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 iB3+KR
function z = zernfun(n,m,r,theta,nflag) pd>a6 lI`
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. {qWG^Db
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N f76|
% and angular frequency M, evaluated at positions (R,THETA) on the KHus/ M&0
% unit circle. N is a vector of positive integers (including 0), and h!N&gZ[0
% M is a vector with the same number of elements as N. Each element D^s0EW-E
% k of M must be a positive integer, with possible values M(k) = -N(k) fd"~[z [
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, }
o"_#\6
% and THETA is a vector of angles. R and THETA must have the same ,q@(L
% length. The output Z is a matrix with one column for every (N,M) /9+A97{
% pair, and one row for every (R,THETA) pair. O mh&)|Iql
% !bV(VRbu
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike a H|OA\<
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), tbzvO<~
% with delta(m,0) the Kronecker delta, is chosen so that the integral Pv-V7`{
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, K# i*9sM
% and theta=0 to theta=2*pi) is unity. For the non-normalized l&sO?P[ /
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. A[ECa{v
% 0"<;You
% The Zernike functions are an orthogonal basis on the unit circle. %M)oHX1p
% They are used in disciplines such as astronomy, optics, and W3V{Xk|
% optometry to describe functions on a circular domain. 'oiD#\t4
% g.Caapy
% The following table lists the first 15 Zernike functions. x$5nLS2.
%
)47j8jL
% n m Zernike function Normalization LJNie*
% -------------------------------------------------- gjegzKU
% 0 0 1 1 So\| Ye
% 1 1 r * cos(theta) 2 -m'3L7:
% 1 -1 r * sin(theta) 2 Nzi/3r7m
% 2 -2 r^2 * cos(2*theta) sqrt(6) R#7+
% 2 0 (2*r^2 - 1) sqrt(3) (LT\
IJSM
% 2 2 r^2 * sin(2*theta) sqrt(6) tY$ty0y-e
% 3 -3 r^3 * cos(3*theta) sqrt(8) n#^?X
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) zsMw5C
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) "'}v 0*[
% 3 3 r^3 * sin(3*theta) sqrt(8) b ?2X>QJ
% 4 -4 r^4 * cos(4*theta) sqrt(10) lGnql 1(
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Q 9gFTLQ
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) yrE,,N%I
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ny
% 4 4 r^4 * sin(4*theta) sqrt(10) V:F+HMBk
% -------------------------------------------------- tgvpf/cQ
% S1az3VJI\
% Example 1: o3i,B),K
% L VU)W^
% % Display the Zernike function Z(n=5,m=1) -l40)^ E}
% x = -1:0.01:1; /_:T\`5uO
% [X,Y] = meshgrid(x,x); FU(}=5n
% [theta,r] = cart2pol(X,Y); 4l%?mvA^m
% idx = r<=1; tJh3$K\
% z = nan(size(X)); ;vI*ThzdD
% z(idx) = zernfun(5,1,r(idx),theta(idx)); EBIa%,
% figure *_{l
% pcolor(x,x,z), shading interp !rsa4t@t
% axis square, colorbar w`.T/
% title('Zernike function Z_5^1(r,\theta)') N[a ljC-R
% 47C(\\
% Example 2: *< $c
=
% s}[A4`EWH
% % Display the first 10 Zernike functions 5!SoN}$
% x = -1:0.01:1; GTp?)nh^
% [X,Y] = meshgrid(x,x); qlz9&w
% [theta,r] = cart2pol(X,Y); rF8W(E_=
% idx = r<=1; }rKJeOo^x?
% z = nan(size(X)); <uBhi4
% n = [0 1 1 2 2 2 3 3 3 3]; -40'[a9E
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; wuBlFUSg
% Nplot = [4 10 12 16 18 20 22 24 26 28]; GCP{Z]u
% y = zernfun(n,m,r(idx),theta(idx)); _uO!N(k.
% figure('Units','normalized') z\Pe{J
% for k = 1:10 xs2,t*
% z(idx) = y(:,k); 55>" R{q
% subplot(4,7,Nplot(k)) .Ca"$2
% pcolor(x,x,z), shading interp &J lpA<^s;
% set(gca,'XTick',[],'YTick',[]) ,c,Xd
% axis square `N|U"s;
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) _C< 6349w
% end RjR&D?dc
% IdV,%d{
% See also ZERNPOL, ZERNFUN2. .])>A')r
cX|[WT0[I
% Paul Fricker 11/13/2006 zp7V\W;
&
iA55yT+
$zk^yumdE
% Check and prepare the inputs: ,2 zt.aqB
% ----------------------------- Sk6b`W7$
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) sorSyuGr
error('zernfun:NMvectors','N and M must be vectors.') Q vv\+Jp^
end !G)mjvEe
la
G$v-r
if length(n)~=length(m) F\-B3i%0
error('zernfun:NMlength','N and M must be the same length.') 5u2{n rc
end Vl5SL{+D
j!zA+hF(
n = n(:); EX`"z(L
m = m(:); P &;y]
,)E
if any(mod(n-m,2)) T-L|Q,-{-
error('zernfun:NMmultiplesof2', ... ${A5-
'All N and M must differ by multiples of 2 (including 0).') pP|,7c5
end kZV^F*7
CCbkxHMf|!
