Properties of Some Integral Transforms Arising in Tomography

This dissertation deals with several types of imaging: radio tomography, single scattering optical tomography, photoacoustic tomography, and Compton camera imaging. Each of these tomographic techniques leads to a Radon-type transform: radio tomography brings about an elliptical Radon transform, single scattering optical tomography reduces to the V-line Radon transform, and photoacoustic tomography with line detectors boils down to a cylindrical Radon transform. We also introduce a different Radon-type transform arising in photo acoustic tomography with circular detectors, and study mathematically similar object, a toroidal Radon transform. We also consider the cone transform arising in Compton camera imaging as well as the windowed ray transform. We provide inversion formulas for all these transforms. When given some Radon- type transform, we are interested not only in inversion formulas, but also in range conditions, and stability. We thus address range conditions, a stability estimate for some of the Radon-type transforms above

Trkalian fields: ray transforms and mini-twistors

We study X-ray and Divergent beam transforms of Trkalian fields and their relation with Radon transform. We make use of four basic mathematical methods of tomography due to Grangeat, Smith, Tuy and Gelfand-Goncharov for an integral geometric view on them. We also make use of direct approaches which provide a faster but restricted view of the geometry of these transforms. These reduce to well known geometric integral transforms on a sphere of the Radon or the spherical Curl transform in Moses eigenbasis, which are members of an analytic family of integral operators. We also discuss their inversion. The X-ray (also Divergent beam) transform of a Trkalian field is Trkalian. Also the Trkalian subclass of X-ray transforms yields Trkalian fields in the physical space. The Riesz potential of a Trkalian field is proportional to the field. Hence, the spherical mean of the X-ray (also Divergent beam) transform of a Trkalian field over all lines passing through a point yields the field at this point. The pivotal point is the simplification of an intricate quantity: Hilbert transform of the derivative of Radon transform for a Trkalian field in the Moses basis. We also define the X-ray transform of the Riesz potential (of order 2) and Biot-Savart integrals. Then, we discuss a mini-twistor respresentation, presenting a mini-twistor solution for the Trkalian fields equation. This is based on a time-harmonic reduction of wave equation to Helmholtz equation. A Trkalian field is given in terms of a null vector in C3 with an arbitrary function and an exponential factor resulting from this reduction.Comment: 37 pages, http://dx.doi.org/10.1063/1.482610

Inversion of circular means and the wave equation on convex planar domains

We study the problem of recovering the initial data of the two dimensional wave equation from values of its solution on the boundary \partial \Om of a smooth convex bounded domain \Om \subset \R^2. As a main result we establish back-projection type inversion formulas that recover any initial data with support in \Om modulo an explicitly computed smoothing integral operator \K_\Om. For circular and elliptical domains the operator \K_\Om is shown to vanish identically and hence we establish exact inversion formulas of the back-projection type in these cases. Similar results are obtained for recovering a function from its mean values over circles with centers on \partial \Om. Both reconstruction problems are, amongst others, essential for the hybrid imaging modalities photoacoustic and thermoacoustic tomography.Comment: [14 pages, 2 figures

An inversion formula for transport equation in 3-dimensions using several complex variable analysis

In this paper, the photon stationary transport equation has been extended from $\mathbb{R}^3$ to $\mathbb{C}^3$. A solution of the inverse problem is obtained on a hyper-sphere and a hyper-cylinder as X-ray and Radon transform, respectively. We show that these results can be transformed into each other and they agree with known results