下面这个函数大家都不会陌生,计算zernike函数值的,并根据此可以还原出图像来, ;): 8yBMk
我输入10阶的n、m,r,theta为38025*1向量,最后得到的z是29525*10阶的矩阵, z[WC7hvU
这个,跟我们用zygo干涉仪直接拟合出的36项zernike系数,有何关系呢? "sFW~Y
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? Oamv9RyDvC
VYL@RL'
_L$)2sl1R
x7vq?fP0n
Lf5%M|o.)
function z = zernfun(n,m,r,theta,nflag) 1Z\(:ab13
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. +n@f'a">
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N x^zdTMNhw
% and angular frequency M, evaluated at positions (R,THETA) on the Bs_S.JP<`
% unit circle. N is a vector of positive integers (including 0), and %GM>u2baw
% M is a vector with the same number of elements as N. Each element n"(7dl?
% k of M must be a positive integer, with possible values M(k) = -N(k) A;odVaH7
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, &J\B\`
% and THETA is a vector of angles. R and THETA must have the same bBA$}bv
% length. The output Z is a matrix with one column for every (N,M) =Nw2;TkB[
% pair, and one row for every (R,THETA) pair. `2>XH:+7F
% 0LS-i% 0
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike $kD7y5
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), f_oq1 W)9
% with delta(m,0) the Kronecker delta, is chosen so that the integral ||R0U@F,
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, @/9>=#4c
% and theta=0 to theta=2*pi) is unity. For the non-normalized U$A/bEhw
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. I|H,)!Z
% D0f*eSXE{
% The Zernike functions are an orthogonal basis on the unit circle. ,oBlJvm
% They are used in disciplines such as astronomy, optics, and OWqrD@
% optometry to describe functions on a circular domain. B,4q>KQA
% JRD8Lz]Q3
% The following table lists the first 15 Zernike functions. z9^c]U U)E
% $+7 ci~gs
% n m Zernike function Normalization D`en%Lf!m
% -------------------------------------------------- f(!E!\&n^
% 0 0 1 1 p Z"o@';!
% 1 1 r * cos(theta) 2 xtOx|FkYcl
% 1 -1 r * sin(theta) 2 BlL|s=dlQV
% 2 -2 r^2 * cos(2*theta) sqrt(6) :=y0'f
V(@
% 2 0 (2*r^2 - 1) sqrt(3) -RGPtD@
% 2 2 r^2 * sin(2*theta) sqrt(6) 'c#IMlv
% 3 -3 r^3 * cos(3*theta) sqrt(8) pG(Fz0b{
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) it~Z|$
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) expxp#S
% 3 3 r^3 * sin(3*theta) sqrt(8) =PV/`I_h
% 4 -4 r^4 * cos(4*theta) sqrt(10) A1Ka(3"
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) *vb ^N0P
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) K|US~Hgv
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) JfbKf~g
% 4 4 r^4 * sin(4*theta) sqrt(10) %Mh Q
% -------------------------------------------------- U{"f.Z:Ydo
% FW_G\W.
% Example 1: MvBD@`&7
% Mxo6fn6-46
% % Display the Zernike function Z(n=5,m=1) 7 %3<~'v[
% x = -1:0.01:1; bQ<b[
% [X,Y] = meshgrid(x,x); l^ARW
E
% [theta,r] = cart2pol(X,Y); ln fm0
% idx = r<=1; s1{[{L3
% z = nan(size(X)); +GYS26
% z(idx) = zernfun(5,1,r(idx),theta(idx)); A])OPqP{
% figure kymn)Ea
% pcolor(x,x,z), shading interp \2j|=S6
% axis square, colorbar %Z7%jma
% title('Zernike function Z_5^1(r,\theta)') `os8;`G
% BY$[ g13
% Example 2: 5Q|sta!
% _PV*lK=
% % Display the first 10 Zernike functions G)8ChnJa!m
% x = -1:0.01:1; +>4^mE" \
% [X,Y] = meshgrid(x,x); D;jK/2
% [theta,r] = cart2pol(X,Y); sXiv,
% idx = r<=1; l0Y?v 4
% z = nan(size(X)); f|#8qiUS
% n = [0 1 1 2 2 2 3 3 3 3]; Rjq a_hxrS
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; ./7v",#*.'
% Nplot = [4 10 12 16 18 20 22 24 26 28]; p-,Iio+
% y = zernfun(n,m,r(idx),theta(idx)); ;T>+,
% figure('Units','normalized') qi&D+~Gv!
