Differential Phase-contrast Interior Tomography

Differential phase contrast interior tomography allows for reconstruction of a refractive index distribution over a region of interest (ROI) for visualization and analysis of internal structures inside a large biological specimen. In this imaging mode, x-ray beams target the ROI with a narrow beam aperture, offering more imaging flexibility at less ionizing radiation. Inspired by recently developed compressive sensing theory, in numerical analysis framework, we prove that exact interior reconstruction can be achieved on an ROI via the total variation minimization from truncated differential projection data through the ROI, assuming a piecewise constant distribution of the refractive index in the ROI. Then, we develop an iterative algorithm for the interior reconstruction and perform numerical simulation experiments to demonstrate the feasibility of our proposed approach

Solving the interior problem of computed tomography using a priori knowledge

A case of incomplete tomographic data for a compactly supported attenuation function is studied. When the attenuation function is a priori known in a subregion, we show that a reduced set of measurements are enough to uniquely determine the attenuation function over all the space. Furthermore, we found stability estimates showing that reconstruction can be stable near the region where the attenuation is known. These estimates also suggest that reconstruction stability collapses quickly when approaching the set of points that is viewed under less than 180°. This paper may be seen as a continuation of the work \u27Truncated Hilbert transform and image reconstruction from limited tomographic data\u27 (Defrise et al 2006 Inverse Problems 22 1037). This continuation tackles new cases of incomplete data that could be of interest in applications of computed tomography

Lie groups of conformal motions acting on null orbits

Space-times admitting a 3-dimensional Lie group of conformal motions $C_3$ acting on null orbits are studied. Coordinate expressions for the metric and the conformal Killing vectors (CKV) are provided (irrespectively of the matter content) and then all possible perfect fluid solutions are found, although none of these verify the weak and dominant energy conditions over the whole space-time manifold.Comment: 5 pages, Late

The state space and physical interpretation of self-similar spherically symmetric perfect-fluid models

The purpose of this paper is to further investigate the solution space of self-similar spherically symmetric perfect-fluid models and gain deeper understanding of the physical aspects of these solutions. We achieve this by combining the state space description of the homothetic approach with the use of the physically interesting quantities arising in the comoving approach. We focus on three types of models. First, we consider models that are natural inhomogeneous generalizations of the Friedmann Universe; such models are asymptotically Friedmann in their past and evolve fluctuations in the energy density at later times. Second, we consider so-called quasi-static models. This class includes models that undergo self-similar gravitational collapse and is important for studying the formation of naked singularities. If naked singularities do form, they have profound implications for the predictability of general relativity as a theory. Third, we consider a new class of asymptotically Minkowski self-similar spacetimes, emphasizing that some of them are associated with the self-similar solutions associated with the critical behaviour observed in recent gravitational collapse calculations.Comment: 24 pages, 12 figure

General non-rotating perfect-fluid solution with an abelian spacelike C_3 including only one isometry

The general solution for non-rotating perfect-fluid spacetimes admitting one Killing vector and two conformal (non-isometric) Killing vectors spanning an abelian three-dimensional conformal algebra (C_3) acting on spacelike hypersurfaces is presented. It is of Petrov type D; some properties of the family such as matter contents are given. This family turns out to be an extension of a solution recently given in \cite{SeS} using completely different methods. The family contains Friedman-Lema\^{\i}tre-Robertson-Walker particular cases and could be useful as a test for the different FLRW perturbation schemes. There are two very interesting limiting cases, one with a non-abelian G_2 and another with an abelian G_2 acting non-orthogonally transitively on spacelike surfaces and with the fluid velocity non-orthogonal to the group orbits. No examples are known to the authors in these classes.Comment: Submitted to GRG, Latex fil

A Lorentzian Gromov-Hausdoff notion of distance

This paper is the first of three in which I study the moduli space of isometry classes of (compact) globally hyperbolic spacetimes (with boundary). I introduce a notion of Gromov-Hausdorff distance which makes this moduli space into a metric space. Further properties of this metric space are studied in the next papers. The importance of the work can be situated in fields such as cosmology, quantum gravity and - for the mathematicians - global Lorentzian geometry.Comment: 20 pages, 0 figures, submitted to Classical and quantum gravity, seriously improved presentatio

A physical application of Kerr-Schild groups

The present work deals with the search of useful physical applications of some generalized groups of metric transformations. We put forward different proposals and focus our attention on the implementation of one of them. Particularly, the results show how one can control very efficiently the kind of spacetimes related by a Generalized Kerr-Schild (GKS) Ansatz through Kerr-Schild groups. Finally a preliminar study regarding other generalized groups of metric transformations is undertaken which is aimed at giving some hints in new Ans\"atze to finding useful solutions to Einstein's equations.Comment: 18 page

Hypersurface homogeneous locally rotationally symmetric spacetimes admitting conformal symmetries

All hypersurface homogeneous locally rotationally symmetric spacetimes which admit conformal symmetries are determined and the symmetry vectors are given explicitly. It is shown that these spacetimes must be considered in two sets. One set containing Ellis Class II and the other containing Ellis Class I, III LRS spacetimes. The determination of the conformal algebra in the first set is achieved by systematizing and completing results on the determination of CKVs in 2+2 decomposable spacetimes. In the second set new methods are developed. The results are applied to obtain the classification of the conformal algebra of all static LRS spacetimes in terms of geometrical variables. Furthermore all perfect fluid nontilted LRS spacetimes which admit proper conformal symmetries are determined and the physical properties some of them are discussed.Comment: 15 pages; to appear in Classical Quantum Gravity; some misprints in Tables 3,5 and in section 4 correcte

Convex optimization problem prototyping for image reconstruction in computed tomography with the Chambolle-Pock algorithm

The primal-dual optimization algorithm developed in Chambolle and Pock (CP), 2011 is applied to various convex optimization problems of interest in computed tomography (CT) image reconstruction. This algorithm allows for rapid prototyping of optimization problems for the purpose of designing iterative image reconstruction algorithms for CT. The primal-dual algorithm is briefly summarized in the article, and its potential for prototyping is demonstrated by explicitly deriving CP algorithm instances for many optimization problems relevant to CT. An example application modeling breast CT with low-intensity X-ray illumination is presented.Comment: Resubmitted to Physics in Medicine and Biology. Text has been modified according to referee comments, and typos in the equations have been correcte

Bi-conformal vector fields and their applications

We introduce the concept of bi-conformal transformation, as a generalization of conformal ones, by allowing two orthogonal parts of a manifold with metric \G to be scaled by different conformal factors. In particular, we study their infinitesimal version, called bi-conformal vector fields. We show the differential conditions characterizing them in terms of a "square root" of the metric, or equivalently of two complementary orthogonal projectors. Keeping these fixed, the set of bi-conformal vector fields is a Lie algebra which can be finite or infinite dimensional according to the dimensionality of the projectors. We determine (i) when an infinite-dimensional case is feasible and its properties, and (ii) a normal system for the generators in the finite-dimensional case. Its integrability conditions are also analyzed, which in particular provides the maximum number of linearly independent solutions. We identify the corresponding maximal spaces, and show a necessary geometric condition for a metric tensor to be a double-twisted product. More general ``breakable'' spaces are briefly considered. Many known symmetries are included, such as conformal Killing vectors, Kerr-Schild vector fields, kinematic self-similarity, causal symmetries, and rigid motions.Comment: Replaced version with some changes in the terminology and a new theorem. To appear in Classical and Quantum Gravit