Time-of-flight Compton Scatter Imaging for Cargo Security · 2017-12-11 · Time-of-ight Compton Scatter Imaging for Cargo Security Marta M. Betcke [email protected] Department of

Time-of-flight Compton Scatter Imagingfor Cargo Security

Marta M. [email protected]

Department of Computer ScienceUniversity College London

joint work with Nicholas Calvert (Christie), Edward J. Morton(Rapiscan) and Robert D. Speller (UCL)

6th December 2017, CambridgeM.M. Betcke ToF CS Imaging

Cargo inspection

120 million shipping containers moved a year, estimated cargovalue of more than 4 trillion US$

at entry points the goods need to be inspected to match thecontainer against the declared manifest

manual inspection impossible

imaging challenge: penetrate through the container and thedensely packed contents

deployed technology: x-ray transmission, Compton backscatter

Rapiscan Eagel M45 / M60 Rapiscan Eagel P60 Rapiscan Eagel R60

M.M. Betcke ToF CS Imaging

X-ray transmission cargo imaging systems

Rapiscan Systems, 2012

high-energy (up to 9 MeV)polyenergetic x-ray source and alinear array of detectors

to form an image the container istranslated w.r.t. the x-raysource/detectors assembly,resulting in a standard twodimensional x-ray image

sensitivity is geared towardshigh-density objects

Detectors

D

e

t

e

c

t

o

r

sSource


Backscatter cargo imaging systems

AS&E, 2012

Source

Detectors

Detectors

Rotating

pencil beam x-ray source is swept over the container andscattered photons recorded

backscatter: +one sided access, -carry low energy resulting ineffective low penetration

contrast mainly from low absorption objects in particular lowatomic number materials (commonly encountered incontraband items)

complementary to the x-ray transmission


Compton scatter

Compton scatter dominant x-rayinteraction for low-to-medium atomicnumber materials at high energies

Incident photon interacts with anelectron (at rest): photon is deflectedand portion of the energy is transferredto the electron

Scattered photon energy

E =E ′

1 + α (1− cos θ),

α = E ′/mec2 = E ′/511keV, E ′ - incident photon energy,

mec2 = 511keV - rest mass energy of an electron.

Llimit on the backscattered energy (smallest at θ = π)

limE ′→∞

E (E ′, θ) =511keV

1− cos θ


Scattering cross-section

Klein-Nishina differential cross-section

d (eσ)

dΩ=

r20

2

(1 + cos2 θ

)( 1

1 + α (1− cos θ)

)2(

1 +α2 (1− cos θ)2

(1 + cos2 θ) [1 + α (1− cos θ)]

),

r0 is the classical radius of the electron (2.8× 10−15 m),α = E ′/511keV.

0 50 100 150 200 250 300 3500.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Scatter Angle

Scatt

er

Cro

ss S

ection,

norm

alis

ed

to r

02(s

tera

dia

n

¢

1ele

ctr

on

¢

1)

1 keV

150 keV

1 MeV

150 kVp

KN assumes that photon interacts with an unbound electron(i.e. binding energy of the electron insignificant compared to theenergy of the photon). This is accurate for high-energy photons.


Radiative transfer

Boltzmann transport equation

1

v

∂I (r, s, t,E )

∂t+ s · ∇I (r, s, t,E ) + µ(r,E ) I (r, s, t,E ) (RTE)

= ρ(r)

∫ π

0

∫ 2π

0

dσ

dΩ(E ′, s′ · s︸︷︷︸

=cos θ

)I(r, s, t,E ′

)d s′︸︷︷︸

=sin θdφdθ

+1

vQ(r, s, t,E ) .

with boundary conditions

I (r, s, t,E ) = 0, (r, s, t,E ) ∈ Γ × (0,Tmax)× (0,Emax) ,

I (r, s, 0,E ) = 0, (r, s, 0,E ) ∈ X × Sd−1 × (0,Emax) ,

with Tmax, Emax <∞, the spatial domain X ⊂ Rd (d ≥ 2) isconvex, bounded, open subset with boundary ∂X , and

Γ = (r, s) ∈ ∂X × Sd−1, s.t.− s · ν (x) > 0,

the set of incoming conditions with ν (x) is the outward normal.M.M. Betcke ToF CS Imaging

Single scatter approximation

We assume an isotropic instantaneous point source at rs and t = 0with source energy distribution γ(E )

Q(r, t,E ) =γ(E )v

4πδ(r − rs)δ(t).

