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 VhNz8)
function z = zernfun(n,m,r,theta,nflag) EbdfV-E
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. cra+T+|>Kc
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ma((2My'H
% and angular frequency M, evaluated at positions (R,THETA) on the tuhA
9}E
% unit circle. N is a vector of positive integers (including 0), and GxKqD;;u?=
% M is a vector with the same number of elements as N. Each element _~T!9
% k of M must be a positive integer, with possible values M(k) = -N(k) >>5NX"{
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, kbMYMx.[
% and THETA is a vector of angles. R and THETA must have the same QPfc(Z
% length. The output Z is a matrix with one column for every (N,M) >2Kh0rIH
% pair, and one row for every (R,THETA) pair. PoT`}-9
% QV&D l_
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 9J?wO9rI
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), X3V'Cy/sy
% with delta(m,0) the Kronecker delta, is chosen so that the integral 6C+"`(u%V
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 8f3vjK'
% and theta=0 to theta=2*pi) is unity. For the non-normalized J52
o
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% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. :at$HCaK
% Ba/Yl
% The Zernike functions are an orthogonal basis on the unit circle. ]~E0gsq
% They are used in disciplines such as astronomy, optics, and
4A2?Uhpy
% optometry to describe functions on a circular domain. l@ap]R
% nTz6LVF
% The following table lists the first 15 Zernike functions. <Ce2r"U1e
% 7IjQi=#:
% n m Zernike function Normalization 9s_,crq5
% -------------------------------------------------- yfC^x%d7G
% 0 0 1 1 k+DR]icv
% 1 1 r * cos(theta) 2 zBe8,, e
% 1 -1 r * sin(theta) 2 QJ7L7S
% 2 -2 r^2 * cos(2*theta) sqrt(6) G3{=@Z1
% 2 0 (2*r^2 - 1) sqrt(3) |K|h+fgG6*
% 2 2 r^2 * sin(2*theta) sqrt(6) 7%{ |
% 3 -3 r^3 * cos(3*theta) sqrt(8) =F;.l@:
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8)
Yl.0aS
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) &[;HYgp
% 3 3 r^3 * sin(3*theta) sqrt(8) <E0UK^-}
% 4 -4 r^4 * cos(4*theta) sqrt(10) 4X*>H
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Z" uY}P3
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) MC{
2X
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) j7)Ao*WN
% 4 4 r^4 * sin(4*theta) sqrt(10) ?:L:EW8
% -------------------------------------------------- qvv2O1c"A
% T
N!=@Gy
% Example 1: +fnK/%b
% tT79p.z B
% % Display the Zernike function Z(n=5,m=1) izx#3u$P
% x = -1:0.01:1; Yp:KI7
% [X,Y] = meshgrid(x,x); jvQ*t_L
% [theta,r] = cart2pol(X,Y); xSBc-u#< G
% idx = r<=1; Bdu&V*0g
% z = nan(size(X)); //4Xq8y
% z(idx) = zernfun(5,1,r(idx),theta(idx)); u3o#{~E/#
% figure NZ3/5%We/
% pcolor(x,x,z), shading interp $e /^u[~:
% axis square, colorbar gL3"Gg3
% title('Zernike function Z_5^1(r,\theta)') !0dNQ[$82
% }nMPSerE
% Example 2: Zw~+Pb
% MXyaE~LK
% % Display the first 10 Zernike functions }@^4,FKJ
% x = -1:0.01:1; Q"7Gy<
% [X,Y] = meshgrid(x,x); d`/tE?Gw
% [theta,r] = cart2pol(X,Y); is@b&V]
% idx = r<=1; _{ZqO;[u
% z = nan(size(X)); -@Uqz781
% n = [0 1 1 2 2 2 3 3 3 3]; }YHX-e<Yx]
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; 25&J7\P*
% Nplot = [4 10 12 16 18 20 22 24 26 28]; A<B=f<N3gV
% y = zernfun(n,m,r(idx),theta(idx)); "kA*Vc#
% figure('Units','normalized') UDL
RCS8i
% for k = 1:10 A.5i"Ci[ie
% z(idx) = y(:,k); 3ux0Jr2yT
% subplot(4,7,Nplot(k)) \{EpduwZ
% pcolor(x,x,z), shading interp "XT"|KF|D
% set(gca,'XTick',[],'YTick',[]) R+7oRXsu
% axis square Z*FrB58
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) %b^OeWip
% end 1NcCy!+
% U.@*`Fg
% See also ZERNPOL, ZERNFUN2. IO/4.m-aN#
@e'5E^
% Paul Fricker 11/13/2006 E(i[o?
