www.biomathforum.org/biomath/index.php/biomath

ORIGINAL ARTICLE

New properties of the attenuated V-line
transform for breast cancer detection with

Compton cameras
Hanqiu Tan1, Rim Gouia-Zarrad2

1Department of Mathematics and Applied Mathematics, Virginia Commonwealth University
PO Box 842014, Richmond, Virginia, USA

tanh4@mymail.vcu.edu
2Department of Mathematics and Statistics, American University of Sharjah

Sharjah 26666, UAE
rgouia@aus.edu

Received: 21 July 2017, accepted: 14 November 2017, published: 11 December 2017

Abstract—In the recent past, Compton camera be-
came an attractive alternative to the Anger camera
in scintimammography, known as nuclear medicine
breast imaging or molecular breast imaging. This
novel imaging modality leads to the use of the V-line
transform, which integrates a function along coupled
rays with a common vertex. In previous works the
attenuation phenomena was mostly neglected. How-
ever, in scintimammography ignoring the effect of
the attenuation of photon can significantly degrade
the quality of the reconstruction image. In this
paper, we introduce the attenuated V-line transform
and establish a new integral relation between the
attenuated V-line transform and the exponential
Radon transform. The results are not only interest-
ing as original mathematical discoveries, but also
can be useful in challenging applications e.g., in
breast imaging for tumor detection close to the chest
wall.

Keywords-V-line transform; attenuated V-line

transform; imaging of breast cancer; breast cancer
detection; Compton camera.

I. INTRODUCTION

According to the American Cancer Society, in
the United States, breast cancer is one of the most
commonly diagnosed cancer among women and it
has the second highest mortality rate [1]. Statistic
shows that one in eight women will have breast
cancer in her lifetime [1]. Despite the high chance
of breast cancer related death, early diagnosis and
timely treatment can decrease the fatality rate [2].

Breast cancer screening has made significant
advancement in recent years. Currently, X-ray
mammography and ultrasonography are the two
main techniques applied for breast cancer detec-
tion. According to Lee et al., the widespread use
of mammographic screening has led to nearly 30%
decrease in breast cancer mortality since 1990

Copyright: c© 2017 Tan et al. This article is distributed under the terms of the Creative Commons Attribution License (CC
BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source
are credited.
Citation: Hanqiu Tan, Rim Gouia-Zarrad, New properties of the attenuated V-line transform for breast
cancer detection with Compton cameras, Biomath 6 (2017), 1711147,
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Hanqiu Tan, Rim Gouia-Zarrad, New properties of the attenuated V-line transform for ...

Fig. 1. Single-head Compton camera.

[3]. Ultrasound is an important clinical adjunct
technique to mammography by providing details
of the lesions. However, these two techniques are
not free from limitations. Smith and Andropoulou
claim in [4] that despite its effectiveness for de-
tection of breast cancer among women 50 and 69
years old, mammography suffers radiation risk and
diminished sensitivity in dense breasts of younger
women.

Several new techniques have been developed to
improve the preciseness of breast cancer screening.
One effective invention is the scintimammography.
Scintimammography, known as nuclear medicine
breast imaging has the benefit of enabling breast
cancer detection among young women by using
radioactive materials and electronically collimated
camera called Compton camera. The Compton
camera utilizes the Compton scattering effect to
locate the radioactive source located inside the
breast. A Compton camera consists of two parallel-
positioned detectors (see Fig. 1). Incident gamma
rays are scattered in the scatter detector and sub-
sequently detected by the absorb detector. In both
interactions, the energies E1 and E2 and positions
u1 and u2 are recorded. The angle φ can be found
as follows (e.g. see [5], [6])

cosφ = 1−
mc2E1

(E1 + E2)E2
,

where m is the mass of the electron and c is
the speed of light. The information data u1, u2
and φ can be used to locate the gamma source
somewhere on the cone surface in 3D (see Fig. 2)
and the two semilines with common vertex in 2D
(see Fig. 3).

