Rim Gouia-Zarrad

Inversion of the circular Radon transform on an annulus

The representation of a function by its circular Radon transform (CRT) and various related problems arise in many areas of mathematics, physics and imaging science. There has been a substantial spike of interest toward these problems in the last decade mainly due to the connection between the CRT and mathematical models of several emerging medical imaging modalities. This paper contains some new results about the existence and uniqueness of the representation of a function by its CRT with partial data. A new inversion formula is presented in the case of the circular acquisition geometry for both interior and exterior problems when the Radon transform is known for only a part of all possible radii. The results are not only interesting as original mathematical discoveries, but can also be useful for applications, e.g., in medical imaging.

Exact inversion of the conical Radon transform with a fixed opening angle

We study a new class of Radon transforms defined on circular cones called the conical Radon transform. In it maps a function to its surface integrals over circular cones, and in it maps a function to its integrals along two rays with a common vertex. Such transforms appear in various mathematical models arising in medical imaging, nuclear industry and homeland security. This paper contains new results about inversion of conical Radon transform with fixed opening angle and vertical central axis in and . New simple explicit inversion formulae are presented in these cases. Numerical simulations were performed to demonstrate the efficiency of the suggested algorithm in 2D.

Analytical reconstruction formula for n -dimensional conical Radon transform

In recent years, Radon type transforms that integrate functions along families of curves or surfaces, have been intensively studied due to their applications to inverse scattering, synthetic aperture radar (SAR), imaging science, nuclear industry, etc. In this paper, we consider the transform that integrates a function f ( x ) in R n over a family of cones invariant to translation. A new exact inversion formula is presented in the case of fixed opening angle and vertical central axis. In addition, the results of numerical simulations are presented for the case n = 2 .

Approximate inversion algorithm of the elliptical Radon transform

We address the fundamental question of image reconstruction in bistatic regime in which the measurements represent line integrals over a family of ellipses with foci at the source and receiver locations. An integral transform, the elliptical Radon transform is introduced and used to model the data. This paper presents some new numerical results about the inversion of the elliptical Radon in 2D. A new approximate inversion formula is presented in the case of circular acquisition geometry when the source and the receiver are rotating around the origin at a fixed distance from each other. We demonstrate the efficiency of the suggested algorithm by presenting a computational implementation of the method on a numerical phantom. This novel algorithm can be efficiently implemented as a numerical method in several bistatic imaging modalities e.g. in biomedical imaging.

Existence and uniqueness of the inversion of a generalized Radon transform

In this paper, we study a new class of Radon transform defined on circular cones called the conical Radon transform (CRT). There has been a substantial spike of interest towards this integral transform in the last decade mainly due to its connection to the mathematical models of several applications used in different domains such as nuclear industry, homeland security, astrophysics etc. This paper contains new results about the existence and uniqueness of the representation of a function by its CRT with fixed opening angle and fixed axis direction in R3. The results are relevant not only as a new generalized Radon transform but also as a new reconstructing method for imaging systems in applied sciences.

Reconstruction of a Function from its Elliptical Radon Transform

The talk discusses the fundamental question of image reconstruction in bistatic regime in which the measurements represent line integrals over a family of ellipses with foci at the source and receiver locations. An integral transform, the elliptical Radon transform is introduced and used to model the data. This talk presents some new numerical results about the inversion of the elliptical Radon in 2D. A new approximate inversion formula is presented in the case of circular acquisition geometry when the source and the receiver are rotating around the origin at a fixed distance from each other. We demonstrate the efficiency of the suggested algorithm by presenting a computational implementation of the method on a numerical phantom. This novel algorithm can be efficiently implemented as a numerical method in several bistatic imaging modalities e.g. in biomedical imaging.