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 r1^m#!=B
function z = zernfun(n,m,r,theta,nflag) LZZ:P
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. Tye$na&$}
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 2l\D~ y
% and angular frequency M, evaluated at positions (R,THETA) on the YU ]G5\UU
% unit circle. N is a vector of positive integers (including 0), and ,6%hu|Y*
% M is a vector with the same number of elements as N. Each element 3.K{T
% k of M must be a positive integer, with possible values M(k) = -N(k) aHVdClD2o
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, =+SVzK,+3
% and THETA is a vector of angles. R and THETA must have the same O,V6hU/ *
% length. The output Z is a matrix with one column for every (N,M) 1DI"LIL
% pair, and one row for every (R,THETA) pair. /:
\V wH
% Mo?t[]L
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike B~'VDOG$Z
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), buxI-wv
% with delta(m,0) the Kronecker delta, is chosen so that the integral <?=mLOo=
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ^R8U-V8:
% and theta=0 to theta=2*pi) is unity. For the non-normalized O[5_9W
4
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. pJ)+}vascR
% yccuTQvz
% The Zernike functions are an orthogonal basis on the unit circle. 6S&=OK^
% They are used in disciplines such as astronomy, optics, and \h'E5LO
% optometry to describe functions on a circular domain. GWA!Ab'<U
% 7B:ZdDj
% The following table lists the first 15 Zernike functions. 8R??J>h5\
% Ndug9j\2
% n m Zernike function Normalization [iO$ c]!H
% -------------------------------------------------- XYxm8ee"j
% 0 0 1 1 F`ZIc7(.{
% 1 1 r * cos(theta) 2 ftI+#0?[!
% 1 -1 r * sin(theta) 2 kS\.
% 2 -2 r^2 * cos(2*theta) sqrt(6) |)72E[lL
% 2 0 (2*r^2 - 1) sqrt(3) 7S~9E2N
% 2 2 r^2 * sin(2*theta) sqrt(6) 44fq1<.K
% 3 -3 r^3 * cos(3*theta) sqrt(8) >`rNT|rg
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) +~i+k~{`H
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) mC[U)` ey
% 3 3 r^3 * sin(3*theta) sqrt(8) _WjETyh
[H
% 4 -4 r^4 * cos(4*theta) sqrt(10) /i~^LITH
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) kT }'"
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) _c(C;s3o
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) s2kZZP8-
% 4 4 r^4 * sin(4*theta) sqrt(10) Rm\'];
% -------------------------------------------------- ,Q /nS$
% /Vm}+"BCS
% Example 1: L/iVs`qF
% kvgs $
% % Display the Zernike function Z(n=5,m=1) V^$rH<
% x = -1:0.01:1; >$S,>d_k`
% [X,Y] = meshgrid(x,x);
1N$gE
% [theta,r] = cart2pol(X,Y); ^Mvsq)
% idx = r<=1; ?:''VM.
% z = nan(size(X)); (HrkUkw
% z(idx) = zernfun(5,1,r(idx),theta(idx)); ";S*[d.2tA
% figure ch,Zk )y:_
% pcolor(x,x,z), shading interp N>nvt.`P
% axis square, colorbar ?lwQne8/
% title('Zernike function Z_5^1(r,\theta)') EDidg"0p
% 3!oQmG_T
% Example 2: :@@A
% va/4q+1GfH
% % Display the first 10 Zernike functions I\uB"Z{9
% x = -1:0.01:1; ,<P[CUD&&
% [X,Y] = meshgrid(x,x); _l{5'm
% [theta,r] = cart2pol(X,Y); |gRgQGeB
% idx = r<=1; n-b<vEZw#
% z = nan(size(X)); % 6hw
% n = [0 1 1 2 2 2 3 3 3 3]; S_ -QvG2
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; ME10dr
% Nplot = [4 10 12 16 18 20 22 24 26 28]; `7qp\vYL
% y = zernfun(n,m,r(idx),theta(idx)); /jn3'q_,
% figure('Units','normalized') lKhh=Pc2
% for k = 1:10 QH' [(
% z(idx) = y(:,k); 6[h$r/GXh"
% subplot(4,7,Nplot(k)) &<P^Tvqq&
% pcolor(x,x,z), shading interp }
Ved
% set(gca,'XTick',[],'YTick',[]) wAOVH].
