A joint reconstruction and lambda tomography regularization technique for energy-resolved X-ray imaging

Abstract

We present new joint reconstruction and regularization techniques inspired by ideas in microlocal analysis and lambda tomography, for the simultaneous reconstruction of the attenuation coefficient and electron density from x-ray transmission (i.e., x-ray CT) and backscattered data (assumed to be primarily Compton scattered). To demonstrate our theory and reconstruction methods, we consider the 'parallel line segment' acquisition geometry of Webber J and Miller E (2020 Inverse Problems 36 025007), which is motivated by system architectures currently under development for airport security screening. We first present a novel microlocal analysis of the parallel line geometry which explains the nature of image artefacts when the attenuation coefficient and electron density are reconstructed separately. We next introduce a new joint reconstruction scheme for low effective Z (atomic number) imaging (Z < 20) characterized by a regularization strategy whose structure is derived from lambda tomography principles and motivated directly by the microlocal analytic results. Finally we show the effectiveness of our method in combating noise and image artefacts on simulated phantoms.

Full text

A JOINT RECONSTRUCTION AND LAMBDA TOMOGRAPHY
REGULARIZATION TECHNIQUE FOR ENERGY-RESOLVED X-RAY
IMAGING
08/07/2020 22:08
JAMES WEBBER, ERIC TODD QUINTO, AND ERIC L. MILLER
Abstract. We present new joint reconstruction and regularization techniques inspired by
ideas in microlocal analysis and lambda tomography, for the simultaneous reconstruction
of the attenuation coefficient and electron density from X-ray transmission (i.e., X-ray CT)
and backscattered data (assumed to be primarily Compton scattered). To demonstrate our
theory and reconstruction methods, we consider the “parallel line segment” acquisition ge-
ometry of [54], which is motivated by system architectures currently under development for
airport security screening. We first present a novel microlocal analysis of the parallel line
geometry which explains the nature of image artefacts when the attenuation coefficient and
electron density are reconstructed separately. We next introduce a new joint reconstruction
scheme for low effective Z(atomic number) imaging (Z < 20) characterized by a regulariza-
tion strategy whose structure is derived from lambda tomography principles and motivated
directly by the microlocal analytic results. Finally we show the effectiveness of our method
in combating noise and image artefacts on simulated phantoms.
1. Introduction
In this paper we introduce new joint reconstruction and regularization techniques based
on ideas in microlocal analysis and lambda tomography [13, 14] (see also [32] for related
work). We consider the simultaneous reconstruction of the attenuation coefficient µEand
electron density nefrom joint X-ray CT (transmission) and Compton scattered data, with
particular focus on the parallel line segment X-ray scanner displayed in figures 1 and 2. The
acquisition geometry in question is based on a new airport baggage scanner currently in
development, and has the ability to measure X-ray CT and Compton data simultaneously.
The line segment geometry was first considered in [54], where injectivity results are derived in
Compton Scattering Tomography (CST). We provide a stability analysis of the CST problem
of [54] here, from a microlocal perspective. The scanner depicted in figure 1 consists of a
row of fixed, switched, monochromatic fan beam sources (S), a row of detectors (DA) to
measure the transmitted photons, and a second (slightly out of plane) row of detectors (DC)
to measure Compton scatter. The detectors are assumed to energy-resolved, a common
assumption in CST [35, 51, 41, 43, 55], and the sources are fan-beam (in the plane) with
opening angle π(so there is no restriction due to cropped fan-beams).
The attenuation coefficient relates to the X-ray transmission data by the Beer-Lambert
law, log I0
IA=RLµEdl[31, page 2] where IAis the photon intensity measured at the
detector, I0is the initial source intensity and µEis the attenuation coefficient at energy
E. Here Lis a line through Sand DA, with arc measure dl. Hence the transmission
data determines a set of integrals of µEover lines, and the problem of reconstructing µEis
1

DAat {z= 2 −rm}
DC
S
x2
x1
O
LCR
a
Figure 1. Parallel line X-ray CT geometry. Here S,DCand DAdenote the
source and detector rows. The length of the detector (and source) array is
2a. A cone CR⊂S1is highlighted in orange. We will refer to CRlater for
visualisation in section 4.
equivalent to inversion of the line Radon transform with limited data (e.g., [31, 33]). Note
that we need not account for the energy dependence of µEin this case as the detectors are
energy-resolved, and hence there are no issues due to beam-hardening. See figure 1.
When the attenuation of the incoming and scattered rays is ignored, the Compton scat-
tered intensity in two-dimensions can be modelled as integrals of neover toric sections
[35, 51, 55]
(1.1) IC=ZT
nedt,
where ICis the Compton scattered intensity measured at a point on DC. A toric section
T=C1∪C2is the union of two intersecting circles of the same radii (as displayed in figure
2), and dtis the arc measure on T. The recovery of neis equivalent to inversion of the
toric section Radon transform [34, 35, 51, 55]. See figure 2. See also [41, 43] for alternative
reconstruction methods. We now discuss the approximation made above to neglect the
attenuative effects from the CST model. When the attenuation effects are included, the
inverse scattering problem becomes nonlinear [42]. We choose to focus on the analysis of
the idealised linear case here, as this allows us to apply the well established theory on linear
Fourier Integral Operators (FIO) and microlocal analysis to derive expression for the image
artefacts. Such analysis will likely give valuable insight into the expected artefacts in the
nonlinear case. The nonlinear models and their inversion properties are left for future work.
2

DAat {z= 2 −rm}
DC
1
1
1
S
a
x2
Ox1
x0
r
s
x
ξCT
C1(s, x0)C2(s, x0)
β
Figure 2. Parallel line CST geometry. S,DCand DAdenote the source and
detector rows. The remaining labels are referenced in the main text. A cone
CT⊂S1is highlighted in orange. We will refer to CTlater for visualisation
in section 3. Note that we have cropped out part of the left side (left of O) of
the scanner of figure 1 in this picture.
The line and circular arc Radon transforms with full data, are known [35, 30] to have
inverses that are continuous in some range of Sobolev norms. Hence with adequate regular-
ization we can reconstruct an image free of artefacts. With limited data however [31, 51], the
solution is unstable and the image wavefront set (see Definition 2.2) is not recovered stably in
all directions. We will see later in section 3 through simulation that such data limitations in
the parallel line geometry cause a blurring artefact over a cone in the reconstruction. There
may also be nonlocal artefacts specific to the geometry (as in [55]), which we shall discover
later in section 3 in the geometry of figure 2.
The main goal of this paper is to combine limited datasets in X-ray CT and CST with
new lambda tomography regularization techniques, to recover the image edges stably in all
directions. We focus particularly on the geometry of figure 1. In lambda tomography the
image reconstruction is carried out by filtered backprojection of the Radon projections, where
the filter is chosen to emphasize boundaries. This means that the jump singularities in the
lambda reconstruction have the same location and direction to those of the target function,
but the smooth parts are undetermined. A common choice of filter is a second derivative in
the linear variable [7, 43]. The application of the derivative filter emphasizes the singularities
3

