| Cécilia Tarpau

Modalities of Compton Scattering Tomography: concept, modelling and associated inverse problems

Cécilia Tarpau1,2,3

Supervised by Prof. M. K. Nguyen1, Prof L. Dumas2 and Dr. G. Rollet3

1ETIS, CY Cergy Paris Université, ENSEA, CNRS UMR 8051, France 2LMV, Université Versailles Saint Quentin, CNRS UMR 8100, France 3LPTM, CY Cergy Paris Université, CNRS UMR 8089, France

PhD Defense, December 7th 2021

Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Motivations of this work

Computerized Tomography (CT) [Hounsfield 1 and Cormack 2**, Nobel prize 1979]** :

Imaging technique based on the acquisition of transmitted rays in straight lines

−→ Objective : reconstruct the considered object

\1. Hounsfield, “Computerized transverse axial scanning (tomography). 1. Description of the system”, Br. J. Radiol., vol. 46, pp. 1016-1022, 1973. 2. Cormack, “Representation of a function by its line integrals, with some radiological applications”, PAGE \* romaniii J. App. Phys., vol.35, pp. 2908-2913., 1964.

2/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Motivations of this work

Computerized Tomography (CT) [Hounsfield 1 and Cormack 2**, Nobel prize 1979]** :

Imaging technique based on the acquisition of transmitted rays in straight lines

−→ Objective : reconstruct the considered object

Beer’s law : physical principle behind CT

I0 Z d µ(z)dz ln =

I 0

2/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Motivations of this work

Computerized Tomography (CT) [Hounsfield 1 and Cormack 2**, Nobel prize 1979]** :

Imaging technique based on the acquisition of transmitted rays in straight lines

−→ Objective : reconstruct the considered object

Beer’s law : physical principle behind CT

I Z d

ln 0 = µ(z)dz

I 0

Image formation based on the forward Radon transform (RT) on straight linesZ Z

Rf (ρ,ϕ) = f (r,θ)δ(ρ − r cos(θ − ϕ))drdθ (1)

(r,θ)∈R×[0,π[

2/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Motivations of this work

Computerized Tomography (CT) [Hounsfield 1 and Cormack 2**, Nobel prize 1979]** :

Imaging technique based on the acquisition of transmitted rays in straight lines

−→ Objective : reconstruct the considered object

Beer’s law : physical principle behindZ dCT

ln = µ(z)dz

I 0

Image formation based on the forward Radon transform (RT)

on straight linesZ Z

Rf (ρ,ϕ) = f (r,θ)δ(ρ − r cos(θ − ϕ))drdθ (1)

(r,θ)∈R×[0,π[

Image reconstruction 2 p.v. : Cauchy principal value.

−1 π on itsZinverse dρ ∂

Z ∞

f (r,θ) = dϕ p.v. Rf (ρ,ϕ)

2π2 0 ∞ ρ − r cos(θ − ϕ) ∂ρ

(2)

2/43

Let H be the Hilbert transform. Since

H{u}(t) = p.v. ∞ u(τ) ∂ u −1(|ν| · F (u)(ν))(t),

dτ and H (t) = F (3) π ∞ t − τ ∂ t

Eq. (2) becomes 1 Z π

f (x,y) = dϕ F −1 (|ν| · F (R(ρ,ϕ))(ν))(x cos ϕ + y sin ϕ) (4)

2π 0

Let H be the Hilbert transform. Since

1 ∞ u(τ) ∂ u

H{u}(t) = p.v. dτ and H (t) = F −1(|ν| · F (u)(ν))(t), (3)

π ∞ t − τ ∂ t

Eq. (2) becomes 1 Z π

f (x,y) = dϕ F −1 (|ν| · F (R(ρ,ϕ))(ν))(x cos ϕ + y sin ϕ) (4)

2π 0

Reconstruction algorithm : Filtered back-projection

•

Let H be the Hilbert transf1orm. SinceZ ∞ u(τ) ∂ u

H{u}(t) = p.v. dτ and H (t) = F −1(|ν| · F (u)(ν))(t), (3)

π ∞ t − τ ∂ t

Eq. (2) becomes 1 π

f (x,y) = dϕ F −1 (|ν| · F (R(ρ,ϕ))(ν))(x cos ϕ + y sin ϕ) (4)

2π 0

Reconstruction algorithm : Filtered back-projection

Weight the projections Rf (ρ,ϕ) by the ramp filter in the Fourier domain Filtering

Let H be the HHilber{u}(t ttr)ansf= 1or pm..v. SinceZ u(τ) dτ and H ∂ u (t) = F −1(|ν| · F (u)(ν))(t),

∞

(3) π ∞ t − τ ∂ t

Eq. (2) becomes 1 Z π

f (x,y) = dϕ F −1 (|ν| · F (R(ρ,ϕ))(ν))(x cos ϕ + y sin ϕ) (4)

2π 0

Reconstruction algorithm : Filtered back-projection

Weight the projections Rf (ρ,ϕ) by the ramp filter in the Fourier domain Filtering

For each ϕ, interpolated the filtered data on the considered lines,

Sum the weighted interpolations on all directions ϕ, Back-projection Normalize by 1/2π.

Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

An inversion formula leading to the filtered back-projection algorithm

Let H be the Hilbert transform. Since

1 ∞ u(τ) ∂ u

H{u}(t) = p.v. dτ and H (t) = F −1(|ν| · F (u)(ν))(t), (3)

π ∞ t − τ ∂ t

Eq. (2) becomes 1 Z π

f (x,y) = dϕ F −1 (|ν| · F (R(ρ,ϕ))(ν))(x cos ϕ + y sin ϕ) (4)

2π 0

Reconstruction algorithm : Filtered back-projection

Weight the projections Rf (ρ,ϕ) by the ramp filter in the Fourier domain Filtering

For each ϕ, interpolated the filtered data on the considered lines,

Sum the weighted interpolations on all directions ϕ, Back-projection Normalize by 1/2π.

Original object Data acquisition

PAGE4/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

An inversion formula leading to the filtered back-projection algorithm

Let H be the Hilbert transform. Since

1 ∞ u(τ) ∂ u −1

H{u}(t) = p.v. dτ and H (t) = F (|ν| · F (u)(ν))(t), (3)

π ∞ t − τ ∂ t

Eq. (2) becomes 1 Z π

f (x,y) = dϕ F −1 (|ν| · F (R(ρ,ϕ))(ν))(x cos ϕ + y sin ϕ) (4)

2π 0

Reconstruction algorithm : Filtered back-projection

Weight the projections Rf (ρ,ϕ) by the ramp filter in the Fourier domain Filtering

For each ϕ, interpolated the filtered data on the considered lines,

Sum the weighted interpolations on all directions ϕ, Back-projection Normalize by 1/2π.