Denote the right hand side term in RTE

I (r, s, t,E ) =

∫4π

dσ

dΩ

(E ′, s′ · s

)I(r, s′, t,E ′

)d s′,

and the angle-averaged radiance

I(r, t,E ′

)=

1

4π

∫4π

I(r, s′, t,E ′

)d s′.


Scatter order expansion

The radiance and angle-averages intensities can be represented asa series in order of scatterings

I(r, s, t,E ′

)=∞∑

N=0

IN(r, s, t,E ′

),

I(r, t,E ′

)=∞∑

N=0

IN(r, t,E ′

),

and the recursive equations for partial intensities are(1

v

∂

∂t+ s · ∇+ µ(r,E )

)IN(r, s, t,E ) = ρ(r) IN−1(r, s, t,E ) , (1)

for N = 1, 2, . . ., and(1

v

∂

∂t+ s · ∇+ µ(r,E )

)I0(r, s, t,E ) =

1

vQ(r, s, t,E ) . (2)


Between scattering events the photon moves along a straight line

r = r′ + v(t − t ′)s,

r′ photon position at the time t ′ of the scattering event,r = ‖r − r′‖ = v(t − t ′) path length of the photon after scatteringin the direction s.Express the variables at the time of scattering event as

t ′ = t − r/v ,

r′ = r − r s

Integrate equations (1), (2) along the new variable r

I0(r, s, t,E ) =1

v

∫ ∞0

e−∫ r0

0 µ(r−r ′ s,E)dr ′Q(r − r0s, s, t − r0/v ,E ) dr0,

(3a)

IN(r, s, t,E ) =

∫ ∞0

e−∫ rN

0 µ(r−r ′ s,E)dr ′ρ(r−rN s)IN−1(r − rN s, s, t − rN/v ,E ) drN .

(3b)


Substituting the source Q into (3a) yields the ballistic peak

I0(r, s, t,E ) =γ(E )

4π

∫ ∞0

e−∫ r0

0 µ(r−r ′s,E)dr ′δ(r − r0s− rs)δ(t − r0/v)dr0

=γ(E )

4πe−

∫ vt0 µ(r−r ′s,E)dr ′δ(r − vt s− rs). (4)

Integrating (4) over the solid angle yields

I0(r, t,E ) =γ(E )

4π‖r − rs‖2e−

∫ vt0 µ(r−r ′sr,s ,E)dr ′δ(‖r − rs‖ − vt),

where sr,s = (r − rs)/‖r − rs‖ is the unit vector pointing from thesource to r.


For i = 1, (3b) describes the single scatter radiance

I1(r, s, t,E ) =

∫ ∞0

e−∫ r1

0 µ(r−r ′s,E)dr ′ρ(r−r1s)I0(r − r1s, s, t − r1/v ,E ) dr1.

(5)Substituting (4) into (5) we obtain

I1(r, s, t,E ) =

∫ ∞0

e−∫ r1

0 µ(r−r ′ s,E)dr ′ρ(r − r1s)

×∫

4π

dσ(E ′, s′ · s)dΩ

I0(r − r1s, s′, t − r1/v ,E

′) d s′dr1

=

∫4π


γ(E ′)

4π

∫ ∞0

ρ(r − r1s)e−

∫ r10 µ(r−r ′ s,E)dr ′

×∫ ∞

0

e−∫ r0

0 µ(r−r1 s−r ′ s′,E ′)dr ′δ(r − r1s− r0s′ − rs)δ(vt − r1 − r0)dr0dr1d s

′

=

∫4π


γ(E ′)

4π

∫ ∞0

ρ(r − r1s)e−

∫ r10 µ(r−r ′ s,E)dr ′

×e−∫ vt−r1

0 µ(r−r1 s−r ′ s′,E ′)dr ′δ(r − r1s− (vt − r1)s′ − rs)dr1d s′


Integrating both sides over the solid angle we obtain the singlescattering intensity

