Stability estimates for the regularized inversion of the truncated Hilbert transform

In limited data computerized tomography, the 2D or 3D problem can be reduced to a family of 1D problems using the differentiated backprojection (DBP) method. Each 1D problem consists of recovering a compactly supported function $f \in L^2(\mathcal F)$, where $\mathcal F$ is a finite interval, from its partial Hilbert transform data. When the Hilbert transform is measured on a finite interval $\mathcal G$ that only overlaps but does not cover $\mathcal F$ this inversion problem is known to be severely ill-posed [1]. In this paper, we study the reconstruction of $f$ restricted to the overlap region $\mathcal F \cap \mathcal G$. We show that with this restriction and by assuming prior knowledge on the $L^2$ norm or on the variation of $f$, better stability with H\"older continuity (typical for mildly ill-posed problems) can be obtained.Comment: added one remark, larger fonts for axis labels in figure

Distribution and dynamics of Tc-99m-pertechnetate uptake in the thyroid and other organs assessed by single-photon emission computed tomography in living mice

Background: Tc-99m pertechnetate is a well-known anion, used for clinical imaging of thyroid function. This gamma emitter is transported by the sodium iodide symporter but is not incorporated into thyroglobulin. Scintigraphy using Tc-99m pertechnetate or 123 iodide represents a powerful tool for the study of sodium iodide symporter activity in different organs of living animal models. However, in many studies that have been performed in mice, the thyroid could not be distinguished from the salivary glands. In this work, we have evaluated the use of a clinically dedicated single-photon emission computed tomography (SPECT) camera for thyroid imaging and assessed what improvements are necessary for the development of this technique. Methods: SPECT of the mouse neck region, with pinhole collimation and geometric calibration, was used for the individual measurement of Tc-99m pertechnetate uptake in the thyroid and the salivary glands. Uptake in the stomach was studied by planar whole-body imaging. Uptake kinetics and biodistribution studies were performed by sequential imaging. Results: This work has shown that thyroid imaging in living mice can be performed with a SPECT camera originally built for clinical use. Our experiments indicate that Tc-99m pertechnetate uptake is faster in the thyroid than in the salivary glands and the stomach. The decrease in Tc-99m pertechnetate uptake after injection of iodide or perchlorate as competitive inhibitors was also studied. The resulting rate decreases were faster in the thyroid than in the salivary glands or the stomach. Conclusions: We have shown that a clinically dedicated SPECT camera can be used for thyroid imaging. In our experiments, SPECT imaging allowed the analysis of Tc-99m pertechnetate accumulation in individual organs and revealed differences in uptake kinetics

Terres agricoles de la ville d'Antananarivo, une disparition inéluctable ?

A Madagascar, la capitale Antananarivo semble demeurer fortement agricole malgré une croissance démographique intense. Cependant, en l'absence de données récentes à l'échelle de l'agglomération, il est difficile de juger de l'évolution conjointe de l'urbanisation et de l'agriculture urbaine. Dans ce contexte, cette communication analyse les déterminants de l'évolution de l'emprise agricole dans l'agglomération d'Antananarivo. Sur base d'enquêtes qualitatives menées dans 38 communes de l'agglomération d'Antananarivo et de l'analyse diachronique d'image s satellitaires, trois dynamiques agricoles contrastées sont mises en évidence : 1) le déclin d'un espace agricole ancien qui disparaît peu à peu devant la pression du bâti dans l'ultra centre et le long des axes routiers, mais également, contre toute attente : 2) la conquête e de l'espace urbain par de nouvelles formes d'agriculture et enfin, 3) l'extension des superficies cultivées sur les tanety, auparavant réservé aux pâturages. Au-delà de facteurs tels que la topographie, la présence de nouvelles infrastructures routières ou la proximité à des services " urbains ", l'évolution du système agro-urbain dépend fortement de l'évolution démographique, de la demande alimentaire urbaine et des faibles créations d'emploi dans le secondaire et tertiaire. Cela renforce en effet l'intérêt pour les ménages de garder et investir dans une activité agricole source d'aliments et de revenus, garante d'un minimum de sécurité économique et alimentaire, et insérée dans un portefeuille d'activité plus large valorisant les opportunités économiques de la ville. Dans des contextes où les plans et les politiques d'urbanisme peinent à se concrétiser, la recherche montre que le maintien de l'agriculture résulte et dépend en partie de volontés d'acteurs locaux. En démontrant la place de cette agriculture, la conclusion pointe la diversité des leviers d'action et invite à débattre de ses enjeux à différentes échelles de gouvernance

Image Reconstruction from Truncated Data in Single-Photon Emission Computed Tomomgraphy with Uniform Attenuation

International audienceWe present a mathematical analysis of the problem of image reconstruction from truncated data in two-dimensional (2D) single-photon emission computed tomography (SPECT). Recent results in classical tomography have shown that accurate reconstruction of some parts of the object is possible in the presence of truncation. We have investigated how these results extend to 2D parallel-beam SPECT, assuming that the attenuation map is known and constant in a convex region $\Omega$ that includes all activity sources. Our main result is a proof that, just like in classical tomography accurate SPECT reconstruction at a given location x ∈ $\Omega$,does not require the data on all lines passing through $\Omega$; some amount of truncation can be tolerated. Experimental reconstruction results based on computer-simulated data are given in support of the theory

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

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

Tiny a priori knowledge solves the interior problem in computed tomography

Based on the concept of differentiated backprojection (DBP) (Noo et al 2004 Phys. Med. Biol. 49 3903, Pan et al 2005 Med. Phys. 32 673, Defrise et al 2006 Inverse Problems 22 1037), this paper shows that the solution to the interior problem in computed tomography is unique if a tiny a priori knowledge on the object f(x, y) is available in the form that f(x, y) is known on a small region located inside the region of interest. Furthermore, we advance the uniqueness result to obtain more general uniqueness results which can be applied to a wider class of imaging configurations. We also develop a reconstruction algorithm which can be considered an extension of the DBP-POCS (projection onto convex sets) method described by Defrise et al (2006 Inverse Problems 22 1037), where we not only extend this method to the interior problem but also introduce a new POCS algorithm to reduce computational cost. Finally, we present experimental results which show evidence that the inversion corresponding to each obtained uniqueness result is stable

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

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