Original object Data acquisition Reconstr uction

= PAGE3 + 14/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Limitations of conventional computed tomography

Considers Compton effect as noise

= PAGE4 + 15/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Limitations of conventional computed tomography

Considers Compton effect as noise

Incident radiation Recoiled electron

S• •

M ω

Scattered radiation

E(ω)

•D

4/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Limitations of conventional computed tomography

Considers Compton effect as noise

Incident radiation Recoiled electron

S• •

M ω

Scattered radiation

E(ω)

•D

Compton formula : one to one correspondence between angle and energy

E(ω) = 0

1 + E0 (1 − cos ω) (5)

Considers Compton effect as noise

Incident radiation Recoiled electron

E0 • Compton effect non-negligible : •

S M ω in the energy range 0.1-5 MeV

70% of emitted photon are scattered

Scattered radiation

E(ω)

•D

Compton formula : one to one correspondence between angle and energy

1 + E0 (1 − cos ω) (5) E(ω) =

Considers Compton effect as noise

Incident radiation Recoiled electron

E0 • Compton effect non-negligible : •

S M ω in the energy range 0.1-5 MeV

70% of emitted photon are scattered

Scattered radiation

Scattered and detected photons :

E(ω)

cause blur, loss of contrast and •

D occur false detections

Compton formula : one to one correspondence between angle and energy

E(ω) = 0

1 + E0 (1 − cos ω) (5)

mc2

4/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Limitations of conventional computed tomography

Considers Compton effect as noise

Incident radiation Recoiled electron

E0 • Compton effect non-negligible : •

S M ω in the energy range 0.1-5 MeV

70% of emitted photon are scattered

Scattered radiation

Scattered and detected photons :

E(ω)

cause blur, loss of contrast and •

D occur false detections

Compton formula : one to one correspondence between angle and energy

1 + E0 (1 − cos ω) (5) E(ω) =

mc2

Idea : use information carried by Compton scattered radiation to reconstruct the object

=⇒ Starting point of the development of systems of Compton Scattering Tomograph**y**** (CST)**

= PAGE3 + 25/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Data acquisition with CST modalities

Idea of CST : exploit wisely scattered radiation to reconstruct electronic density of the object

Computed Tomography (CT) Compton Scattering Tomography (CST)

CT data : • CST data

Line integral of attenuation function Circular arc integral of electronic density

= PAGE4 + 26/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Data acquisition with CST modalities

Idea of CST : exploit wisely scattered radiation to reconstruct electronic density of the object

Computed Tomography (CT) Compton Scattering Tomography (CST)

CT data : • CST data

Line integral of attenuation function Circular arc integral of electronic density

**Image acquisition with a CST modality leads to a Circular Arc Radon Transform (CART) **Image reconstruction requires the inversion of the corresponding generalized CART

5/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Data acquisition with CST modalities

6/43

Norton’s CST modality (1994) 1

\1. S. J. Norton, “Compton Scattering Tomography”, J. Appl. Phys., 76(4), pp. 2007-2015, 1994.

Norton’s CST modality (1994) 1 Nguyen and Truong’s CST modality (2010) 2 3

S. J. Norton, “Compton Scattering Tomography”, J. Appl. Phys., 76(4), pp. 2007-2015, 1994.
M. K. Nguyen and T. T. Truong, “Inversion of a new circular-arc Radon transform for Compton Scattered Tomography”,
___, “Recent Developments on Compton Scatter Tomography […]”, Numerical Simulation - From Theory to Industry

Inv. Prob., 26, 065005, 2010. , 978-953-51-0749-1, 2012.

Norton’s CST modality (1994) 1 Nguyen and Truong’s CST modality (2010) 2 3

Truong and Nguyen’s CST modality (2011) 3 4

S. J. Norton, “Compton Scattering Tomography”, J. Appl. Phys., 76(4), pp. 2007-2015, 1994.
M. K. Nguyen and T. T. Truong, “Inversion of a new circular-arc Radon transform for Compton Scattered Tomography”, Inv. Prob., 26, 065005, 2010.
___, “Recent Developments on Compton Scatter Tomography […]”, Numerical Simulation - From Theory to Industry, 978-953-51-0749-1, 2012.
T. T. Truong and M. K. Nguyen, “Radon transforms on generalized Cormack’s curves […]”, Inv. Prob., 27, 125001, 2011.

Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Contributions of my thesis

Norton’s CST modality (1994) 1 Nguyen and Truong’s CST modality (2010) 2 3

Truong and Nguyen’s CST modality (2011) 3 4 Webber and Miller’s CST modality (2020) 5

S. J. Norton, “Compton Scattering Tomography”, J. Appl. Phys., 76(4), pp. 2007-2015, 1994.
M. K. Nguyen and T. T. Truong, “Inversion of a new circular-arc Radon transform for Compton Scattered Tomography”, Inv. Prob., 26, 065005, 2010.
___, “Recent Developments on Compton Scatter Tomography […]”, Numerical Simulation - From Theory to Industry, 978-953-51-0749-1, 2012.
T. T. Truong and M. K. Nguyen, “Radon transforms on generalized Cormack’s curves […]”, Inv. Prob., 27, 125001, 2011.
J. Webber and E. L. Miller, “Compton scattering tomography in translationnal geometries”, Inverse problems, 36**, 0250007, (#_page23_x27.35_y-1.00)2020. (#_page31_x27.35_y-1.00) (#_page0_x27.35_y-1.00) (#_page119_x27.35_y-1.00)

Contributions of my thesis around three modalities :

Fixed source rotating detector Circular CST modality

CST system

= PAGE3 + 58/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Contributions of my thesis

Contributions of my thesis around three modalities :

Fixed source rotating detector Circular CST modality

Provide a forward mo- CST system del for data acquisition

(Corresponding CART)

8/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Contributions of my thesis

Contributions of my thesis around three modalities :

Fixed source rotating detector Circular CST modality

Provide a forward mo- Derive mathematical CST system del for data acquisition strategies to invert the

(Corresponding CART) CART

8/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Contributions of my thesis

Contributions of my thesis around three modalities :

Fixed source rotating detector Circular CST modality

Provide a forward mo- Carry out simulations Derive mathematical

CST system del for data acquisition to show the efficiency

strategies to invert the

(Corresponding CART) of the proposed algo-

CART

rithms

Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Contributions of my thesis

Contributions of my thesis around three modalities :

8/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Contributions of my thesis

Fixed source rotating detector Circular CST modality

Webber and Miller’s modality

8/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Contributions of my thesis

Carry out simulations delProvidefor dataa forwacquisitionard mo- to show the efficiency

Derive mathematical

CST system (Corresponding CART) of the proposed algo-

strategies to invert the

CART

rithms

8/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Contributions of my thesis

Contributions of my thesis around three modalities :

8/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Contributions of my thesis

Fixed source rotating detector Circular CST modality

Webber and Miller’s modality

8/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Contributions of my thesis

Provide a forward mo- Carry out simulations Derive mathematical

CST system del for data acquisition to show the efficiency

strategies to invert the

(Corresponding CART) of the proposed algo-

CART

rithms

Working assumptions (commonly used in the literature) :

We exploit first order scattering. Attenuation is neglected.
Ideal source and detectors, assumed to be point-like, with perfect energy resolution. (#_page30_x27.35_y-1.00)

8/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Outline of the presentation

The proposition of the Circular Compton Scattering Tomography

Presentation of CCST and its advantages

Measurement model for the proposed CST system

Three approaches for inverting the corresponding Radon transform

The proposition of a fixed source rotating detector CST

Presentation of the CST system

Measurement model and study of the corresponding Radon transform An analytic inversion formula for the corresponding Radon transform

The proposition of an alternative inversion formula for image reconstruction with Webber’s modality

Presentation of the CST system

Measurement model and study of the corresponding Radon transform An analytic inversion formula with efficient implementation

Concluding remarks and perspectives

9/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

10/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Presentation of CCST and its advantages

Circular Compton Scattering Tomography (CCST)

Setup of the proposed system :

Fixed source S
ND fixed detectors Dk,k ∈ {1,ND } placed on a ring passing through the source

Advantages : System compact and completely fixed but able to scan small objects

First system to combine these two requirements of biomedical

imaging

=⇒ Such a system allows reducing acquisition time and time to ex- posure.