I1(r, t,E ) =1

4π

∫ ∞0

∫ ∞0

ρ(r − r1)dσ(E ′, r0 · r1)

dΩγ(E ′)

× r−20 r−2

1 e−∫ r1

0 µ(r−r ′ r1,E)dr ′e−∫ r0

0 µ(r−r1−r ′ r0,E ′)dr ′

× δ(r − r1 − r0 − rs)δ(vt − r1 − r0)dr0dr1,

with r0 = r0/r0, r1 = r1/r1 unit vectors.Integration over the surface of a prolate ellipsoid with foci in rsand r and principal axis vt:

r0 + r1 = r − rs and r0 + r1 = vt.

rs r

r0 r1

vt


Modelling of µ(E )

Assuming that Compton scatter is the only interaction the scattercoefficient µs(Z ,E ′) is equal to

µs(Z ,E ′) = ρ (Z )

∫ 4π

0

d (eσ)

dΩ

(E ′, θ

)d s′, (6)

= ρ (Z ) eσ(E ′).

From (6) we have

ρ (Z ) =µs(Z ,E ′)

eσ (E ′),

and hence the energy dependence of µs(Z ,E ) reduces to eσ (E )

µs (Z ,E ) =µs (Z ,E ′) eσ (E )

eσ (E ′).

leaving µ(r) := µs(r,E ′) as the only one unknown in the RTE.


ToF Compton Scatter

Defining the time dependent data on the boundary as

F (µ( r)) = yQ (r, t) =

∫ ∞0

∫ 4π

0I (r, s, t,E ) d sdE . (7)

In practice ToFCS, measurements are made at a discrete set ofdetectors, integrating over time bin photons emitted from a sourceat position rs . The data measured by a point detector at rd withangle dependent sensitivity function wd (s) in the kth time of flightbin [tk−1, tk)

ys,d ,k =

∫ tk

tk−1

∫ ∞0

∫4π

wd (s)I (rd , s, t,E )d sdEdt. (8)

The aperture function wd is supported on a half of the unit sphereto restrict to photons hitting the front face of the detector. It canalso be used to model the directional sensitivity of the detector.


Jacobian

I1(rd , s, t,E ) =

∫4π

deσ(E ′, s′ · s)dΩ

γ(E ′)

4π

∫ ∞0

=ρ(rd−r1)︷︸︸︷µ(rd − r1,E

′)

eσ(E ′)

× e−∫ r1

0

=µ(rd−r′ s,E)︷︸︸︷c1µ(rd − r ′s,E ′) dr ′e−

∫ vt−r10 µ(rd−r1s−r ′s′,E ′)dr ′

× δ(rd − r1s− (vt − r1)s′ − rs)dr1d s′.

with

c1 =eσ (E )

eσ (E ′). (9)

Jacobian of ToF CS problem using SS approximation

∂ys,d∂µ

=

∫ ∞0

∫4π

wd (s)∂I1(rd , s, t,E )

∂µd sdE , (10)

using the SS approximation I1(rd , s, t,E ).


Denoting

M0(µ) = e−∫ vt−r1

0 µ(rd−r1s−r ′s′,E)dr ′ , M1(µ) = e−∫ r1

0 c1µ(rd−r ′s,E)dr ′ ,

and their partial derivatives

∂M0(µ)

∂µ= M0(µ)

(−∫ vt−r1

0

1dr ′),

∂M1(µ)

∂µ= M1(µ)

(−c1

∫ r1

0

1dr ′)

we obtain

∂I1(rd , s, t,E )

∂µ=

∫4π

deσ(E ′, s′ · s)dΩ

γ(E ′)

4π

∫ ∞0

[1

eσ(E ′)M0(µ)M1(µ)

+µ(rd − r1,E

′)

eσ(E ′)M0(µ)

∂M1(µ)

∂µ

+µ(rd − r1,E

′)

eσ(E ′)M1(µ)

∂M0(µ)

∂µ

]× δ(rd − r1s− (vt − r1)s′ − rs)dr1d s

′. (11)


Discretization (in-plane scatter)

