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We present a novel analysis of a Radon transform, $R$, which maps an $L^2$ function of compact support to its integrals over smooth surfaces of revolution with centers on an embedded hypersurface in $\mathbb{R}^n$. Using microlocal analysis, we derive necessary and sufficient conditions relating to $R$ for the Bolker condition to hold, which has implications regarding the existence and location of image artifacts. We present a general inversion framework based on Volterra equation theory and known results on the spherical Radon transform, and we prove injectivity results for $R$. Several example applications of our theory are discussed in the context of, e.g., Compton Scatter Tomography (CST) and Ultrasound Reflection Tomography (URT). In addition, using the proposed inversion framework, we validate our microlocal theory via simulation, and present simulated image reconstructions of image phantoms with added noise., Comment: 23 pages, 7 figures

We present novel microlocal and injectivity analyses of ellipsoid and hyperboloid Radon transforms. We introduce a new Radon transform, $R$, which defines the integrals of a compactly supported $L^2$ function, $f$, over ellipsoids and hyperboloids with centers on a smooth connected surface, $S$. $R$ is shown to be a Fourier Integral Operator (FIO) and in our main theorem we prove that $R$ satisfies the Bolker condition if the support of $f$ is connected and not intersected by any plane tangent to $S$. Under certain conditions, this is an equivalence. We give examples where our theory can be applied. Focusing specifically on a cylindrical geometry of interest in Ultrasound Reflection Tomography (URT), we prove injectivity results and investigate the visible singularities. In addition, we present example reconstructions of image phantoms in two-dimensions, and validate our microlocal theory., Comment: 26 pages, 11 figures

We present a microlocal analysis of two novel Radon transforms of interest in Compton Scattering Tomography (CST), which map compactly supported $L^2$ functions to their integrals over seven-dimensional sets of apple and lemon surfaces. Specifically, we show that the apple and lemon transforms are elliptic Fourier Integral Operators (FIO), which satisfy the Bolker condition. After an analysis of the full seven-dimensional case, we focus our attention on $n$-D subsets of apple and lemon surfaces with fixed central axis, where $n<7$. Such subsets of surface integrals have applications in airport baggage and security screening. When the data dimensionality is restricted, the apple transform is shown to violate the Bolker condition, and there are artifacts which occur on apple-cylinder intersections. The lemon transform is shown to satisfy the Bolker condition, when the support of the function is restricted to the strip $\{0

Here we present a novel microlocal analysis of generalized Radon transforms which describe the integrals of $L^2$ functions of compact support over surfaces of revolution of $C^{\infty}$ curves $q$. We show that the Radon transforms are elliptic Fourier Integral Operators (FIO) and provide an analysis of the left projections $\Pi_L$. Our main theorem shows that $\Pi_L$ satisfies the semi-global Bolker assumption if and only if $g=q'/q$ is an immersion. An analysis of the visible singularities is presented, after which we derive novel Sobolev smoothness estimates for the Radon FIO. Our theory has specific applications of interest in Compton Scattering Tomography (CST) and Bragg Scattering Tomography (BST). We show that the CST and BST integration curves satisfy the Bolker assumption and provide simulated reconstructions from CST and BST data. Additionally we give example "sinusoidal" integration curves which do not satisfy Bolker and provide simulations of the image artefacts. The observed artefacts in reconstruction are shown to align exactly with our predictions., Comment: 24 pages, 9 figures

Here we present new joint reconstruction and regularization techniques inspired by ideas in microlocal analysis and lambda tomography, for the simultaneous reconstruction of the attenuation coefficient and electron density from X-ray transmission (i.e., X-ray CT) and backscattered data (assumed to be primarily Compton scattered). To demonstrate our theory and reconstruction methods, we consider the "parallel line segment" acquisition geometry of Webber and Miller ("Compton scattering tomography in translational geometries." Inverse Problems 36, no. 2 (2020): 025007), which is motivated by system architectures currently under development for airport security screening. We first present a novel microlocal analysis of the parallel line geometry which explains the nature of image artefacts when the attenuation coefficient and electron density are reconstructed separately. We next introduce a new joint reconstruction scheme for low effective $Z$ (atomic number) imaging ($Z<20$) characterized by a regularization strategy whose structure is derived from lambda tomography principles and motivated directly by the microlocal analytic results. Finally we show the effectiveness of our method in combating noise and image artefacts on simulated phantoms., Comment: 32 pages, 15 figures