11/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

One system, two possible uses :

A double scanning configuration for both small

and large objects

Internal scanning for small objects

External scanning for large objects

12/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

One system, two possible uses :

A double scanning configuration for both small A bi-imaging system combining fan-beam CT

and large objects and CCST

CT mode

Internal scanning for small objects Cross section of attenuation map (E = E0)

CCST mode

External scanning for large objects Cross section of electron density map (E < E0)

= PAGE3 + 1013/43

Measurement model for the proposed CST system

Setup of the fixed system

Source S placed at the origin of the coordinates system

ND detectors Dk,k ∈ {1,ND } equally placed

on the ring of diameter P of polar coordinates

(rD ,θD ) = P sin θD ,π 1 + (6)

k k k N + 1

Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Measurement model for the proposed CST system

Setup of the fixed system

Source S placed at the origin of the coordinates system

ND detectors Dk,k ∈ {1,ND } equally placed

on the ring of diameter P of polar coordinates

(rD ,θD ) = P sin θD ,π 1 + (6)

k k k N + 1

Parameterisation of the scanning arcs

Ai(ρi,ϕi) : ri = ρi cos(θi − ϕi), i ∈ {1, 2} (7)

where

ρ1(ω,θDk 2(−ω,θDk ) = P sin θDk / sin(ω),

) = ρ

ϕ1(ω,θDk 2(−ω,θDk ) = θDk + ω − π/2.

) = ϕ

and θ1 ∈ [θD ,θD + ω],θ2 ∈ [θD − ω,θD ]. k k k k

PAGE14/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Measurement model for the proposed CST system

Setup of the fixed system

Source S placed at the origin of the coordinates system

ND detectors Dk,k ∈ {1,ND } equally placed

on the ring of diameter P of polar coordinates

(rD ,θD ) = P sin θD ,π 1 + (6)

k k k ND + 1

Parameterisation of the scanning arcs

Ai(ρi,ϕi) : ri = ρi cos(θi − ϕi), i ∈ {1, 2} (7)

where

ϕρ1((ωω,,θθDk ) = ϕ2(−ω,θDk ))== Pθ sin+θDωk−/ sin(π/2ω.),

) = ρ2(−ω,θD

1 Dk k Dk

and θ1 ∈ [θD ,θD + ω],θ2 ∈ [θD − ω,θD ].

k k k k

Corresponding Radon transform : Z

RAf (ω,θD ) = f (M ) dl(M ). (8)

M ∈(A1∪A2)(ω,θDk )

Three approaches for inverting the corresponding Radon transform

Corresponding Radon transform :

RAf (ω,θD ) = f (M ) dl(M ).

k M ∈(A ∪A2)(ω,θ )

1 Dk

Approaches 1 and 2 : Suppose collimated detectors

RAf (ω,θD ) = f (M ) dl(M ).

M ∈A(ω,θDk )

= PAGE4 + 1115/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Three approaches for inverting the corresponding Radon transform

Corresponding Radon transform :

RAf (ω,θD ) = f (M ) dl(M ).

k M ∈(A1∪A2)(ω,θDk )

Approach 3 : Supposes uncollimated detectors Approaches 1 and 2 : Suppose collimated detectors

Z f (M ) dl(M ). RAf (ω,θDk ) = f (M ) dl(M ). RA Dk ) = M ∈(A1∪A2)(ω,θD )

f (ω,θ

M ∈A(ω,θD ) k

= PAGE3 + 1215/43

Approach n°1 : Inversion of RA considering the family of circles supporting the family of circle arcs

After rearranging data according to (ρ,ϕ), we can consider, as scanning manifold, the circle supporting the considered circles arcs, assuming that the exterior part gives no contribution in data acquistion.

This leads to the following Radon transform :

Z ∞ Z 2π

RAf (ρ,ϕ) = RCir f (ρ,ϕ) = ρf (r,θ) δ(r − ρ cos(θ − ϕ)) drdθ. (9)

0 π

Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Approach n°1 : Inversion of RA considering the family of circles supporting the family of circle arcs

This leads to the following Radon transform :

Z ∞ Z 2π

RAf (ρ,ϕ) = RCir f (ρ,ϕ) = ρf (r,θ) δ(r − ρ cos(θ − ϕ)) drdθ. (9)

0 π

This family of circles corresponds to a special case of Cormack’s β-curves 1. The inversion of the corresponding

RT is given in 2 :

Z 2π Z ∞

f (r,θ) = 1 dϕ p.v. dρ ∂RCf (ρ,ϕ) ρ . (10)

2π2r 0 0 ∂ρ r − ρ cos(θ − ϕ)

A. M. Cormack, “The Radon transform on a family of curves in the plane”, In Proc. Amer. Math. Soc., vol. 83, no. 2, pp. 325-330, 1981.
A. M. Cormack, “Radon’s problem - old and new”, In Proc. SIAM-AMS, vol. 14, pp. 33-39, 1984. (#_page39_x27.35_y-1.00) (#_page37_x27.35_y-1.00) (#_page30_x27.35_y-1.00) (#_page0_x27.35_y-1.00) (#_page119_x27.35_y-1.00)

The proposition of a suitable formulation for numerical simulations :

Since 1 Z ∞ u(τ)

H{u}(t) = p.v. dτ and H{u} (t) = F −1(−isign(ν) · F (u)(ν))(t), (11)

π ∞ t − τ

Eq. (10) becomesZ f (ρ,ϕ) x2 + y2

f (x,y) = 2π dϕ 1 · F −1 −i ·sign(ν) · F ∂RCir ·ρ (ν)

2π 0 x cos ϕ + y sin ϕ ∂ρ cos ϕ + y sin ϕ (12)

A filtered back-projection type reconstruction algorithm :

= PAGE3 + 1316/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

TheSincepropositionH{u}of(t)a =suitab1 p.v.le Zform∞ uulation(τ) dτfor andn Humerical{u} (t) =sim F −ulations1(−isign(**:**ν) · F (u)(ν))(t),

(11) π ∞ t − τ

Eq. (10) becomesZ 2π 1 f (ρ,ϕ) x2 + y2

f (x,y) = 1 dϕ · F −1 −i ·sign(ν) · F ∂RCir ·ρ (ν)

2π 0 x cos ϕ + y sin ϕ ∂ρ cos ϕ + y sin ϕ (12)

A filtered back-projection type reconstruction algorithm :

Discrete derivation of the projections and multiply by ρ,

Filtering

Filter the result by −i ·sign(ν) in the Fourier domain,

PAGE16/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

The proposition of a suitable formulation for numerical simulations :

Since Z ∞

H{u}(t) = 1 p.v. u(τ) dτ and H{u} (t) = F −1(−isign(ν) · F (u)(ν))(t), (11)

π ∞ t − τ

Eq. (10) becomes1 Z 2π

f (x,y) = dϕ 1 ·F −1 −i ·sign(ν) · F ∂RCir f (ρ,ϕ) ·ρ (ν) x2 + y2

2π 0 x cos ϕ + y sin ϕ ∂ρ cos ϕ + y sin ϕ (12)

A filtered back-projection type reconstruction algorithm :

Discrete derivation of the projections and multiply by ρ,

Filtering

Filter the result by −i ·sign(ν) in the Fourier domain,

For each ϕ, interpolate the data on the considered circles,

Weight the result by 1 ,

(x cos ϕ+y sin ϕ) Back-projection

Sum the weighted interpolations, Normalize by 1/2π.

