163 research outputs found
Exact reconstruction formulas for a Radon transform over cones
Inversion of Radon transforms is the mathematical foundation of many modern
tomographic imaging modalities. In this paper we study a conical Radon
transform, which is important for computed tomography taking Compton scattering
into account. The conical Radon transform we study integrates a function in
over all conical surfaces having vertices on a hyperplane and symmetry
axis orthogonal to this plane. As the main result we derive exact
reconstruction formulas of the filtered back-projection type for inverting this
transform.Comment: 8 pages, 1 figur
A series solution and a fast algorithm for the inversion of the spherical mean Radon transform
An explicit series solution is proposed for the inversion of the spherical
mean Radon transform. Such an inversion is required in problems of thermo- and
photo- acoustic tomography. Closed-form inversion formulae are currently known
only for the case when the centers of the integration spheres lie on a sphere
surrounding the support of the unknown function, or on certain unbounded
surfaces. Our approach results in an explicit series solution for any closed
measuring surface surrounding a region for which the eigenfunctions of the
Dirichlet Laplacian are explicitly known - such as, for example, cube, finite
cylinder, half-sphere etc. In addition, we present a fast reconstruction
algorithm applicable in the case when the detectors (the centers of the
integration spheres) lie on a surface of a cube. This algorithm reconsrtucts
3-D images thousands times faster than backprojection-type methods
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