnorm = false; T'6MAxEZUq
end jxc^OsYj
L5[{taZ,
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ?iXN..6x
% Compute the Zernike Polynomials KBC?SxJSJc
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Gxh r0'
i,V,0{$
% Determine the required powers of r: J2ZV\8t
% ----------------------------------- 76oJCNY
m_abs = abs(m); G0%},Q/
rpowers = []; 9{*$[%d1
for j = 1:length(n) gOy;6\/
rpowers = [rpowers m_abs(j):2:n(j)]; wn-1fz<d
end /SW*y@R2l
rpowers = unique(rpowers);
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S
% Pre-compute the values of r raised to the required powers, =\`iC6xP}
% and compile them in a matrix: ,ZV>"'I:
% -----------------------------
/\.[@]
if rpowers(1)==0 .Gt_~x
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 'fjouO
rpowern = cat(2,rpowern{:}); I+{2DY/}
rpowern = [ones(length_r,1) rpowern]; V O\g"Yc
else %*
k`z#b
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); @WCA7DW!
rpowern = cat(2,rpowern{:}); FUVp}>#U
end C4aAPkcp2$
=u-q#<h4;
% Compute the values of the polynomials: h6b(FTC^
% -------------------------------------- AqiH1LAE
y = zeros(length_r,length(n)); F|a'^:Qs
for j = 1:length(n) 9-+N;g!q
s = 0:(n(j)-m_abs(j))/2; [XE\2Qa8e
pows = n(j):-2:m_abs(j); $35C1"
for k = length(s):-1:1 1/f{1k
p = (1-2*mod(s(k),2))* ... =Y- .=}jp;
prod(2:(n(j)-s(k)))/ ... Y&<]:)
prod(2:s(k))/ ...
NDUH10Y:[
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... D7r&z?
prod(2:((n(j)+m_abs(j))/2-s(k))); P$x9Z3d_
idx = (pows(k)==rpowers); j1rR3)oP
y(:,j) = y(:,j) + p*rpowern(:,idx); g=/!Ry=
end {'p <
o$(S
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if isnorm $cWt^B'
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); _\.4ofK(
end s:k?-u@
end
jF-:e;-
% END: Compute the Zernike Polynomials <Umr2Vw-
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Q=61.lP6
5Gs>rq" #
% Compute the Zernike functions: 7YxVtN
% ------------------------------ YkFAu8b>
idx_pos = m>0; RFLfvD<
idx_neg = m<0; BRy3D\}
+%f6{&q$
z = y; "} "/d(
if any(idx_pos) +T&YYO8>5
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); boI&q>-6Re
end &) 64:l&
if any(idx_neg) '?jsH+j+
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ^TD%l8o6
end UEx13!iFo
#M||t|9iu?
% EOF zernfun