if any(m>n) B#HV20\?v
error('zernfun:MlessthanN', ... k1ipvKxp:8
'Each M must be less than or equal to its corresponding N.') ^=eq .(>
end 9
9Ba{qj
cZNi~
if any( r>1 | r<0 ) 0lX)Cl
error('zernfun:Rlessthan1','All R must be between 0 and 1.') pyUNRqp
end I#"t'=9H
j2RRSz&9
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) vS7/ ~:C
error('zernfun:RTHvector','R and THETA must be vectors.') |HrM_h<X
end K^"w]ii=
cU7rq j_
r = r(:); Hze-Ob8
theta = theta(:); Z} c'Bm(
length_r = length(r); El~-M`Gf
if length_r~=length(theta) xj0cgK|!
error('zernfun:RTHlength', ... cqeR<len
'The number of R- and THETA-values must be equal.') k/df(cs
end 4rI:1yGt@
g d z
% Check normalization: ;*y|8od
B
% -------------------- X Y~;)<s_
if nargin==5 && ischar(nflag) %4j&H!y-w;
isnorm = strcmpi(nflag,'norm'); LYp'vZ!
if ~isnorm D`~JbKV5@^
error('zernfun:normalization','Unrecognized normalization flag.') HbNYP/MN3
end #2h+dk$1
else _e6a8
isnorm = false; ?3`q+[:
end sa_R$ /H
Ej'
7h~ =v
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% #MUiL=
% Compute the Zernike Polynomials %moJF1
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% vKU`C?,L
3AK(dC[ri
% Determine the required powers of r: c\M#5+ 1j
% ----------------------------------- ,
Hn7(^t
m_abs = abs(m); D,k(~
rpowers = []; I[ai:
for j = 1:length(n) HeCcF+
rpowers = [rpowers m_abs(j):2:n(j)]; :v`o6x8
end =K:(&6f<t
rpowers = unique(rpowers); IeVLn^?+:
J2r1=5HS
% Pre-compute the values of r raised to the required powers, *9J1$Wa
% and compile them in a matrix: WM/#.
% ----------------------------- $'^&\U~?
if rpowers(1)==0 kGm:VYf%
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); #Rc5c+/(
rpowern = cat(2,rpowern{:}); )%s +?
rpowern = [ones(length_r,1) rpowern]; )!cI|tovs
else |=\91fP68`
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); .8]Y-
rpowern = cat(2,rpowern{:}); X:Q$gO?[4
end Y&s2C%jT
6)ycmu;!$
% Compute the values of the polynomials: U h.Sc:trA
% -------------------------------------- ;+
G9-
y = zeros(length_r,length(n)); s;J\Kc?"|
for j = 1:length(n) va5FxF*%
s = 0:(n(j)-m_abs(j))/2; 4b4QbJ$
pows = n(j):-2:m_abs(j); CN/IH
for k = length(s):-1:1 Vu[:A
p = (1-2*mod(s(k),2))* ... _S"f_W
prod(2:(n(j)-s(k)))/ ... R uLvG+
prod(2:s(k))/ ... |q_
!.
a
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... a}>Dz 1R
prod(2:((n(j)+m_abs(j))/2-s(k))); ]ZTcOf
idx = (pows(k)==rpowers); =Ey`M#t;
y(:,j) = y(:,j) + p*rpowern(:,idx); g]|_
`
end 7UKYmJk.
kM!V.e[g
if isnorm B;!f<"a8
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); )r9b:c\
end y? "@v.
end [Uli>/%JB
% END: Compute the Zernike Polynomials H?uukmZl
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ANMYX18M
Gy!P,a)z
% Compute the Zernike functions: .Pw%DZ'
% ------------------------------ 3sFeP&
idx_pos = m>0; wVqd$nsY"
idx_neg = m<0; Kd3QqVJBz1
Q.k
:\m*h
z = y; )p8I@E
if any(idx_pos) "}b'E#
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 1zjaR4Tf
end .
uR M{Bs
if any(idx_neg) =XT)J6z^"
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); xMI+5b8
end aV>aiR=
m&IsDAn
% EOF zernfun