= PAGE3 + 1417/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Proposed simulation results with approach n°1

Parameter choices :

Object of size 512 × 512 pixels placed inside a ring of diameter P = 1024.
Number of detectors : ND = 3217 −→ one detector per arc length
Number of scanning circles per detector : Nϕ = 3000

Original object

Proposed simulation results with approach n°1

Parameter choices :

Object of size 512 × 512 pixels placed inside a ring of diameter P = 1024.
Number of detectors : ND = 3217 −→ one detector per arc length
Number of scanning circles per detector : Nϕ = 3000

Data acquisition Original object R f (θ ,ϕ)

A Dk

Proposed simulation results with approach n°1

Parameter choices :

Object of size 512 × 512 pixels placed inside a ring of diameter P = 1024.
Number of detectors : ND = 3217 −→ one detector per arc length
Number of scanning circles per detector : Nϕ = 3000

17/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Data acquisition Original object RAf (θD ,ϕ)

Rearranged data RAf (ρ,ϕ)

17/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Proposed simulation results with approach n°1

Parameter choices :

Object of size 512 × 512 pixels placed inside a ring of diameter P = 1024.
Number of detectors : ND = 3217 −→ one detector per arc length
Number of scanning circles per detector : Nϕ = 3000

17/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Data acquisition Original object R f (θ ,ϕ)

A Dk

Rearranged data

RAf (ρ,ϕ) Reconstruction

17/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Proposed simulation results with approach n°1

Parameter choices :

Object of size 512 × 512 pixels placed inside a ring of diameter P = 1024.
Number of detectors : ND = 3217 −→ one detector per arc length
Number of scanning circles per detector : Nϕ = 3000

= PAGE2 + 1517/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Data acquisition Original object R f (θ ,ϕ)

A Dk

Rearranged data

RAf (ρ,ϕ) Reconstruction

= PAGE3 + 1518/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Approach n°2 : Inversion of RA via an intermediate RT

Proposed method : Proceed via geometric

inversion (GI) with the change of variables RT on circle arcs RA RT on half-lines RH r′ = q2/r (q is the inversion parameter). GI

RAf (ρ,ϕ) = RHfapp(q2/ρ,ϕ) where fapp(r,θ) = q2 f q2 ,θ

r2 r

= PAGE4 + 1519/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Approach n°2 : Inversion of RA via an intermediate RT

Proposed method : Proceed via geometric

inversion (GI) with the change of variables RT on circle arcs RA RT on half-lines RH r′ = q2/r (q is the inversion parameter). GI

RAf (ρ,ϕ) = RHfapp(q2/ρ,ϕ) where fapp(r,θ) = q2 f q2 ,θ

r2 r

PAGE18/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Approach n°2 : Inversion of RA via an intermediate RT

Proposed method : Proceed via geometric

inversion (GI) with the change of variables RT on circle arcs RA RT on half-lines RH r′ = q2/r (q is the inversion parameter). GI

RAf (ρ,ϕ) = RHfapp(q2/ρ,ϕ) where fapp(r,θ) = q2 f q2 ,θ

r2 r

Analytic inversion RH is established in this work, generalising a previous work of Nguyen 1.

−→ Leads also to a back-projection type reconstruction algorithm

\1. T. T. Truong, and M. K. Nguyen. “New properties of the V-line Radon transform and their imaging applications.” Journal of Physics A : Mathematical

and Theoretical 4840, 2015.

PAGE19/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Bonus : possibility of an external scanning

19/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Proposed general algorithm for image formation and reconstruction

Original **Projection on **object circular arcs

f (x,y) - R f (ρ,ϕ)

Proposed general algorithm for image formation and reconstruction Original Projection on Reconstruction

object circular arcs

Projection on half-lines

f (x,y) - R f (ρ,ϕ) - R f (ρ,ϕ)

A H app

Proposed general algorithm for image formation and reconstruction Original Projection on Reconstruction

object circular arcs

Projection on Apparent half-lines object

f (x,y) - R f (ρ,ϕ) - R f (ρ,ϕ) - f (x′,y′)

A H app app

GI R−1

Back-projection type reconstruction algorithm

20/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Proposed general algorithm for image formation and reconstruction

Original Projection on Reconstruction Reconstructed

object circular arcs object

Projection on Apparent half-lines object

f (x,y) - R f (ρ,ϕ) - R fapp(ρ,ϕ) - f (x′,y′) - f (x,y)

A H app

GI RH−1 GI−1

Back-projection type reconstruction algorithm

= PAGE3 + 1821/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Proposed simulation results with approach n°2 - Internal scanning

Same parameter choices :

Object of size 512 × 512 pixels placed inside a ring of diameter P = 1024. ND = 3217 and Nϕ = 3000.

Original object

= PAGE4 + 1822/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Proposed simulation results with approach n°2 - Internal scanning

Same parameter choices :

Object of size 512 × 512 pixels placed inside a ring of diameter P = 1024. ND = 3217 and Nϕ = 3000.

Original object Data acquisition on circle arcs Rearranged data

21/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Proposed simulation results with approach n°2 - Internal scanning

Same parameter choices :

Object of size 512 × 512 pixels placed inside a ring of diameter P = 1024. ND = 3217 and Nϕ = 3000.

Original object Data acquisition on circle arcs Rearranged data

Data acquisition on half-lines

21/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Proposed simulation results with approach n°2 - Internal scanning

Same parameter choices :

Object of size 512 × 512 pixels placed inside a ring of diameter P = 1024. ND = 3217 and Nϕ = 3000.

Original object Data acquisition on circle arcs Rearranged data

Data acquisition on half-lines Reconstruction

21/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Proposed simulation results with approach n°2 - Internal scanning

Same parameter choices :

Object of size 512 × 512 pixels placed inside a ring of diameter P = 1024. ND = 3217 and Nϕ = 3000.

Original object Data acquisition on circle arcs Rearranged data

Data acquisition on half-lines Reconstruction Comp. with approach 1

= PAGE3 + 1922/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Proposed simulation results with approach n°2 - External scanning

Same parameter choices except the size of the object :

Object of size 1200 × 360 pixels placed inside a ring of diameter P = 1024. ND = 3217 and Nω = Nϕ = 3000.

Original objet

= PAGE5 + 1924/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Proposed simulation results with approach n°2 - External scanning

Same parameter choices except the size of the object :

Object of size 1200 × 360 pixels placed inside a ring of diameter P = 1024. ND = 3217 and Nω = Nϕ = 3000.

Data acquisition on circle Rearranged data Data acquisition on

arcs half-lines

Original objet

PAGE22/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Proposed simulation results with approach n°2 - External scanning

Same parameter choices except the size of the object :

Object of size 1200 × 360 pixels placed inside a ring of diameter P = 1024. ND = 3217 and Nω = Nϕ = 3000.

Data acquisition on circle Rearranged data Data acquisition on

arcs half-lines

Original objet Reconstruction

Approach n°3 : Study of the Radon transform corresponding to the case of CCST without collimators

Corresponding RT for CCST with uncollimated detectors :

RAf (ω,θD ) = f (M ) dl(M ).

The inversion of this RT is still an open problem. k )

M ∈(A1∪A2)(ω,θD

= PAGE3 + 2023/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Approach n°3 : Study of the Radon transform corresponding to the case of CCST without collimators

Corresponding RT for CCST with uncollimated detectors :

RAf (ω,θD ) = f (M ) dl(M ).

M ∈(A1∪A2)(ω,θD )

The inversion of this RT is still an open problem. k

Equivalent family of V-lines by GI : V-lines with ver- tices on a line and axes passing through the origin

PAGE24/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Approach n°3 : Study of the Radon transform corresponding to the case of CCST without collimators

Corresponding RT for CCST with uncollimated detectors :

RAf (ω,θD ) = f (M ) dl(M ).

The inversion of this RT is still an open problem. M ∈(A1∪A2)(ω,θ k )

= PAGE2 + 2123/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Equivalent family of V-lines by GI : V-lines with ver- tices on a line and axes passing through the origin

Proposed reconstruction results using Tikhonov regularization

Shepp-Logan phantom of size 128 × 128 placed inside a ring of diameter P = 256 with ND = 805 detectors.

Data acquisition Image reconstruction

= PAGE3 + 2124/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Summary of the contributions of this part

Proposition of the Circular Compton Scattering Tomography

Fixed and compact CST modality able to scan both small and large objects

=⇒ Enlarges the applications for the same CST setup from biomedical imaging to non-destructive testing.