in the Radon projections, and this is a key idea behind lambda tomography [13, 14, 52], and
the microlocal view on lambda CT (e.g., [7, 39, 43]). The regularization penalty we propose
aims to minimize the difference kdm
dsmR(µE−ne)kL2(R×S1)for some m≥1, where Rdenotes the
Radon line transform. Therefore, with a full set of Radon projections, the lambda penalties
enforce a similarity in the locations and direction of the image singularities (edges) of µEand
ne. Further dm
dsmRfor m≥1 is equivalent to taking m−1/2 derivatives of the object (this
operation is continuous of positive order m−1/2 in Sobolev scales), and hence its inverse is
a smoothing operation, which we expect to be of aid in combatting the measurement noise.
In addition, the regularized inverse problem we propose is linear (similarly to the Tikhonov
regularized inverse [21, page 99]), which (among other benefits of linearity) allows for the
fast application of iterative least squares solvers in the solution.
The literature considers joint image reconstruction and regularization in for example,
[1, 19, 40, 46, 47, 6, 50, 12, 20, 11, 48, 10, 45, 4]. See also the special issue [3] for a
more general review of joint reconstruction techniques. In [12] the authors consider the
joint reconstruction from Positron Emission Tomography (PET) and Magnetic Resonance
Imaging (MRI) data and use a Parallel Level Set (PLS) prior for the joint regularization. The
PLS approach (first introduced in [10]) imposes soft constraints on the equality of the image
gradient location and direction, thus enforcing structural similarity in the image wavefront
sets. This follows a similar intuition to the “Nambu” functionals of [48] and the “cross-
gradient” methods of [16, 17] in seismic imaging, the latter of which specify hard constraints
that the gradient cross products are zero (i.e. parallel image gradients). The methods of [12]
use linear and quadratic formulations of PLS, denoted by Linear PLS (LPLS) and Quadratic
PLS (QPLS). The LPLS method will be a point of comparison with the proposed method.
We choose to compare with LPLS as it is shown to offer greater performance than QPLS in
the experiments conducted in [12].
In [20] the authors consider a class of techniques in joint reconstruction and regularization,
including inversion through correspondence mapping, mutual information and Joint Total
Variation (JTV). In addition to LPLS, we will compare against JTV as the intuition of
JTV is similar to that of lambda regularization (and LPLS), in the sense that a structural
similarity is enforced in the image wavefronts. Similar to standard Total Variation (TV),
which favours sparsity in the (single) image gradient, the JTV penalties (first introduced in
[45] for colour imaging) favour sparsity in the joint gradient. Thus the image gradients are
more likely to occur in the same location and direction upon minimization of JTV. The JTV
penalties also have generalizations in colour imaging and vector-valued imaging [4].
In [54] the authors introduce a new toric section transform Tin the geometry of figure 2.
Here explicit inversion formulae are derived, but the stability analysis is lacking. We aim to
address the stability of Tin this work from a microlocal perspective. Through an analysis of
the canonical relations of T, we discover the existence of nonlocal artefacts in the inversion,
similarly to [55]. In [40] the joint reconstruction of µEand neis considered in a pencil
beam scanner geometry. Here gradient descent solvers are applied to nonlinear objectives,
derived from the physical models, and a weighted, iterative Tikhonov type penalty is applied.
The works of [6] improve the wavefront set recovery in limited angle CT using a partially
learned, hybrid reconstruction scheme, which adopts ideas in microlocal analysis and neural
networks. The fusion with Compton data is not considered however. In our work we assume
an equality in the wavefront sets of neand µE(in a similar vein to [47]), and we investigate
4

the microlocal advantages of combining Compton and transmission data, as such an analysis
is lacking in the literature.
The remainder of this paper is organized as follows. In section 2, we recall some definitions
and theorems from microlocal analysis. In section 3 we present a microlocal analysis of Tand
explain the image artefacts in the nereconstruction. Here we prove our main theorem (Theo-
rem 3.2), where we show that the canonical relation Cof Tis 2–1. This implies the existence
of nonlocal image artefacts in a reconstruction from toric section integral data. Further we
find explicit expressions for the nonlocal artefacts and simulate these by applying the normal
operations T∗Tto a delta function. In section 4 we consider the microlocal artefacts from
X-ray (transmission) data. This yields a limited dataset for the Radon transform, whereby
we have knowledge the line integrals for all Lwhich intersect Sand DA(see figure 1). We
use the results in [5] to describe the resulting artifacts in the X-ray CT reconstruction. In
section 5, we detail our joint reconstruction method for the simultaneous reconstruction of
µEand ne. Later in section 5.4 we present simulated reconstructions of µEand neusing the
proposed methods and compare against JTV [20] and LPLS [12] from the literature. We
also give a comparison to a separate reconstruction using TV.
2. Microlocal definitions
We next provide some notation and definitions. Let Xand Ybe open subsets of Rn.
Let D(X) be the space of smooth functions compactly supported on Xwith the standard
topology and let D0(X) denote its dual space, the vector space of distributions on X. Let
E(X) be the space of all smooth functions on Xwith the standard topology and let E0(X)
denote its dual space, the vector space of distributions with compact support contained in
X. Finally, let S(Rn) be the space of Schwartz functions, that are rapidly decreasing at ∞
along with all derivatives. See [44] for more information.
Definition 2.1 ([25, Definition 7.1.1]).For a function fin the Schwartz space S(Rn), we
define the Fourier transform and its inverse as
(2.1) Ff(ξ) = ZRn
e−ix·ξf(x)dx,F−1f(x) = (2π)−nZRn
eix·ξf(ξ)dξ.
We use the standard multi-index notation: if α= (α1, α2, . . . , αn)∈ {0,1,2, . . . }nis a
multi-index and fis a function on Rn, then
∂αf=∂
∂x1α1∂
∂x2α2
···∂
∂xnαn
f.
If fis a function of (y,x,σ) then ∂α
yfand ∂α
σfare defined similarly.
We identify cotangent spaces on Euclidean spaces with the underlying Euclidean spaces,
so we identify T∗(X) with X×Rn.
If φis a function of (y,x,σ)∈Y×X×RNthen we define dyφ=∂φ
∂y1,∂ φ
∂y2,··· ,∂ φ
∂yn, and
dxφand dσφare defined similarly. We let dφ= (dyφ, dxφ, dσφ).
The singularities of a function and the directions in which they occur are described by the
wavefront set [9, page 16]:
Definition 2.2. Let XLet an open subset of Rnand let fbe a distribution in D0(X). Let
(x0,ξ0)∈X×(Rn\{0}). Then fis smooth at x0in direction ξ0if exists a neighbourhood
5

Uof x0and Vof ξ0such that for every φ∈ D(U) and N∈Rthere exists a constant CN
such that
(2.2) |F(φf )(λξ)| ≤ CN(1 + |λ|)−N.
The pair (x0,ξ0) is in the wavefront set, WF(f), if fis not smooth at x0in direction ξ0.
This definition follows the intuitive idea that the elements of WF(f) are the point–normal
vector pairs above points of Xwhere fhas singularities. For example, if fis the characteristic
function of the unit ball in R3, then its wavefront set is WF(f) = {(x, tx) : x∈S2, t 6= 0},
the set of points on a sphere paired with the corresponding normal vectors to the sphere.
The wavefront set of a distribution on Xis normally defined as a subset the cotangent
bundle T∗(X) so it is invariant under diffeomorphisms, but we will continue to identify
T∗(X) = X×Rnand consider WF(f) as a subset of X×Rn\ {0}.
Definition 2.3 ([25, Definition 7.8.1]).We define Sm(Y×X×RN) to be the set of a∈
E(Y×X×RN) such that for every compact set K⊂Y×Xand all multi–indices α, β, γ
the bound
∂γ
y∂β
x∂α
σa(y,x,σ)≤CK,α,β (1 + |σ|)m−|α|,(y,x)∈K, σ∈RN,
holds for some constant CK,α,β >0. The elements of Smare called symbols of order m.
Note that these symbols are sometimes denoted Sm
1,0.
Definition 2.4 ([26, Definition 21.2.15]).A function φ=φ(y,x,σ)∈ E(Y×X×RN\{0})
is a phase function if φ(y,x, λσ) = λφ(y,x,σ), ∀λ > 0 and dφis nowhere zero. A phase
function is clean if the critical set Σφ={(y,x,σ) : dσφ(y,x,σ)=0}is a smooth manifold
with tangent space defined by d(dσφ) = 0.
By the implicit function theorem the requirement for a phase function to be clean is satisfied
if d (dσφ) has constant rank.
Definition 2.5 ([26, Definition 21.2.15] and [27, Section 25.2]).Let Xand Ybe open subsets
of Rn. Let φ∈ E Y×X×RNbe a clean phase function. In addition, we assume that φ
is nondegenerate in the following sense:
dy,σφand dx,σφare never zero.
The critical set of φis
Σφ={(y,x,σ)∈Y×X×RN\ {0}: dσφ= 0}.
The canonical relation parametrised by φis defined as
(2.3) C={((y,dyφ(y,x,σ)) ; (x,−dxφ(y,x,σ))) : (y,x,σ)∈Σφ},
Definition 2.6. Let Xand Ybe open subsets of Rn. A Fourier integral operator (FIO) of
order m+N/2−n/2 is an operator A:D(X)→ D0(Y) with Schwartz kernel given by an
oscillatory integral of the form
(2.4) KA(y,x) = ZRN
eiφ(y,x,σ)a(y,x,σ)dσ,
where φis a clean nondegenerate phase function and a∈Sm(Y×X×RN) is a symbol. The
canonical relation of Ais the canonical relation of φdefined in (2.3).
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This is a simplified version of the definition of FIO in [8, Section 2.4] or [27, Section 25.2]
that is suitable for our purposes since our phase functions are global. For general information
about FIOs see [8, 27, 26].
Definition 2.7. Let C ⊂ T∗(Y×X) be the canonical relation associated to the FIO A:
E0(X)→ D0(Y). Then we let πLand πRdenote the natural left- and right-projections of C,
πL:C → T∗(Y) and πR:C → T∗(X).
Because φis nondegenerate, the projections do not map to the zero section. If a FIO
Fsatisfies our next definition, then F∗F(or F∗φFif Fdoes not map to E0(Y)) is a
pseudodifferential operator [18, 37].
Definition 2.8. Let F:E0(X)→ D0(Y) be a FIO with canonical relation Cthen F(or C)
satisfies the semi-global Bolker Assumption if the natural projection πY:C → T∗(Y) is an
embedding (injective immersion).
3. Microlocal properties of translational Compton transforms
Here we present a microlocal analysis of the toric section transform in the translational
(parallel line) scanning geometry. Through an analysis of two separate limited data problems
for the circle transform (where the integrals over circles with centres on a straight line are
known) and using microlocal analysis, we show that the canonical relation of the toric section
transform is 2–1. The analysis follows in a similar way to the work of [55]. We discuss the
nonlocal artefacts inherent to the toric section inversion in section 3.1, and then go on to
explain the artefacts due to limited data in section 3.2.
We first define our geometry and formulate the toric section transform of [54] in terms of
δfunctions, before proving our main microlocal theory.
Let rm>1 and define the set of points to be scanned as
X:= {(x1, x2)∈R2: 2 −rm< x2<1}.
Note that rmcontrols the depth of the scanning tunnel as in figures 1 and 2. Let
Y:= (0,∞)×R
then for j= 1,2, and (s, x0)∈Y, we define the circles Cjand their centers cj
(3.1) r=√s2+ 1,cj(s, x0) = ((−1)js+x0,2)
Cj(s, x0) = {x∈R2:|x−cj(s, x0)|2−s2−1 = 0}.
Note that r=√s2+ 1 is the radius of the circle Cj. The union of the reflected circles C1∪C2
is called a toric section. Let f∈L2
0(X) be the electron charge density. To define the toric
section transform we first introduce two circle transforms
(3.2) T1f(s, x0) = ZC1
fds, T2f(s, x0) = ZC2
fds.
Now we have the definition of the toric section transform [54]
(3.3) Tf(s, x0) = ZC1∪C2
fds=T1(f)(s, x0) + T2(f)(s, x0)
where dsdenotes the arc element on a circle and (s, x0)∈Y.
7

