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

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.

Geometry of Stationary Sets for the Wave Equation in R^n, The Case of Finitely Supported Initial Data, An Announcement

Remarks on stationary sets for the wave equation

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 $\mathcal{F}$ as a Fourier integral operator with fold/blowdown singularities. Next we analyze the normal operator $\mathcal{F}^*\mathcal{F}$ in both cases (where $\mathcal{F}^{*}$ is the $L^{2}$-adjoint of $\mathcal{F}$). When the transmitter and receiver move in the same direction, we prove that $\mathcal{F}^*\mathcal{F}$ belongs to a class of operators associated to two cleanly intersecting Lagrangians, $I^{p,l} (\Delta, C_1)$. When they move in opposite directions, $\mathcal{F}^*\mathcal{F}$ 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 ...

Singular FIOs in SAR Imaging, II: Transmitter and Receiver at Different Speeds

Following from the previous chapter Motion compensation strategies in tomography, this article provides a complementary study on the overall information content in dynamic tomographic data using the framework of microlocal analysis and Fourier integral operators. Based on this study, we further analyze which characteristic features of the studied specimen can be reliably reconstructed from dynamic tomographic data and which additional artifacts have to be expected in a dynamic image reconstruction. Our theoretical results, in particular the affect of the dynamic behavior on the measured data and the reconstruction result, is then illustrated in detail at various numerical examples from dynamic photoacoustic tomography.

Microlocal Properties of Dynamic Fourier Integral Operators

The three-dimensional elliptic Radon transform (eRT) averages distributions over ellipsoids of revolution. It thus serves as a linear model in seismic imaging where one wants to recover the earth's...

Eric Todd Quinto

Papers

Geometry of Stationary Sets for the Wave Equation in R^n, The Case of Finitely Supported Initial Data, An Announcement

Remarks on stationary sets for the wave equation

Singular FIOs in SAR Imaging, II: Transmitter and Receiver at Different Speeds

Microlocal Properties of Dynamic Fourier Integral Operators

Imaging with the Elliptic Radon Transform in Three Dimensions from an Analytical and Numerical Perspective