In this paper, we derive new asymptotic expansions for the solutions of higher order elliptic equations in the presence of small inclusions. As a byproduct, we derive a topological derivative based algorithm for the reconstruction of piecewise smooth functions. This algorithm can be used for edge detection in imaging, topological optimization, and inverse problems, such as quantitative photoacoustic tomography, for which we demonstrate the effectiveness of our asymptotic expansion method numerically.

Total variation flow, total variation regularization, and the taut string algorithm are known to be equivalent filters for one-dimensional discrete signals. In addition, the filtered signal simultaneously minimizes a large number of convex functionals in a certain neighborhood of the data. In this article we study the question of to what extent this situation remains true in a more general setting, namely for data given on the vertices of an oriented graph and the total variation being $J(f) = \sum_{i,j} |f(v_i) - f(v_j)|$. Relying on recent results on invariant $\varphi$-minimal sets we prove that the minimizer to the corresponding Rudin--Osher--Fatemi (ROF) model on the graph has the same universal minimality property as in the one-dimensional setting. Interestingly, this property is lost if $J$ is replaced by the discrete isotropic total variation. Next, we relate the ROF minimizer to the solution of the gradient flow for $J$. It turns out that, in contrast to the one-dimensional setting, these two problems are not equivalent in general, but conditions for equivalence are available.

We consider a problem of quantitative static elastography, the estimation of the Lamé parameters from internal displacement field data. This problem is formulated as a nonlinear operator equation. To solve this equation, we investigate the Landweber iteration both analytically and numerically. The main result of this paper is the verification of a nonlinearity condition in an infinite dimensional Hilbert space context. This condition guarantees convergence of iterative regularization methods. Furthermore, numerical examples for recovery of the Lamé parameters from displacement data simulating a static elastography experiment are presented.

To effectively use photoacoustic tomography to obtain cross-sectional images, a combination of a focused laser illumination and focusing acoustic detectors is used. In this work, we discuss how to incorporate cylindrically shaped focusing detectors in the mathematical modelling of photoacoustics and derive approximative reconstruction formulas. Moreover, we show how such focusing detectors combined with a focused illumination can yield a quantitative reconstruction of the material properties (the transport coefficient and the product of the Grüneisen parameter and the absorption coefficient) of a weakly scattering medium.

The goal of quantitative photoacoustic tomography is to determine optical and acoustical material properties from initial pressure maps such as those obtained, for instance, from photoacoustic imaging. The most relevant parameters are absorption, diffusion, and Grüneisen coefficients, all of which can be heterogeneous. Recent work by Bal and Ren shows that, in general, unique reconstruction of all three parameters is impossible, even if multiple measurements of the initial pressure (corresponding to different laser excitation directions at a single wavelength) are available. Here, we propose a restriction to piecewise constant material parameters. We show that in the diffusion approximation of light transfer, piecewise constant absorption, diffusion, and Grüneisen coefficients can be recovered uniquely from photoacoustic measurements at a single wavelength. In addition, we implemented our ideas numerically and tested them on simulated three-dimensional data.

We investigate the applicability of integral invariants as geometrical shape descriptors in the context of ill-posed inverse problems. We propose the use of a Tikhonov functional, where the penalty term is based on the difference of integral invariants. As a case example, we consider the problem of inverting the Radon transform of an object with only limited angular data available. We approximate the ill-posed operator equation by a minimization problem involving a Tikhonov functional and show existence of minimizers of the functional. Because of its nondifferentiability, we derive for the numerical minimization smooth approximations, which converge in the sense of $\Gamma$-limits.

The covariance matrix adaptation evolution strategy (CMA-ES) has been successfully used to minimize functionals on vector spaces. We generalize the concept of the CMA-ES to Riemannian manifolds and evaluate its performance in two experiments. First, we minimize synthetic functionals on the 2-D sphere. Second, we consider the reconstruction of shapes in 3-D voxel data. A novel formulation of this problem leads to the minimization of edge and region-based segmentation functionals on the Riemannian manifold of parametric 3-D medial axis representation. We compare the results to gradient-based methods on manifolds and particle swarm optimization on tangent spaces and differential evolution.

Snow avalanche victims have only a good chance to survive when they are located within a short time. This requires an active beacon for them to wear or a very rapid deployment of a search-and-rescue team with dogs. Customary ground-penetrating radar (GPR) instruments used on the snow surface are not able to reduce fatality numbers because they are slow to search a field. A potential alternative could be an airborne search using radar. An airborne radar search is technologically challenging because a very large data stream has to be processed and visualized in real time, and the interaction of the electromagnetic waves with snow, subsurface, and objects must be understood. We studied a two-step algorithm to locate avalanche victims in real time. The algorithm was validated using realistic test arrangements and conditions using an aerial tramway. The distance dependence of the reflection energy with increased flight heights, the coherence between the use of more antennas and the detectable range, and the reflection images of different avalanche victims were measured. The algorithm detected an object for each investigated case, where the reflection energy of the scans was higher than for the scans of pure snow. Airborne GPR has a large potential to become a rapid search method in dry snow avalanches. However, a fully operational version still requires substantial improvements in hardware and software.

Fourier reconstruction algorithms significantly outperform conventional backprojection algorithms in terms of computation time. In photoacoustic imaging, these methods require interpolation in the Fourier space domain, which creates artifacts in reconstructed images. We propose a novel reconstruction algorithm that applies the one-dimensional nonuniform fast Fourier transform to photoacoustic imaging. It is shown theoretically and numerically that our algorithm avoids artifacts while preserving the computational effectiveness of Fourier reconstruction.

We study the evolution of T-spline level sets (i.e, implicitly defined T-spline curves and surfaces). The use of T-splines leads to a sparse representation of the geometry and allows for an adaptation to the given data, which can be unorganized points or images. The evolution process is governed by a combination of prescribed, data-driven normal velocities, and additional distance field constraints. By incorporating the distance field constraints we are able to avoid additional branches and singularities of the T-spline level sets without having to use re-initialization steps. Experimental examples are presented to demonstrate the effectiveness of our approach.