Can be combined with Fan-beam CT to have a bi-imaging system

=⇒ Such a bi-imaging system is able to provide two physical properties of the object to scan : attenuation and electronic density.

= PAGE4 + 2125/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Summary of the contributions of this part

Proposition of the Circular Compton Scattering Tomography

Fixed and compact CST modality able to scan both small and large objects

=⇒ Enlarges the applications for the same CST setup from biomedical imaging to non-destructive testing.

Can be combined with Fan-beam CT to have a bi-imaging system

=⇒ Such a bi-imaging system is able to provide two physical properties of the object to scan : attenuation and electronic density.

Study of CCST with and without collimation

With collimation :

Proposition of two equivalent methods for image reconstruction from analytic inversion formulas =⇒ Lead to two filtered back-projection type algorithms never introduced before

Without collimation :

In absence of inversion formula, we proposed reconstruction results using Tikhonov regularisation.

PAGE24/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Summary of the contributions of this part

Proposition of the Circular Compton Scattering Tomography

Fixed and compact CST modality able to scan both small and large objects

=⇒ Enlarges the applications for the same CST setup from biomedical imaging to non-destructive testing.

Can be combined with Fan-beam CT to have a bi-imaging system

=⇒ Such a bi-imaging system is able to provide two physical properties of the object to scan : attenuation and electronic density.

Study of CCST with and without collimation

With collimation :

Proposition of two equivalent methods for image reconstruction from analytic inversion formulas =⇒ Lead to two filtered back-projection type algorithms never introduced before

Without collimation :

In absence of inversion formula, we proposed reconstruction results using Tikhonov regularisation.

Extension towards a class of cones and study of the corresponding Radon transform (see Chapter 6)

Proof of the invertibility of this Radon transform in two and three dimensions
Proposition of analytic inversion formulas

=⇒ inversion formulas not usable in the context of a CST imaging system

PAGE25/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

25/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Proposition of a second CST modality

Fixed source rotating detector CST

Setup of the proposed system

Fixed source S placed at the origin
One detector D moving on a circle of radius R around the source localized by φ : D(φ) = R(cos φ, sin φ)
Object placed outside of the circle =⇒ Large object scanning

PAGE27/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Proposition of a second CST modality

Fixed source rotating detector CST

Setup of the proposed system

Fixed source S placed at the origin
One detector D moving on a circle of radius R around the source localized by φ : D(φ) = R(cos φ, sin φ)
Object placed outside of the circle =⇒ Large object scanning Advantages :

Uncollimated detector −→ data acquisition on double circle arcs No synchronisation needed between the source and the detector Simple and compact system

PAGE26/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Proposition of a second CST modality

Fixed source rotating detector CST

Setup of the proposed system

Fixed source S placed at the origin
One detector D moving on a circle of radius R around the source localized by φ : D(φ) = R(cos φ, sin φ)
Object placed outside of the circle =⇒ Large object scanning

Advantages :

Uncollimated detector −→ data acquisition on double circle arcs No synchronisation needed between the source and the detector Simple and compact system

Parameterisation of the scanning arcs Polar coordinates of the scan- ning arcs AC and AC according to ρ the diameter and φ the angular

position of the1detector :2

AC (ρ,φ) : r = ρ cos θ − φ − (−1)i cos−1 R , (13)

i ρ

θ ∈ [φ,φ + 2ω − π] for A , θ ∈ [φ − 2ω + pi,φ] for A

C1 C2

and ρ = R/ sin ω.

−→ family of double circle arcs invariant by rotation

= PAGE3 + 2427/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Measurement model and study of the corresponding Radon transform

Corresponding Radon transform :

RA f (ρ,φ) = f (r,θ) ds.

(AC1 ∪AC2 )(ρ,φ)

−→ A similar procedure as Cormack work 1 2 can be done to derive the inverse transform of RA . C

\1. A. M. Cormack, “The Radon transform on a family of curves in the plane”, In Proc. Amer. Math. Soc., vol. 83, no. 2, pp. 325-330, 1981.

\2. A. M. Cormack, “Radon’s problem - old and new”, In Proc. SIAM-AMS, vol. 14, pp. 33-39, 1984.

= PAGE5 + 2429/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Measurement model and study of the corresponding Radon transform

Corresponding Radon transform :

RA f (ρ,φ) = f (r,θ) ds.

(AC1 ∪AC2 )(ρ,φ)

−→ A similar procedure as Cormack work 1 2 can be done to derive the inverse transform of RA . Proposed stable inverse formula of RAC f (RAC f )n(ρ) C

Gn 2 cos n arccos R , Denoting G(ρ,ϕ) the projections of circular harmonic components (ρ) =

f (r,θ) can be completely and directly1 Zreco dφveredwith Zthe ∞follo ∂wingG(ρ,equationϕ) : ρ

2π

f (r,θ) = p.v. dρ . (14)

2π 0 2π R ∂ρ r − ρ cos(θ − φ)

where p.v. denotes the Cauchy principal value.

−→ leads to a filtered back-projection type algorithm

\1. A. M. Cormack, “The Radon transform on a family of curves in the plane”, In Proc. Amer. Math. Soc., vol. 83, no. 2, pp. 325-330, 1981.

\2. A. M. Cormack, “Radon’s problem - old and new”, In Proc. SIAM-AMS, vol. 14, pp. 33-39, 1984.

= PAGE3 + 2528/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Proposed simulation results

Parameter choices :

Object of size 512 × 512 pixels.
The detector is moving on a circle of radius R = 256 with a constant step of arc length between two adjacent positions, i.e. Nφ = 1609.
Number of scanning circles per detector position : Nρ = 815 and ρmax = 7000.

Original objet

= PAGE5 + 2530/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Proposed simulation results

Parameter choices :

Object of size 512 × 512 pixels.
The detector is moving on a circle of radius R = 256 with a constant step of arc length between two adjacent positions, i.e. Nφ = 1609.
Number of scanning circles per detector position : Nρ = 815 and ρmax = 7000.

Original objet Data acquisition

PAGE28/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Proposed simulation results

Parameter choices :

Object of size 512 × 512 pixels.
The detector is moving on a circle of radius R = 256 with a constant step of arc length between two adjacent positions, i.e. Nφ = 1609.
Number of scanning circles per detector position : Nρ = 815 and ρmax = 7000.

Original objet Data acquisition Reconstruction

= PAGE3 + 2629/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Summary of the contributions of this part

Proposition of the Fixed source rotating detector CST modality

Simple and compact CST modality able to scan large objects
Un-collimated deector

Study of the data measurement model

Proposition of a method for image reconstruction from an analytic inversion formula =⇒ Lead to a filtered back-projection type algorithm never introduced before

= PAGE4 + 2630/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Summary of the contributions of this part

Proposition of the Fixed source rotating detector CST modality

Simple and compact CST modality able to scan large objects
Un-collimated deector

Study of the data measurement model

Proposition of a method for image reconstruction from an analytic inversion formula =⇒ Lead to a filtered back-projection type algorithm never introduced before

Extension of this modality in three dimensions (see Chapter 7 of the thesis)

Proposition of a three dimensional extension for the CST modality
Proof of the invertibility of the corresponding Radon transform

29/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

30/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Presentation of Webber and Miller’s CST modality

A CST modality with translational geometry 1

Setup of the system

A source S and a uncollimated detector D, separated by a fixed distance and labelled by their horizontal position x0 move respectively on a horizontal line of equation z = 3 and z = 1.
Object placed below of the detector path =⇒ Large objects scanning

Parameterisation of the scanning arcs

Sj (x0,r),j ∈ {1, 2, 3, 4} :

p q x1 = r2 − 1 + r2 − (z − 2)2,

p q x2 = r2 − 1 − r2 − (z − 2)2,

p q x3 = − r2 − 1 + r2 − (z − 2)2, p q

x4 = − r2 − 1 − r2 − (z − 2)2 where r = 1/ sin(π − ω) and z ∈]2 − r, 1[ .