We express Tin terms of delta functions as is done for the generalized Funk-Radon
transforms studied by Palamodov. [36]
Tf(s, x0) = T1f(s, x0) + T2f(s, x0)
=1
2r
2
X
j=1 ZR2
δ(|x−cj(s, x0)|2−s2−1)f(x)dx
=1
2r
2
X
j=1 Z∞
−∞ ZR2
e−iσ(|x−((−1)js+x0,2)|2−s2−1)f(x)dxdσ.
(3.4)
Note that the factor in front of the integrals comes about using the change of variables formula
and that Tjf=Rδ|x−cj(s, x0)| − √s2+ 1f(x) dx. So the toric section transform is the
sum of two FIO’s with phase functions
φj(s, x0,x, σ) = σ(|x−((−1)js+x0,2)|2−s2−1)
for j= 1,2. Our distributions fare supported away from the intersection points of C1and
C2, and hence we can consider the microlocal properties of T1and T2separately to describe
the microlocal properties of T.
Proposition 3.1. For j= 1,2, the circle transform Tjis an FIO or order −1/2with
canonical relation
(3.5)
Cj=ns, x0,(−1)j−1σ(x1−x0),−σ((−1)j−1s+x1−x0); (x,−σ(x−cj(s, x0))) :
(s, x0)∈Y, σ ∈R\ {0},x∈Cj(s, x0)∩ {x2<1}o.
Furthermore Cjsatisfies the semi-global Bolker assumption for j= 1,2.
Proof. First, one can check that φjand Tjboth satisfy the restrictions in Definition 2.6 so
Tjis a FIO. Using this definition again and the fact that its symbol is order zero [37], one
sees that it has order −1/2.
A straightforward calculation using Definition 2.5 shows that the canonical relation of Tj
is as given in (3.5). Note that we have absorbed a factor of 2 into σin this calculation.
Global coordinates on Cjare given by
(3.6)
(s, x0, x1, σ)7→s, x0,(−1)j−1σ(x1−x0),−σ((−1)j−1s+x1−x0);
(x1, x2),−σ((x1, x2)−cj(s, x0))
where x2= 2 −ps2+ 1 −(x1−(x0+ (−1)js))2
because x2<1. Recall that cjis given in (3.1).
We now show that Cjsatisfies the semiglobal Bolker assumption by finding a smooth inverse
in these coordinates to the projection ΠL:Cj→T∗(Y). Let λ= (s, x0, τ1, τ2)∈ΠL(Cj). We
solve for x1and σin the equation ΠL(s, x0, x1, σ) = λ. Then, sand x0are known as are
(3.7) τ1= (−1)j−1σ(x1−x0)τ2=−σ((−1)j−1s+x1−x0).
A straightforward linear algebra exercise shows that the unique solutions for σand x1are
(3.8) σ=(−1)jτ2−τ1
s, x1=sτ1
(−1)jτ1−τ2
+x0
8

This gives a smooth inverse to ΠLon the image ΠL(Cj) and finishes the proof. 
Because Cjsatisfies the Bolker Assumption, the composition C∗
j◦ Cj⊂∆, where ∆ is the
diagonal in T∗(X). Hence in a reconstruction from circular integral data with centres on a
line we would not expect to see image artefacts for functions supported in x−2>0 unless
one uses a sharp cutoff on the data.
The canonical relation Cof Tcan be written as the disjoint union C=C1∪ C2since
(C1(s, x0)∩C2(s, x0)) ∩supp(f) = ∅for any (s, x0)∈Y.
For convenience, we will sometimes label the coordinate x0in (3.6) as (x0)1it is associated
with C1and (x0)2if it is associated with C2.
Theorem 3.2. For j= 1,2, the projection πR:Cj→T∗(X)is bijective onto the set
(3.9) D={(x,ξ)∈T∗(X) : ξ26= 0}.
In addition, πR:C → T∗(X)is two-to one onto D.
Proof. Let µ= (x;ξ)∈T∗(X)\ {0}and let x= (x1, x2) and ξ= (ξ1, ξ2). If µ∈πR(Cj) for
either j= 1 or j= 2, then ξ26= 0 by (3.5) since x2<2. For the rest of the proof, assume µ
is in the set Dgiven by (3.9)
We will now describe the preimage of µin Cj. The covector µis conormal to a unique
circle centered on x2= 2, and its center is on the line through xand parallel ξ. If the center
has coordinates (c, 2), then a calculation shows that cis given by
(3.10) c=c(x,ξ) = x1−ξ1(x2−2)
ξ2
.
Using this calculation, one sees that the radius of the circle and coordinate sare given by
(3.11) r=r(x,ξ) = (2 −x2)|ξ|
|ξ2|, s =s(x,ξ) = √r2−1
and the coordinate (x0)jis given by
(3.12) (x0)j= (x0)j(x,ξ) = x1+ξ1(2 −x2)
ξ2
+ (−1)j−1sfor j= 1,2.
A straightforward calculation shows that
(3.13) σ=σ(x,ξ) = −ξ2
2−x2
.
This gives the coordinates (3.6) on Cjand shows that πR:Cj→Dis injective with smooth
inverse.
Now, we consider the projection from C. Given (x,ξ)∈D, our calculations show that the
preimage in C, in coordinates (3.6) is given by two distinct points
(s(x,ξ),(x0)j(x,ξ), x1, σ(x,ξ)) for j= 1,2.
The coordinates are given by (3.11), (3.12) and (3.13) respectively. 
The abstract adjoint Tt
jcannot be composed with Tifor i= 1,2, because the support of
Tifcan be unbounded in r, even for f∈ E0(X) and Tt
jis not defined for such distributions.
9

