Tuong T. Truong

On the V-line radon transform and its imaging applications

Radon transforms defined on smooth curves are well known and extensively studied in the literature. In this paper, we consider a Radon transform defined on a discontinuous curve formed by a pair of half-lines forming the vertical letter V. If the classical two-dimensional Radon transform has served as a work horse for tomographic transmission and/or emission imaging, we show that this V-line Radon transform is the backbone of scattered radiation imaging in two dimensions. We establish its analytic inverse formula as well as a corresponding filtered back projection reconstruction procedure. These theoretical results allow the reconstruction of two-dimensional images from Compton scattered radiation collected on a one-dimensional collimated camera. We illustrate the working principles of this imaging modality by presenting numerical simulation results.

On an integral transform and its inverse in nuclear imaging

In a nuclear imaging modality, the goal is to reconstruct the object under study from photon intensity distributions on a detector. However, photon scattering, mainly as a consequence of the Compton effect, considerably affects the image quality of the object. This is why most image reconstruction methods operate only with primary or non-scattered photons. Nevertheless the restored image remains noisy and weak in intensity. In this paper a new relation between the object and photon intensity distributions, generated by photons scattered at various deflection angles is established. It takes the form of an integral transform, compounded from Fourier and Hankel transforms. Most importantly this new transformation is invertible. As a result a novel principle for image reconstruction using scattered photons is derived and may lead to the conception of a new type of imaging device.

The Mathematical Foundations of 3D Compton Scatter Emission Imaging

The mathematical principles of tomographic imaging using detected (unscattered) X- or gamma-rays are based on the two-dimensional Radon transform and many of its variants. In this paper, we show that two new generalizations, called conical Radon transforms, are related to three-dimensional imaging processes based on detected Compton scattered radiation. The first class of conical Radon transform has been introduced recently to support imaging principles of collimated detector systems. The second class is new and is closely related to the Compton camera imaging principles and invertible under special conditions. As they are poised to play a major role in future designs of biomedical imaging systems, we present an account of their most important properties which may be relevant for active researchers in the field.

Inversion of a new circular-arc Radon transform for Compton scattering tomography

A new circular-arc Radon transform arising from the mathematical modeling of image formation in a new modality of Compton scattering tomography is introduced. We describe some of its properties and establish its analytic inverse formula. This result demonstrates the feasibility of image reconstruction from Compton scattered radiation in Compton scattering tomography. We also show that it belongs to a larger class of Radon transforms on algebraic curves, which remain invariant under a specific geometric inversion.

On new {\mathfrak V}-line Radon transforms in \mathbb {R}^{2} and their inversion

Radon transforms on piecewise smooth curves in are rather unfamiliar and have not been so far widely investigated. In this paper we consider three types of Radon transforms defined on a pair of half-lines in the shape of a V, with a fixed axis direction. These three Radon transforms arise from recently suggested tomographic procedures. Our main result consists in obtaining their analytic inverse formulas, which may serve as mathematical foundation for new imaging systems in engineering and physics.

On a Novel Approach to Compton Scattered Emission Imaging

Imaging processes built on the Compton scattering effect are currently under intense investigation. However, despite many innovative contributions, this topic still pose a formidable mathematical and technical challenge. In this work, we argue that, in the framework of single-photon emission imaging, collecting Compton scattered radiation from an emitting object, allows to image the radiotracer distribution in vivo. Data is acquired by a stationary collimated gamma camera under the form of compounded conical projections of the activity density function. Mathematically, the image formation process is described by the so-called compounded conical radon transform (CCRT) and three-dimensional object reconstruction is based on an inversion formula of the CCRT. We perform numerical simulations to show the feasibility of this new imaging modality, which offers the remarkable advantage of operating in stationary mode without the need of bulky and cumbersome spatial rotational mechanism of conventional gamma cameras. This is highly attractive for applications in medical imaging, industrial non-destructive evaluation, nuclear waste storage surveillance and homeland security monitoring. Finally, to improve drastically the sensitivity, we introduce a new feature allowing to acquire data without mechanical collimation and support the findings with some preliminary simulation results.

Radon transforms on generalized Cormack?s curves and a new Compton scatter tomography modality

In his seminal work of 1981, Cormack established that Radon transforms defined on two remarkable families of curves in the plane are invertible and admit explicit inversion formulas via circular harmonic decomposition. A sufficient condition for finding larger classes of curves enjoying the same property is given in this paper. We show that these generalized Cormack’s curves are given by the solutions of a nonlinear first-order differential equation, which is invariant under geometric inversion. A derivation of the analytic inverse formula of the corresponding Radon transforms, as well as some of their main properties, are worked out. Interestingly, among these generalized Cormack’s curves are circles orthogonal to a circle of fixed radius centered at the origin of coordinates. It is suggested that a novel Compton scatter tomography modality may be modeled by a Radon transform defined on these circles. (Some figures may appear in colour only in the online journal)

Stark effect of interactive electron-hole pairs in spherical semiconductor quantum dots.

We present a theoretical approach, based on the effective mass approximation model, on the quantum-confinement Stark effects for spherical semiconducting quantum dots in the regime of strong confinement of interactive electron-hole pairs and limiting weak electric field. The respective roles of Coulomb potential and polarization energy are investigated in detail. Under reasonable physical assumptions, analytical calculations can be performed. They show that the Stark shift is a quadratic function of the electric field amplitude in this regime. The computed numerical values obtained from this approach are found to be in good agreement with experimental data over a significant domain of quantum dot sizes.

Scattered radiation emission imaging: principles and applications

Imaging processes built on the Compton scattering effect have been under continuing investigation since it was first suggested in the 50s. However, despite many innovative contributions, there are still formidable theoretical and technical challenges to overcome. In this paper, we review the state-of-the-art principles of the so-called scattered radiation emission imaging. Basically, it consists of using the cleverly collected scattered radiation from a radiating object to reconstruct its inner structure. Image formation is based on the mathematical concept of compounded conical projection. It entails a Radon transform defined on circular cone surfaces in order to express the scattered radiation flux density on a detecting pixel. We discuss in particular invertible cases of such conical Radon transforms which form a mathematical basis for image reconstruction methods. Numerical simulations performed in two and three space dimensions speak in favor of the viability of this imaging principle and its potential applications in various fields.

Modeling and simulation results on high sensitivity scattered gamma-ray emission imaging

A new modality in gamma-ray emission imaging, based on the use of scattered radiation detected with an uncollimated gamma camera, is put forward. Recently, we have shown that scattered radiation by Compton effect registered on a collimated gamma camera can be in principle used to reconstruct an object in three dimensions. To improve drastically the sensitivity of this process, we propose that data acquisition should be performed without mechanical collimation. As a first step, image formation in two dimensions is derived and validated by Monte Carlo simulations. Then, numerical reconstructions are presented to support the feasibility and attractiveness of this new concept.