SourceDetector

time-of-flight bin t

µ sampled on an equidistant cartesian grid

time-of-flight bins of equal size [t − δt, t + δt) correspondingto scattering from an elliptical annulus2a = (t − δt/2) . . . 2a = (t + δt/2)

intersection between the pixel and the annulus approximatedby a polygon and its centroid used as the scattering point r0integral along the broken ray (rs, r0, rd) was approximatedusing Siddon’s algorithm for both parts of the ray, withfractions of the pixel used inside the scattering voxel


Discretization (in-plane scatter)

A0/A1 ∈ Rnbr×m: attenuation of the incident/scattered ray in voxelAs ∈ Rnbr×m: scatter from a voxel to a point detector,nbr #broken rays, m #voxels.

Integration using mid-point rule:

A0,i ,j = lij ,

A1,i ,j = lijeσ (E )

eσ (E0),

A s,k,j =γ(E0)

eσ(E0)

deσ(E0, θk,j)

dΩ

Tk,jΩ(rs, j) sin θk,j|rd − r0|2

lij : intersection of ith broken-ray and jth pixel,Tk,j : width of kth elliptical annulus ell(rs, rd, [tk − δt, tk + δt)), atthe scattering point in the jth pixel, θk,j is the correspondingscattering angle,Ω(rs, j): solid angle presented by the pixel to the source.


Discretized forward operator A(µ)µ

The discretised nonlinear forward operator has the form

F (µ) = A(µ)µ

with µ ∈ Rm scatter coefficients in each pixel.A(µ) ∈ Rn×m is obtained by summing over all pixels in eachelliptical annulus, n: #annuli = #detectors x #sources x #ToFbins: kth row of A

Ak(µ) =∑

r :r∈ell(rs,rd,tk )

e−A0,r,:·µAs,re−A1,r,:·µ

e−A0,r,:·µ: attenuation of the primary beam (before scatter) alongthe broken ray defined by rs , the scatter point r ∈ ell(rs, rd, tk),and rdAs,r scattering probability from rs into rd in the voxel containing re−A1,r,:·µ attenuation of the broken ray after scatter


Discretized Jacobian J

With the same assumptions and discretization as for the forwardproblem (point source, point detector, scattering voxel), kth row ofJ

Jk =∑

r :r∈ell(rs,rd,tk )

wd (s)(e−A0,r ·µAs,re

−A1,r ·µ

− A0,r · 1 · e−A0,r ·µAs,re−A1,r ·µ · µ

− A1,r · 1 · e−A0,r ·µAs,re−A1,r ·µ · µ

).

As before, the subscript r is used integrate over all voxels in theToF elliptical annulus.A0,r · 1 is the integral of 1 along the primary ray,A1,r · 1 is the integral of 1 along the scattered ray.


Geometry

ROI: 112× 96 pixel grid with 37.5 mm x 37.5 mm pixelsToF bin δt = 0.5 ns i.e. 75 mm spacing between ellipses100 detectors placed at the boundary of square ROI, spaced every150 mmsingle monoenergetic (1MeV) source placed in the bottom leftcorner of the ROI.

x (m)

-2 -1 0 1 2

y (

m)

0

0.5

1

1.5

2

2.5

3

3.5

x (m)

-2 -1 0 1 2

y (

m)

0

0.5

1

1.5

2

2.5

3

3.5


Phantom

The values of µs correspond to Lithium (triangle), Plastic (smallellipse), Carbon (large ellipse), Sodium (cross) at 10% densitypacking and at 0.1 of scatter coefficient of Aluminium (containerwalls) to account for thickness (approximately 3.75 mm.) vs pixelsize.

x (m)

-2 -1 0 1 2

y (

m)

0

0.5

1

1.5

2

2.5

3

3.5

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

x (m)

-2 -1 0 1 2

y (

m)

0

0.5

1

1.5

2

2.5

3

3.5

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Figure: µs of the simulated container [m−1] (left) hi-resolution (122 x96): data simulation, (right) low-recolution (56 x 48): reconstruction


Reconstruction

InvertF (µ) = A(µ)µ+ ε,

where ε is noise vector.

ill-posed

nonlinear with special product structure

Solve using variational methods

arg minµ≥0

1

2‖y − A(µ)µ‖2

2 + λTV (µ)

.

non-convex data fitting functional even for Gaussian noise.