Another limitation of the conventional tomog-
raphy imaging system is its low efficiency when
close to the detector edge, which leads to failure
in detecting tumors close to the chest wall (e.g.
see [7]). New Compton camera, such as C-SMM
system, overcomes this problem by placing the
detector directly on the chest. It allows the higher
efficiency and sensitivity towards the area close to
the chest wall [7]. On the other hand, Compton
cameras have the flexibility to use a wide range of
radio-pharmaceutical energies. In many detecting
process, two Compton cameras are used with one
below and one above the breast. This system is
called the dual-head Compton camera system. In
[8], Hruska et al mentioned that the dual-head
Compton camera system simultaneously acquires
the superior and inferior views, thus it provides the
views of the breast in both the craniocaudal po-
sition and mediolateral oblique position. Theoreti-
cally speaking, dual-head Compton camera is more
sensitive in detecting breast tumors than the single-
head Compton camera, as it gives two separate
images [9]. However, the preliminary results from
[8] explained that due to the symmetry position,
the results from the two cameras are very similar.
Several works, e.g. [7], [8], [9], [10], concentrated
on the single-head Compton camera to detect
breast cancer (see Fig. 1). In all these works the
attenuation phenomena was neglected. However,
in scintimammography ignoring the effect of the
attenuation of photon can significantly degrade the
quality of the reconstruction image. This problem
has been recently studied by [11]. The authors
studied the attenuated V-line transform in 2 di-
mensions using circular harmonic expansions and
derived an analytic inversion approach in the case
of vertices on a circle.

In this paper we introduce the attenuated V-line
transform and present some of its new properties
on a class of V-lines with vertical central axis and
vertex on the x-axis. We start by detailing the
problem in section 2 before providing the defi-
nitions and recalling the important results about
the V-line Radon transform in section 3. At last,
section 4 introduces the concept of the attenuated

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Hanqiu Tan, Rim Gouia-Zarrad, New properties of the attenuated V-line transform for ...

Fig. 2. The Compton camera.

V-line transform in R2 and present an integral
relations between the attenuated V-line transform
and the exponential Radon transform.

II. FORMULATION OF THE PROBLEM

In 2D scintimammography, f represents the
distribution of radiotracer concentration inside the
breast. Before the γ rays emitted from the radioac-
tive source arrive at the detector, they are attenu-
ated by an attenuation coefficient µ which is a real
function on R2. In the case of uniformly attenuat-
ing medium µ can be approximated as a constant
in the domain of the function f. Therefore the data
may be modeled by a set of exponential weighted
V-line integrals of f over two semilines L+ and
L− with common vertex (u,0) , vertical central
axis and a half opening angle φ, called attenuated
V-line projections V [f]µ(u,φ) (see Fig. 3). After
making all the measurements for all possible φ and
all vertexes (u,0), one obtains a two-dimensional
family of V [f]µ(u,φ) data. The problem of image
reconstruction in 2D scintimammography requires
the inversion of V [f]µ, i.e. finding f from the
measured data V [f]µ. Our goal is to present some
new properties of the attenuated V-line transform
V [f]µ on a class of V-lines with vertical central
axis and vertex on the x-axis.

III. NOTATIONS AND PRELIMINARIES

Let f, be compactly supported function in the
half space R × (0,∞) ∈ S(R2) and let θ =
(cosφ,sinφ)ᵀ ∈ S1 := {v ∈ R2, |v| = 1}.
V (u,φ) simply consists of two half-lines L+ and
L− with common vertex (u,0). We denote by φ

the half opening angle of the V-lines and µ ∈ R
the attenuation constant.

Definition 3.1: The attenuated V-line transform
of f at point (u,φ) ∈ R× (0, π

2
) is defined by

V [f]µ(u,φ) =

∫
V (u,φ)

f(v)eµl(u,v)dl,

where l(u, v) = |(u,0) − (x,y)| is the distance
from the vertex (u,0) to the point with coordinate
(x,y), dl is the length element on the V-line
V (u,φ).

Fig. 3. Geometrical setup of the V-line V (u,φ).

The V-line transform V [f](u,φ) ∈ R × (0, π
2
) is

thus given by V [f]µ=0 (e.g. see [12], [13], [14],
[15], [16], [17], [18], [19], [20], [21], [22]).

Definition 3.2: The exponential Radon trans-
form Tµf on the unit cylinder Z = S1 × R is
defined by

Tµf(θ,s) =

∫
R
eµtf(sθ + tθ⊥)dt

where θ⊥ = (−sinφ,cosφ) ∈ S1.
The classical Radon transform Rf(θ,s) ∈ Z =
S1 ×R is thus given by Tµ=0.

