Gaik Ambartsoumian

A Range Description for the Planar Circular Radon Transform

The transform considered in the paper integrates a function supported in the unit disk on the plane over all circles centered at the boundary of this disk. Such a circular Radon transform arises in several contemporary imaging techniques, as well as in other applications. As is common for transforms of Radon type, its range has infinite codimension in standard function spaces. Range descriptions for such transforms are known to be very important for computed tomography—for instance, when dealing with incomplete data, error correction, and other issues. A complete range description for the circular Radon transform is obtained. Range conditions include the recently found set of moment‐type conditions, which happens to be incomplete, as well as other conditions that have less standard form. In order to explain the procedure better, a similar (nonstandard) treatment of the range conditions is described first for the usual Radon transform on the plane.

On the injectivity of the circular Radon transform

The circular Radon transform integrates a function over the set of all spheres with a given set of centres. The problem of injectivity of this transform (as well as inversion formulae, range descriptions, etc) arises in many fields from approximation theory to integral geometry, to inverse problems for PDEs and recently to newly developing types of tomography. A major breakthrough in the 2D case was made several years ago in a work by Agranovsky and Quinto. Their techniques involved microlocal analysis and known geometric properties of zeros of harmonic polynomials in the plane. Since then there has been an active search for alternative methods, especially those based on simple PDE techniques, which would be less restrictive in more general situations. This paper provides some new results that one can obtain by methods that essentially involve only the finite speed of propagation and domain dependence for the wave equation.

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.

Inversion of the V-line Radon transform in a disc and its applications in imaging

Several novel imaging modalities proposed during the last couple of years are based on a mathematical model, which uses the V-line Radon transform (VRT). This transform, sometimes called broken-ray Radon transform, integrates a function along V-shaped piecewise linear trajectories composed of two intervals in the plane with a common endpoint. Image reconstruction problems in these modalities require inversion of the VRT. While there are ample results about inversion of the regular Radon transform integrating along straight lines, very little is known for the case of the V-line Radon transform. In this paper, we derive an exact inversion formula for the VRT of functions supported in a disc of arbitrary radius. The formula uses a two-dimensional restriction of VRT data, namely the incident ray is normal to the boundary of the disc, and the breaking angle is fixed. Our method is based on the classical filtered back-projection inversion formula of the Radon transform, and has similar features in terms of stability, speed, and accuracy.

A series formula for inversion of the V-line Radon transform in a disc

The paper presents an exact formula for a Fourier series reconstruction of a function from its V-line Radon transform in a disc. This transform (often also called broken-ray Radon transform) appears in mathematical models of several imaging modalities, e.g. single-scattering optical tomography and @c-ray emission tomography. Our inversion formula relaxes the support restriction on the image function required in the previously discovered inversion technique (Ambartsoumian, 2012) [8], and uses data from only half of the set of broken rays required before. The general strategy of the current approach was outlined in (Ambartsoumian, 2012) [8].

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.

Thermoacoustic tomography - Numerical results

Filtered backprojection (FBP) reconstruction is the method of choice for diagnostic xray CT, despite the fact that backprojection is computationally costly. FBP image quality is superior over fast Fourier reconstruction techniques because interpolation errors are localized and the backprojector applies the Radon transform, annihilating all measurement errors orthogonal to its range. We discuss computational complexity, sampling rates, and quadrature techniques for FBP reconstruction of thermo/photo/optoacoustic data.

Numerical Inversion of a Broken Ray Transform Arising in Single Scattering Optical Tomography

The article presents an efficient image reconstruction algorithm for single scattering optical tomography (SSOT) in circular geometry of data acquisition. This novel medical imaging modality uses photons of light that scatter once in the body to recover its interior features. The mathematical model of SSOT is based on the broken ray (or V-line Radon) transform (BRT), which puts into correspondence to an image function its integrals along V-shaped piecewise linear trajectories. The process of image reconstruction in SSOT requires inversion of that transform. We implement numerical inversion of a broken ray transform in a disc with partial radial data. Our method is based on a relation between the Fourier coefficients of the image function and those of its BRT recently discovered by Ambartsoumian and Moon. The numerical algorithm requires solution of ill-conditioned matrix problems, which is accomplished using a half-rank truncated singular value decomposition method. Several numerical computations validating the inversion formula are presented, which demonstrate the accuracy, speed, and robustness of our method in the case of both noise-free and noisy data.

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.

Reconstruction algorithms for interior and exterior spherical radon transform-based ultrasound imaging

This work is concerned with the numerical implementation of a reconstruction algorithm developed to recover a function from its spherical means over spheres centered on a circle. The algorithm is experimentally verified by simulations using numerical phantoms. In the scheme of tomography, acoustic waves are generated by illuminating an object with a short burst of radio-frequency waves. In applications, like breast cancer imaging, which use modalities like photo-acoustic tomography (PAT) that model the acoustic pressures as spherical means, data are measured on the detectors located in a circle surrounding the object. This is then used to reconstruct the absorption density inside the object. In contrast, applications like bore hole tomography and improved Intravascular Ultra Sound (IVUS) imaging for prostate cancer, which use modalities like Radial Reflection Diffraction Tomography (RRDT), a ring of detectors placed exterior to the object, collect the acoustic waves as back-scattered field. This work uses a single algorithm to reconstruct functions from data collected using these two different techniques - one, by placing the object inside the ring of detectors, and the other, by placing the object exterior to the ring of detectors. The algorithm then draws a comparison between the two reconstructions. The case of bistatic ultrasound imaging, where the elliptical Radon transform is appropriate, is also discussed.