Therefore, we introduce a smooth cutoff function. Choose rM>2 and let ψ(s) be a smooth
compactly supported function equal to one for s∈h1,p1−r2
Miand define
(3.14) T∗
jg=Tt
j(ψg)
for all g∈ D0(Y) because our bound on rintroduces a bound on x0so the integral is over a
bounded set for each x∈X.
3.1. The nonlocal artefacts. Now, we can state our next theorem, which describes the
artifacts that can be added to the reconstruction using the normal operator, T∗T.
Theorem 3.3. If f∈ E0(X)then
(3.15) WF (T∗Tf)⊂(WF(f)∩D)∪Λ12(f)∪Λ21(f)
where Dis given by (3.9), and the sets Λij are given for (x,ξ)∈Dby
(3.16) Λij (f) = {λij (x,ξ):(x,ξ)∈WF(f)∩D}
where the functions λ12 and λ21 are given by (3.19) and (3.20) respectively. Note that the
functions λij are defined for only some (x,ξ)∈Dand singularities at other points do not
generate artifacts.
Therefore, T∗Trecovers most singularities of f, as indicated in the first term in (3.15),
but it adds two sets of nonlocal singularities, as given by Λ12(f) and Λ21(f). Note that, even
if T∗
jand Tjare both elliptic above a covector (x,ξ), artifacts caused by other points could
mask singularities of fthat “should” be visible in T∗Tf.
Proof. Let f∈ E0(X). By the H¨ormander-Sato Lemma [25, Theorem 8.2.13] We have the
expansion
(3.17)
WF(T∗T(f)) ⊂(C∗◦ C)◦WF(f)
= [(C∗
1◦ C1)∪(C∗
2◦ C2)] ◦WF(f)
∪(C∗
2◦ C1)◦WF(f)∪(C∗
1◦ C2)◦WF(f)
The first term in brackets in (3.17) is {(x,ξ;x,ξ):(x,ξ)∈D}◦WF(f) = WF(f)∩D. This
proves the first part of the inclusion (3.15).
We now analyze the other two terms to define the functions λij and finish the proof. Let
(x,ξ)∈WF(f)∩D. First, consider λ12 (x,ξ) = C∗
2◦ C1◦(x,ξ).1Using the calculations in
the proof of Theorem 3.2 one sees that C1◦(x,ξ) is given by
(3.18)
(s, x0, τ1, τ2) where s=s(x,ξ) = s(2 −x2)2|ξ|2
ξ2
2−1
x0= (x0)1(x,ξ) = x1+ξ1(2 −x2)
ξ2
+s σ =σ(x,ξ) = −ξ2
2−x2
τ1=σ(x1−x0)τ2=−σ(s+x1−x0)
where we have taken these from the proof of Theorem 3.2. To find C∗
2◦C1◦(x,ξ) we calculate
the composition of the covector described in (3.18) with C∗
2. Note that the values of x0and
1For convenience, we will abbreviate the set theoretic composition Ci◦ {(x,ξ)}by Ci◦(x,ξ).
10

sare the same in both calculations and are given by (3.18). After using (3.8) and that
ξ1(2−x2)
ξ2=x0−s−x1, one sees that
(3.19)
λ12(x,ξ) = ((y1, y2),η) where
y1=y1(x,ξ) = s(x1−x0)
2(x1−x0) + s+x0
y2=y2(x,ξ)=2−s(2 −x2)2|ξ|2
ξ2
2−(y1−(x0+s))2
η=−2ξ1−sξ2
2−x2(y−c2(s(x,ξ)(x0)1(x,ξ)))
where x0= (x0)1(x,ξ) and s=s(x,ξ) are given in (3.18) and ηis calculated using the
expression (3.8) with j= 2.
Note that the function λ12 is defined for only some (x,ξ)∈D; for example if the argument
for the square root defining y2(x,ξ) is negative, then y2(x,ξ) is not defined and the point
(x,ξ) will not generate artifacts in Λ12.
A similar calculation shows for (y,η)∈Dthat
(3.20)
λ21(y,η) = ((x1, x2),ξ) where
x1=x1(y,η) = s(y1−x0)
−2(y1−x0) + s+x0
x2=x2(y,η) = 2 −ps2(y,η)+1−(x1−(x0−s))2
ξ=2η1−sη2
2−y2(x−c1(s(y,η),(x0)2(y,η)))
where
s=s(y,η) = s(2 −y2)2|η|2
η2
2−1x0= (x0)2(y,η) = y1+η1(2 −y2)
η2−s.
Note that the function λ21 is not defined for all (y,η)∈D, and other points (y,η) do not
generate artifacts. This is for the same reason as for λ12.
Remark 3.4.The artefacts caused by a singularity of fare as strong as the reconstruction
of that singularity. To see this, first note that each T∗
jTismooths of order one in Sobolev
scale since it an FIO of order −1 [24, Theorem 4.3.1].
The visible singularities in the reconstruction come from the compositions T∗
1T1and T∗
2T2
since these are pseudodifferential operators of order −1. The artefacts come from the “cross”
compositions T∗
2T1and T∗
1T2, and they are FIO of order −1. Therefore, since the terms that
preserve the real singularities of f,T∗
iTi,i= 1,2, are also of order −1, T∗Tsmooths each
singularity of fby one order in Sobolev scale and the composition T∗
2T1(corresponding to
the artifact λ12, if defined at this covector) can create an artefact from that singularity that
are also one order smoother than that singularity, and similarly with the composition T∗
1T2.
Second, our results are valid, not only for the normal operator T∗Tbut for any filtered
backprojection method T∗PTwhere Pis a pseudodifferential operator. This is true since
pseudodifferential operators have canonical relation ∆ and they do not move singularities,
11

so our microlocal calculations are the same. If Phas order k, then T∗PTdecreases the
Sobolev order of each singularity of fby order (k−1) in Sobolev norm and can create an
artefact from that singularity of the same order.
3.2. Artifacts for T∗Tdue to limited data. In practice we do not have access to
Tf(s, x0) for all s∈(0,∞) (or r∈(1,∞)) and x0∈R, and will have knowledge of
x0∈(−a, a) and r∈(1, rM) for some a > 0 (see figures 1 and 2) and maximum radius
rM>1.
We now evaluate which wavefront directions (x,ξ) will be visible from this limited data.
Let us consider the pair (x,ξ)∈C2(s, x0)×S1and let βbe the angle of ξfrom the vertical
as depicted in figure 2. Then c2(s, x0) = ((2 −x2) tan β+x1,2) and
|x−c2(s, x0)|2=r2=⇒(1 + tan2β)(2 −x2)2=r2=⇒tan β=sr2
(2 −x2)2−1.
Let βm=βm(x)∈(0, π/2) be defined by
(3.21) tan βm=sr2
M
(2 −x2)2−1
(noting that we only consider xsuch that 1 > x2>2−rM). Then the maximum directional
coverage of the singularities (wavefront set) at a given x∈Xwhich are resolved by the
Compton data are described by the open cone of ξ∈S1
(3.22) CT={±(sin β, −cos β) : −βm< β < βm},
and the opening angle of the cone depends on the depth of x(i.e. x2). See figure 2. The
cone CTillustrated corresponds to the case when β=βm.
In all of our numerical experiments, we set the tunnel height as rm−1 = 6 and the detector
line width is 2a= 8. We let rM> rmbe large enough to penetrate the entire scanning tunnel
(up to the line {x2= 2 −rm}as highlighted in figures 1 and 2), so as to imply a unique
reconstruction [55]. Specifically we set the maximum radius rM= 9 and simulate T(r, x0)
for r∈ {1+0.02j: 1 ≤j≤400}and x0∈ {−4+0.04j: 1 ≤j≤200}. Further the densities
considered are represented on [−2,2] ×[−3,1] (200 ×200 pixel grid) in the reconstructions
shown. The machine design considered is such that for any x∈[−2,2] ×[−1.5,1] we have
the maximal directional coverage in CTallowed for the limited r < rM(see figure 7). With
the exception of the horizontal bar phantom depicted in figure 14, all objects considered for
reconstruction are approximately in this region.
To demonstrate the artefacts, we apply a discrete form of T∗Tto a delta function. We
have the expansion
(3.23) T∗T= (T1+T2)∗(T1+T2) = T∗
1T1+T∗
2T2+T∗
1T2+T∗
2T1.
Using equations (3.18), (3.19), and (3.20) one can show for g∈L2(Y) that the backprojection
operators T∗
j,j= 1,2 can be written
(3.24) T∗
jg(x) = Zβm
−βm
g√r2−1, x1+ (−1)j−1√r2−1 + rsin βr=(2−x2)
cos β
dβ.
12

