David Finch

Inversion of spherical means and the wave equation in even dimensions

We establish inversion formulas of the so-called filtered back-projection type to recover a function supported in the ball in even dimensions from its spherical means over spheres centered on the boundary of the ball. We also find several formulas to recover initial data of the form

Cone Beam Reconstruction with Sources on a Curve

(f,0)

The spherical mean value operator with centers on a sphere

(or

The range of the spherical mean value operator for functions supported in a ball

(0,g)

Uniqueness for the attenuated x-ray transform in the physical range

) for the free space wave equation in even dimensions from the trace of the solution on the boundary of the ball, provided that the initial data has support in the ball.

The x-ray transform for a non-Abelian connection in two dimensions

An inversion procedure is developed for reconstructing a function of compact support in

Local Reconstruction Applied to X-Ray Microtomography

\mathbb{R}^3

A Characterization of the Range of the Divergent Beam x-Ray Transform

from its divergent beam x-ray transform with sources on a curve. The results apply to many curves not satisfying the conditions of Tuy [SIAM J. Appl. Math., 43 (1983), pp. 546–552), for example, planar circles of sufficiently large radius. In some cases, estimates in Sobolev norms are established for the inversion operator.

Sums of homogeneous functions and the range of the divergent beam x-ray transform

Let B represent the ball of radius ρ in Rn and S its boundary; consider the map , where represents the mean value of f on a sphere of radius r centered at p. We summarize and discuss the results concerning the injectivity of , the characterization of the range of , and the inversion of . There is a close connection between mean values over spheres and solutions of initial value problems for the wave equation. We also summarize the results for the corresponding wave equation problem.

Stability and Consistency for the Divergent Beam X-Ray Transform

Suppose n > 1 is an odd integer, f is a smooth function supported in a ball B with boundary S, and u is the solution of the initial value problem u tt - Δ x u = 0, (x, t) ∈ R n x [0, ∞); u(x,t = 0) = 0, u t (x, t = 0) = f(x), x ∈ R n . We characterize the range of the map f → u| S×[0,∞) and give a stable scheme for the inversion of this map. This also characterizes the range of the map sending f to its mean values over spheres centred on S.