25 research outputs found
Artifacts in incomplete data tomography - with applications to photoacoustic tomography and sonar
We develop a paradigm using microlocal analysis that allows one to
characterize the visible and added singularities in a broad range of incomplete
data tomography problems. We give precise characterizations for photo- and
thermoacoustic tomography and Sonar, and provide artifact reduction strategies.
In particular, our theorems show that it is better to arrange Sonar detectors
so that the boundary of the set of detectors does not have corners and is
smooth. To illustrate our results, we provide reconstructions from synthetic
spherical mean data as well as from experimental photoacoustic data
Sparse regularization in limited angle tomography
AbstractWe investigate the reconstruction problem of limited angle tomography. Such problems arise naturally in applications like digital breast tomosynthesis, dental tomography, electron microscopy, etc. Since the acquired tomographic data is highly incomplete, the reconstruction problem is severely ill-posed and the traditional reconstruction methods, e.g. filtered backprojection (FBP), do not perform well in such situations.To stabilize the reconstruction procedure additional prior knowledge about the unknown object has to be integrated into the reconstruction process. In this work, we propose the use of the sparse regularization technique in combination with curvelets. We argue that this technique gives rise to an edge-preserving reconstruction. Moreover, we show that the dimension of the problem can be significantly reduced in the curvelet domain. To this end, we give a characterization of the kernel of the limited angle Radon transform in terms of curvelets and derive a characterization of solutions obtained through curvelet sparse regularization. In numerical experiments, we will show that the theoretical results directly translate into practice and that the proposed method outperforms classical reconstructions
Joint Image Reconstruction and Segmentation Using the Potts Model
We propose a new algorithmic approach to the non-smooth and non-convex Potts
problem (also called piecewise-constant Mumford-Shah problem) for inverse
imaging problems. We derive a suitable splitting into specific subproblems that
can all be solved efficiently. Our method does not require a priori knowledge
on the gray levels nor on the number of segments of the reconstruction.
Further, it avoids anisotropic artifacts such as geometric staircasing. We
demonstrate the suitability of our method for joint image reconstruction and
segmentation. We focus on Radon data, where we in particular consider limited
data situations. For instance, our method is able to recover all segments of
the Shepp-Logan phantom from angular views only. We illustrate the
practical applicability on a real PET dataset. As further applications, we
consider spherical Radon data as well as blurred data
On Artifacts in Limited Data Spherical Radon Transform: Curved Observation Surface
In this article, we consider the limited data problem for spherical mean
transform. We characterize the generation and strength of the artifacts in a
reconstruction formula. In contrast to the third's author work [Ngu15b], the
observation surface considered in this article is not flat. Our results are
comparable to those obtained in [Ngu15b] for flat observation surface. For the
two dimensional problem, we show that the artifacts are orders smoother
than the original singularities, where is vanishing order of the smoothing
function. Moreover, if the original singularity is conormal, then the artifacts
are order smoother than the original singularity. We provide
some numerical examples and discuss how the smoothing effects the artifacts
visually. For three dimensional case, although the result is similar to that
[Ngu15b], the proof is significantly different. We introduce a new idea of
lifting the space
A new 3D model for magnetic particle imaging using realistic magnetic field topologies for algebraic reconstruction
We derive a new 3D model for magnetic particle imaging (MPI) that is able to
incorporate realistic magnetic fields in the reconstruction process. In real
MPI scanners, the generated magnetic fields have distortions that lead to
deformed magnetic low-field volumes (LFV) with the shapes of ellipsoids or
bananas instead of ideal field-free points (FFP) or lines (FFL), respectively.
Most of the common model-based reconstruction schemes in MPI use however the
idealized assumption of an ideal FFP or FFL topology and, thus, generate
artifacts in the reconstruction. Our model-based approach is able to deal with
these distortions and can generally be applied to dynamic magnetic fields that
are approximately parallel to their velocity field. We show how this new 3D
model can be discretized and inverted algebraically in order to recover the
magnetic particle concentration. To model and describe the magnetic fields, we
use decompositions of the fields in spherical harmonics. We complement the
description of the new model with several simulations and experiments.Comment: 27 pages, 11 figure, 3 table
Data-proximal null-space networks for inverse problems
Inverse problems are inherently ill-posed and therefore require
regularization techniques to achieve a stable solution. While traditional
variational methods have well-established theoretical foundations, recent
advances in machine learning based approaches have shown remarkable practical
performance. However, the theoretical foundations of learning-based methods in
the context of regularization are still underexplored. In this paper, we
propose a general framework that addresses the current gap between
learning-based methods and regularization strategies. In particular, our
approach emphasizes the crucial role of data consistency in the solution of
inverse problems and introduces the concept of data-proximal null-space
networks as a key component for their solution. We provide a complete
convergence analysis by extending the concept of regularizing null-space
networks with data proximity in the visual part. We present numerical results
for limited-view computed tomography to illustrate the validity of our
framework