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 |UR.7rOV
function z = zernfun(n,m,r,theta,nflag) &]vd7Q.t
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. sUbZVPDr
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 'a"<uk3DT
% and angular frequency M, evaluated at positions (R,THETA) on the 3\D jV2t
% unit circle. N is a vector of positive integers (including 0), and wau81rSd
% M is a vector with the same number of elements as N. Each element 9=<
Z>
% k of M must be a positive integer, with possible values M(k) = -N(k) S~6<'N&[
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, j*xens$)
% and THETA is a vector of angles. R and THETA must have the same dc?Yk3(Y
% length. The output Z is a matrix with one column for every (N,M) oTx#e[8f{
% pair, and one row for every (R,THETA) pair. P9%9/ B:-
% OvK_CN{
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike gXw\_ue<
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 9wWjl}%
% with delta(m,0) the Kronecker delta, is chosen so that the integral DMs|Q$XB
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, *Z/B\nb
% and theta=0 to theta=2*pi) is unity. For the non-normalized ,Y!T!o}1
% polynomials, max(Znm(r=1,theta))=1 for all [n,m].
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% *$l8H[
% The Zernike functions are an orthogonal basis on the unit circle. bJBx~
% They are used in disciplines such as astronomy, optics, and Y$Uvt_
% optometry to describe functions on a circular domain. v"$; aJ
% )YnB6@=nyk
% The following table lists the first 15 Zernike functions. !J2Lp
% P_ZguNH
% n m Zernike function Normalization Vq<|DM3z<
% -------------------------------------------------- KqtI^qC8
% 0 0 1 1 n$=n:$`q
% 1 1 r * cos(theta) 2 qx!IlO
% 1 -1 r * sin(theta) 2 Rwy:.)7B$q
% 2 -2 r^2 * cos(2*theta) sqrt(6) 'GW@P
% 2 0 (2*r^2 - 1) sqrt(3) Hpsg[d)!
% 2 2 r^2 * sin(2*theta) sqrt(6)
{'r*Jb0
% 3 -3 r^3 * cos(3*theta) sqrt(8) ^NnZYr.
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) 9f"6Jw@F
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) ?tSY=DK\n
% 3 3 r^3 * sin(3*theta) sqrt(8) Y":hb;&
% 4 -4 r^4 * cos(4*theta) sqrt(10) ZjI^0D8
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ]}5jX^j
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) !8A5Y[(XD
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) _`D_0v(X
% 4 4 r^4 * sin(4*theta) sqrt(10) :YV!;dKJ
% -------------------------------------------------- m=}kGzIY4
% d+^4;Hv4
% Example 1: Jp,ohVRNq
% igo7F@_,
% % Display the Zernike function Z(n=5,m=1) I}8F3_b,#
% x = -1:0.01:1; !.w S+
% [X,Y] = meshgrid(x,x); (ZI&'"H
% [theta,r] = cart2pol(X,Y); t(_XB|AKm
% idx = r<=1; YInW)My.h
% z = nan(size(X)); j`tUx#
h
% z(idx) = zernfun(5,1,r(idx),theta(idx)); X G^
% figure {< wq }~
% pcolor(x,x,z), shading interp ev@1+7(
% axis square, colorbar 2]C0d8=*?
% title('Zernike function Z_5^1(r,\theta)') 0<Pe~i_=
% .#}SK!"B
% Example 2: )1]C%)zn
% ?=T&|pp
% % Display the first 10 Zernike functions hZJ Nh,,w
% x = -1:0.01:1; v~xG*e
% [X,Y] = meshgrid(x,x); iFDQnt
[t
% [theta,r] = cart2pol(X,Y); (>Yii_Cd
% idx = r<=1; k1cBMDSokO
% z = nan(size(X)); X F40;urm
% n = [0 1 1 2 2 2 3 3 3 3]; +22[ h@
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; '"KK|]vJ
% Nplot = [4 10 12 16 18 20 22 24 26 28]; AV&ege
% y = zernfun(n,m,r(idx),theta(idx)); jBB<{VV|
% figure('Units','normalized') x*)Wl!
