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 vGnFX0?h
function z = zernfun(n,m,r,theta,nflag) kWacc&*|
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. X`(fJ',
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N MH|F<$42
% and angular frequency M, evaluated at positions (R,THETA) on the [1Aoj|
% unit circle. N is a vector of positive integers (including 0), and I)kc[/^j$
% M is a vector with the same number of elements as N. Each element [C/{ ru&E
% k of M must be a positive integer, with possible values M(k) = -N(k) !9{hbmF#
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, {r~=mQ
% and THETA is a vector of angles. R and THETA must have the same WH"'Ju5}
% length. The output Z is a matrix with one column for every (N,M) {;|pcx\L6~
% pair, and one row for every (R,THETA) pair. {b'
% =CW> ;h]
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike n2~WUK
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), dC;&X
g`
% with delta(m,0) the Kronecker delta, is chosen so that the integral /:^nG+
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, +\*b?x
% and theta=0 to theta=2*pi) is unity. For the non-normalized }Q*J!OH
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. '"+Gn52#
% A.mFa1lH
% The Zernike functions are an orthogonal basis on the unit circle. &8pGq./lr=
% They are used in disciplines such as astronomy, optics, and 6oq5CD oq
% optometry to describe functions on a circular domain. l =t/"M=
% cs7^#/3<
% The following table lists the first 15 Zernike functions. -\USDi(
% ?lfyC/
% n m Zernike function Normalization I o"3wL)2
% -------------------------------------------------- kBLFK3i
% 0 0 1 1 +!W:gA
% 1 1 r * cos(theta) 2 y@,PTF
% 1 -1 r * sin(theta) 2 S?6-I,]h
% 2 -2 r^2 * cos(2*theta) sqrt(6) aOw#]pB|
% 2 0 (2*r^2 - 1) sqrt(3) *~YdL7f)J
% 2 2 r^2 * sin(2*theta) sqrt(6) \#]C !JQ
% 3 -3 r^3 * cos(3*theta) sqrt(8) <Y6zJ#BD
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) o>nw~_ H\
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) ,(-V<>/*.|
% 3 3 r^3 * sin(3*theta) sqrt(8) ]l C2YD}
% 4 -4 r^4 * cos(4*theta) sqrt(10) 7M
_
mR Vh
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) :iLRCK3C
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) "G~!J\
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Tr s2M+r)
% 4 4 r^4 * sin(4*theta) sqrt(10) /qJC p![X
% -------------------------------------------------- .p.(
\5Fo
% 2 S~( P
% Example 1: V?'p E
% {]cr.y]\
% % Display the Zernike function Z(n=5,m=1) =+UtAf<n
% x = -1:0.01:1; ,*dLE
% [X,Y] = meshgrid(x,x); ,Jh#$mil
% [theta,r] = cart2pol(X,Y); .>#O'Z&q9
% idx = r<=1; jl>TZ)4}V
% z = nan(size(X)); BgD3P.;[
% z(idx) = zernfun(5,1,r(idx),theta(idx)); a]7g\rg)
% figure Ww60-d}}Q
% pcolor(x,x,z), shading interp 71 %$&6
% axis square, colorbar =+K?@;?
% title('Zernike function Z_5^1(r,\theta)') ,`RX~ H=C
% cD6 ^7QF
% Example 2: j{r@>g;3
% #;~HoOK*#
% % Display the first 10 Zernike functions ^"D^D`$@
% x = -1:0.01:1; Hi=</ Wy;
% [X,Y] = meshgrid(x,x); 7M4J{}9
% [theta,r] = cart2pol(X,Y); e ><0crb
% idx = r<=1; AX$r,KmE
% z = nan(size(X)); L%(NXSfu7
% n = [0 1 1 2 2 2 3 3 3 3]; Z'j[N4%BK
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; S<NK!89
% Nplot = [4 10 12 16 18 20 22 24 26 28]; ,mHUo4h1O
% y = zernfun(n,m,r(idx),theta(idx)); .{c7 I!8
% figure('Units','normalized') FG[rH]
% for k = 1:10 i0$*):b
% z(idx) = y(:,k); KpYezdPF)
% subplot(4,7,Nplot(k)) -z+,j(@
% pcolor(x,x,z), shading interp ,dTmI{@O
% set(gca,'XTick',[],'YTick',[]) yc~<h/}#
% axis square B~ i
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) $l"%o9ICG
% end xSd&xwP
% k9OGnCW\
% See also ZERNPOL, ZERNFUN2. RZV6;=/
d1\nMm}v
