Table of contents

We consider region of interest (ROI) tomography of piecewise constant functions. Additionally, an algorithm is developed for ROI tomography of piecewise constant functions using a Haar wavelet basis. A weighted ℓ_p–penalty is used with weights that depend on the relative location of wavelets to the region of interest. We prove that the proposed method is a regularization method, i.e., that the regularized solutions converge to the exact piecewise constant solution if the noise tends to zero. Tests on phantoms demonstrate the effectiveness of the method.

In this paper we consider the inverse acoustic scattering (in ${{\mathbb{R}}^{3}}$) or electromagnetic scattering (in ${{\mathbb{R}}^{2}}$, for the scalar TE-polarization case) problem of reconstructing possibly multiple defective penetrable regions in a known anisotropic material of compact support. We develop the factorization method for a non-absorbing anisotropic background media containing penetrable defects. In particular, under appropriate assumptions on the anisotropic material properties of the media we develop a rigorous characterization for the support of the defective regions from the given far field measurements. Finally we present some numerical examples in the two-dimensional case to demonstrate the feasibility of our reconstruction method including examples for the case when the defects are voids (i.e. subregions with refractive index the same as the background outside the inhomogeneous hosting media).

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 seven angular views only. We illustrate the practical applicability on a real positron emission tomography dataset. As further applications, we consider spherical Radon data as well as blurred data.

The reconstruction of dynamic images from few projection data is a challenging problem, especially when noise is present and when the dynamic images are vary fast. In this paper, we propose a variational model, sparsity enforced matrix factorization (SEMF), based on low rank matrix factorization of unknown images and enforced sparsity constraints for representing both coefficients and bases. The proposed model is solved via an alternating iterative scheme for which each subproblem is convex and involves the efficient alternating direction method of multipliers (ADMM). The convergence of the overall alternating scheme for the nonconvex problem relies upon the Kurdyka–Łojasiewicz property, recently studied by Attouch et al (2010 Math. Oper. Res.35 438) and Attouch et al (2013 Math. Program.137 91). Finally our proof-of-concept simulation on 2D dynamic images shows the advantage of the proposed method compared to conventional methods.

We are concerned with Tikhonov regularization of linear ill-posed problems with ℓ₁ coefficient penalties. Griesse and Lorenz (2008 Inverse Problems24 035007) proposed a semismooth Newton method for the efficient minimization of the corresponding Tikhonov functionals. In the class of high-precision solvers for such problems, semismooth Newton methods are particularly competitive due to their superlinear convergence properties and their ability to solve piecewise affine equations exactly within finitely many iterations. However, the convergence of semismooth Newton schemes is only local in general. In this work, we discuss the efficient globalization of B(ouligand)-semismooth Newton methods for ℓ₁ Tikhonov regularization by means of damping strategies and suitable descent with respect to an associated merit functional. Numerical examples are provided which show that our method compares well with existing iterative, globally convergent approaches.

We suggest a prospective method for detecting and visualizing defects in fibre-reinforced composites by computing external volume forces from measurements acquired by sensors that are integrated on the surface of the structure. Anisotropic materials like carbon fibre-reinforced composites are widely used in light weight construction which can exhibit damages that are not optically detectable. The key idea of our method is the interpretation of defects in such structures as if they were induced by an external volume force. This idea is based on the observation that a propagating elastic wave interferes with a damaged area by reflecting the wave. In that sense a damage can be seen as an additional source. Thus identifying the external volume force which has caused this wave is supposed to reveal the location of the defect. This approach leads to the inverse problem of determining the inhomogeneity of a hyperbolic initial-boundary value problem. We tackle this ill-posed problem by minimizing a Tikhonov functional which takes the oberservation points of our surface measurements into account. In the article we address the solvability of the direct problem, state and analyze the PDE-based optimization problem that aims for computing the external force and develop a numerical realization of its solution using the conjugate gradient method. First numerical results for a simple model case with different sensor adjustments show that the defects in fact are detectable. In that sense this article might be seen as starting point of future research which should comprehend deeper numerical studies and analysis of the problem.

This paper is concerned with a method of image reconstruction and feature extraction for the attenuated Radon transform in two dimensions based on the decomposition in circular harmonics of the integral kernel in Novikovʼs inversion formula for an arbitrary known attenuation. This analytical decomposition of the reconstruction kernel provides an alternative reconstruction algorithm. Besides, we propose to use our formula to directly extract features of the object with no need for process imaging techniques. Numerical results attest to the strengths and limitations of our reconstruction method in terms of accuracy and robustness for image and feature reconstruction.

We provide extensions of the classical Ambarzumian theorem for bounded C³ domains of any dimension. The simple proof is based on classical spectral function asymptotics. We prove a stability property by showing that if the perturbation of the eigenvalues of the zero potential is small in some sense then the L₂-norm of the potential is also small. The problem is motivated by connections to a number of applications.

Using measure theoretic arguments, we provide a general framework for describing and studying the general linear inverse dispersion problem where no a priori assumptions on the source function has been made (other than assuming that it is indeed a source, i.e. not a sink). We investigate the source-sensor relationship and rigorously state solvability conditions for when the inverse problem can be solved using a least-squares optimization method. That is, we derive conditions for when the least-squares problem is well-defined.