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, :$0yp`k
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, kc(m.k!|f\
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? &gKDw!al
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? xkv%4H>
)F Nn
p=odyf1hK
V>/,&~0
05g %5vHF
function z = zernfun(n,m,r,theta,nflag) BoxtP<C"
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. +abb[
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 7Mk>`4D'c
% and angular frequency M, evaluated at positions (R,THETA) on the &>&6OV]P'
% unit circle. N is a vector of positive integers (including 0), and x-]:g&5T
% M is a vector with the same number of elements as N. Each element VXW*LEk
% k of M must be a positive integer, with possible values M(k) = -N(k) 8i5S
}
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, 6l[v3l"t
% and THETA is a vector of angles. R and THETA must have the same '~
H`Ffd.
% length. The output Z is a matrix with one column for every (N,M) zw+RDo
% pair, and one row for every (R,THETA) pair. XwFTAaZ
% Va?i#<a
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike C+g}+
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), %P D}VF/Y
% with delta(m,0) the Kronecker delta, is chosen so that the integral 4.^T~n G
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, dr c-5{M
% and theta=0 to theta=2*pi) is unity. For the non-normalized c@Br_-
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. H6{Bx2J1*
% K_~kL0=4
% The Zernike functions are an orthogonal basis on the unit circle. OGIv".~s4
% They are used in disciplines such as astronomy, optics, and {@F'BB\
% optometry to describe functions on a circular domain. z~3GgR"1d
% /_})7I52
% The following table lists the first 15 Zernike functions. :9av]Yv&
% %S%IW
% n m Zernike function Normalization <b.p/uA
% -------------------------------------------------- Hqs!L`oW)
% 0 0 1 1 i1XRBC9
% 1 1 r * cos(theta) 2 tH4q*\U
% 1 -1 r * sin(theta) 2 w^Yo)"6
% 2 -2 r^2 * cos(2*theta) sqrt(6) A]TEs)#*7)
% 2 0 (2*r^2 - 1) sqrt(3) wN58uV '
% 2 2 r^2 * sin(2*theta) sqrt(6) _cE_\Ay
% 3 -3 r^3 * cos(3*theta) sqrt(8)
ti (Hx
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) f;Oh"Yt
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) `g3AM%3
% 3 3 r^3 * sin(3*theta) sqrt(8) tcT=a@
% 4 -4 r^4 * cos(4*theta) sqrt(10)
EQ ee5}
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) `FQ]ad Fz
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) a6j& po
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 1O<6=oH
% 4 4 r^4 * sin(4*theta) sqrt(10) T[)!7@4r
% -------------------------------------------------- *asv^aFpS
% Sc/l.]k+
% Example 1: \
a,}1FS
% c8YbBdk'
% % Display the Zernike function Z(n=5,m=1) '~Cn+xf4]
% x = -1:0.01:1; p]EugLEmG
% [X,Y] = meshgrid(x,x); Q"C*j'n
% [theta,r] = cart2pol(X,Y); YI
?P@y
% idx = r<=1; |Z94@uB
% z = nan(size(X)); "gJ.mhHX
% z(idx) = zernfun(5,1,r(idx),theta(idx)); ~abyjM
% figure `_)H aF>/
% pcolor(x,x,z), shading interp Vy
I\Jmr
% axis square, colorbar Te
L&6F$
% title('Zernike function Z_5^1(r,\theta)') g i6s+2
% n"T ^
% Example 2: Bh'fkW3
% 'E9{qPLk(
% % Display the first 10 Zernike functions Q5T3
% x = -1:0.01:1; n)"JMzjQ<
% [X,Y] = meshgrid(x,x); !#_2 ![
% [theta,r] = cart2pol(X,Y); c0'ryS_Z9
% idx = r<=1;
,AweHUEn
% z = nan(size(X)); !IdVg $7
% n = [0 1 1 2 2 2 3 3 3 3]; rAfz?
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; XQ9W
y
% Nplot = [4 10 12 16 18 20 22 24 26 28]; xws{"m,NX~
% y = zernfun(n,m,r(idx),theta(idx)); :\P@c(c{^C
% figure('Units','normalized') ~Ym_ {
% for k = 1:10 -
[h[
% z(idx) = y(:,k); i7-~"g
% subplot(4,7,Nplot(k)) OU/}cu
% pcolor(x,x,z), shading interp $
mE*=
% set(gca,'XTick',[],'YTick',[]) G~8BND[."