\1. J. Webber and E. L. Miller, “Compton scattering tomography in translationnal geometries”, Inverse problems, 36**, 0250007, 2020.

31/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Measurement model and study of the corresponding Radon transform

Corresponding RadonZ r transf1orm**:**XDenoting2 pf1 − 1 + (−1)j r 1 − z 2 + x0,z!

(x,z) = f (x, 2r− z),

RD f (x0,r) = q f1 r2 r +

1 1 − zr j=1

r !!

f1 − r2 − 1 + (−1)j r 1 − z 2 + x ,z dz, (15)

r 0

PAGE32/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Measurement model and study of the corresponding Radon transform

Corresponding RadonZ transform : Denoting2 f r z 2

1(x,z) = f (x, 2 − z),

RD f (x0,r) = r q 1  f1 r2 − 1 + (−1)j r 1 − + x0,z!

X p

1 1 − z 2 r +

r j=1

r !!

f1 − r2 − 1 + (−1)j r 1 − z 2 + x ,z dz, (15)

r 0

In the original study of Webber and Miller 1 :

Proof of invertibility of RD using the theory of integral equation
Proposition of an inversion formula as an integral tranformation with a kernel computed iteratively.

−→ A numerical calculation of such kind of kernel may require high computational time and/or memory.

The authors underlined that their approach for inversion is severely ill-posed, in terms of stablity (cf Remark 3.4). −→ The implementation of this inversion formula can lead to large instabilities.

\1. J. Webber and E. L. Miller, “Compton scattering tomography in translationnal geometries”, Inverse problems, 36**, 0250007, 2020.

PAGE33/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Measurement model and study of the corresponding Radon transform

Corresponding Radon transform :

0 1 1 − z 2 XDenoting2 pfr12(x,z) = f (x,j2r−1z)−, z 2 !

Z r 

RD f (x ,r) = q f1 − 1 + (−1) r r + x0,z +

r j=1

r !!

f1 − r2 − 1 + (−1)j r 1 − z 2 + x ,z dz, (15)

r 0

In the original study of Webber and Miller 1 :

Proof of invertibility of RD using the theory of integral equation
Proposition of an inversion formula as an integral tranformation with a kernel computed iteratively.

−→ A numerical calculation of such kind of kernel may require high computational time and/or memory.

The authors underlined that their approach for inversion is severely ill-posed, in terms of stablity (cf Remark 3.4). −→ The implementation of this inversion formula can lead to large instabilities.

Objective of my work : propose an alternative inversion formula suitable for a faster and efficient

reconstruction algorithm.

\1. J. Webber and E. L. Miller, “Compton scattering tomography in translationnal geometries”, Inverse problems, 36**, 0250007, 2020.

= PAGE3 + 3033/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

An alternative reconstruction formula with efficient implementation

Prvaroposediable isinversion formula : Gbf (ξ,0r) = Rb D f√(ξ,r) ,

Denoting Gf (x ,r) the operator whose Fourier transform according to the first

(16) 2r cos(ξ r2 − 1)

Gbf as follows

ifr > 1 and 0 when r ∈ [0, 1], the1 Zunkno∞ ewnixξ Zfunction∞ H0 f iscompletelypξ2 reco2cosvered(σ(2from− z))σdσdξ, (17)

f (x,z) = Gbf ξ, + σ

4π ∞ 0

where H0 is the zero-order Hankel transform.

= PAGE5 + 3035/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

An alternative reconstruction formula with efficient implementation

Proposed inversion formula : Denoting Gf (x0,r) the operator whose Fourier transform according to the first variable is Gbf (ξ,r) = RD f√(ξ,r) , (16)

2r cos(ξ r2 − 1)

Gbf as follows

ifr > 1 and 0 when r ∈ [0, 1], the1 Zunkno∞ ewnixξ Zfunction∞ 0 f iscompletelyξ, p 2 + σrecocosvered(σ(2from− z))σdσdξ, (17)

f (x,z) = H Gbf ξ 2

4π ∞ 0

where H0 is the zero-order Hankel transform.

Proposed formulation suitable for simulations : From the inversion formula (17), it follows in the Fourier domain fb (ξ,σ) = 2π2|σ|Gb‡f (ξ,σ), (18)

where fb1 ‡ G‡f (x,z) = Z ∞ x , q(x − x ) f1 and G‡f , defined as follows (19)

and Gb f are the respective two-dimensional Fourier transforms of

Gf 2 + z2 dx0.

0 0

∞

= PAGE3 + 3134/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Proposed reconstruction algorithm :

Part 1 : Obtain Gbrearrangf (ξ,r) =ed dataRb DGff√(ξfr,rom)− 1)R D 2r ϵ2 + cos(ξ√2r2 − 1)2 .

D f :

Rb f (ξ,r) cos(ξ√r − 1)

−→ Gbf (ξ,r) = (20) 2r cos(ξ r2

Compute the 1D Fourier transform of RD f (x0,r) according to the first variable using FFT.

Compute Gbf (ξ,r) according to (20).

Perform the inverse FFT to obtain Gf (x0,r).

= PAGE4 + 3135/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Proposed reconstruction algorithm :

Part 1 : Obtain Gbrearrangf (ξ,r) =ed dataRDGff√(ξfr,rom) RD f : Rb f (ξ,r) cos(ξ√√r2 − 1) .

2r cos(ξ r2 − 1) −→ Gbf (ξ,r) = D 2r ϵ2 + cos(ξ r − 1)2 (20)

Compute the 1D Fourier transform of RD f (x0,r) according to the first variable using FFT.

Compute Gbf (ξ,r) according to (20).

Perform the inverse FFT to obtain Gf (x0,r).

Part 2 : Use relation (19) to obtain G‡f (x,z)

Z q

G‡f (x,z) = ∞ Gf x , (x − x0)2 + z2 dx0.

∞

For each x0, interpolate the obtained data and sum on all values of x0 to have the back-projected data

G‡f (x,z).

PAGE35/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Proposed reconstruction algorithm :

Part 1 : Obtain rearranged data Gf from RD f :

b Rb D f (ξ,r) b Rb D f (ξ,r) cos(ξ√√r − 1) 2 .

Gf (ξ,r) = √ −→ Gf (ξ,r) = (20)

2r cos(ξ r2 − 1) 2r ϵ2 + cos(ξ r2 − 1)

Compute the 1D Fourier transform of RD f (x0,r) according to the first variable using FFT.

Compute Gbf (ξ,r) according to (20).

Perform the inverse FFT to obtain Gf (x0,r).

Part 2 : Use relation (19) to obtain G‡f (x,z)

Z q

G‡f (x,z) = ∞ Gf x0, (x − x )2 + z2 dx0.

∞

For each x0, interpolate the obtained data and sum on all values of x0 to have the back-projected data

G‡f (x,z).

Part 3 : Use equation (18) to finally recover the object f : fb1(ξ,σ) = 2π2|σ|Gb‡f (ξ,σ)

Perform the 2D FFT of G‡f (x,z) and weight by 2π2|σ|.

Compute the inverse FFT of the result to recover f .

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Proposed simulation results

Parameter choices :

Object of size 256 × 256 pixels.
Distance between S and D : 2 pixels
Number of positions NSD for the pair source - detector : NSD = 2048, Number of scanning circles per position : Nr = 1024.
Regularization parameter ϵ arbitrarily chosen : ϵ = 0.01.