(a) T∗Tδ.(b) (T∗
1T1+T∗
2T2)δ.(c) (T∗
1T2+T∗
2T1)δ.
(d) χS12∪S21 .(e) Λij artefacts. (f) Λij artefacts on [−2,2] ×
[−3,1].
Figure 3. T∗Tδ(the δfunction is centered at O= (0,−1)) images with the
predicted artefacts due to the limited data backprojection (on S12 ∪S21) and
those induced by Λ12 and Λ21.
Note that we are not restricting x0to [−a, a] but we are restricting sto 0,pr2
M−1, and
hence the cutoff function ψof equation 3.14 is equal to one on the bounds of integration.
Now, let fbe a delta function at y. We calculate the artifacts
(3.25) T∗
1T2f(x)6= 0 ⇐⇒ ∃β∈[−βm, βm] s.t. |y−c2(s, x0)|=r,
where r=2−x2
cos β,s=√r2−1 and x0=x1+s+rsin β. Similarly
(3.26) T∗
2T1f(x)6= 0 ⇐⇒ ∃β∈[−βm, βm] s.t. |y−c1(s, x0)|=r,
where r=2−x2
cos β,s=√r2−1 and x0=x1−s+rsin β. Hence the only contributions to the
backprojection from T∗
1T2and T∗
2T1are on the following sets:
(3.27) S12 ={x:∃β∈[−βm, βm] s.t. |y−c2(s, x0)|=r}
where r=2−x2
cos βand x0=x1+s+rsin βand
(3.28) S21 ={x:∃β∈[−βm, βm] s.t. |y−c1(s, x0)|=r}.
where r=2−x2
cos βand x0=x1−s+rsin β. This means that all Λij artifacts will be in these
sets. Note that besides the Λij artifacts shown in figure 4e and 4f there are limited data
artifacts caused by circles meeting yof radius rM(figures 4a-4c) and these are of higher
strength in Sobolev norm.
To simulate a δfunction discretely we assign a value of 1 to nine neighbouring pixels in
a 200–200 grid (which will represent [−2,2] ×[−3,1]) and set all other pixel values to zero.
13

(a) T∗Tδ.(b) (T∗
1T1+T∗
2T2)δ.(c) (T∗
1T2+T∗
2T1)δ.
(d) χS12∪S21 .(e) Λij artefacts. (f) Λij artefacts on [−2,2] ×
[−3,1].
Figure 4. T∗Tδ(the δfunction is centered at (0,0.9)) images with the pre-
dicted artefacts due to the limited data backprojection (on S12 ∪S21) and those
induced by Λ12 and Λ21.
Letting our discrete delta function be denoted by xδ, we approximate T∗Tδ≈ATAxδ, where
Ais the discrete form of T. For comparison we show images of
(T∗
1T2+T∗
2T1)δ≈(AT
1A2+AT
2A1)xδ,
a characteristic function on the set S12 ∪S21, and the artefacts induced by Λ12 and Λ21.
Here Ajis the discrete form of Tj, for j= 1,2. See figure 3. For example, in figure 3b we
see a butterfly wing type artefact in (AT
1A1+AT
2A2)xδ. This is due to the limited rand x0
data inherent to our acquisition geometry (there are unresolved wavefront directions). In the
(AT
1A2+AT
1A2)xδimage of figure 3c we see artefacts appearing on the set S12 ∪S21 as shown
in figure 3d. This is as predicted by our theory. The artefacts induced by the Λij in this case
lie outside the scanning region ([−2,2] ×[−3,1]), and hence they are not observed in the
reconstruction. See figures 3e and 3f. In figure 4 the artefact curves intersect [−2,2]×[−3,1]
in the top left and right-hand corners respectively. See figures 4e and 4f. In this case the
artefacts are observed faintly in the reconstructions (their magnitude is small compared to
the delta function), and it is unclear whether they align with our predictions.
To show the artefacts induced by the Λij more clearly, we repeat the analysis above using
filtered backprojection, and a second derivative filter Φ = d2
dr2.That is we show images of
T∗ΦTδ. Note that Φis applied in the variable r(the torus radius). The application of
derivative filters is a common idea in lambda tomography [13, 14], and is known to highlight
the image contours (singularities or edges) in the reconstruction [43, Theorem 3.5]. See
figure 5. As the artefacts induced by Λij appeared to be largely outside the scanning region
14

(a) T∗ΦTδ(location 1). (b) (T∗
1ΦT2+T∗
2ΦT1)δ.(c) Λij artefacts.
(d) T∗ΦTδ(location 2). (e) (T∗
1ΦT2+T∗
2ΦT1)δ.(f) Λij artefacts.
Figure 5. T∗ΦTδand (T∗
1ΦT2+T∗
2ΦT1)δimages with the predicted arte-
facts induced by Λ12 and Λ21. We give examples for two δfunction locations.
Location 1 is (0,0.85) and location 2 is (-2.8,0.9).
([−2,2] ×[−3,1]) in our previous simulations, we have increased the scanning region size to
[−3,3] ×[−4,2], to show more the effects of the Λij in the observed reconstruction. Here
Φ suppresses the artefacts due to limited data, and the Λij artefacts appear as additional
contours in the reconstruction. The observed artefacts appear most clearly in figures 5b and
5e, and align exactly with our predictions in figures 5c and 5f.
Remark 3.5.With precise knowledge of the locations of the artefacts induced by the Λij
we can assist in the design of the proposed parallel line scanner. That is we can choose a,
rMand the scanning tunnel size to minimize the presence of the nonlocal artefacts in the
reconstruction (i.e., those from Λij (f)). Such advice would be of benefit to our industrial
partners in airport screening to remove the concern for nonlocal artefacts in the image
reconstruction of baggage. Indeed the machine dimensions we have chosen seem to be suitable
as the artefacts appear largely outside the reconstruction space (see figures 3 and 4).
4. The transmission artefacts
The detector row DCcollects Compton (back) scattered data, which determines Tf(s, x0)
for a range of sand x0, where f=neis the electron charge density. The forward detector
array DAcollects transmission (standard X-ray CT) data, which determine a set of straight
line integrals over the attenuation coefficient f=µE, for some photon energy E.The data
is limited to the set of lines which intersect S(the source array) and DA.This limited data
can cause artefacts in the X-ray reconstruction, and we will analyze these artefacts using
the theory in [5]. Let Ls,θ ={x∈R2:x·Θ = s}be the line parameterized by a rotation
15

θ∈[0, π] and a directed distance from the origin s∈R. Here Θ = (cos θ, sin θ) and Ls,θ is
the line containing sΘ and perpendicular to Θ.
In the scanning geometry of this article, the set Sof X-ray transmitters is the segment
between (−4,3) and (4,3) and the set of X-ray detectors, DA, is the segment between
(−4,−5) and (4,−5) as in figure 1. For this reason, the cutoff in the sinogram space is
described by the set
(4.1) H={(s, θ)∈R×[−π/2, π/2] : Ls,θ ∩S6=∅and Ls,θ ∩DA6=∅}.
The characteristic function of Happears as a skewed diamond shape in sinogram space.
To illustrate the added artifacts inherent in this incomplete data problem, we simulate
reconstructions of delta functions with transmission CT data on H.That is, we apply the
backprojection operator R∗
LRLto δfunctions, where RLfdenotes Rf for (s, θ)∈H. See
figure 6. By the theory in [5], artefacts caused by the incomplete data occur on lines Ls,θ for
Figure 6. RT
LRLδ(t,−1) images at varying δfunction translations along the
line x2=−1.
(s, θ) in the boundary of H. Each delta function in Figure 6 is at a point (t, −1) for some
twith 0 < t < 2, so the lines that meet the support of the delta function, (t, −1) that are
in the boundary of Hmust also contain either (4,3) or (4,−5). This is true because Sand
DAare mirror images about the line y=−1 and t∈(0,2).
Furthermore, by symmetry (the δfunctions are on the center line of the scanning region),
these artifact lines will be reflections of each other in the vertical line x1=t. This is
illustrated in our reconstructions in Figure 6. The opening angle of the cone in the delta
reconstructions decreases (fewer wavefront set directions are stably resolved) as we translate
δto the right on the line x1=−1.
Example 4.1.We now use these ideas to analyze the visible wavefront directions for the joint
problem. Let S= [−4,4] ×[−5,3] be the square between Sand DAand let O= (0,−1), the
center of S. We consider wavefronts at points (x1, x2)∈[−2,2] ×[−3,1] which is a square
centered at Oand the region in which our simulated reconstructions are done.
By (4.1), lines in the data set must intersect both Sand DA, so lines through Oin the
data set are all lines through Othat are more vertical than the diagonals of S. Because
visible wavefront directions are normal to lines in the X-ray CT data set [38], the wave-
front directions which are resolved lie in the horizontal open cone between normals to these
diagonals. Therefore, they are in the cone
CR={±(cos α, sin α) : −π/4< α < π/4},
16

