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

Méthode de correction d'atténuation en imagerie SPECT 3D basée sur les conditions de rang de la transformée exponentielle

La reconstruction de l'activité, en imagerie nucléaire, nécessite, pour qu'elle soit quantitative, une correction de l'atténuation subit par les photons émis, due au milieu traversé (par exemple les tissus et os du corps). Une méthode, basée sur les conditions de rang de la transformée exponentielle tridimentionnelle (3D), est proposée pour compenser des effets de l'atténuation. Elle ne nécéssite pas d'images scanner qui serviraient à connaître la fonction d'atténuation. Par ailleurs, elle permet de considérer toutes les géométries SPECT 3-D, parallèles. Cette méthode a été mise en oeuvre pour la reconstruction de données RSH SPECT. À notre connaissance, elle est la première tantative d'utilisation des conditions de rang pour la correction d'atténuation en imagerie pleinement 3D

Data Consistency for Linograms and Planograms

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Centers and centroids of the cone-beam projection of a ball

International audienceIn geometric calibration of cone-beam (CB) scanners, point-likemarker objects such as small balls are imaged to obtain positioning information from which the unknown geometric parameters are extracted. The procedure is sensitive to errors in the positioning information, and one source of error is a small bias which can occur in estimating the detector locations of the CB projections of the centers of the balls. We call these detector locations the center projections. In general, the CB projection of a ball of uniform density onto a flat detector forms an ellipse. Inside the ellipse lie the center projection M, the ellipse center C and the centroid G of the intensity values inside the ellipse. The center projection is invariably estimated from C or G which are much easier to extract directly from the data. In this work, we quantify the errors incurred in using C or G to estimateM. We prove mathematically that the points C, G,M and O are always distinct and lie on the major axis of the ellipse, where O is the detector origin, defined as the orthogonal projection of the cone vertex onto the detector. (The ellipse can only degenerate to a circle if the ball is along the direct line of sight to O, and in this case all four points coincide.) The points always lie in the same order: O, M, G, C which establishes that the centroid has less geometric bias than the ellipse center for estimating M. However, our numerical studies indicate that the centroid bias is only 20% less than the ellipse center bias so the benefit in using centroid estimates is not substantial. For the purposes of quantifying the bias in practice, we show that the ellipse center bias ||CM|| can be conveniently estimated by eA/(πf˜) where A is the area of the elliptical projection, e is the eccentricity of the ellipse and f˜ is an estimate of the focal length of the system. Finally, we discuss how these results are affected by physical factors such as beam hardening, and indicate extensions to balls of non-uniform density

Data consistency conditions for truncated fanbeam and parallel projections

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Quantification of Tomographic Incompleteness in Cone-Beam Reconstruction

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Tomographic Reconstruction in the 21st Century [Region-of-interest reconstruction from incomplete data]

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Equal parallel and cone-beam projections: a curious property of D-symmetric object functions

International audienceIn tomographic image reconstruction, the object density function is the unknown quantity whose projections are measured by the scanner. In the three-dimensional (3D) case, we define the D-reflection of such a density function as the object obtained by a particular weighted reflection about the plane z = D, and a D-symmetric function as one whose D-reflection is equal to itself. D-symmetric object functions have the curious property that their parallel projection onto the detector plane z = D is equal to their cone-beam projection onto the same detector with x-ray source location at the origin. Much more remarkable is the additional fact that for any fixed D-symmetric object, every oblique parallel projection onto this same detector plane equals the cone-beam projection for a corresponding source location. The mathematical proof is straight forward but not particularly enlightening, and we also provide here an alternative physical demonstration that explains the various weighting terms in the context of classical tomosynthesis. Furthermore, we clarify the distinction between the new formulation presented here, and the original formulation of Edholm and co-workers who obtained similar properties but for a pair of objects whose divergent and parallel projections matched, but with no D-symmetry. We do not claim any immediate imaging application or useful physics from these notions, but we briefly comment on consequences for methods that apply data consistency conditions in image reconstruction

Tomographic Reconstruction in the 21st Century, Region-of-interest reconstruction from incomplete data

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