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 2 0tO#{Li
function z = zernfun(n,m,r,theta,nflag) .WX,Nd3@
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. ~Y7dH
Dn
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N T2=HG Z
% and angular frequency M, evaluated at positions (R,THETA) on the @0NJ{
% unit circle. N is a vector of positive integers (including 0), and p=Y>i 'CG
% M is a vector with the same number of elements as N. Each element N|K4{Frm
% k of M must be a positive integer, with possible values M(k) = -N(k) (dqCa[
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, ,DQjDMjrf
% and THETA is a vector of angles. R and THETA must have the same <jA105U"m>
% length. The output Z is a matrix with one column for every (N,M) [syj#
% pair, and one row for every (R,THETA) pair. j}f[W [2
%
c5% 6Y2W0
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike wRvb8F0
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ,<`)>2 'o
% with delta(m,0) the Kronecker delta, is chosen so that the integral QkQ!Ep(
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ~F!,PM/
% and theta=0 to theta=2*pi) is unity. For the non-normalized
]Oeh=gq
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. BPv>$
m+.
% w0lT%CPx
% The Zernike functions are an orthogonal basis on the unit circle. np9dM
% They are used in disciplines such as astronomy, optics, and + ulagE|7
% optometry to describe functions on a circular domain. vScjq5"p
% -c*\o3)
% The following table lists the first 15 Zernike functions. 5,)vJ,fs
% ZDt?j
% n m Zernike function Normalization G~,:2
o3
% -------------------------------------------------- "ju'UOcS/
% 0 0 1 1 KP]"P*?
?
% 1 1 r * cos(theta) 2 uLR<FpM
% 1 -1 r * sin(theta) 2 (?0`d
% 2 -2 r^2 * cos(2*theta) sqrt(6) 'b&yrBFD
% 2 0 (2*r^2 - 1) sqrt(3) P8Qyhc
% 2 2 r^2 * sin(2*theta) sqrt(6) :-T*gqj|
% 3 -3 r^3 * cos(3*theta) sqrt(8) o\VUD
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) 3gEMRy*+
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) u]*0;-tz
% 3 3 r^3 * sin(3*theta) sqrt(8) UL$}{2N,_
% 4 -4 r^4 * cos(4*theta) sqrt(10) #xh_
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) tc+WWDP#"
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) LeOP;#
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 88s/Q0l
% 4 4 r^4 * sin(4*theta) sqrt(10) 49H+(*@v@
% -------------------------------------------------- 4T6 {Y
% aB~S?.l
% Example 1: qet>1<
% @(g_<@Jz
% % Display the Zernike function Z(n=5,m=1) saf&dd
% x = -1:0.01:1; W~1~k{A
% [X,Y] = meshgrid(x,x); $'rG-g!f\
% [theta,r] = cart2pol(X,Y); &ANP`=
% idx = r<=1; >$WQxbwM(
% z = nan(size(X)); tBbOY}.VD
% z(idx) = zernfun(5,1,r(idx),theta(idx));
]:M0Kj&h
% figure 46@{5)Tq
% pcolor(x,x,z), shading interp :Qt
% axis square, colorbar D\dWt1n
% title('Zernike function Z_5^1(r,\theta)') EOj"V'!
% Z<[<n0o1
% Example 2: u$#Wv2| mk
% @mP]*$00
% % Display the first 10 Zernike functions N["W Ir
% x = -1:0.01:1; S-Y=-"
% [X,Y] = meshgrid(x,x); PXzsj.
% [theta,r] = cart2pol(X,Y); E>'a,!QPv
% idx = r<=1; 2Y\
d<.M
% z = nan(size(X)); `]Fx.)C#
% n = [0 1 1 2 2 2 3 3 3 3]; EP'h@zdz
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; X|f7K
% Nplot = [4 10 12 16 18 20 22 24 26 28]; fWfk[(M'9
% y = zernfun(n,m,r(idx),theta(idx)); t7n(Qkrv
% figure('Units','normalized') Z>c3
% for k = 1:10 wI]"U2L5
% z(idx) = y(:,k); Un`^jw#_
% subplot(4,7,Nplot(k)) R'EUV0KX>Y
% pcolor(x,x,z), shading interp %,Sf1fUJ
% set(gca,'XTick',[],'YTick',[]) c0B|F
% axis square voP7"Dl[
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ('wY9kvL&
% end <h%O?mkC
% poGc a1
% See also ZERNPOL, ZERNFUN2. Nkxmm/Z
;<yd^Xs
% Paul Fricker 11/13/2006 m8'C_U^89
UcBe'r}G
aRG2@5
% Check and prepare the inputs: fkf1m:Ckh
% ----------------------------- \^ghdU
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) *.L81er5~
error('zernfun:NMvectors','N and M must be vectors.') 1)
ta
end -F'b8:m
4wC+S9I#E^
if length(n)~=length(m) ?]D"k4
error('zernfun:NMlength','N and M must be the same length.') \fA{1
end d>;&9;)H
I}Nd$P)>
n = n(:); }ci#>
m = m(:); u[nyW3MZ
if any(mod(n-m,2)) (Yp+bS(PU*
error('zernfun:NMmultiplesof2', ... ?A(QyaKz
'All N and M must differ by multiples of 2 (including 0).') DXz}YIEC
end 'F>'(XWWQ
XGP6L 0j
if any(m>n) t]7&\ihZi~
error('zernfun:MlessthanN', ... X[f=h=|
'Each M must be less than or equal to its corresponding N.') O@p]KSfk
end :Ad&$eg+
0a-:<zm
if any( r>1 | r<0 ) :_!8
WB
error('zernfun:Rlessthan1','All R must be between 0 and 1.') FQ
g~l4WX
end `PY>Hgb
v) vkn/:
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) yAoe51h?
