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 =R 4]Kf
function z = zernfun(n,m,r,theta,nflag) GA`
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%ZERNFUN Zernike functions of order N and frequency M on the unit circle. !4gHv4v;
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N s-(c-E09
% and angular frequency M, evaluated at positions (R,THETA) on the -sdzA6dp
% unit circle. N is a vector of positive integers (including 0), and S.`hl/
% M is a vector with the same number of elements as N. Each element ;&f(7 Q+T_
% k of M must be a positive integer, with possible values M(k) = -N(k) e6H}L:;
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, ]dl.~;3~~
% and THETA is a vector of angles. R and THETA must have the same O"kb*//
% length. The output Z is a matrix with one column for every (N,M) 1zG6^U
% pair, and one row for every (R,THETA) pair. *93=}1gN
% w8jpOvj
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ,)!%^~v
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), yi Xb<g+B
% with delta(m,0) the Kronecker delta, is chosen so that the integral R]Z#VnL@qz
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, S!x;w7j
% and theta=0 to theta=2*pi) is unity. For the non-normalized #`U?,>2q
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. t6`(9o@}
% cTn(Tv9s
% The Zernike functions are an orthogonal basis on the unit circle. #`=>Mza
% They are used in disciplines such as astronomy, optics, and M
#0v# {o
% optometry to describe functions on a circular domain. . XbDb
% n[qnrk*3
%
% The following table lists the first 15 Zernike functions. lKU{jWA
% )?B-en\
% n m Zernike function Normalization $bF+J8%D
% -------------------------------------------------- 7\$ b%A
% 0 0 1 1 .I]v
D#o
% 1 1 r * cos(theta) 2 .HGK 3
% 1 -1 r * sin(theta) 2 ])bgUH
% 2 -2 r^2 * cos(2*theta) sqrt(6) $s]&92
% 2 0 (2*r^2 - 1) sqrt(3) p\#;(pf}s
% 2 2 r^2 * sin(2*theta) sqrt(6) *SI,K)BP
% 3 -3 r^3 * cos(3*theta) sqrt(8) ]]`[tVaFr
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) yw%ES
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) ",[ /pb
% 3 3 r^3 * sin(3*theta) sqrt(8) `1Md1e:J
% 4 -4 r^4 * cos(4*theta) sqrt(10) b "}ya/
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) @MFEBc}
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) $sb@*K}:4
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) >Wx9a"H^(
% 4 4 r^4 * sin(4*theta) sqrt(10) uh#E^~5S
% -------------------------------------------------- {|j-e{*
% V4CA*FEA
% Example 1: Mh3L(z]/E
% sAs`O@
% % Display the Zernike function Z(n=5,m=1) Au/'|%2#(
% x = -1:0.01:1; -iW>T5f
% [X,Y] = meshgrid(x,x); fpjFO&ML
% [theta,r] = cart2pol(X,Y); 8'fF{C
% idx = r<=1; J|o<;9dg1
% z = nan(size(X)); |a /cw"
% z(idx) = zernfun(5,1,r(idx),theta(idx)); Uvi@HB HJ
% figure -Gl!W`$I`
% pcolor(x,x,z), shading interp 0pB'^Q{
% axis square, colorbar <&B]p
% title('Zernike function Z_5^1(r,\theta)') &`>dY
/Y
% ,If"4C!w
% Example 2: [xGL0Z%)t
% Z$m&F0g
% % Display the first 10 Zernike functions _U*1D*kLI[
% x = -1:0.01:1; DAtAc(05)
% [X,Y] = meshgrid(x,x); &Q\k`0vzVB
% [theta,r] = cart2pol(X,Y); EL2z&
% idx = r<=1; B=X_c5
% z = nan(size(X)); 8(A
k
% n = [0 1 1 2 2 2 3 3 3 3]; yTe25l{QaF
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; ntL%&wY
% Nplot = [4 10 12 16 18 20 22 24 26 28]; c^&:':Z%'
% y = zernfun(n,m,r(idx),theta(idx)); 7<2?NLE8*
% figure('Units','normalized') ,g|ht%"
% for k = 1:10 _jxysFl=
% z(idx) = y(:,k); |qf9-36
% subplot(4,7,Nplot(k)) (f#{<^ gd
% pcolor(x,x,z), shading interp -wNhbV2
% set(gca,'XTick',[],'YTick',[]) u+jx3aP:
% axis square 7-9HCP
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) P;&U3i
% end {cw+kY]m4-
% w#bdb;
% See also ZERNPOL, ZERNFUN2. ,>TDxI;
