93 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
Deep learning versus -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
-minimization. Recently, deep learning appeared as a new paradigm for
CS and other inverse problems. In this paper, we compare a recently invented
joint -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 -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
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