What is Cone-Beam CT and How Does it Work?

A duality in integral geometry A duality for symmetric spaces The fourier transform on a symmetric space The Radon transform on $X$ and on $X_o$. Range questions Differential equations on symmetric spaces Eigenspace representations Solutions to exercises Bibliography Symbols frequently used Index.

Geometric Analysis on Symmetric Spaces

This paper treats the linearized inverse scattering problem for the case of variable background velocity and for an arbitrary configuration of sources and receivers. The linearized inverse scattering problem is formulated in terms of an integral equation in a form which covers wave propagation in fluids with constant and variable densities and in elastic solids. This integral equation is connected with the causal generalized Radon transform (GRT), and an asymptotic expansion of the solution of the integral equation is obtained using an inversion procedure for the GRT. The first term of this asymptotic expansion is interpreted as a migration algorithm. As a result, this paper contains a rigorous derivation of migration as a technique for imaging discontinuities of parameters describing a medium. Also, a partial reconstruction operator is explicitly derived for a limited aperture. When specialized to a constant background velocity and specific source–receiver geometries our results are directly related to some known migration algorithms.

Imaging of discontinuities in the inverse scattering problem by inversion of a causal generalized Radon transform

The theory of pseudo differential operators, discussed in § 1, is well suited for investigating various problems connected with elliptic differential equations. However, this theory fails to be adequate for studying equations of hyperbolic type, and one is then forced to examine a wider class of operators, the so-called Fourier integral operators (Egorov [1975], Hormander [1968, 1971, 1983, 1985], Kumano-go [1982], Shubin [1978], Taylor [1981], Treves [1980]).

Fourier Integral Operators

Suppose D is a bounded, connected, open set in Rn and f is a smooth function on Rn with support in $\overD$. We study the recovery of f from the mean values of f over spheres centered on a part or the whole boundary of D. For strictly convex $\overline{D}$, we prove uniqueness when the centers are restricted to an open subset of the boundary. We provide an inversion algorithm (with proof) when the mean values are known for all spheres centered on the boundary of D, with radii in the interval [0, diam(D)/2]. We also give an inversion formula when D is a ball in Rn, $n \geq 3$ and odd, and the mean values are known for all spheres centered on the boundary.

http://www.math.udel.edu/~rakesh/RESEARCH/meanvalue.pdf

Determining a function from its mean values over a family of spheres

This special issue of Inverse Problems honours the achievements of Nobel Laureate Allan Cormack (1924–1998) whose pioneering work provided mathematical foundations of computerized tomography. It gathers top articles on inverse problems in imaging, including a broad range of tomographic modalities, mathematics, and applications. Inverse problems, typically image reconstruction from indirect data, now play a crucial role in many aspects of human life such as healthcare, national security, non-destructive testing, and remote sensing. The relentless progress in imaging technology opens myriad possibilities, pushes researchers to overcome new theoretical and practical challenges, and has an enormous impact on everyday life. Twenty-nine high-quality original research papers have been collected in this special issue. They represent a broad range of inverse problems, from theoretical and computational perspectives, and are all related to modern challenges in imaging. Most inverse problems involved in physical processes are ill-posed, requiring suitable regularization schemes to stabilize the inverse and the reconstruction process. The emergence of numerous novel applications as well as the evolution of constraints such as costs, time, access, etc and hence the need for more efficient, smaller, and cheaper systems, lead to new challenges in terms of regularization and reconstruction schemes. This special issue offers many answers to these challenges. [19] provides a deep and fresh look at l1 Tikhonov regularization. [13] adapts the ADMM method for total-variation myopic deconvolution. [3] proposes bilevel optimization neural networks based on the fractional Laplacian to solve inverse problems. [17] also addresses the question of edge-preserving regularization method and works out an inner-outer approach that improves Tikhonov regularization by constructing adaptive smoothing operators. [11] studies lp-lq constrained minimization problems and develops associated modulus-based iterative schemes. In [23], the authors develop the theory of Bayesian

Foreword to special issue of Inverse Problems on modern challenges in imaging

This article provides a mathematical analysis of singular (nonsmooth) artifacts added to reconstructions by filtered backprojection (FBP) type algorithms for X-ray 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 region-of-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.

Theorems that Characterize Artifacts for Arbitrary Limited X-ray CT Data

We prove Morera theorems for the Radon transform integrating on spheres through a point in ℂ n . The proofs use spherical functions and integral equations techniques as well as a support theorem for a generalized Radon transform on hyperplanes.

Morera Theorems for Spheres through a Point in CN

We consider the generalized Radon transform (defined in terms of smooth weight functions) on hyperplanes in $\mathbb{R}^n$. We analyze general filtered backprojection type reconstruction methods for limited data with filters given by general pseudodifferential operators. We provide microlocal characterizations of visible and added singularities in $\mathbb{R}^n$ and define modified versions of reconstruction operators that do not generate added artifacts. We calculate the symbol of our general reconstruction operators as pseudodifferential operators, and provide conditions for the filters under which the reconstruction operators are elliptic for the visible singularities. If the filters are chosen according to those conditions, we show that almost all visible singularities can be recovered reliably. Our work generalizes the results for the classical line transforms in $\mathbb{R}^2$ and the classical reconstruction operators (that use specific filters). In our proofs, we employ a general paradigm that is based on the calculus of Fourier integral operators. Since this technique does not rely on explicit expressions of the reconstruction operators, it enables us to analyze more general imaging situations.

Eric Todd Quinto

Papers

Foreword to special issue of Inverse Problems on modern challenges in imaging

Theorems that Characterize Artifacts for Arbitrary Limited X-ray CT Data

Morera Theorems for Spheres through a Point in CN

Limited data problems for the generalized Radon transform in $\mathbb{R}^n$