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

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

Reconstruction of 3D Whole-Body PET Data Using Blurred Anatomical Labels

The diagnostic utility of whole-body PET is often limited by the high level of statistical noise in the images. An improvement in image quality can be obtained by incorporating correlated anatomical information during the reconstruction of the PET data. The combined PET/CT (SMART) scanner allows the acquisition of accurately aligned PET and CT whole-body data. The authors present results of incorporating aligned anatomical information from the CT during the reconstruction of 3D whole-body PET data. They use the FORE+PWLS method for the reconstruction and a label model to incorporate anatomical information via penalty weights. Since in practice mismatches between anatomical and functional data are unavoidable, the labels are “blurred” to reflect the uncertainty associated with the anatomical information. Results show the potential advantage of incorporating anatomical information by using a blurred labels with the penalty weights.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/85864/1/Fessler153.pd

Asymptotic Analysis of The SVD For The Truncated Hilbert Transform With Overlap

The truncated Hilbert transform with overlap H-T is an operator that arises in tomographic reconstruction from limited data, more precisely in the method of differentiated back-projection. Recent work [R. Al-Aifari and A. Katsevich, SIAM J. Math. Anal., 46 (2014), pp. 192213] has shown that the singular values of this operator accumulate at both zero and one. To better understand the properties of the operator and, in particular, the ill-posedness of the inverse problem associated with it, it is of interest to know the rates at which the singular values approach zero and one. In this paper, we exploit the property that H-T commutes with a second-order differential operator L-S and the global asymptotic behavior of its eigenfunctions to find the asymptotics of the singular values and singular functions of H-T

Etude de l'acquisition des données en TEP 2D

La Tomographie par Emissions de Positons (TEP) est une modalité d'imagerie nucléaire: on injecte au patient un traceur radioactif qui se fixe sur des constituants de l'organisme jouant un rôle important dans un processus physiologique dont on désire acquérir des images. En TEP, le radiotraceur est un émetteur β+ ou positron. Après avoir perdu son énergie cinétique au terme d'un parcours de l'ordre d'un millimètre, le positron s'annihile avec un électron pour donner deux photons γ, émis dans des directions opposées. La détection se fait à l'aide de plusieurs couronnes de détecteurs qui enregistrent en coincidence l'arrivée des deux photons. La trajectoire linéaire formé par le trajet des deux photons dans le cas idéal, appelée LOR (Line Of Response) constitue la donnée principale qu'acquiert le système. Le programme de reconstruction d'images s'intéresse au problème inverse: à partir des données de toutes les LOR issues des différentes émissions, il localise le lieu d'émission et ainsi fournit la distribution spatiale du radiotraceur dans le corps du patient. Le but de ce travail est de définir un schéma de compression de données, i.e. réduire le nombre de LOR, en étudiant le système d'échantillonnage des LOR dans le but d'accélerer le temps de reconstruction des images. On se limite à l'acquisition en mode 2D ( ou 3D à coupes parallèles): uniquement les LOR orthogonales à l'axe du scanner, dont les extrémités appartiennent à une même couronne, sont prises en compte. Pour cela, on s'appuie sur les travaux [1] et [2] qui, dans une géométrie similaire en tomographie X, ont établi les conditions d'échantillonnage efficaces en utilisant la paramétrisation fan beam. Dans ce travail, on utilise une autre paramétrisation plus adaptée à la TEP et on en déduit des schémas d'échantillonnage. Cependant, le schéma efficace n'étant pas compatible avec la géométrie des détecteurs, nous le mettons à profit dans un schéma de compression du nombre de LOR, appelé mashing

Clinically feasible reconstruction of 3D whole-body PET/CT data using blurred anatomical labels

We present the results of utilizing aligned anatomical information from CT images to locally adjust image smoothness during the reconstruction of three-dimensional (3D) whole-body positron emission tomography (PET) data. The ability of whole-body PET imaging to detect malignant neoplasms is becoming widely recognized. Potentially useful, however, is the role of whole-body PET in quantitative estimation of tracer uptake. The utility of PET in oncology is often limited by the high level of statistical noise in the images. Reduction in noise can be obtained by incorporating a priori image smoothness information from correlated anatomical information during the reconstruction of PET data. A combined PET/CT scanner allows the acquisition of accurately aligned PET and x-ray CT whole-body data. We use the Fourier rebinning algorithm (FORE) to accurately convert the 3D PET data to two-dimensional (2D) data to accelerate the image reconstruction process. The 2D datasets are reconstructed with successive over-relaxation of a penalized weighted least squares (PWLS) objective function to model the statistics of the acquisition, data corrections, and rebinning. A 3D voxel label model is presented that incorporates the anatomical information via the penalty weights of the PWLS objective function. This combination of FORE + PWLS + labels was developed as it allows for both reconstruction of 3D whole-body data sets in clinically feasible times and also the inclusion of anatomical information in such a way that convergence can be guaranteed. Since mismatches between anatomical (CT) and functional (PET) data are unavoidable in practice, the labels are ‘blurred’ to reflect the uncertainty associated with the anatomical information. Simulated and experimental results show the potential advantage of incorporating anatomical information by using blurred labels to calculate the penalty weights. We conclude that while the effect of this method on detection tasks is complicated and unclear, there is an improvement on the estimation task.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/48971/2/m20101.pd

SU(n) symmetry and ideal mixing of mesons

We study mass formulae for 1/2+ baryons and mesons in SU(n) symmetry and show how the ideal mixing of mesons fits into this scheme.Nous étudions les formules de masse des baryons 1/2+ et des mésons dans le cadre des symétries SU(n) et montrons comment le mélange idéal des mésons entre dans ce modèle.Defrise Michel. SU(n) symmetry and ideal mixing of mesons. In: Bulletin de la Classe des sciences, tome 62, 1976. pp. 742-751