Image reconstruction from radially incomplete spherical Radon data (original) (raw)

Local reconstruction from the spherical Radon transform in 3D

arXiv (Cornell University), 2022

In this article we study the spherical mean Radon transform in R 3 with detectors centered on a plane. We use the consistency method suggested by the author of this article for the inversion of the transform in 3D. A new iterative inversion formula is presented. This formula has the benefit of being local and is suitable for practical reconstructions. The inversion of the spherical mean Radon transform is required in mathematical models in thermo-and photo-acoustic tomography, radar imaging, and others.

Inverting the spherical Radon transform for physically meaningful functions

2003

In this paper we refer to the reconstruction formulas given in Andersson's On the determination of a function from spherical averages, which are often used in applications such as SAR 1 and SONAR 2 . We demonstrate that the first one of these formulas does not converge given physically reasonable assumptions. An alternative is proposed and it is shown that the second reconstruction formula is well-defined but might be difficult to compute numerically.

A series solution and a fast algorithm for the inversion of the spherical mean Radon transform

Inverse Problems, 2007

An explicit series solution is proposed for the inversion of the spherical mean Radon transform. Such an inversion is required in problems of thermo-and photoacoustic tomography. Closed-form inversion formulae are currently known only for the case when the centers of the integration spheres lie on a sphere surrounding the support of the unknown function, or on certain unbounded surfaces. Our approach results in an explicit series solution for any closed measuring surface surrounding a region for which the eigenfunctions of the Dirichlet Laplacian are explicitly knownsuch as, for example, cube, finite cylinder, half-sphere etc. In addition, we present a fast reconstruction algorithm applicable in the case when the detectors (the centers of the integration spheres) lie on a surface of a cube. This algorithm reconsrtucts 3-D images thousands times faster than backprojection-type methods.

On artifacts in limited data spherical Radon transform: curved observation surface

Inverse Problems, 2015

We study the limited data problem of the spherical Radon transform in two and three dimensional spaces with general acquisition surfaces. In such situations, it is known that the application of filtered-backprojection reconstruction formulas might generate added artifacts and degrade the quality of reconstructions. In this article, we explicitly analyze a family of such inversion formulas, depending on a smoothing function that vanishes to order k on the boundary of the acquisition surfaces. We show that the artifacts are k orders smoother than their generating singularity. Moreover, in two dimensional space, if the generating singularity is conormal satisfying a generic condition then the artifacts are even k + 1 2 orders smoother than the generating singularity. Our analysis for three dimensional space contains an important idea of lifting up a space. We also explore the theoretical findings in a series of numerical experiments. Our experiments show that a good choice of the smoothing function might lead to a significant improvement of reconstruction quality.

Explicit inversion formulas for the spherical mean Radon transform

arXiv (Cornell University), 2006

We derive explicit formulas for the reconstruction of a function from its integrals over a family of spheres, or for the inversion of the spherical mean Radon transform. Such formulas are important for problems of thermo-and photo-acoustic tomography. A closed-form inversion formula of a filtration-backprojection type is found for the case when the centers of the integration spheres lie on a sphere in R n surrounding the support of the unknown function. An explicit series solution is presented for the case when the centers of the integration spheres lie on a general closed surface.

Using Radon Transform in Image Reconstruction

The optical tomography problem presents some interesting difficulties for both experimental and theoretical work. This paper has attempted an overview of the theoretical problems for image reconstruction. In this paper we review some general approaches to inverse problems to set the context for optical tomography. An essential requirement is to treat the problem in a nonlinear fashion, by using an iterative method. The inverse problem is approached by numerical solutions methods using MathCad program. The Radon transform is the basic tool of the computerized tomography. In the sequel we introduce this transform, review some properties and present a numerical program for its inversion. We show some results that represent the most complex and realistic simulations of optical tomography yet developed.

Explicit inversion formulae for the spherical mean Radon transform

Inverse Problems, 2007

We derive explicit formulas for the reconstruction of a function from its integrals over a family of spheres, or for the inversion of the spherical mean Radon transform. Such formulas are important for problems of thermo-and photo-acoustic tomography. A closed-form inversion formula of a filtration-backprojection type is found for the case when the centers of the integration spheres lie on a sphere in R n surrounding the support of the unknown function. An explicit series solution is presented for the case when the centers of the integration spheres lie on a general closed surface.

An approach to the spherical mean Radon transform with detectors on a line

2017

The article suggests a new approach what is called a consistency method for the inversion of the spherical Radon transform in 2D with detectors on a line. It is known that there is not an exact inversion formula in 2D. By means of the method was proved that the reconstruction has a local description and found a new iteration formula which give an practical algorithm to recover an unknown function supported completely on one side of a line LLL from its spherical means over circles centered on the line LLL. Such an inversion is required in problems of thermo- and photo-acoustic tomography.

On the exactness of the universal backprojection formula for the spherical means Radon transform

Inverse Problems

The spherical means Radon transform is defined by the integral of a function f in R n over the sphere S ( x , r ) of radius r centered at a x, normalized by the area of the sphere. The problem of reconstructing f from the data where x belongs to a hypersurface Γ ⊂ R n and r ∈ ( 0 , ∞ ) has important applications in modern imaging modalities, such as photo- and thermo- acoustic tomography. When Γ coincides with the boundary ∂ Ω of a bounded (convex) domain Ω ⊂ R n , a function supported within Ω can be uniquely recovered from its spherical means known on Γ. We are interested in explicit inversion formulas for such a reconstruction. If Γ = ∂ Ω , such formulas are only known for the case when Γ is an ellipsoid (or one of its partial cases). This gives rise to a question: can explicit inversion formulas be found for other closed hypersurfaces Γ? In this article we prove, for the so-called ‘universal backprojection inversion formulas’, that their extension to non-ellipsoidal domains Ω is...