The Fourier transform generated by a function
f(x,y) with respect to the first argument is de-
noted by

f̂(λ,y) =

∫
R
f(x,y)e−iλxdx

where (λ,y) ∈ R×R.

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IV. THE V-LINE TRANSFORM AND ITS
INVERSIONS

The inversion problem of the V-line transform
has been studied by many authors. In the following
we only recall some of the important results useful
for breast cancer detection with Compton cameras.

Theorem 4.1: (Projection-slice theorem for the
V-line Radon transform) Consider a function f ∈
C∞(R2) compactly supported in the half space
R× (0,∞). Then we have

cosφV̂ [f](λ,φ) = 2F(c)[f̂](λ,λtanφ),

where F(c) is the Fourier-cosine transform with
respect to the second argument, f̂ is the Fourier
transform of f with respect to the first argument
and V̂ [f](λ,φ) is the Fourier transform of V [f]
with respect to the first argument.

Theorem 4.2: Consider a function f ∈ C∞(R2)
compactly supported in the half space R×(0,∞)
∈S(R2). We can write

f̂(λ,y) =
|λ|
π

∫ ∞
0

costy
√
t2 + 1

V̂ [f](λ,t) dt,

where f̂ and V̂ [f](λ,φ) are the Fourier transforms
of f and V [f] respectively in the first argument
and t = tanφ.

Proof: See [22], [23].
An alternative inversion formula is based on the

results of the classical Radon transform.
Theorem 4.3: Consider a function f ∈ C∞(R2)

compactly supported in the half space R×(0,∞)
∈S(R2). Then we have

V [f](u,φ) = Rfs(θ,ucosφ)

where fs is an even extension of f obtained by
symmetry with respect to the x-axis and θ =
(cosφ,sinφ)ᵀ ∈ S1.

Proof: We can write the V-line transform

V [f](u,φ) =

∫ ∞
0

f(u + tsinφ,tcosφ)dt

+

∫ ∞
0

f(u− tsinφ,tcosφ)dt.

We use the substitution

y = tcosφ

to obtain

V [f](u,φ) =

∫ ∞
0

f(u + y tanφ,y)
dy

cosφ

+

∫ ∞
0

f(u−y tanφ,y)
dy

cosφ
.

We change y′ = −y in the first integral

V [f](u,φ) =

∫ 0
−∞

f(u−y tanφ,−y)
dy

cosφ

+

∫ ∞
0

f(u−y tanφ,y)
dy

cosφ
.

Using an approach similar to [24], we consider an
even extension obtained by symmetry with respect
to the x-axis denoted by fs:

fs(x,y) =

{
f(x,y) 0 ≤ y
f(x,−y) 0 > y

or we can write it as

fs(x,y) = f(x, |y|).

Now we can combine the two integrals

V [f](u,φ) =
1

cosφ

∫ ∞
−∞

fs(u−y tanφ,y)dy.

We recognize the integral as the integral along the
line perpendicular to (cosφ,sinφ)ᵀ with signed
distance ucosφ from the origin. In fact, we can
explain the above conclusion using the definition
of the classical Radon transform,

Rfs(θ,s) =

∫
R
fs(sθ + tθ

⊥)dt,

where (θ,s) ∈ S1 ×R, we can write

Rfs ((cosφ,sinφ)
ᵀ,ucosφ) =∫

R
fs(u+cos

2 φ−tsinφ,ucosφsinφ+tcosφ)dt.

With the change of variables

y = ucosφsinφ + tcosφ,

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we can write

Rfs ((cosφ,sinφ)
ᵀ,ucosφ)

=

∫
R
fs(u−y tanφ,y)

dy

cosφ

=
1

cosφ

∫
R
fs(u−y tanφ,y)dy.

Corollary 4.4: An exact solution of the inver-
sion problem for the V-line transform is given by
the formula

fs(v)=
1

2πi

∫ π
0
H(∂tV [f])

(
< v.θ >
cosφ

,φ

)
dφ, (1)

where H is the Hilbert transform defined by

Hg(t) =
1

2π

∫
R
sgn(r)ĝ(r)eirtdr,

ĝ(r) is the Fourier transform of g(t) and sgn(r)
is the sign function.

Proof: The filtered backprojection formula is
used to invert the classical Radon transform (see
[25]).