which is shown in figure 1.
An analysis of the singularities that are visible by the Compton data was done in section
3.2. For the point O, the angle defined by (3.21) gives βm= 1.23 and the cone of visible
directions given by (3.22) is the vertical cone with angles from the vertical between −βmand
βmsince the parameter rM= 9. A calculation shows that 2(π/4 + βm)> π and this implies
CR∪CT=S1and we have a full resolution of the image singularities at O.
Figure 7. Picture of the range of wavefront directions that are visible at
points in [−2,2] ×[−3,1] from the joint data. Angles are measured from
0◦= no coverage to 360◦= full coverage. We let Γ denote the solid light-
colored (yellow) region (roughly the top 3/4 of the figure) in which all wavefront
directions are recovered.
However, for points near the bottom x2=−3, there are invisible singularities that are not
visible from either the Compton or X-ray data. For example, the vertical direction (0,1) is
not normal to any circle in the Compton data set at any point (0, x2) for x2∈[−2.5,−3].
Figure 7 shows the points for which all wavefront directions are visible at those points (yellow
color–roughly for points (x1, x2) for x2>−2) and near the bottom of the reconstruction
region, there are more missing directions.
5. A joint reconstruction approach and results
In this section we detail our joint reconstruction scheme and lambda tomography regu-
larization technique, and show the effectiveness of our methods in combatting the artefacts
observed and predicted by our microlocal theory. We first explain the physical relationship
between µEand ne, which will be needed later in the formulation of our regularized inverse
problem.
5.1. Relating µEand ne.The attenuation coefficient and electron density satisfy the for-
mula [53, page 36]
(5.1) µE(Z) = ne(Z)σE(Z),
where σEdenotes the electron cross section, at energy E. Here Zdenotes the effective
atomic number. In the proposed application in airport baggage screening (among many
other applications such as medical CT) we are typically interested in the materials with low
effective Z. Hence we consider the materials with Z < 20 in this paper. For large enough
Eand Z < 20, σE(Z) is approximately constant as a function of Z. Equivalently µEand
17

nEare approximately proportional for low Zand high Eby equation 5.1. See figure 8. We
see a strong correlation between neand µEwhen E= 100keV and Z < 20, and even more
so when Eis increased to E= 1MeV. The sample of materials considered consists of 153
compounds (e.g. wax, soap, salt, sugar, the elements) taken from the NIST database [29].
In this case σE≈νfor some ν∈Ris approximately constant and we have µE≈νne. For a
Figure 8. Scatter plot of nevs µEfor E= 100keV (left) and E= 1MeV
(right), for 153 compounds with effective Z < 20 taken from the NIST [28]
database. The correlation is R=0.93 (left) and R=0.98 (right).
given energy E,νis the slope of the straight line fit as in figure 8. Throughout the rest of
this paper, we set νas the slope of the straight line in the left hand of figure 8 (i.e. ν≈0.57),
and present reconstructions of µEfor E= 100keV.
5.2. Lambda regularization; the idea. In sections 3 and 4 we discovered that the RLµE
and Tnedata provide complementary information regarding the detection and resolution of
edges in an image. More specifically the line integral data resolved singularities in an open
cone CRwith central axis x1and the toric section integral data resolved singularities in a
cone CTwith central axis x2. So the overlapping cones CR∪CTgive a greater coverage of S1
than when considered separately. In figures 3, 4 and 6, this theory was later verified through
reconstructions of a delta functions by (un)filtered backprojection.
For a further example, let us consider a more complicated phantom than a delta function,
one which is akin to densities considered later for testing our joint reconstruction and lambda
regularization method. In figure 9 we have presented reconstructions of an image phantom
f(with no noise) from RLf(transmission data–middle figure) using FBP, and from Tf
(Compton data–right figure) by an application of T∗d2
dr2(a contour reconstruction). In the
reconstruction from Compton data, we see that the image singularities are well resolved in
the vertical direction (x2), and conversely in the horizontal direction (x1) in the reconstruc-
tion from transmission data. In the middle picture (reconstruction from RL), the visible
singularities of the object are tangent to lines in the data set (normal wavefront set) and
the artifacts are along lines at the end of the data set that are tangent to boundaries of the
objects. In the right-hand reconstruction from Compton data, the visible boundaries are
tangent to circles in the data set and the streaks are along circles at the end of the data set.
Note that the visible boundaries in each picture complement each other and together, image
the full objects. This is all as predicted by the theory of sections 3 and 4 (and is consistent
with the theory in [5] and [15]) and highlights the complementary nature of the Compton
and transmission data in their ability to detect and resolve singularities.
18

Given the complementary edge resolution capabilities of RLµEand Tne, and given the
approximate linear relationship between µEand ne, we can devise a joint linear least squares
reconstruction scheme with the aim to recover the image singularities stably in all directions
in the neand µEimages simultaneously. To this end we employ ideas in lambda tomography
Figure 9. Image phantom f(left), a reconstruction from RLfusing FBP
(middle) and T∗d2
dr2Tf(right).
and microlocal analysis.
Let f∈ E(Rn) and let Rf (s, θ) = Rfθ(s) = RLs,θ fdldenote the hyperplane Radon
transform of f, where Ls,θ is as defined in section 4. The Radon projections Rfθdetect
singularities in fin the direction Θ = (cos θ, sin θ) (i.e. the elements (x,Θ) ∈WF(f)).
Applying a derivative filter dm
dsmRθ, for some m≥1, increases the strength of the singularities
in the Θ direction by order min Sobolev scale. Given f, g ∈ E(Rn), we aim to enforce
a similarity in WF(f) and WF(g) through the addition of the penalty term kdm
dsmR(f−
g)kL2(R×S1)to the least squares solution. Note that we are integrating over all directions
in S1to enforce a full directional similarity in WF(f) and WF(g). Specifically in our case
f=µE,g=neand we aim to minimize the quadratic functional
(5.2) arg min
µE,ne






wRL0
0T
α[dm
dsmR−νdm
dsmR]
µE
ne−

wb1
b2
0






2
2
,
where RLdenotes a discrete, limited data Radon operator, Ris the discrete form of the
full data Radon operator, Tis the discrete form of the toric section transform, b1is known
transmission data and b2is the Compton scattered data. Here αis a regularization parameter
which controls the level of similarity in the image wavefront sets. The lambda regularizers
enforce the soft constraint that µE=νne(since dm
dsmRf = 0 ⇐⇒ f= 0 for f∈L2
c(X)),
but with emphasis on the location, direction and magnitude of the image singularities in the
comparison. Further we expect the lambda regularizers to have a smoothing effect given the
nature of dm
dsmRas a differential operator (i.e. the inverse is a smoothing operation). Hence we
expect αto also act as a smoothing parameter. The weighting w=kT k2/kRLk2is included
so as to give equal weighting to the transmission and scattering datasets in the inversion. We
denote the joint reconstruction method using lambda tomography regularizers as “JLAM”.
A common choice for min lambda tomography applications is m= 2 [7, 43] (hence the name
“lambda regularizers”). With complete X-ray data, the application of a Lambda term yields
this R∗d2
ds2Rf =−4π√−∆f[31, Example 9], so the singularities of fare preserved and
emphasized by order 1 in Sobolev scale, so they will dominate the Lambda reconstruction.
19

Figure 10. Top row – Simple density (left) and attenuation (right) phan-
toms. Bottom row – Complex density (left) and attenuation (right) phantoms.
The associated materials are labelled in each case.
TV JLAM JTV LPLS
ne.26 .14 .05 .03
µE.40 .15 .05 .03
F-score TV JLAM JTV LPLS
supp(ne).81 .99 .99 ∼1
∇ne.76 .87 .82 .87
supp(µE).78 ∼1∼1∼1
∇µE.49 .87 .82 .86
Table 1. Simple phantom and F-score comparison using TV, JLAM, JTV
and LPLS.
±JLAM JTV LPLS
ne.15 ±.01 .05 ±.005 .06 ±.002
µE.16 ±.02 .05 ±.005 .06 ±.03
F±JLAM JTV LPLS
supp(ne).99 ±.01 .99 ±.004 ∼1±.005
∇ne.86 ±.01 .84 ±.02 .86 ±.04
supp(µE).98 ±.04 .99 ±.005 ∼1±.005
∇µE.86 ±.01 .85 ±.02 .87 ±.04
Table 2. Randomized simple phantom ±and F±comparison over 100 runs
using JLAM, JTV and LPLS.
Hence choosing m= 2 is sufficient for a full recovery of the image singularities. Since the
singularities are dominant in the lambda term, they are matched accurately in (5.2). Indeed
we have already seen the effectiveness of such a filtering approach in recovering the image
20