% for k = 1:10 ZjS(ad*.2
% z(idx) = y(:,k); srK53vKMHW
% subplot(4,7,Nplot(k)) IM=+3W;ak
% pcolor(x,x,z), shading interp x#mtS-sw2Q
% set(gca,'XTick',[],'YTick',[]) +;dXDZ2
% axis square };r|}v !~_
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) @(>XOj?+
% end &wjB{%
% DT\ym9
% See also ZERNPOL, ZERNFUN2. /&(1JqzlB
&{ ZSE^
)|MJnx9
% Paul Fricker 11/13/2006 DvYwCgLR
{fU?idY)c
ybE[B}pOeZ
jgq{pZ#E
_=EZ `!%
% Check and prepare the inputs: ^)0 9OV+hF
% ----------------------------- 5)`h0TK
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) /c#l9&,
error('zernfun:NMvectors','N and M must be vectors.') .,M;huRg
end Y@%`ZPJ
;&
|qSa'
a+Ab]m8`
if length(n)~=length(m) *ik/p
error('zernfun:NMlength','N and M must be the same length.') ,{8v4b-
end Kam]Mn'
mxp Y&Y
|dk[cX>
n = n(:);
)Gb,^NGr
m = m(:); xA #H0?a]
if any(mod(n-m,2)) M{E{N K
error('zernfun:NMmultiplesof2', ... 2h q>T&8
'All N and M must differ by multiples of 2 (including 0).') k>5 O`Y:
end uPLErO9Es[
mU@xcN
bX2"89{
if any(m>n) Fw"$A0
error('zernfun:MlessthanN', ... 6 P*O&1hv
'Each M must be less than or equal to its corresponding N.') 9i%9
end 'N6 S}w7
S4bBafj[I
z"UPyW1?
if any( r>1 | r<0 ) B+"g2Y
error('zernfun:Rlessthan1','All R must be between 0 and 1.') HnU Et/
end e&1\'Zq?>
AVFjBybu9
!h: Q
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) m@\ZHbq
error('zernfun:RTHvector','R and THETA must be vectors.') ,S!w'0k|n
end Gx'TkU=
l r~gG3
@;Y~frT
r = r(:); o`6|ba
theta = theta(:); cj
g.lzYH
length_r = length(r); Vz"u>BP3~
if length_r~=length(theta) /;oqf4MF
error('zernfun:RTHlength', ... 8\Hr5FqB(
'The number of R- and THETA-values must be equal.') T)SbHp Y
end JE;+T[I
zOB=aG?/
&HDP!SLS
% Check normalization: :2y"3azxk
% -------------------- v}[dnG
if nargin==5 && ischar(nflag) 6+`tn
isnorm = strcmpi(nflag,'norm'); +iA=y=;blH
if ~isnorm z-,VnhLx
error('zernfun:normalization','Unrecognized normalization flag.') L`[z[p{?
end 1%`7.;!i
else GwLFL.Ke
isnorm = false; }V`mp
end ]'h; {;ug
VKW|kU7Cs$
TH?9< C-C
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% !')y&7a~
% Compute the Zernike Polynomials '\~^TFi
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% qf8[!5GM
{{!Y]\2S
L?RF;jf
% Determine the required powers of r: SEd5)0X^
% ----------------------------------- =6T
4>rP
m_abs = abs(m); q_t4OrLr=
rpowers = []; P Sx304
for j = 1:length(n) \Fb| {6+
rpowers = [rpowers m_abs(j):2:n(j)]; R_kQPP
end i8PuC^]
rpowers = unique(rpowers); =Ho"N`Qy
-<f;l_(
%XTA;lrz
% Pre-compute the values of r raised to the required powers, }!s!;BOx
% and compile them in a matrix: OB^Tq~i
% ----------------------------- nH[+n `{o
if rpowers(1)==0 g,kzQ}_
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); )^O-X.1
rpowern = cat(2,rpowern{:}); %fjuG
rpowern = [ones(length_r,1) rpowern]; q/gB<p9
else N "Wqy
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); `-UJ /{
rpowern = cat(2,rpowern{:}); hOk00az
end |$+5@+Zz
xWX*tJ4
|m
G7XL,
% Compute the values of the polynomials: P0GeZ02]
% -------------------------------------- :Vuf6,
y = zeros(length_r,length(n)); Q^_/By@
for j = 1:length(n) KL?) akk
s = 0:(n(j)-m_abs(j))/2; o>lmst%<
pows = n(j):-2:m_abs(j); F%/h*
for k = length(s):-1:1 xN0*8
p = (1-2*mod(s(k),2))* ... l!~
mxUb
prod(2:(n(j)-s(k)))/ ... Bl;KOR
prod(2:s(k))/ ... z2yJ#
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 0$vj!-Mb^j
prod(2:((n(j)+m_abs(j))/2-s(k))); 0pgY1i7
idx = (pows(k)==rpowers); 'mM jjG9
y(:,j) = y(:,j) + p*rpowern(:,idx); (ywo
a
end 6][1<}8
x"4%(xBu
if isnorm 5#JJ?
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); y'M#z_.z
end >cR)?P/o
end ,?-\
x6
% END: Compute the Zernike Polynomials |M~ON=
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 2#5,MP~r
xytr2V ]aV
=y]$0nh
% Compute the Zernike functions: ?.bnIwQe
% ------------------------------ [`_io>*g
idx_pos = m>0; F[`ZqW
idx_neg = m<0; eC`pnE
M8;lLcgu.
F
# YPOH
z = y; ]B4}eBt5)@
if any(idx_pos) oQ2KW..q
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ,^s
end edC4BHE
if any(idx_neg) 4&X*pL2;
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); c6AWn>H
end 5,KWprb
vK'?:}~
+Ov2`O8?
% EOF zernfun 0t.p1