We can conclude that the knowledge of the V-
line transform can be transferred into the knowl-
edge of the classical Radon transform of fs the
original function and its mirror with respect to x-
axis. Using the uniqueness inversion of the classi-
cal Radon transform, fs can be uniquely recovered
and consequently f (see [24] for numerical simu-
lations).

V. THE ATTENUATED V-LINE TRANSFORM

Theorem 5.1: Consider a function f ∈ C∞(R2)
compactly supported in the half space R×(0,∞).
Then we have

cosφV̂ [f]µ(λ,φ)=

∫ ∞
−∞
f̂(λ, |y|)e

µ|y|
cos φe−iλy tanφdy

where f̂ is the Fourier transform of f with respect
to the first argument and V̂ [f]µ(λ,φ) the Fourier
transform of V [f]µ with respect to the first argu-
ment.

Proof: V (u,φ) simply consists of two half-
lines L+ and L− with common vertex (u,0), so

we can write the attenuated V-line transform

V [f]µ(u,φ) =

∫
L+
f(x,y)eµl((u,0),(x,y))dl

+

∫
L−
f(x,y)eµl((u,0),(x,y))dl,

V [f]µ(u,φ) =

∫ ∞
0

f(u + tsinφ,tcosφ)eµtdt

+

∫ ∞
0
f(u−tsinφ,tcosφ)eµtdt.

We use the substitution

y = tcosφ

to obtain

V [f]µ(u,φ) =

∫ ∞
0

f(u + y tanφ,y)e
µy

cos φ
dy

cosφ

+

∫ ∞
0
f(u−y tanφ,y)e

µy

cos φ
dy

cosφ
.

We change y′ = −y in the first integral

cosφV [f]µ(u,φ)=

∫ ∞
−∞

f(u−y tanφ, |y|)e
µ|y|
cos φ dy.

This result is further simplified using the Fourier
transform with respect to the first argument to
obtain

cosφV̂ [f]µ(λ,φ)=

∫ ∞
−∞
f̂(λ, |y|)e

µ|y|
cos φe−iλy tanφdy.

Theorem 5.2: Consider a function f ∈ C∞(R2)
compactly supported in the half space R×(0,∞).
Then we have∫ π

0
V [f]µ(u,φ)dφ =

eµ|u|

|u|
∗f(u),

where u = (u,0).

Proof:∫ π
0
V [f]µ(u,φ)dφ

=

∫ π
0

∫ ∞
0
f(u+tsinφ,tcosφ)eµtdtdφ

+

∫ π
0

∫ ∞
0

f(u−tsinφ,tcosφ)eµtdtdφ.

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Changing φ to −φ in the second integral and using
2π-periodicity of sine and cosine functions, we
obtain∫ π
0
V [f]µ(u,φ)dφ

=

∫ 2π
0

∫ ∞
0
f(u+tcosφ,tsinφ)eµtdtdφ.

Let y = (tcosφ,tsinφ), we get∫ π
0
V [f]µ(u,φ)dφ =

∫
R2
f(u + y)eµ|y|

dy
|y|
.

We can write it in the convolution form∫ π
0
V [f]µ(u,φ)dφ =

eµ|u|

|u|
∗f(u).

This formula is the starting point for reconstruction
method of ρ-filtered layergram type, see ([25],
Chapter V.6). The authors plan to address this
problem in future work.

Using the dual operator T]µ as defined

T]µg(u) =
∫
S1
eµu·x

⊥
g(x, u · x)dx,

we derive a new integral relation between the
attenuated V-line transform and the exponential
Radon transform.

Corollary 5.3: Consider a function f ∈
C∞(R2) compactly supported in the half space
R× (0,∞). Then we have∫ π

0
(V [f]µ + V [f]−µ) (u,φ)dφ = T

]
−µTµf(u).

Proof:∫ π
0

(V [f]µ + V [f]−µ) (u,φ)dφ

=

(
2 cosh(µ|u|)
|u|

)
∗f(u).

∫ π
0

(V [f]µ + V [f]−µ) (u,φ)dφ = T
]
−µTµf(u).

The last equality is due to ([25], Chapter II.6).

ACKNOWLEDGMENT
A part of this paper was written during the first

author’s one semester long visit to the Ameri-
can University of Sharjah (AUS). The work was
supported by the American University of Sharjah
(AUS) research grant FRG2.

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	Introduction
	Formulation of the problem
	Notations and preliminaries
	The V-line transform and its inversions
	The attenuated V-line transform
	References