The limited-view problem is studied for thermoacoustic tomography, which is also referred to as photoacoustic or optoacoustic tomography depending on the type of radiation for the induction of acoustic waves. We define a ''detection region,'' within which all points have sufficient detection views. It is explained analytically and shown numerically that the boundaries of any objects inside this region can be recovered stably. Otherwise some sharp details become blurred. One can identify in advance the parts of the boundaries that will be affected if the detection view is insufficient. If the detector scans along a circle in a two-dimensional case, acquiring a sufficient view might require covering more than a -, or less than a -arc of the trajectory depending on the position of the object. Similar results hold in a three-dimensional case. In order to support our theoretical conclusions, three types of reconstruction methods are utilized: a filtered backprojection ͑FBP͒ approximate inversion, which is shown to work well for limited-view data, a local-tomography-type reconstruction that emphasizes sharp details ͑e.g., the boundaries of inclusions͒, and an iterative algebraic truncated conjugate gradient algorithm used in conjunction with FBP. Computations are conducted for both numerically simulated and experimental data. The reconstructions confirm our theoretical predictions.
The circular Radon transform integrates a function over the set of all spheres with a given set of centers. The problem of injectivity of this transform (as well as inversion formulas, range descriptions, etc.) arises in many fields from approximation theory to integral geometry, to inverse problems for PDEs, and recently to newly developing types of tomography. A major breakthrough in the 2D case was made several years ago in a work by M. Agranovsky and E. T. Quinto. Their techniques involved microlocal analysis and known geometric properties of zeros of harmonic polynomials in the plane. Since then there has been an active search for alternative methods, especially the ones based on simple PDE techniques, which would be less restrictive in more general situations. The article provides some new results that one can obtain by methods that essentially involve only the finite speed of propagation and domain dependence for the wave equation.
The transform considered in the paper integrates a function supported in the unit disk on the plane over all circles centered at the boundary of this disk. Such circular Radon transform arises in several contemporary imaging techniques, as well as in other applications. As it is common for transforms of Radon type, its range has infinite codimension in standard function spaces. Range descriptions for such transforms are known to be very important for computed tomography, for instance when dealing with incomplete data, error correction, and other issues. A complete range description for the circular Radon transform is obtained. Range conditions include the recently found set of moment type conditions, which happens to be incomplete, as well as the rest of conditions that have less standard form. In order to explain the procedure better, a similar (non-standard) treatment of the range conditions is described first for the usual Radon transform on the plane.
Abstract. We study a new class of Radon transforms defined on circular cones called the conical Radon transform. In R 3 it maps a function to its surface integrals over circular cones, and in R 2 it maps a function to its integrals along two rays with a common vertex. Such transforms appear in various mathematical models arising in medical imaging, nuclear industry and homeland security. This paper contains new results about inversion of conical Radon transform with fixed opening angle and vertical central axis in R 2 and R 3 . New simple explicit inversion formulae are presented in these cases. Numerical simulations were performed to demonstrate the efficiency of the suggested algorithm in 2D.
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