Density neAttenuation µE,E= 100keV
TVJLAMJTVLPLS
Figure 11. Simple phantom reconstructions, noise level η= 0.1. Comparison
of methods TV, JLAM, JTV and LPLS.
21

contours earlier in the right hand of figure 9. We find that setting m= 2 here works well as
a regularizer on synthetic image phantoms and simulated data with added pseudo random
noise, as we shall now demonstrate. We note that the derivative filters for m6= 2 are also
worth exploration but we leave such analysis for future work.
5.3. Proposed testing and comparison to the literature. To test our reconstruction
method, we first consider two test phantoms, one simple and one complex (as in [55]). See
figure 11. The phantoms considered are supported on Γ, the region in figure 7 in which there
is full wavefront coverage from joint X-ray and Compton scattered data. The simple density
phantom consists of a Polyvinyl Chloride (PVC) cuboid and an Aluminium sphere with an
approximate density ratio of 1:2 (PVC:Al). The complex density phantom consists of a water
ellipsoid, a Sulfur ellipsoid, a Calcium sulfate (CaSO4) right-angled-triangle and a thin film
of Titanium dioxide (TiO2) in the shape of a cross. The density ratio of the materials which
compose the complex phantom is approximately 1:2:3:4 (H2O:S:CaSO4:TiO2). The density
values used are those of figure 8 taken from the NIST database [28], and the background
densities (≈0) were set to the density of dry air (near sea level). The corresponding attenu-
ation coefficient phantoms are simulated similarly. The materials considered are widely used
in practice. For example CaSO4is used in the production of plaster of Paris and stucco (a
common construction material) [57], and TiO2is used in the making of decorative thin films
(e.g. topaz) and in pigmentation [56, page 15].
To simulate data we set
(5.3) b=b1
b2=RLµE
Tne,
and add a Gaussian noise
(5.4) bη=b+ηkbk2
vG
√l,
for some noise level η, where lis the length of band vGis a vector of length lof draws
from N(0,1). For comparison we present separate reconstructions of µEand neusing Total
Variation (TV regularizers). That is we will find
(5.5) arg min
µEkRLµE−b1k2
2+αTV(µE)
to reconstruct µEand
(5.6) arg min
nekT ne−b2k2
2+αTV(ne)
for ne, where TV(f) = k∇fk1and α > 0 is a regularization parameter. We will denote
this method as “TV”. In addition we present reconstructions using the state-of-the-art joint
reconstruction and regularization techniques from the literature, namely the Joint Total
Variation (JTV) methods of [20] and the Linear Parallel Level Sets (LPLS) methods of [12].
To implement JTV we minimize
(5.7) arg min
µE,ne


wRL0
0TµE
ne−wb1
b2



2
2
+αJTVβ(µE, ne),
22

where w=kT k2/kRLk2as before, and
(5.8) JTVβ(µE, ne) = Z[−2,2]×[−3,1] k∇µE(x)k2
2+k∇ne(x)k2
2+β21
2dx,
where β > 0 is an additional hyperparameter included so that the gradient of JTVβis defined
at zero. This allows one to apply techniques in smooth optimization to solve (5.7).
To implement LPLS we minimize
(5.9) arg min
µE,ne


wRL0
0TµE
ne−wb1
b2



2
2
+αLPLSβ(µE, ne),
where
(5.10) LPLSβ(µE, ne) = Z[−2,2]×[−3,1] k∇µE(x)kβk∇ne(x)kβ− |∇µE(x)· ∇ne(x)|β2dx,
where |x|β=p|x|2+β2and kxkβ=pkxk2
2+β2for β > 0. The JTV and LPLS penalties
seek to impose soft constraints on the equality of the image wavefront sets of µEand ne. For
example setting β= 0 in the calculation of LPLSβyields
LPLSβ(µE, ne) = k∇µEk2k∇nek2− |∇µE· ∇ne|
=k∇µEk2k∇nek2(1 − | cos θ|),
(5.11)
where θis the angle between ∇neand ∇µE. Hence (5.11) is minimized for the gradients
which are parallel (i.e. when θ= 0, π), and thus using LPLSβas regularization serves to
enforce equality in the image gradient direction and location (i.e. when LPLSβis small, the
gradient directions are approximately equal).
We wish to stress that the comparison with JTV and LPLS is included purely to illustrate
the potential advantages (and disadvantages) of the lambda regularizers when compared to
the state-of-the-art regularization techniques. Namely is the improvement in image quality
due to joint data, lambda regularizers or are they both beneficial? We are not claiming a
state-of-the-art performance using JLAM, but our results show JLAM has good performance,
and it is numerically easier to implement, requiring only least squares solvers. There are
two hyperparameters (αand β) to be tuned in order to implement JTV and LPLS, which
is more numerically intensive (e.g. using cross validation) in contrast to JLAM with only
one hyperparameter (α). Moreover the LPLS objective is non-convex [12, appendix A], and
hence there are potential local minima to contend with, which is not an issue with JLAM,
being a simple quadratic objective.
TV JLAM JTV LPLS
ne.36 .24 .16 .09
µE.63 .30 .19 .13
F-score TV JLAM JTV LPLS
supp(ne).78 .98 .97 .99
∇ne.73 .83 .84 .84
supp(µE).65 .94 .98 .99
∇µE.39 .83 .84 .85
Table 3. Complex phantom and F-score comparison using TV, JLAM,
JTV and LPLS.
To minimize (5.2), we store the discrete forms of RL,Rand Tas sparse matrices and
apply the Conjugate Gradient Least Squares (CGLS) solvers of [22, 23] (specifically the
23

Density neAttenuation µE,E= 100keV
TVJLAMJTVLPLS
Figure 12. Complex phantom reconstructions, noise level η= 0.1. Compar-
ison of methods TV, JLAM, JTV and LPLS.
24

±JLAM JTV LPLS
ne.25 ±.02 .13 ±.03 .13 ±.03
µE.31 ±.03 .16 ±.02 .17 ±.04
F±JLAM JTV LPLS
supp(ne).98 ±.01 .98 ±.005 .99 ±.004
∇ne.76 ±.06 .78 ±.05 .75 ±.05
supp(µE).90 ±.06 .96 ±.03 .98 ±.007
∇µE.73 ±.07 .77 ±.05 .74 ±.05
Table 4. Randomized complex phantom ±and F±comparison over 100 runs
using JLAM, JTV and LPLS.
“IRnnfcgls” code) with non-negativity constraints (since the physical quantities neand µE
are known a-priori to be nonnegative). To solve equations (5.5) and (5.6) we apply the
heuristic least squares solvers of [22, 23] (specifically the “IRhtv” code) with TV penalties
and non-negativity constraints. To solve (5.7) and (5.9) we apply the codes of [12], modified
so as to suit a Gaussian noise model (a Poisson model is used in [12, equation 3]). The
relative reconstruction error is calculated as =kx−yk2/kxk2, where xis the ground
truth image and yis the reconstruction. For all methods compared against we simulate data
and added noise as in equations (5.3) and (5.4), and the noise level added for each simulation
is η= 0.1 (10% noise). We choose αfor each method such that is minimized for a noise
level of η= 0.1. That is we are comparing the best possible performance of each method.
We set βfor JTV and LPLS to the values used on the “lines2” data set of [12]. We do not
tune βto the best performance (as with α) so as to give fair comparison between TV, JLAM,
JTV and LPLS. After the optimal hyperparameters were selected, we performed 100 runs of
TV, JLAM, JTV and LPLS on both phantoms for 100 randomly selected sets of materials.
That is, for 100 randomly chosen sets of values from figure 8 and the NIST database, with
the NIST values corresponding to the nonzero parts of the phantoms. We present the mean
(µ) and standard deviation (σ) relative errors over 100 runs in the left-hand of tables 2
and 4 for the simple and complex phantom respectively. The results are given in the form
±=µ±σfor each method. In addition to the relative error , we also provide metrics
to measure the structural accuracy of the results. Specifically we will compare F-scores on
the image gradient and support, as is done in [49, 2]. The gradient F-score [2] measures the
wavefront set reconstruction accuracy, and the support F-score [49, page 5] (see DICE score)
is a measure of the geometric accuracy. That is, the support F-score checks whether the
reconstructed phantoms are the correct shape and size. The F-score takes values on [0,1].
For this metric, values close to one indicate higher performance, and conversely for values
close to zero. Similarly to , we present the F-scores of the randomized tests in the form
F±=µF±σF, where µFand σFare the mean and standard deviation F-scores respectively.
In all tables, the support F-scores are labelled by supp(ne) and supp(µE), and by ∇neand
∇µEfor the gradient F-scores.
5.4. Results and discussion. See figure 11 for image reconstructions of the simple phan-
tom using TV, JLAM, JTV and LPLS, and see table 1 for the corresponding and F-score
values. See table 2 for the ±and F±values calculated over 100 randomized simple phantom
reconstructions. For the complex phantom, see figure 12 for image reconstructions, and table
3 for the and F-score values. See table 4 for ±and F±. In the separate reconstruction of ne
(using method TV) we see a blurring of the ground truth image edges (wavefront directions)
in the horizontal direction and there are artefacts in the reconstruction due to limited data,
25