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% Paul Fricker 11/13/2006 xFp9H'j{
M[R'
RcI0n"Gi_
% Check and prepare the inputs: (t,|FkVLV
% ----------------------------- *iPBpEWC
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) IY(;:#l
error('zernfun:NMvectors','N and M must be vectors.') :ZM=P3QZ
end ,qdZ6bv,]|
j2n
4; m
if length(n)~=length(m) ~%: TE}
error('zernfun:NMlength','N and M must be the same length.') 9!D
c=
end =A"z.KfV
G#'G9/Tm
n = n(:); AIA4c"w.EO
m = m(:); f_QZql
if any(mod(n-m,2)) cavzXz
error('zernfun:NMmultiplesof2', ... sNC~S%[
'All N and M must differ by multiples of 2 (including 0).') S8]YS@@D
end uv7tbI"r
|X~vsM0
if any(m>n) w/CD-
error('zernfun:MlessthanN', ... oR<;Tr~{q
'Each M must be less than or equal to its corresponding N.') %-NG eN8
end $[(FCS
qKuHd~M{ 1
if any( r>1 | r<0 ) mi sPJO&QD
error('zernfun:Rlessthan1','All R must be between 0 and 1.') M;@/697G
end 8RVeKnpXTV
-9"[/
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 0Jm)2@
error('zernfun:RTHvector','R and THETA must be vectors.') x^UAtKSy
end v%Su#xq/
[>kzQYT[
r = r(:); YzAGhAyw
theta = theta(:); @'?7au ''
length_r = length(r); ')E4N+h/
if length_r~=length(theta) UTuOean ]'
error('zernfun:RTHlength', ... 3:!5 ]
'The number of R- and THETA-values must be equal.') {=E,.%8
end QPsvc6ds
BUEV+SZ4
% Check normalization: y@]:7
% -------------------- 6J\A%i
if nargin==5 && ischar(nflag) K%>3ev=y.s
isnorm = strcmpi(nflag,'norm'); dxWG+S
if ~isnorm D4QLlP
error('zernfun:normalization','Unrecognized normalization flag.') i}ti
end xgB-m[Xi
else "NO*(<C.R
isnorm = false; Jb`yK@x
end f<2<8xS
[Z+E_Lbz
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ,6^V)F
% Compute the Zernike Polynomials _~E_#cNn
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% JuXuS
NE!]
% Determine the required powers of r: &0@AM_b
% ----------------------------------- 1BA5|
m_abs = abs(m); VxGR[kq$]
rpowers = []; )m$i``*<
for j = 1:length(n) <o&\/uO~H
rpowers = [rpowers m_abs(j):2:n(j)]; C Z/:(sOJ
end q8fnUK?i
rpowers = unique(rpowers); l#%G~c8x
YU%U
% Pre-compute the values of r raised to the required powers, >W@3_{0
% and compile them in a matrix: 5B+I\f&
% ----------------------------- e5.sqft
if rpowers(1)==0 &GLe4zEh
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); Ng<oz*>U
rpowern = cat(2,rpowern{:}); {}v<2bS
rpowern = [ones(length_r,1) rpowern]; _,]@xFCOH
else \"hP*DJ"
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); G_n~1?
rpowern = cat(2,rpowern{:}); )u(Dq u\t
end :jioF{,
I r;Z+}4>Y
% Compute the values of the polynomials: q#!c6lG
% -------------------------------------- Lk]/{t0
y = zeros(length_r,length(n)); e.pq6D5
for j = 1:length(n) c9N5c
s = 0:(n(j)-m_abs(j))/2; )qV&sru.$
pows = n(j):-2:m_abs(j); g7U>G=,;?U
for k = length(s):-1:1 S.A|(?x
p = (1-2*mod(s(k),2))* ... 5Gsjt+
o
prod(2:(n(j)-s(k)))/ ... ~l>2NY
prod(2:s(k))/ ... $79-)4;z4
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... _i+7O^=d6X
prod(2:((n(j)+m_abs(j))/2-s(k))); -
-H%FYF`
idx = (pows(k)==rpowers); 92S,W?(
y(:,j) = y(:,j) + p*rpowern(:,idx); QF`o%mI
end B< BS>(Nr>
9$:+5f,%a
if isnorm E'4dI:
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); y@Q?
guB
end B(|dT66K
end 8ORr
% END: Compute the Zernike Polynomials H@hHEzO
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Z ms
Di8;Tq
% Compute the Zernike functions: ^5d9n<_xnQ
% ------------------------------ _Zs]za.#)|
idx_pos = m>0; 4 Z<
idx_neg = m<0; \H5{[ZUn
T hLR<\
z = y; PFnq:G^L
if any(idx_pos) E?Ofkc$q
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); f.y~ Sew
end K+s
xO/}h
if any(idx_neg) w_eUU)z
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); |;6l1]hk6
end !u=,b fyH
z("Fy
% EOF zernfun