% axis square ~q T1<k
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) U1HD~
% end Nb!6YY=Ez-
% F3 l^^Mc
% See also ZERNPOL, ZERNFUN2. j]l}K*8(
PUZXmnB
% Paul Fricker 11/13/2006 \;:@=9`
pn%|;
vwH7/+
% Check and prepare the inputs: u r.T YKF
% ----------------------------- n`T[eb~
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) =O'%)Y&
error('zernfun:NMvectors','N and M must be vectors.') AUjTcu>i
end 'kg]|"M
#Xw[i
if length(n)~=length(m) L%O8vn^3
error('zernfun:NMlength','N and M must be the same length.') (:HbtrI
end Cz);mOb%M%
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n = n(:); ;$L!`"jn
m = m(:); ;ld~21#m
if any(mod(n-m,2)) Nl<,rD+KSD
error('zernfun:NMmultiplesof2', ... No&[ \;
'All N and M must differ by multiples of 2 (including 0).') iN4'jD^oP
end V\`="
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if any(m>n) MVv^KezD
error('zernfun:MlessthanN', ... >;r05,mc
'Each M must be less than or equal to its corresponding N.') 2-c0/?_4
end 2T%f~yQ^
y^46z(I
if any( r>1 | r<0 ) Cl.T'A$
error('zernfun:Rlessthan1','All R must be between 0 and 1.') _%Ld
Ez
end 1HWJxV"
r4ttEJ-jG
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Ahbu >LPk
error('zernfun:RTHvector','R and THETA must be vectors.') L.:QI<n
end \J:T]
gI5nWEM0{
r = r(:); N&h!14]{Z
theta = theta(:); UYrzsUjg&
length_r = length(r); 'I>#0VRr
if length_r~=length(theta) sK/"
error('zernfun:RTHlength', ... D=sc41]
'The number of R- and THETA-values must be equal.') _";pk _
end }~'Wz*Gm
+vSE}
% Check normalization: .);:K
% -------------------- A y[L{!)2{
if nargin==5 && ischar(nflag) T|2%b*/
isnorm = strcmpi(nflag,'norm'); U*:'/.
if ~isnorm 9:w,@Phe
error('zernfun:normalization','Unrecognized normalization flag.') .
\0=1P:
end I8]NY !'cW
else .%Q Ea_\
isnorm = false; $[CA#AXE
end iQ"F`C
`#8R+c=$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% gK\7^95
% Compute the Zernike Polynomials azc:C
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% V>92/w.fe
u`@FA?+E1
% Determine the required powers of r: X
hX'*{3k
% ----------------------------------- dKTAc":-}
m_abs = abs(m); 9,eR=M]+:
rpowers = []; !QS<;)N@
for j = 1:length(n) " z'!il#
rpowers = [rpowers m_abs(j):2:n(j)]; 'k Z1&_{
end /- 4B)mL
rpowers = unique(rpowers); J4 #]8!A
S5a<L_
% Pre-compute the values of r raised to the required powers, + qqN
% and compile them in a matrix: wT yM9wz&
% ----------------------------- JW'acD
if rpowers(1)==0 a\_,_psK
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); lFY8^#@
rpowern = cat(2,rpowern{:}); j1+Y=@MA
rpowern = [ones(length_r,1) rpowern]; 3*2pacHpE
else U/o}{,$A
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); s2=X>,kz?
rpowern = cat(2,rpowern{:}); =W*`HV-w
end Qo *]l_UO;
!PIdw~YC
% Compute the values of the polynomials: 5305N!
% -------------------------------------- eJp-s" %
y = zeros(length_r,length(n)); y<d#sv(s
for j = 1:length(n) w/6@R 4)p
s = 0:(n(j)-m_abs(j))/2; 'FFc"lqj
pows = n(j):-2:m_abs(j); <U pjAuG8
for k = length(s):-1:1 Fsj[J E
p = (1-2*mod(s(k),2))* ... 3y ,?>-
prod(2:(n(j)-s(k)))/ ... Ps\^OJR
prod(2:s(k))/ ... 26K~m@
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... >;W(Jb7e
prod(2:((n(j)+m_abs(j))/2-s(k))); <5~>.DuE
idx = (pows(k)==rpowers); @ R Bw T
y(:,j) = y(:,j) + p*rpowern(:,idx); 6|}mTG^
end 7*"LW
N@0scfO6<
if isnorm h
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y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); [Cp{i<C
end =
g}yA=.
end zUqDX{I8
% END: Compute the Zernike Polynomials ht9b=1wd%s
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% $8r:&Iw
3k^jR1
% Compute the Zernike functions: ?9TogW>W
% ------------------------------ 64fG,b
idx_pos = m>0; -m/4\D
idx_neg = m<0; K^\9R
3IFU{0a`
z = y; E76:}(
if any(idx_pos) S
&u94hlC
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); E: k?*l
end F9W5x=EK\
if any(idx_neg) 4PQWdPv;
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); .vMi<U;
end kM`#U
*j
!&[4T#c
% EOF zernfun