Linear problem

Make use of the special product structure, assumptionA(µ)µ ≈ A(µ0)µ for some µ0. Solve linear problem

arg minµ≥0

1

2||BA(µ0)y − BA(µ0)A(µ0)µ||22 + λTV (µ)

,

with normalization / preconditioning

BA = diag

√∑

jA2i ,j

θell(i)/θsrc

−1

(12)

The denominator weights ellipses by their intersection with the grid(for outer ToF bins only fraction of the ellipse is involved): θell(i)angle that ith ellipse presents to the source, θsrc fan angle of thesource.Solve using ADMM using TV+ functional [Beck, Teboulle’12].


Linear reconstruction results

5% Gaussian noise. Solution for µ0 = µtrue and µ0 = mean(µtrue)after 30 ADMM iterations.

x(pixels)

10 20 30 40

y(pixels)

5

10

15

20

25

30

35

40

45

50

550

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

x(pixels)

10 20 30 40

y(pixels)

5

10

15

20

25

30

35

40

45

50

55-1.5

-1

-0.5

0

0.5

1

x(pixels)

10 20 30 40

y(pixels)

5

10

15

20

25

30

35

40

45

50

550

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

x(pixels)

10 20 30 40

y(pixels)

5

10

15

20

25

30

35

40

45

50

55-1.5

-1

-0.5

0

0.5

1


Nonlinear ADMM

Apply Gauss-Newton to the unregularized problem

arg minµ‖F (µ)− y‖2

2.

In each GN step regularise the linearised problem using TV+ andsolve with linear ADMM:

minimise1

2||J(µj)δµj − rj ||22 + λTV+(µj+1) , rj = y − F (µj)

s.t. µj+1 = µj + δµj .

ADMM with x = δµj , z = µj+1, A = −I , B = I , c = µj :

xk+1 := arg minx

1

2

∣∣∣∣∣∣∣∣[ J√ρI

]x −

[rj

−√ρ(µj − zk − uk

) ]∣∣∣∣∣∣∣∣22

,

zk+1 := arg minz

(λTV+(z) + (ρ/2) ||z − (xk+1 + µj − uk)||22

),

uk+1 := uk +(zk+1 − µj − xk+1

).

In practice we precondition the linearisation using initial guess µ0

i.e. solve min 1/2‖BJ(µ0)J(µj)δµj − BJ(µ0)rj‖22 + λTV+(µj+1).


NADMM reconstruction results

Iteration, k

1 1.5 2 2.5 3 3.5 4 4.5 5

1/2

||y-F

(µ

k)|

|22

+

TV

(µ

k)

0

0.25

0.5

0.75

1

1.25

MS

E(µ

k)

0

0.015

0.03

0.045

0.06

0.075

µ0

= µm

, f( µk)

µ0

= µc, f( µ

k)

µ0

= µm

, MSE(µk)

µ0

= µc, MSE(µ

k)

µ0

= µm

, NL ADMM, k = 4

x(pixels)

10 20 30 40

y(p

ixels

)

5

10

15

20

25

30

35

40

45

50

550

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Initialized with µ0 = mean(µtrue). For ρ = 0.53 and λ = 0.003,the minimal MSE was achieved after 4 iterations.


vs linear ADMM with A(µtrue)

x(pixels)

10 20 30 40

y(pixels)

5

10

15

20

25

30

35

40

45

50

550

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

µ0

= µm

, NL ADMM, k = 4

x(pixels)

10 20 30 40

y(p

ixels

)

5

10

15

20

25

30

35

40

45

50

550

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

(left) linear ADMM reconstruction with A(µtrue). (right) NADMMinitialized with µ0 = mean(µtrue). For ρ = 0.53 and λ = 0.003, theminimal MSE was achieved after 4 iterations.



Documents

Time-of-flight Compton Scatter Imaging for Cargo Security · 2017-12-11 · Time-of-ight Compton Scatter Imaging for Cargo Security Marta M. Betcke [email protected] Department of