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.