Tomography of Highly Scattering Media with the Method of ...€¦ · J.C.Schotland and V.A.Markel ,...
Transcript of Tomography of Highly Scattering Media with the Method of ...€¦ · J.C.Schotland and V.A.Markel ,...
Tomography of Highly Scattering Media with the Method of Rotated
Reference Frames
Manabu MachidaGeorge Panasyuk
John SchotlandVadim Markel
Radiology/BioengineeringUPenn, Philadelphia
Outline
Motivation(The Forward Problem Perspective)
Plane Wave Modes The Weyl Expanson
The Helmholtz Equation
The Diffusion Equation
RTE Not known Not known
2 20( ) 0
constk u
k∇ + ==
( ) 0, const
D uD
αα
−∇ ⋅ ∇ + ==
20
iek
⋅
⋅ =
k r
k k
01 ( ) 2
2 20
2
ik ri Qze i Q e d q
rQ k q
π− ⋅ +=
= −
∫ q ρ
20
/
ek Dα
− ⋅
⋅ = =
k r
k k
01 2
2 20
12
k ri Qze Q e d q
rQ k q
π
−− ⋅ −=
= +
∫ q ρ
ˆ ˆ( ) ( , )
ˆ ˆ ˆ = ( , ) ( , )
, const
t
s
t s
I
A I
μ
μ
μ μ
⋅∇ + =
′ ′
=∫
s r s
s s r s
MOTIVATION (The Inverse Problems Perspective)
6
Given a data function ( , )which is measured formultiple pairs ( , ),find the absorption coefficient ( ) inside the slab
s d
s d
φ
α
ρ ρ
ρ ρ
r
S
D
α(r)
zsρ
dρ
sz dz
Linearized Integral Equation3( , ) ( , ; ) ( )ds d s d rφ δα= Γ∫ρ ρ ρ ρ r r
( ) ( )α α δα0= +r r
0
0
( , ; , ) ( , ; , )( , ) ( , ; , )
(measurable data-function)
s s d d s s d ds d
s s d d
I z z I z zI z z
φ −=
ρ ρ ρ ρρ ρρ ρ
0 0( , ) ( , ; ) ( ; , ) (first Born approximation)
s d s s d dG z G zΓ =ρ ρ ρ r r ρ
8
( )
( ) 2 2
0
2
2
0 2
( , ) ( , ) d d
/ 2 , / 2 ;
Data function: ( , ) ( / 2 , / 2 )
( , ) ( / 2 ; ) ( / 2 ; ) ( ; )d
( ; ) ( , ) d
d( , ; , )2
s s d dis d s d s d
s d
L
s d
i
s s s
e
g z g z z z
z dz z e
qG z z g
φ φ ρ ρ
ψ φ
ψ δα
δα δα ρ
π
⋅ + ⋅
⋅
=
= + = −
= + −
= + −
=
=
∫
∫
∫
∫
q ρ q ρ
ρ q
q q ρ ρ
q q p q q p
q p q p q p
q p q p q p q
q ρ
ρ ρ
( )
( )
2( )
0 2
( ; )
d( , ; , ) ( ; )2
s
d
i
id d d
z e
qG z z g z eπ
⋅ −
⋅ −= ∫
q ρ ρ
q ρ ρ
q
ρ ρ q
Analytical SVD approach: Making use of the translational invariance
6cm
S.D.Konecky, G.Y.Panasyuk, K.Lee, V.Markel, A.G.Yodh and J.C.SchotlandImaging complex structures with diffuse lightOptics Express 16(7), 5048-5060 (2008)
( )
2( )
0 2
In the case of RTE, we need the plane-wave decomposition of theGreens function, of the form
dˆ ˆ ˆ ˆ( , ; , ) ( ; , ; , )2
and the integral kernel
ˆˆ( , ; ) ( / 2 ; , ; , ) ( / 2 ;
i
s
qG g z z e
z g z z g
π′⋅ −′ ′ ′ ′=
Γ = + −
∫
∫
q ρ ρr s r s q s s
q p q p z s q p 2ˆ ˆ, ; , )d
We then get the integral equations of the form
( , ) ( , ; ) ( ; )dd
s
d
z
z
z z s
z z zψ δα= Γ∫
s z
q p q p q
THE METHOD
Rotated Reference FramesThe usual sperical harmonics are defined in the laboratoryreference frame. Then and are the polar angles of the
ˆunit vector in that frame.θ ϕ
s
THE MAIN IDEA:For each value of the Fourier variable ,use spherical harmonics defined in areference frame whose z-axis is aligned withthe direction of .
k
k
We call such frames "rotated".Spherical harmonics defined in the rotaded frame are
ˆˆdenoted by ( ; ).Y s k
Rotation of the Laboratory Frame (x,y,z).
x
y
zz'
y'
x'ϕ
kθ
θ
ˆˆ ˆ( ; ) ( , ,0) ( )l
llm m m lm
m l
Y D Yϕ θ′ ′′=−
= ∑ k ks k s
Wigner D-functionsEuler angles
Spherical functionsin the laboratoryframe
Advantages of the Method ● Numerical problem is reduced to diagonalization
of a set of tridiagonal matrices● Once the eigenvalues and eigenvectors are
computed, the Green’s function, the plane-wave modes and the Weyl expansion for the GF can be obtained in analytical form
● The Weyl expansion can also be obtained in a slab with appropriate boundary conditions
RESULTS: SIMULATIONS
A set of 5 point absorbersin an L=6l* slab
The field of view is 16l*
RTE Diffusion approximation
A bar target in the center of the same slab
RTE Diffusion approximation
RESULTS: SIMULATIONS
Two thin vertical wires in a 1cm thick slab filled with intralipid solution(looks like milk)
RTE Diffusion approximation
1. V.A.Markel, “Modified spherical harmonics method for solving the radiative transport equation,” Waves in Random Media 14(1), L13-L19 (2004).
2. G.Y.Panasyuk, J.C.Schotland, and V.A.Markel, “Radiative transport equation in rotated reference frames,” Journal of Physics A, 39(1), 115-137 (2006).
3. J.C.Schotland and V.A.Markel , “Fourier-Laplace structure of the inverse scattering problem for the radiative transport equation,” Inverse Problems and Imaging 1(1), 181-188 (2007).
Publications:
Available on the web athttp://whale.seas.upenn.edu/vmarkel/papers.html
CONCLUSIONS● The method of rotated reference frames can be used in
optical tomography of mesoscopic samples
● The images are of superior quality compared to those obtained by using the diffusion approximation
● The quality of images can be comparable to that in X-ray tomography because the RTE retains some information about ballistic rays, single-scattered rays, etc.