as predicted by our microlocal theory. In the TV reconstruction of µEwe see a similar effect,
but in this case we fail to resolve the wavefront directions in the vertical direction due to
limited line integral data. This is as predicted by the theory of section 4 and [5]. In section
Figure 13. Horizontal Al bar density (left) and attenuation (right) phantoms.
3 we discovered the existence also of nonlocal artefacts in the nereconstruction, which were
induced by the mappings λij. However these were found to lie largely outside the imaging
space unless the singularity in question (x,ξ)∈WF(ne) were such that xis close to the
detector array (see figures 3 and 4). Hence why we do not see the effects of the λij in the
phantom reconstructions, as the phantoms are bounded sufficiently away from the detector
array. The added regularization may smooth out such artefacts also, which was found to be
the case in [55].
Using the joint reconstruction methods (i.e. JLAM, JTV and LPLS) we see a large
reduction in the image artefacts in neand µE, since with joint data we are able to stably
resolve the image singularities in all directions. The improvement in and the F-score is
also significant, particularly in the µEreconstruction. Thus it seems that the joint data is
the greater contributor (over the regularization) to the improvement in the image quality,
as the approaches with joint data each perform well. Upon comparison of JLAM, JTV and
LPLS, the metrics are significantly improved when using JTV and LPLS over JLAM, but
the image quality and F-scores are highly comparable. This indicates that, while the noise
in the reconstruction is higher using JLAM, the recovery of the image edges and support is
similar using JLAM, JTV and JLAM. As theorized, the lambda regularizers were successful
in preserving the wavefront sets of µEand ne. However there is a distortion present in the
nonzero parts of the JLAM reconstruction. This is the most notable difference in JLAM
and JTV/LPMS, and is likely the cause of the discrepancy. So while the edge preservation
and geometric accuracy of JLAM is of a high quality (and this was our goal), the smoothing
properties of JLAM are not up to par with the state-of-the-art currently. We note however
that the JTV and LPLS objectives are nonlinear (with LPLS also non-convex) and require
significant additional machinery (e.g. in the implementation of the code of [12] used here) in
the inversion when compared to JLAM, which is a straight forward implementation of linear
least squares solvers.
5.5. Reconstructions with limited data. The simple and complex phantoms considered
thus far are supported within Γ (the yellow region of figure 7) so as to allow for a full
26

Density neAttenuation µE,E= 100keV
TVJLAMJTVLPLS
Figure 14. Horizontal bar phantom reconstructions, noise level η= 0.1.
Comparison of methods TV, JLAM, JTV and LPLS.
27

TV JLAM JTV LPLS
ne.28 .09 .04 .02
µE.68 .11 .09 .07
F-score TV JLAM JTV LPLS
supp(ne).71 .99 ∼1∼1
∇ne.64 .82 .79 .83
supp(µE).54 ∼1∼1∼1
∇µE.53 .82 .80 .90
Table 5. Al bar phantom and F-score comparison using TV, JLAM, JTV
and LPLS.
±JLAM JTV LPLS
ne.10 ±.02 .04 ±.004 .03 ±.02
µE.12 ±.03 .12 ±.03 .08 ±.03
F±JLAM JTV LPLS
supp(ne).99 ±.02 ∼1±.003 ∼1±.001
∇ne.83 ±.02 .79 ±.02 .84 ±.02
supp(µE).98 ±.06 .99 ±.02 ∼1±.008
∇µE.81 ±.03 .77 ±.02 .85 ±.02
Table 6. Randomized bar phantom ±and F±comparison over all NIST
materials considered (153 runs) using JLAM, JTV and LPLS.
wavefront coverage in the reconstruction. To investigate what happens when the object is
supported outside of Γ, we present additional reconstructions of an Aluminium bar phantom
with support towards the bottom (close to x2=−3) of the reconstruction space. See
figure 13. In this case we have limited data and the full wavefront coverage is not available
with the combined X-ray and Compton data sets. Image reconstructions of the Al bar
phantom are presented in figure 14, and the corresponding and F-score values are displayed
in table 5. See table 4 for the ±and F±values corresponding to the randomized bar
phantom reconstructions. In this case ±and F±were calculated from reconstructions of
153 bar phantoms (we used 100 runs previously), replacing the Al density value of figure 13
with one of each NIST value considered (153 in total). The reconstruction processes and
hyperparameter selection applied here were exactly the same as for the simple and complex
phantom. In this case we see artefacts in the Compton reconstruction along curves which
follow the shape of the boundary of Γ, and the X-ray artefacts constitute a vertical blurring
as before. The error when using JLAM, JTV and LPLS is more comparable in this example
(compared to tables 1 and 3), particularly in the case of the µEphantom. The image quality
and F-scores are again similar as with the simple and complex phantom examples. All joint
reconstruction methods were successful in removing the image artefacts observed in the
separate reconstructions, and thus can offer satisfactory image quality under the constraints
of limited data. However this is only a single test of the capabilities of JLAM, JTV and
LPLS with limited data and we leave future work to conclude such analysis.
6. Conclusions and further work
Here we have introduced a new joint reconstruction method “JLAM” for low effective Z
imaging (Z < 20), based on ideas in lambda tomography. We considered primarily the “par-
allel line segment” geometry of [54], which is motivated by system architectures for airport
security screening applications. In section 3 we gave a microlocal analysis of the toric section
transform T, which was first proposed in [54] for a CST problem. Explicit expressions were
28

provided for the microlocal artefacts and verified through simulation. Section 4 explained
the X-ray CT artefacts using the theory of [5]. Following the theory of sections 3 and 4,
we detailed the JLAM algorithm in section 5. Here we conducted simulation testing and
compared JLAM to separate reconstructions using TV, and to the nonlinear joint inversion
methods, JTV [20] and LPLS [12] from the literature. The joint inversion methods consid-
ered (i.e. JLAM, JTV and LPLS) were successful in preserving the image contours in the
reconstruction, as predicted. However the smoothing applied by JLAM was not as effective
as JTV and LPLS, and we saw a distortion in the JLAM reconstruction (see figures 11 and
12). JTV and LPLS were thus shown to offer better performance than JLAM, with LPLS
producing the best results overall. The advantages of JLAM over JTV and LPLS are in
the fast, linear inversion, and the reduction in tuning parameters (one for JLAM, two for
JTV/LPLS). Given the linearity of JLAM, the ideas of JTV and LPLS can be easily in-
tegrated with lambda regularization to modify the objectives of the literature and improve
further the edge resolution of the reconstruction. To preserve the linearity of JLAM we could
also combine JLAM with a Tikhonov regularizer. This may help smooth out the distortion
observed in the JLAM reconstruction. We leave such ideas for future work.
Acknowledgements
This material is based upon work supported by the U.S. Department of Homeland Security,
Science and Technology Directorate, Office of University Programs, under Grant Award
2013-ST-061-ED0001. The views and conclusions contained in this document are those of
the authors and should not be interpreted as necessarily representing the official policies,
either expressed or implied, of the U.S. Department of Homeland Security. The work of
the second author was partially supported by U.S. National Science Foundation grant DMS
1712207. The authors thank the referees for thorough reviews and thoughtful comments that
improved the article.
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Appendix A. Generating the plots of figure 8
The generation of the plots of figure 8 is explained in more detail here. We will explain
the generation of the plot for E= 100keV. Refer to figure 15. We first plotted µEfor
E= 100keV against nefor all materials in the NIST database [28] with effective Zless than
31

20. This is the left hand plot of figure 15. The set of materials with effective Z < 20 was
Zeff ={Z:σE(Z)< σE(20), E = 100keV},
where σEis the electron cross section. We noticed a large outlier (coal, or amorphous
Carbon) which corrupts the correlation in our favour, and hence we chose to remove the
material from consideration in simulation. The outlier is highlighted in the left hand plot.
After the outlier was removed we noticed a number of materials located at the origin (with
negligible attenuation coefficient and density, such as air) in the middle scatter plot of figure
15. As such materials again bias the correlation and plot standard deviation in our favour,
these were removed to produce the left hand plot of figure 8 in the right hand of figure 15.
The same points were removed in the generation of the right hand plot of figure 8 also, for
E= 1MeV.
Figure 15. Scatter plot with outlier and origin points included (left,
R=0.98), scatter plot with the outlier removed and origin points included,
the origin points highlighted by an orange circle (middle, R=0.95), and the
scatter plot of figure 8 with outliers and origin points removed (right, R=0.93).
Department of Electrical and Computer Engineering, Tufts University, Medford, MA
USA
Email address:James.Webber@tufts.edu
Department of Mathematics, Tufts University, Medford, MA USA
Email address:Todd.Quinto@tufts.edu
Department of Electrical and Computer Engineering, Tufts University, Medford, MA
USA
Email address:elmiller@ece.tufts.edu
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