## Journals

- Advances in Mathematics of Communications
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- Discrete & Continuous Dynamical Systems - A
- Discrete & Continuous Dynamical Systems - B
- Discrete & Continuous Dynamical Systems - S
- Evolution Equations & Control Theory
- Inverse Problems & Imaging
- Journal of Computational Dynamics
- Journal of Dynamics & Games
- Journal of Geometric Mechanics
- Journal of Industrial & Management Optimization
- Journal of Modern Dynamics
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- Mathematical Biosciences & Engineering
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IPI

In this work we consider optical flow on evolving Riemannian 2-manifolds which can be parametrised from the 2-sphere. Our main motivation is to estimate cell motion in time-lapse volumetric microscopy images depicting fluorescently labelled cells of a live zebrafish embryo. We exploit the fact that the recorded cells float on the surface of the embryo and allow for the extraction of an image sequence together with a sphere-like surface. We solve the resulting variational problem by means of a Galerkin method based on vector spherical harmonics and present numerical results computed from the aforementioned microscopy data.

IPI

In this paper we propose an algorithm for the detection of edges in
images that is based on topological asymptotic analysis. Motivated
from the Mumford--Shah functional, we consider a variational
functional that penalizes oscillations outside some approximate
edge set, which we represent as the union of a finite number of thin
strips, the width of which is an order of magnitude smaller than
their length. In order to find a near optimal placement of these
strips, we compute an asymptotic expansion of the functional with
respect to the strip size. This expansion is then employed for defining
a (topological) gradient descent like minimization method. As
opposed to a recently proposed method by some of the authors, which
uses coverings with balls, the usage of strips includes some
directional information into the method, which can be used for
obtaining finer edges and can also result in a reduction of
computation times.

IPI

In this paper we construct a shape space of medial ball
representations from given shape training data using methods of
Computational Geometry and Statistics. The ultimate goal is to
employ the shape space as prior information in supervised
segmentation algorithms for complex geometries in 3D voxel data.
For this purpose, a novel representation of the shape space (i.e.,
medial ball representation) is worked out and its implications on
the whole segmentation pipeline are studied.
Such algorithms have wide applications for industrial processes
and medical imaging, when data are recorded under varying
illumination conditions, are corrupted with high noise or are
occluded.

IPI

In part I we introduced modified Landweber--Kaczmarz methods and
established a convergence analysis. In the present work we
investigate three applications: an inverse problem related to thermoacoustic
tomography, a nonlinear inverse problem for semiconductor equations, and a
nonlinear problem in Schlieren tomography. Each application is
considered in the framework established in the previous part. The
novel algorithms show robustness, stability, computational
efficiency and high accuracy.

IPI

Integral invariants have been proven to be useful for shape
matching and recognition, but fundamental mathematical questions have not
been addressed in the computer vision literature.
In this article we are concerned with the identifiability and numerical algorithms
for the reconstruction of a star-shaped object from its integral invariants.
In particular we analyse two integral invariants and prove injectivity for one of them.
Additionally, numerical experiments are performed.

IPI

We consider image registration, which is the determination of a geometrical transformation between two data sets.
In this paper we propose constrained variational methods which aim for controlling the change of area or volume under
registration transformation.
We prove an existence result, convergence of a finite element method, and present a simple numerical
example for volume-preserving registration.

IPI

In this article we develop and analyze novel iterative regularization techniques for
the solution of systems of nonlinear ill-posed operator equations. The basic idea consists
in considering separately each equation of this system and incorporating a loping strategy.
The first technique is a Kaczmarz-type method, equipped with a
novel stopping criteria. The second method is obtained using an embedding strategy,
and again a Kaczmarz-type approach.
We prove well-posedness, stability and convergence of both methods.

IPI

In this paper we present a decomposition algorithm for computation of the spatial-temporal optical flow of a dynamic image sequence. We consider several applications, such as the extraction of temporal motion features and motion detection in dynamic sequences under varying illumination conditions, such as they appear for instance in psychological flickering experiments. For the numerical implementation we are solving an **integro-differential** equation by a fixed point iteration. For comparison purposes we use a standard time dependent optical flow algorithm, which in contrast to our method, constitutes in solving a spatial-temporal **differential** equation.

IPI

We study Tikhonov regularization for solving ill-posed operator equations where the solutions are functions defined on surfaces. One contribution of this paper is an error analysis of Tikhonov regularization which takes into account perturbations of the surfaces, in particular when the surfaces are approximated by spline surfaces. Another contribution is that we highlight the analysis of regularization for functions with range in vector bundles over surfaces. We also present some practical applications, such as an inverse problem of gravimetry and an imaging problem for denoising vector fields on surfaces, and show the numerical verification.

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