Here we present a novel microlocal analysis of a new toric section transform which describes a two dimensional image reconstruction problem in Compton scattering tomography and airport baggage screening. By an analysis of two separate limited data problems for the circle transform and using microlocal analysis, we show that the canonical relation of the toric section transform is 2--1. This implies that there are image artefacts in the filtered backprojection reconstruction. We provide explicit expressions for the expected artefacts and demonstrate these by simulations. In addition, we prove injectivity of the forward operator for $L^\infty$ functions supported inside the open unit ball. We present reconstructions from simulated data using a discrete approach and several regularizers with varying levels of added pseudo-random noise., Comment: 31 pages, 18 figures

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.

In this article, we consider two bistatic cases arising in synthetic aperture radar imaging: when the transmitter and receiver are both moving with different speeds along a single line parallel to the ground in the same direction or in the opposite directions. In both cases, we classify the forward operator $\Fc$ as a Fourier integral operator with fold/blowdown singularities. Next we analyze the normal operator $\Fc^*\Fc$ in both cases (where $\Fc^{*}$ is the $L^{2}$-adjoint of $\Fc$). When the transmitter and receiver move in the same direction, we prove that $\Fc^*\Fc$ belongs to a class of operators associated to two cleanly intersecting Lagrangians, $I^{p,l} (\Delta, C_1)$. When they move in opposite directions, $\Fc^*\Fc$ is a sum of such operators. In both cases artifacts appear and we show that they are, in general, as strong as the bona-fide part of the image. Moreover, we demonstrate that as soon as the source and receiver start to move in opposite directions, there is an interesting bifurcation in the type of artifact that appears in the image., Comment: accepted SIAM Journal on Mathematical Analysis

In this paper, we use microlocal analysis to understand what X-ray tomographic data acquisition does to singularities of an object which changes during the measuring process. Depending on the motion model, we study which singularities are detected by the measured data. In particular, this analysis shows that, due to the dynamic behavior, not all singularities might be detected, even if the radiation source performs a complete turn around the object. Thus, they cannot be expected to be (stably) visible in any reconstruction. On the other hand, singularities could be added (or masked) as well. To understand this precisely, we provide a characterization of visible and added singularities by analyzing the microlocal properties of the forward and reconstruction operators. We illustrate the characterization using numerical examples., Comment: 27 pages, 10 figures

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.

We present a paradigm for characterization of artifacts in limited data tomography problems. In particular, we use this paradigm to characterize artifacts that are generated in reconstructions from limited angle data with generalized Radon transforms and general filtered backprojection type operators. In order to find when visible singularities are imaged, we calculate the symbol of our reconstruction operator as a pseudodifferential operator.

We develop a paradigm using microlocal analysis that allows one to characterize the visible and added singularities in a broad range of incomplete data tomography problems. We give precise characterizations for photo- and thermoacoustic tomography and Sonar, and provide artifact reduction strategies. In particular, our theorems show that it is better to arrange Sonar detectors so that the boundary of the set of detectors does not have corners and is smooth. To illustrate our results, we provide reconstructions from synthetic spherical mean data as well as from experimental photoacoustic data.

In this article, we consider a generalized Radon transform that comes up in ultrasound reflection tomography. In our model, the ultrasound emitter and receiver move at a constant distance apart along a circle. We analyze the microlocal properties of the transform $R$ that arises from this model. As a consequence, we show that for distributions with support sufficiently inside the circle, $R^*R$ is an elliptic pseudodifferential operator of order $-1$ and hence all the singularities of such distributions can be recovered.

In this article, we analyze the microlocal properties of the linearized forward scattering operator $F$ and the reconstruction operator $F^{*}F$ appearing in bistatic synthetic aperture radar imaging. In our model, the radar source and detector travel along a line a fixed distance apart. We show that $F$ is a Fourier integral operator, and we give the mapping properties of the projections from the canonical relation of $F$, showing that the right projection is a blow-down and the left projection is a fold. We then show that $F^{*}F$ is a singular FIO belonging to the class $I^{3,0}$.

We consider the Cauchy problem for the wave equation in the whole space, R^n, with initial data which are distributions supported on finite sets. The main result is a precise description of the geometry of the sets of stationary points of the solutions to the wave equation., Comment: 9 pages, AMSTeX version 2.1