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This article provides a mathematical analysis of singular (nonsmooth) artifacts added to reconstructions by filtered backprojection (FBP) type algorithms for X-ray computed tomography (CT) with arbitrary incomplete data. We prove that these singular artifacts arise from points at the boundary of the data set. Our results show that, depending on the geometry of this boundary, two types of artifacts can arise: object-dependent and object-independent artifacts. Object-dependent artifacts are generated by singularities of the object being scanned, and these artifacts can extend along lines. They generalize the streak artifacts observed in limited-angle tomography. Object-independent artifacts, on the other hand, are essentially independent of the object and take one of two forms: streaks on lines if the boundary of the data set is not smooth at a point and curved artifacts if the boundary is smooth locally. We prove that these streak and curve artifacts are the only singular artifacts that can occur for FBP in the continuous case. In addition to the geometric description of artifacts, the article provides characterizations of their strength in Sobolev scale in certain cases. The results of this article apply to the well-known incomplete data problems, including limited-angle and regionof-interest tomography, as well as to unconventional X-ray CT imaging setups that arise in new practical applications. Reconstructions from simulated and real data are analyzed to illustrate our theorems, including the reconstruction that motivated this work a synchrotron data set in which artifacts appear on lines that have no relation to the object

We define a general curvilinear Radon transform in R-3, and we develop its microlocal properties. We prove that singularities can be added (or masked) in any backprojection reconstruction method for this transform. We use the microlocal properties of the transform to develop a local backprojection reconstruction algorithm that decreases the effect of the added singularities and reconstructs the shape of the object. This work was motivated by new models in electron microscope tomography in which the electrons travel over curves such as helices or spirals, and we provide reconstructions for a specific transform motivated by this electron microscope tomography problem., AuthorCount:2

QC 20111108

In this article, we will define local and microlocal Sobolev seminorms and prove local and microlocal inverse continuity estimates for the Radon hyperplane transform in these seminorms. The relation between the Sobolev wavefront set of a function f and of its Radon transform is well-known [18]. However, Sobolev wavefront is qualitative and therefore the relation in [18] is qualitative. Our results will make the relation between singularities of a function and those of its Radon transform quantitative. This could be important for practical applications, such as tomography, in which the data are smooth but can have large derivatives., authorCount :2

Filtered back-projection and weighted back-projection have long been the methods of choice within the electron microscopy com- munity for reconstructing the structure of macromolecular assem- blies from electron tomography data. Here, we describe electron lambda-tomography, a reconstruction method that enjoys the ben- efits of the above mentioned methods, namely speed and ease of implementation, but also addresses some of their shortcomings. In particular, compared to these standard methods, electron lambda- tomography is less sensitive to artifacts that come from structures outside the region that is being reconstructed, and it can sharpen boundaries., QC 20120131

We present a new local tomographic algorithm applicable to electron microscope tomography. Our algorithm applies to the standard data acquisition method, single-axis tilting, as well as to more arbitrary acquisition methods including double axis and conical tilt. Using microlocal analysis we put the reconstructions in a mathematical context, explaining which singularities are stably visible from the limited data given by the data collection protocol in the electron microscope. Finally, we provide reconstructions of real specimens from electron tomography data., QC 20120131

We present a new local tomographic algorithm applicable to electron microscopy tomography. Our algorithm applies to the standard data acquisition method, single-axis tilting, as well as for more arbitrary acquisition methods. Using microlocal analysis we put the reconstructions in a mathematical context, explaining which singularities are stably visible from the limited data given by the data collection protocol in the electron microscope., QC 20120131