% Paul Fricker 11/13/2006 G 3,v'D5
ssx#|InY
K$Vu[!l`
% Check and prepare the inputs: GW'v\O
% ----------------------------- VqV [ @[P
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) O+|C<;K
error('zernfun:NMvectors','N and M must be vectors.') J_Tz\bZ3)
end Q17dcgd
t4#gW$+^?H
if length(n)~=length(m) eGq7+
error('zernfun:NMlength','N and M must be the same length.') yD7}
end YwET.(oo
~qeFSU(
n = n(:); 5Y^"&h[/
m = m(:); F/BR#J1
if any(mod(n-m,2)) O#ZZ PJ"
error('zernfun:NMmultiplesof2', ... X>=`l)ZR
'All N and M must differ by multiples of 2 (including 0).') lTqlQ<`V
end .gDq+~r8O
v.Q#<@B^:
if any(m>n) RYEZ'<
error('zernfun:MlessthanN', ... 9/{ zS3h3
'Each M must be less than or equal to its corresponding N.') >":xnX#
end a24 AmoWx
uStAZ~b\
if any( r>1 | r<0 ) _C?Wk:Y@
error('zernfun:Rlessthan1','All R must be between 0 and 1.') )
yMrET
m
end Y /_CPY
F!EiF&[\J
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) c**&, aL
error('zernfun:RTHvector','R and THETA must be vectors.') q/U-6A[0
end \(P?=] -
B ??07j
r = r(:); &;d
N:F;
theta = theta(:); :}v-+eIQ
length_r = length(r); lUs$I{2_
if length_r~=length(theta) ulIEx~qP
error('zernfun:RTHlength', ... h9ScN(|0y
'The number of R- and THETA-values must be equal.') ZK^cG'^2|
end Yu3S3aRE
W]ca~%r
% Check normalization: Tl2t\z+ps
% -------------------- %|(c?`2|
if nargin==5 && ischar(nflag) ~SQxFAto
isnorm = strcmpi(nflag,'norm'); +n;nvf}(
if ~isnorm lJu^Bcrv
error('zernfun:normalization','Unrecognized normalization flag.') 7amVnR1f
end ?x #K:a?
else dz9U.:C
isnorm = false; TsaQR2J@
end M/Yr0"%Q<.
Xh;.T=/E|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% El<*)
% Compute the Zernike Polynomials ^)gyKl:E'
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% E:pk'G0bZ
`sCaGCp
% Determine the required powers of r: 4Lt9Dx1
% ----------------------------------- nL:&G'd
m_abs = abs(m); ZiJF.(JS
rpowers = []; Kt_oo[ey{
for j = 1:length(n) mgjJNzclL
rpowers = [rpowers m_abs(j):2:n(j)]; `sYFQ+D#O
end sh$-}1 ;
rpowers = unique(rpowers); `3rwqcxA
w'H'o!*/
% Pre-compute the values of r raised to the required powers, SO0\d0?u
% and compile them in a matrix: luf5-XT
% ----------------------------- 46 A sD
if rpowers(1)==0 R#d~a;j
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); C:J;'[,S
rpowern = cat(2,rpowern{:}); `uMEK>b
rpowern = [ones(length_r,1) rpowern]; X=$Jp.
else .c"nDCFVR
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); :]-oo*xP
rpowern = cat(2,rpowern{:}); K.)!qkW-%S
end b0$)G-E/Y
Q*smH-Sw
% Compute the values of the polynomials: 2^WJ1: A
% -------------------------------------- k5S;G"iJ
y = zeros(length_r,length(n)); FXof9fa_B
for j = 1:length(n) j?.F-ar
s = 0:(n(j)-m_abs(j))/2; tUv>1)
[
pows = n(j):-2:m_abs(j); K|7"YNohfG
for k = length(s):-1:1 4qOzjEQ
p = (1-2*mod(s(k),2))* ... >j5\J_(;D
prod(2:(n(j)-s(k)))/ ... R{#< NE
prod(2:s(k))/ ... t/iI!}
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... AFz:%m
prod(2:((n(j)+m_abs(j))/2-s(k))); \Z]+j@9
idx = (pows(k)==rpowers); a$My6Qa#
y(:,j) = y(:,j) + p*rpowern(:,idx); K|P0nJT
end <,]:jgX
$xbC^ k
if isnorm 7=l~fKu
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); p27Dcwov
end Hy.u6Jt*/
end }e[ E
% END: Compute the Zernike Polynomials 0WUBj:@g
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% _ .v G)
,"%C.9a
% Compute the Zernike functions: ^{+ry<rS>
% ------------------------------ pp"X0
idx_pos = m>0; 4era5=
idx_neg = m<0; 5p0~AN)
Q]k<Y
z = y; N"S`9B1eD(
if any(idx_pos) %~LY'cfPse
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); j_8 Y Fz5
end 5PeS/%uT@
if any(idx_neg) 66v,/#K
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); #t+?eye~
end MpCPY"WLL
zwfft
% EOF zernfun