21 research outputs found
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 circular Radon transform arises in several contemporary imaging
techniques, as well as in other applications. As it is common for transforms of
Radon type, its range has infinite co-dimension 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 the rest of
conditions that have less standard form. In order to explain the procedure
better, a similar (non-standard) treatment of the range conditions is described
first for the usual Radon transform on the plane.Comment: submitted for publicatio
Spherical radon transforms and mathematical problems of thermoacoustic tomography
The spherical Radon transform (SRT) integrates a function over the set of all
spheres with a given set of centers. Such transforms play an important role in some
newly developing types of tomography as well as in several areas of mathematics
including approximation theory, integral geometry, inverse problems for PDEs, etc.
In Chapter I we give a brief description of thermoacoustic tomography (TAT or
TCT) and introduce the SRT.
In Chapter II we consider the injectivity problem for SRT. A major breakthrough
in the 2D case was made several years ago by M. Agranovsky and E. T. 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, which would be less restrictive in more general situations. We
provide some new results obtained by PDE techniques that essentially involve only
the finite speed of propagation and domain dependence for the wave equation.
In Chapter III we consider the transform that integrates a function supported
in the unit disk on the plane over circles centered at the boundary of this disk. As
is common for transforms of the Radon type, its range has an in finite co-dimension
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.
In Chapter IV we investigate implementation of the recently discovered exact
backprojection type inversion formulas for the case of spherical acquisition in 3D and
approximate inversion formulas in 2D. A numerical simulation of the data acquisition
with subsequent reconstructions is made for the Defrise phantom as well as for some
other phantoms. Both full and partial scan situations are considered
Inversion and Symmetries of the Star Transform
The star transform is a generalized Radon transform mapping a function of two
variables to its integrals along "star-shaped" trajectories, which consist of a
finite number of rays emanating from a common vertex. Such operators appear in
mathematical models of various imaging modalities based on scattering of
elementary particles. The paper presents a comprehensive study of the inversion
of the star transform. We describe the necessary and sufficient conditions for
invertibility of the star transform, introduce a new inversion formula and
discuss its stability properties. As an unexpected bonus of our approach, we
prove a conjecture from algebraic geometry about the zero sets of elementary
symmetric polynomials
V-line 2-tensor tomography in the plane
In this article, we introduce and study various V-line transforms (VLTs)
defined on symmetric 2-tensor fields in . The operators of
interest include the longitudinal, transverse, and mixed VLTs, their integral
moments, and the star transform. With the exception of the star transform, all
these operators are natural generalizations to the broken-ray trajectories of
the corresponding well studied concepts defined for straight-line paths of
integration. We characterize the kernels of the VLTs and derive exact formulas
for reconstruction of tensor fields from various combinations of these
transforms. The star transform on tensor fields is an extension of the
corresponding concepts that have been previously studied on vector fields and
scalar fields (functions). We describe all injective configurations of the star
transform on symmetric 2-tensor fields and derive an exact, closed-form
inversion formula for that operator.Comment: 26 pages, 2 figure
Reconstructions in limited-view thermoacoustic tomography
The limited-view problem is studied for thermoacoustic tomography, which is also referred to as photoacoustic or optoacoustic tomography depending on the type of radiation for the induction of acoustic waves. We define a “detection region,” within which all points have sufficient detection views. It is explained analytically and shown numerically that the boundaries of any objects inside this region can be recovered stably. Otherwise some sharp details become blurred. One can identify in advance the parts of the boundaries that will be affected if the detection view is insufficient. If the detector scans along a circle in a two-dimensional case, acquiring a sufficient view might require covering more than a π-, or less than a π-arc of the trajectory depending on the position of the object. Similar results hold in a three-dimensional case. In order to support our theoretical conclusions, three types of reconstruction methods are utilized: a filtered backprojection (FBP) approximate inversion, which is shown to work well for limited-view data, a local-tomography-type reconstruction that emphasizes sharp details (e.g., the boundaries of inclusions), and an iterative algebraic truncated conjugate gradient algorithm used in conjunction with FBP. Computations are conducted for both numerically simulated and experimental data. The reconstructions confirm our theoretical predictions
A simple range characterization for spherical mean transform in odd dimensions and its applications
This article provides a novel and simple range description for the spherical
mean transform of functions supported in the unit ball of an odd dimensional
Euclidean space. The new description comprises a set of symmetry relations
between the values of certain differential operators acting on the coefficients
of the spherical harmonics expansion of the function in the range of the
transform. As one application of this range characterization, we construct an
explicit counterexample proving that unique continuation type results cannot
hold for the spherical mean transform in odd dimensional spaces. Finally, as an
auxiliary result of one of our proofs, we derive a remarkable cross product
identity for the spherical Bessel functions of the first and second kind, which
may be of independent interest in the theory of special functions