0V!l,pg
Q3y;$ "
% Check and prepare the inputs: M5trNSL&u
% ----------------------------- DU=dLE6-P;
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 2m~V{mUT!
error('zernfun:NMvectors','N and M must be vectors.') h/,${,}J
end !L95^g
]K*8O<
if length(n)~=length(m) @l0|*lo%
error('zernfun:NMlength','N and M must be the same length.') 8Mbeg
,P
end E[^ {w
gp-T"l
n = n(:); ZoB{x*IH
m = m(:); oY=q4D
if any(mod(n-m,2)) .WQ+AE8Q
error('zernfun:NMmultiplesof2', ... :(_+7N[KA
'All N and M must differ by multiples of 2 (including 0).') $8crN$ye
end 1c@}C+F+
ehA;i.n
if any(m>n) n\ Hs@.
error('zernfun:MlessthanN', ... > MH(0+B*
'Each M must be less than or equal to its corresponding N.') A?*o0I
end ZY56\qcY
)=DGdIEt
if any( r>1 | r<0 ) HQ9X7[3
error('zernfun:Rlessthan1','All R must be between 0 and 1.') )H}#A#ovj7
end :>81BuMvg
BJS-Jy$-
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) W8g'lqc|
error('zernfun:RTHvector','R and THETA must be vectors.') S{K0.<,E
end \` w4|T
')N{wSM9Ft
r = r(:); `.2hjO
theta = theta(:); O0PJ6:9P
length_r = length(r); v~/~@jv
if length_r~=length(theta) 28OWNS
M=
error('zernfun:RTHlength', ... D\ H/
'The number of R- and THETA-values must be equal.') ph2$oO
6,
end {ccIxL
/~
U'*t~x<
% Check normalization: {>bW>RO)
% -------------------- .6~`Ubr}E
if nargin==5 && ischar(nflag) OD=!&LM
isnorm = strcmpi(nflag,'norm'); m~'? /!!
if ~isnorm _Zc%z@}
error('zernfun:normalization','Unrecognized normalization flag.') tV/Z)fpyH
end )RsM!}
else syzdd
an
isnorm = false; Ac|5. ?|N
end LG]3hz9^9
z* <y5
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ?tg
y|
% Compute the Zernike Polynomials *{o UWt
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ~3RC>8*Qw
6/ `.(fL1
% Determine the required powers of r: KL'zXkS
% ----------------------------------- ]h9!ei
[
m_abs = abs(m); X_$a,"'~)
rpowers = []; eb|i3.
for j = 1:length(n) w-$[>R[hw
rpowers = [rpowers m_abs(j):2:n(j)]; G9g6.8*&
end +([!A6:
rpowers = unique(rpowers); ,1/}^f6
NcM>{{8
% Pre-compute the values of r raised to the required powers, |3?
8)z\n
% and compile them in a matrix: 3I 0eW%,
% ----------------------------- )$Z(|M4
if rpowers(1)==0 OJ4SbI
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 4l8BQz}sb
rpowern = cat(2,rpowern{:}); Vc3mp;6"
rpowern = [ones(length_r,1) rpowern]; y/c%+Ca/
else Ov82ibp_1
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); AD('=g J
rpowern = cat(2,rpowern{:}); XUV!C7
end b @;.F!x
IK^~X{I?
% Compute the values of the polynomials: e1q"AOV 6
% -------------------------------------- O3NWXe<
y = zeros(length_r,length(n)); W}'WA
for j = 1:length(n) v0l_w
s = 0:(n(j)-m_abs(j))/2; )$x_!=@1
pows = n(j):-2:m_abs(j); r(2R<A
for k = length(s):-1:1 ;,OfJ'q^
p = (1-2*mod(s(k),2))* ... SJgY
prod(2:(n(j)-s(k)))/ ... /OGA$eP
prod(2:s(k))/ ... v$w++3H
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 7 boJ*
prod(2:((n(j)+m_abs(j))/2-s(k))); KbxR
Lx]w
idx = (pows(k)==rpowers); R,@g7p
y(:,j) = y(:,j) + p*rpowern(:,idx); l)+:4N?iVv
end 1q.(69M
J0220 _
if isnorm 2)/NFZ
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); l!IKUzt)7
end {b!7
.Cd=
end 84&XW
% END: Compute the Zernike Polynomials ,7d|O}B
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% l*7?Y7FK
x|~zHFm6
% Compute the Zernike functions: mxqG-*ch-
% ------------------------------ ]y1fM0
idx_pos = m>0; $;D*
n'8Fx
idx_neg = m<0; '=cKU0
G #
~S(^T9R
z = y; #2%([w
if any(idx_pos) keqcV23k
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); %c6E-4b
end 0-2"FdeQU
if any(idx_neg) s\0K o1
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); m s~8QL
end =K$,E4*
E,*&BDW
% EOF zernfun