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Proposed simulation results

Parameter choices :

Object of size 256 × 256 pixels.
Distance between S and D : 2 pixels
Number of positions NSD for the pair source - detector : NSD = 2048, Number of scanning circles per position : Nr = 1024.
Regularization parameter ϵ arbitrarily chosen : ϵ = 0.01. Original object

35/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Proposed simulation results

Parameter choices :

Object of size 256 × 256 pixels.
Distance between S and D : 2 pixels
Number of positions NSD for the pair source - detector : NSD = 2048, Number of scanning circles per position : Nr = 1024.
Regularization parameter ϵ arbitrarily chosen : ϵ = 0.01. Original object

Data acquisition

35/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Proposed simulation results

Parameter choices :

Object of size 256 × 256 pixels.
Distance between S and D : 2 pixels
Number of positions NSD for the pair source - detector : NSD = 2048, Number of scanning circles per position : Nr = 1024.
Regularization parameter ϵ arbitrarily chosen : ϵ = 0.01.

Data acquisition

Original object

Reconstruction

35/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Summary of the contributions of this part

Proposition of an inversion formula alternative to the one previously proposed by Webber and Miller

Use the translational invariance of the scanning circle arcs to establish a relation between object and data in the Fourier domain
Proposition of a fast and efficient reconstruction algorithm for simulations

36/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

37/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Concluding remarks

Main contributions and perspectives

Proposition of two CST modalities

Circular Compton Scattering Tomography Fixed source rotating detector CST modality

38/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Comparison with other CST modalities

Nguyen and Truong’s modality (2010) Norton’s modality (1994)

Truong and Nguyen’s modality (2011) Webber and Miller’s modality (2020)

CCST modality (2018) Fixed source rotating detector modality (2020)

39/43

Comparison of CST modalities suitable for internal scanning

Nguyen and Truong’s modality (2010)

Compact and simple system
Rotation and sync. are mandatory

Truong and Nguyen’s modality (2011)

Compact and simple system
Rotation and sync. are mandatory
Position of the sensors conditions the photon energy.

CCST modality (2018)

Fixed and compact system
Requires a set of detectors
The design may be complicated in case of collimation at detectors.

Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Comparison of CST modalities suitable for external scanning

PAGE40/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Norton’s modality (1994)

Fixed system
Acquisition based on both forward and back-scattered photons
Linear configuration with a set of detectors

Truong and Nguyen’s modality (2011)

Compact and simple system
Rotation and sync. are mandatory
Position of the sensors conditions the photon energy.

CCST modality (2018)

Fixed and compact system
Requires a set of detectors
The design may be complicated in case of collimation at detectors.

Nguyen and Truong’s modality (2010)

Compact and simple system
Rotation and sync. are mandatory

Webber and Miller’s modality (2020)

Simple configuration with an un-coll. detector
Linear configuration
Translation and sync. are mandatory.

Fixed source rotating detector modality (2020)

Simple and compact configuration with an un-coll. detector
No sync. needed
Rotation of the detector is mandatory.

PAGE41/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Comparison of CST modalities suitable for external scanning

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Norton’s modality (1994)

ω ∈]0,π[

Fixed system
Acquisition based on both forward and back-scattered photons
Linear configuration with a set of detectors

Truong and Nguyen’s modality (2011) ω ∈ [π/2,π[

Compact and simple system
Rotation and sync. are mandatory
Position of the sensors conditions the photon energy.

**CCST modality (2018) **ω ∈]0,π[

Fixed and compact system
Requires a set of detectors
The design may be complicated in case of collimation at detectors.

Nguyen and Truong’s modality (2010) ω ∈ [π/2,π[

Compact and simple system
Rotation and sync. are mandatory

Webber and Miller’s modality (2020)

ω ∈ [π/2,π[

Simple configuration with an un-coll. detector
Linear configuration
Translation and sync. are mandatory.

**Fixed source rotating detector modality (2020) **ω ∈ [π/2,π[

Simple and compact configuration with an un-coll. detector
No sync. needed
Rotation of the detector is mandatory.

= PAGE3 + 3841/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Concluding remarks

Main contributions and perspectives

Proposition of two CST modalities :

Circular Compton Scattering Tomography and Fixed source rotating detector CST modality

Proposition of analytic inversion formulas resulting in reconstruction algorithms giving fast

reconstruction results

Require less memory and computation time in comparison with some methods previously proposed 1 2 (also based on analytic inversion formulas)

\1. J. Webber and E. L. Miller, “Compton scattering tomography in translationnal geometries”, Inverse problems, 36, 0250007, 2020.

\2. G. Rigaud, M. K. Nguyen and A. K. Louis, “Novel numerical inversions of two circular-arc Radon transforms in Compton scattering tomography”,

Inverse Problems in Science and Engineering, 20(6), 2012.

= PAGE4 + 3842/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Concluding remarks

Main contributions and perspectives

Proposition of two CST modalities :

Circular Compton Scattering Tomography and Fixed source rotating detector CST modality

Proposition of analytic inversion formulas resulting in reconstruction algorithms giving fast

reconstruction results

Require less memory and computation time in comparison with some methods previously proposed 1 2 (also based on analytic inversion formulas)
Sensitive to missing data

Main contributions and perspectives

Proposition of two CST modalities :

Circular Compton Scattering Tomography and Fixed source rotating detector CST modality

Proposition of analytic inversion formulas resulting in reconstruction algorithms giving fast

reconstruction results

Require less memory and computation time in comparison with some methods previously proposed 1 2 (also based on analytic inversion formulas)
Sensitive to missing data
Requires a large amount of data to reconstruct the object to scan.

Main contributions and perspectives

Proposition of two CST modalities :

Circular Compton Scattering Tomography and Fixed source rotating detector CST modality

Proposition of analytic inversion formulas resulting in reconstruction algorithms giving fast

reconstruction results

Require less memory and computation time in comparison with some methods previously proposed 1 2 (also based on analytic inversion formulas)
Sensitive to missing data
Requires a large amount of data to reconstruct the object to scan.
We remain in a case where the conditions are ideal as in the literature :

Main contributions and perspectives

Proposition of two CST modalities :

Circular Compton Scattering Tomography and Fixed source rotating detector CST modality

Proposition of analytic inversion formulas resulting in reconstruction algorithms giving fast

reconstruction results

Require less memory and computation time in comparison with some methods previously proposed 1 2 (also based on analytic inversion formulas)
Sensitive to missing data
Requires a large amount of data to reconstruct the object to scan.
We remain in a case where the conditions are ideal as in the literature : −→ only first-order scattering was considered,

Main contributions and perspectives

Proposition of two CST modalities :

Circular Compton Scattering Tomography and Fixed source rotating detector CST modality

Proposition of analytic inversion formulas resulting in reconstruction algorithms giving fast

reconstruction results

Require less memory and computation time in comparison with some methods previously proposed 1 2 (also based on analytic inversion formulas)
Sensitive to missing data
Requires a large amount of data to reconstruct the object to scan.
We remain in a case where the conditions are ideal as in the literature : −→ only first-order scattering was considered,

−→ with point-like source and detectors,

Main contributions and perspectives

Proposition of two CST modalities :

Circular Compton Scattering Tomography and Fixed source rotating detector CST modality

Proposition of analytic inversion formulas resulting in reconstruction algorithms giving fast

reconstruction results

Require less memory and computation time in comparison with some methods previously proposed 1 2 (also based on analytic inversion formulas)
Sensitive to missing data
Requires a large amount of data to reconstruct the object to scan.
We remain in a case where the conditions are ideal as in the literature : −→ only first-order scattering was considered,

−→ with point-like source and detectors,

−→ without attenuation in matter

\1. J. Webber and E. L. Miller, “Compton scattering tomography in translationnal geometries”, Inverse problems, 36, 0250007, 2020.