% axis square H^*AaA9-
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) UjQz
% end M%`CzCL
u
% Z8ea)_{#
% See also ZERNPOL, ZERNFUN2. P?/JyiO}
`6)Qi*Z
3\@2!:>
% Paul Fricker 11/13/2006 B6
(\1
p"p~Bx
(cOe*>L;
d<7b<f"~
wK-VA$;:
% Check and prepare the inputs: +FqD.= 8
% ----------------------------- 'wk,t^)
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) I?}jf?!oM
error('zernfun:NMvectors','N and M must be vectors.') kZz'&xdv'.
end B4RrUA32
]}!@'+=
G-T^1?
if length(n)~=length(m) &M}X$k I
error('zernfun:NMlength','N and M must be the same length.') +Pb:<WT}%
end W :]2Tp
HK/WO jr
E+O{^C=
n = n(:); 'c7nh{F
m = m(:); aYaEy(m
if any(mod(n-m,2)) [[IMf-]
error('zernfun:NMmultiplesof2', ... uKP4ur@1
'All N and M must differ by multiples of 2 (including 0).') uL/wV~g
end 71R,R,
ce\d35x!
qX-ptsQ
if any(m>n) 4n1g4c-
error('zernfun:MlessthanN', ... d=xjLbsZ
'Each M must be less than or equal to its corresponding N.') 1z8"Gk6
end 4tZ *%!I'
adP :{j
UA8hYWRP
if any( r>1 | r<0 ) Njg$~30
error('zernfun:Rlessthan1','All R must be between 0 and 1.') -{cmi,oy
end 7?=^0?a
gQ{ #C'
T"z<D+pN
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) p3U)J&]c6
error('zernfun:RTHvector','R and THETA must be vectors.') {h+8^
end Pz2 b
MT&aH~YB
=tP9n ;D
r = r(:); T ?[28|
theta = theta(:); rQimQ|+
length_r = length(r); fwz:k]vk
if length_r~=length(theta) ,~d0R4)
error('zernfun:RTHlength', ... 4]U=Y>\Sr
'The number of R- and THETA-values must be equal.') (&e!u{I
end SCcvU4`o
XvaIOt>A
tS2&S 6u
% Check normalization: 3
,>M-F
% -------------------- OZxJDg
if nargin==5 && ischar(nflag) ur}'Y^0iR
isnorm = strcmpi(nflag,'norm'); GGuU(sL*
if ~isnorm vdq=F|&
error('zernfun:normalization','Unrecognized normalization flag.') 8${n}}
end f#!+l1GV
else l/G+Xj4M
isnorm = false; S/`#6
end Qfn:5B]tI
f(|k0$EIu
.#QE*<T)]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% dBXiLrEbs
% Compute the Zernike Polynomials @njNP^'Kx
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% s6|'s<x"j
2PlhnU Q7
AZ3T#f![L@
% Determine the required powers of r: Uqel
UL}
% ----------------------------------- _aFe9+y
m_abs = abs(m); '."_TEIF
rpowers = []; d~u=,@FK
for j = 1:length(n) Nnh\FaI
rpowers = [rpowers m_abs(j):2:n(j)]; [MpWvLP"x
end B r#{
rpowers = unique(rpowers); VP#KoX85
d0 )725Ia
|E1U$,s~u
% Pre-compute the values of r raised to the required powers, xT+_JT65
% and compile them in a matrix: 0&,D&y%
% ----------------------------- Lm4`O%
if rpowers(1)==0 fmuh9Z
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); unFRfec{
rpowern = cat(2,rpowern{:}); ^N/d`IAjv
rpowern = [ones(length_r,1) rpowern]; ,&UKsrs_
else bSsg`
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 6MVu"0#
rpowern = cat(2,rpowern{:}); c* ueI5i
end zQyI4RHG[
v])ew|
=5\*Zh1
% Compute the values of the polynomials: cHvm
% -------------------------------------- @ual+=L
y = zeros(length_r,length(n)); kGV:=h
for j = 1:length(n) ?62Im^1/
s = 0:(n(j)-m_abs(j))/2; !.6n=r8d
pows = n(j):-2:m_abs(j); QJ XP-
for k = length(s):-1:1 j,j|'7J%
p = (1-2*mod(s(k),2))* ... a.V5fl0?I@
prod(2:(n(j)-s(k)))/ ... ,\6Vb*G|E>
prod(2:s(k))/ ... t<UJR*R=L
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... M^Sa{S*?
prod(2:((n(j)+m_abs(j))/2-s(k))); p]mN)
idx = (pows(k)==rpowers); G(7%*@SX
y(:,j) = y(:,j) + p*rpowern(:,idx); lbAhP+B
end Z^|N]Ej
N9!L8BBaK
if isnorm g!XC5*}
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); \U$:/#1Oe
end XkA] 9,@
end kO\ O$J^S
% END: Compute the Zernike Polynomials 4Fft[S(
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Nm"P8/-09
01'>[h#_n
/JS_gr@DK
% Compute the Zernike functions: c& ;@i$X(
% ------------------------------ zr|DC] 3
idx_pos = m>0; Xfk
DMh
idx_neg = m<0; ;eYG\uKC{
4k225~GQ:C
^sf,mM~D
z = y; /
dJz?0
if any(idx_pos) Or? )Nlg6x
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); *6?mZ*GYY
end N
(4H}2
if any(idx_neg) 8._uwA<[
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Cx2#
0$
end -Rpra0o.
C
b=5w>*
AIg4u(j
% EOF zernfun :(A&8<}-6