% for k = 1:10 ]v#T'<Nl
% z(idx) = y(:,k); >AfJxdd1
% subplot(4,7,Nplot(k)) ^wHO!$
% pcolor(x,x,z), shading interp :@3d
% set(gca,'XTick',[],'YTick',[]) Z?@07Y[|K
% axis square VEpQT
Qp
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) EgO4:8$h
% end +tA rH
C]
% u*U?VZ5
% See also ZERNPOL, ZERNFUN2. u9&p/qMx2
FUOvH85f
% Paul Fricker 11/13/2006 R.fRQ>rI
0b|!S/*A3
cCeD3CuRA%
% Check and prepare the inputs: @z,'IW74V
% ----------------------------- kOc'@;_O
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) '-~86Q
error('zernfun:NMvectors','N and M must be vectors.') MdKZH\z/
end tJn2:}-s
~cez+VQe
if length(n)~=length(m) \"*l:x-u
error('zernfun:NMlength','N and M must be the same length.') ILpB:g
end 1`uIjXr(
!hc7i=V?
n = n(:); aL`pvsnF
m = m(:); <)&ykcB
if any(mod(n-m,2)) h '}5"m
error('zernfun:NMmultiplesof2', ... ywdNwNJ
'All N and M must differ by multiples of 2 (including 0).') %NBD^gF
end D"Xm9
(
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if any(m>n) o4J K$%
error('zernfun:MlessthanN', ... nxJhK
T
'Each M must be less than or equal to its corresponding N.') *83+!DV|
end Vz#cb5:g
W)"q9(T?%
if any( r>1 | r<0 ) vB,N6~r>
error('zernfun:Rlessthan1','All R must be between 0 and 1.') COT;KC6
n
end hN} X11
9X(bByEO
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) YnMph0\Y^
error('zernfun:RTHvector','R and THETA must be vectors.') x=Ru@n K;
end O{4m-;
UFl*^j_)]
r = r(:); f(C0&"4e
theta = theta(:); HOw][}M_w
length_r = length(r); -R8RAwsLG
if length_r~=length(theta) Vr^wesT\Hx
error('zernfun:RTHlength', ... 'D-imLV<<
'The number of R- and THETA-values must be equal.') %iGME%oXr
end ;EJ6C#}
>7
l^vq'<kI
% Check normalization: s)N1@RBR
% -------------------- OO$<Wgh
if nargin==5 && ischar(nflag) ^NCH)zK]v
isnorm = strcmpi(nflag,'norm'); AV'>
if ~isnorm tQ/w\6{
error('zernfun:normalization','Unrecognized normalization flag.') wS5hXTb"
end dfrq8n]
else -py.YZ
isnorm = false; kSJWQ
end $""[(
d?0
XI0O^[/n{
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% JvUKfsn u{
% Compute the Zernike Polynomials 87HVD Di
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% "<&F=gV
saV3<zgx
% Determine the required powers of r: OVd"'|&6_
% ----------------------------------- hsl8@=_ B
m_abs = abs(m); ;?y?s'>t&
rpowers = []; @ NVq
.z
for j = 1:length(n) vxwctJ&
rpowers = [rpowers m_abs(j):2:n(j)]; S)~h|&A(
end ~E!"YkIr
rpowers = unique(rpowers); p7k0pSt
e88JT_zrO
% Pre-compute the values of r raised to the required powers, Y,8M[UIK
% and compile them in a matrix: F|PYDC
% ----------------------------- FCIT+8K
if rpowers(1)==0 >GjaA1,
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 9+/<[w7
rpowern = cat(2,rpowern{:}); N(
/PJJ~
rpowern = [ones(length_r,1) rpowern]; fLys$*^)^
else x=H*"L=
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); hA"N&v~
rpowern = cat(2,rpowern{:}); ('gjfl
end %xg"e
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sz)3
z
% Compute the values of the polynomials: GJW1|Fk
% -------------------------------------- YZoudX'"
y = zeros(length_r,length(n)); ,og@}gOMB
for j = 1:length(n) $<yb~z7J
s = 0:(n(j)-m_abs(j))/2; <y!BO
pows = n(j):-2:m_abs(j); 5!5P\o
for k = length(s):-1:1 -1< }_*
p = (1-2*mod(s(k),2))* ... ~U^0z|.
prod(2:(n(j)-s(k)))/ ... "'PDreS
prod(2:s(k))/ ... _2TIan}
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... BBp
Hp
prod(2:((n(j)+m_abs(j))/2-s(k))); eAl&[_o|S
idx = (pows(k)==rpowers); @z2RMEC~
y(:,j) = y(:,j) + p*rpowern(:,idx); H,uOshR
end d8x$NW-s
2V
if isnorm W0?yPP=.
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); o30PI
end ~gV|_G
end E7*]t_p"
% END: Compute the Zernike Polynomials SKYS6b
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% B0YY7od
H_$"]iQ
% Compute the Zernike functions: ^&,{
% ------------------------------ KDY~9?}TM
idx_pos = m>0; N.VzA
6C
idx_neg = m<0; `yVJ `}hm
*|4~
0w
z = y; bG5c~
if any(idx_pos) AQFx>:in
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); }XAoMp
end ly{~X
if any(idx_neg) xR%CS`0R
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); u] oS91
end Gud!(5'
!867DX3*
% EOF zernfun