error('zernfun:RTHvector','R and THETA must be vectors.') A&nU]R8S
end 3p0LN'q]A
]\7]%(
r = r(:); 035rPT7-2-
theta = theta(:); f=)2f=
length_r = length(r); ^ f# FI&
if length_r~=length(theta) |SyMngIY
error('zernfun:RTHlength', ... L!=QR8?@E
'The number of R- and THETA-values must be equal.') 8Y5
end kF/9-[]$g,
0v9rv.Y"
% Check normalization: ^tXJj:wtS
% -------------------- P2bZ65>3y
if nargin==5 && ischar(nflag) Yo[;W
vu
isnorm = strcmpi(nflag,'norm'); =JJL[}a|
if ~isnorm )_U<7"~0l
error('zernfun:normalization','Unrecognized normalization flag.') lsJnI|
end Dk?\)lD`
else Nm|!#(L
isnorm = false; ki85!k=Q2
end o865(<p
Th=eNL]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% F7Zwh5W
% Compute the Zernike Polynomials ]\!?qsT3}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Q[nEsYP
w Xsmn1w9
% Determine the required powers of r: ^MVkZ{gtre
% ----------------------------------- ih7/}
m_abs = abs(m); tg{H9tU;
rpowers = []; j$+nKc$
for j = 1:length(n) y\a1iy
rpowers = [rpowers m_abs(j):2:n(j)]; 3D2E?$dX
end 8 XU1/i7N
rpowers = unique(rpowers); 2 $Tj84'X
'_V2!?+RU+
% Pre-compute the values of r raised to the required powers, Y 'ow
% and compile them in a matrix: ;UxP
Kpl
% ----------------------------- p\<u6v ~J
if rpowers(1)==0 ,lyb!k8
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); X-wf:h?i
rpowern = cat(2,rpowern{:}); C+TI]{t
rpowern = [ones(length_r,1) rpowern]; VY3&
else XHK70: i
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); cJrmm2.0kD
rpowern = cat(2,rpowern{:}); l(02W
end +(h\fm7*-
;72T|e
% Compute the values of the polynomials: dxmE3*b`
% -------------------------------------- 6Udov pl
y = zeros(length_r,length(n)); UXdUO@
for j = 1:length(n) >k'c'7/
s = 0:(n(j)-m_abs(j))/2; #W|'1
OX4
pows = n(j):-2:m_abs(j); .,OVzW
for k = length(s):-1:1 ={z*akn,
p = (1-2*mod(s(k),2))* ...
Z
/9>
prod(2:(n(j)-s(k)))/ ... u,UmrR
prod(2:s(k))/ ... %T~ig[GstX
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... Y*14v~\'
prod(2:((n(j)+m_abs(j))/2-s(k))); f\jLqZY
idx = (pows(k)==rpowers); #E&80#Z5
y(:,j) = y(:,j) + p*rpowern(:,idx); A -b
[>}_
end ~x|F)~:0=
,]d,-)KX8
if isnorm dUQDOo
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 8@tPm$
end Ba!J"b]
end +1D+]*t_?[
% END: Compute the Zernike Polynomials L>3x9
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% F~P%AjAx'
;S>])5<
% Compute the Zernike functions: >Vwc3d
% ------------------------------ jJ5W>Q1mK$
idx_pos = m>0; D/Mi^5H)
idx_neg = m<0; Gy6l<:;
WUh$^5W
z = y; fe9LEM8j
if any(idx_pos) _iir<}
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); eb =D/
end ]VD|xm:kj
if any(idx_neg) ayfFVTy1d
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); yp({>{u7
end /|D*w^>
<x<"n t
% EOF zernfun