\2. G. Rigaud, M. K. Nguyen and A. K. Louis, “Novel numerical inversions of two circular-arc Radon transforms in Compton scattering tomography”,

Inverse Problems in Science and Engineering, 20(6), 2012.

41/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Main perspective about CST modalities

Take into account attenuation in matter

−→ Lead to the consideration of attenZuated Radon transforms :

Rf (S,D,E(ω)) = a1(E0,SM ) f (M ) a2(E(ω),MD) dl(M ). (21)

M ∈A(S,D,ω)

42/43 Motivations Circular CST Fixed source rotating detector CST Webber’s modality Conclusion and perspectives

Thank you for your attention!

Associated publications

[J1] C. Tarpau, J. Cebeiro, M. A. Morvidone and M. K. Nguyen, “A new concept of Compton scattering tomography and the development of the corresponding circular Radon transform”, IEEE Transactions on Radiation and Plasma Medical Sciences, vol.4, no.4, pp. 433-440, 2020.

doi : 10.1109/TRPMS.2019.2943555

[J2] C. Tarpau and M. K. Nguyen, “Compton scattering imaging system with two scanning configurations”, Journal of electronic imaging, vol. 29, no. 1, pp. 013005, 2020. doi : 10.1117/1.JEI.29.1.013005

[J3] C. Tarpau, J. Cebeiro, M. K. Nguyen, G. Rollet and M. Morvidone, “Analytic inversion of a Radon transform on Double Circular Arcs with Applications in Compton Scattering Tomography”, IEEE Transactions on Computational Imaging, vol. 6, 2020. doi : 10.1109/TCI.2020.2999672

[J4] J. Cebeiro, C. Tarpau, M. A. Morvidone, D. Rubio and M. K. Nguyen, “On a three-dimensional Compton scattering tomography system with fixed source”, Inverse Problems, vol. 37, no. 5, 054001, 2021. doi : 10.1088/1361-6420/abf0f0.

[J5] C. Tarpau, J. Cebeiro, G. Rollet, M. K. Nguyen and L. Dumas, “An analytical reconstruction formula with efficient implementation for a modality of Compton Scattering Tomography with translational geometry”, accepted for publication in Inverse problems in Imaging, 2021.

Article in preparation

[P1] C. Tarpau, J. Cebeiro, M. K. Nguyen, G. Rollet and L. Dumas, “Analytic inversion of a Radon transform of a class of cones with pivoting axes”, in preparation, 2021.

43/43

If we add a degree of freedom for the family of V-lines…

For example, if the horizontal line is no longer fixed and can move in the plane, the associated RT is invertible.

V-lines of equation :

V(b,φ,ω) = + r cos(φ − ω),b + r sin(φ − ω)

tan φ

, + r cos(φ + ω),b + r sin(φ + ω) ,r ∈ R .

tan φ

Associated Radon transform

Z ∞ b

RVf (b,φ,ω) = f ,b + r(cos(φ − ω), sin(φ − ω)) dr+

tan φ

0 Z ∞

f ,b + r(cos(φ + ω), sin(φ + ω)) dr (22) 0 tan φ

Study of a class of cones with pivoting axes

… the Radon transform is invertible in two dimensions…

Associated Radon transform

Z ∞ b

RVf (b,φ,ω) = f ,b + r(cos(φ − ω), sin(φ − ω)) dr+

tan φ

0 Z ∞

f ,b + r(cos(φ + ω), sin(φ + ω)) dr (22) 0 tan φ

Proposed inversion formula

Denoting Z π

Gf (θD ,a) = RVf (ω,θD ,a)dω, (23)

k k

the unknown function f is related to its projections Gf by

f (x,y) = 1 F −1 kx2 + ky2 · F2(Gf (ρ,ϕ))(k ,k )(x,y). (24)

2π 2 x y

where F2 stands for the two-dimensional Fourier transform and kx and ky are the duals of x and y.

−→ Deconvolution formula.

PAGE3/2 Study of a class of cones with pivoting axes

… the Radon transform is invertible in two dimensions…

Associated Radon transform

Z ∞ b

RVf (b,φ,ω) = f ,b + r(cos(φ − ω), sin(φ − ω)) dr+

tan φ

0 Z ∞

f ,b + r(cos(φ + ω), sin(φ + ω)) dr (22) 0 tan φ

Proposed inversion formula

Denoting Z π

Gf (θD ,a) = RVf (ω,θD ,a)dω, (23)

k k

the unknown function f is related to its projections Gf by

f (x,y) = 1 F −1 kx2 + k2 · F (Gf (ρ,ϕ))(kx,k )(x,y). (24)

2π 2 y 2 y

where F2 stands for the two-dimensional Fourier transform and kx and ky are the duals of x and y.

−→ Deconvolution formula with ➊ a back-projection.

PAGE3/2

Associated Radon transform

Z ∞ b

RVf (b,φ,ω) = f ,b + r(cos(φ − ω), sin(φ − ω)) dr+

tan φ

0 Z ∞

f ,b + r(cos(φ + ω), sin(φ + ω)) dr (22) 0 tan φ

Proposed inversion formula

Denoting Z π

Gf (θD ,a) = RVf (ω,θD ,a)dω, (23)

k k

the unknown function f is related to its projections Gf by

f (x,y) = 21π F2−1 kx2 + ky2 · F2(Gf (ρ,ϕ))(kx,ky) (x,y). (24) where F2 stands for the two-dimensional Fourier transform and kx and ky are the

duals of x and y.

−→ Deconvolution formula with ➊ a back-projection and then ➋ a filtering ope- ration.

Study of a class of cones with pivoting axes

… the Radon transform is invertible in two dimensions…

Associated Radon transform

Z ∞ b

RVf (b,φ,ω) = f ,b + r(cos(φ − ω), sin(φ − ω)) dr+

tan φ

0 Z ∞

f ,b + r(cos(φ + ω), sin(φ + ω)) dr 0 tan φ

Proposed inversion formula

Denoting Z π

Gf (θD ,a) = RVf (ω,θD ,a)dω, k k

Simulation results

(22)

(23)

Original object

PAGE2/2 Study of a class of cones with pivoting axes

… the Radon transform is invertible in two dimensions…

the unknown function f is related to its projections Gf by

f (x,y) = 1 F2−1 k2 + k2 · F2(Gf (ρ,ϕ))(k ,ky) (x,y). (24)

2π x y x

where F2 stands for the two-dimensional Fourier transform and kx and ky are the duals of x and y.

−→ Deconvolution formula with ➊ a back-projection and then ➋ a filtering ope-

Reconstruction ration.

PAGE3/2 Study of a class of cones with pivoting axes

… the Radon transform is invertible in two dimensions…

Associated Radon transform

Z ∞ b

RVf (b,φ,ω) = f ,b + r(cos(φ − ω), sin(φ − ω)) dr+

tan φ

0 Z ∞

f ,b + r(cos(φ + ω), sin(φ + ω)) dr 0 tan φ

Proposed inversion formula

Denoting Z π

Gf (θD ,a) = RVf (ω,θD ,a)dω, k k

Simulation results

(22)

(23)

Original object

PAGE2/2 Study of a class of cones with pivoting axes

… the Radon transform is invertible in two dimensions…

the unknownf (functionx,y) = f1is related−1 qto its+projectionsk Gf by x,ky) (x,y). (24)

2π F2 kx2 y2 · F2(Gf (ρ,ϕ))(k

where F2 stands for the two-dimensional Fourier transform and kx and ky are the

duals of x and y.

−→ Deconvolution formula with ➊ a back-projection and then ➋ a filtering ope-

Reconstruction ration.

… and also in three dimensions PAGE3/2