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 $\R^d$ 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

Deep learning versus ℓ1\ell^1-minimization for compressed sensing photoacoustic tomography

We investigate compressed sensing (CS) techniques for reducing the number of measurements in photoacoustic tomography (PAT). High resolution imaging from CS data requires particular image reconstruction algorithms. The most established reconstruction techniques for that purpose use sparsity and $\ell^1$-minimization. Recently, deep learning appeared as a new paradigm for CS and other inverse problems. In this paper, we compare a recently invented joint $\ell^1$-minimization algorithm with two deep learning methods, namely a residual network and an approximate nullspace network. We present numerical results showing that all developed techniques perform well for deterministic sparse measurements as well as for random Bernoulli measurements. For the deterministic sampling, deep learning shows more accurate results, whereas for Bernoulli measurements the $\ell^1$-minimization algorithm performs best. Comparing the implemented deep learning approaches, we show that the nullspace network uniformly outperforms the residual network in terms of the mean squared error (MSE).Comment: This work has been presented at the Joint Photoacoustics Session with the 2018 IEEE International Ultrasonics Symposium Kobe, October 22-25, 201

Universal inversion formulas for recovering a function from spherical means

The problem of reconstruction a function from spherical means is at the heart of several modern imaging modalities and other applications. In this paper we derive universal back-projection type reconstruction formulas for recovering a function in arbitrary dimension from averages over spheres centered on the boundary an arbitrarily shaped smooth convex domain. Provided that the unknown function is supported inside that domain, the derived formulas recover the unknown function up to an explicitly computed smoothing integral operator. For elliptical domains the integral operator is shown to vanish and hence we establish exact inversion formulas for recovering a function from spherical means centered on the boundary of elliptical domains in arbitrary dimension.Comment: [20 pages, 2 figures] Compared to the previous versions I corrected some typo