##
Multiscale Methods for (Generalized) Sparse
Recovery and Applications in High
Dimensional Imaging

Michael Möller, 2012 (Reviewer: Stanley Osher, UCLA)

This thesis presents various aspects of sparsity related regularization of inverse problems
in high dimensional data. It is divided into two parts.

The first part is of theoretical nature; besides laying the mathematical background,
it discusses the possibility of recovering sparse solutions exactly from highly underdetermined
linear systems. Furthermore, we extend some of the results from the typical
compressed sensing case, i.e., from l1 minimization, to arbitrary polyhedral functions. In
the following chapters, we focus on the development and analysis of numerical methods
for the l1 and polyhedral function minimization. We provide the theory for solving the
so-called inverse scale space flow to any polyhedral function exactly. Furthermore, we
show the equivalence of the popular gradient descent method to an inverse scale space
flow on the dual functional, thus allowing us to apply our convergence theory to this
widely used class of methods. For problems where the inverse scale space
flow is not
directly applicable, we provide a framework for convergence analysis of split Bregman
type methods and show that the application of this method to the primal and dual
problem can yield very di
erent convergence properties. Finally, the variable splitting
approach of split Bregman is taken to its continuous limit, leading again to a polyhedral
inverse scale space
flow.

In the second part of this thesis, we explore multiple applications. The concepts of
sparsity and l1 regularization are applied to the problem of hyperspectral unmixing and
the inverse scale space
ow is shown to yield superior results even in the case of overdetermined
systems with noisy data. The second application is related to hyperspectral
unmixing but can also be seen in the general context of dimension reduction or subset
selection. We propose a new method for selecting a small subset of points from a
dataset that lead to a non-negative sparse description of the whole dataset in terms of
the selected points. Mathematically, we make use of the concept of row-sparsity with
the means of l1;infinity regularization. Finally, we present a project on automatic cell tracking
in phase contrast microscopic videos. With the help of two topology preserving level
set methods we extract the movement of the centroid as well as the change of area and
perimeter of each cell. We validate our model against several manual tracking results.

[pdf-file]

##
Variational Methods using Transport Metrics and Applications

Marzena Franek, 2011 (Reviewer: Daniel Matthes, TU München)

The goal of this thesis is to introduce a new variational approach based on the optimal
transport theory. In particular, we want to investigate the solution of an optimisation
problem consisting of a data term, which is here the quadratic Wasserstein distance
between probability measures and a regularisation term. This novel formulation leads
to a method, which can handle unied discrete measures as well as continuous probability
measures. Moreover, with a special kind of regularisation, namely the TVregularisation,
discontinuities in the data can be preserved. Further, if we consider
the Wasserstein distance in the data term we obtain mass conservation. This is an
improvement to existing models, since the mass conservation must be usually implemented
as an additional constraint.

The second but equivalent method we derive in this thesis is in the spirit of the
uid
dynamic formulation of Benamou and Brenier. By introducing an artificial time variable
we obtain a regularised time dependent optimal transport problem. In addition,
there is a time interpolant between masses given.

Furthermore, the connection to the gradient
floow theory is developed in this thesis and
several evolution equations are identified as Wasserstein gradient
flow equations. This
connection yields a new understanding of the partial differential equations and to an
improved study of existence, uniqueness and the asymptotic behaviour of solutions.

Especially, we study an energy functional which consists of a nonlocal interaction potential
and an internal energy. The associated gradient
ow equation is an aggregation
equation with diffusion term which has applications in chemotaxis or swarming. We
provide a detailed analysis for this aggregation equation. In particular, we give existence
and uniqueness results of nontrivial stationary solutions. The existence versus
nonexistence of such solutions is ruled by a threshold phenomenon, namely nontrivial
stationary solutions exist if and only if the di
usivity constant is strictly smaller than
the total mass of the interaction kernel. Furthermore, we prove that nontrivial stationary
solutions in one space dimension with xed mass and center of mass are unique.
We also provide numerical results.

The central core of this thesis is the discussion of the variational problem with several
standard regularisation functionals, e.g. the logarithmic entropy, the L2-regularisation,
the Dirichlet-regularisation, the Fisher information and the TV-regularisation. Besides
existence and uniqueness results for each regularisation energy we present numerical results which illustrate the different impact on the data. We calculate the self-similar
solutions for the different regularisation functionals to give an idea of the structure of
the solutions. We apply the numerical algorithms to synthetic data as well as on real
life data.

[pdf-file]

##
Mathematical Models for Particle Transport: Crowded Motion

Bärbel Schlake, 2011 (Reviewer: Ansgar Jüngel, TU Wien)

This thesis is concerned with problems arising in the mathematical modelling of
crowded particles. A stochastic, self-consistent model with volume-exclusion is developed,
which asymptotically leads to a nonlinear system of drift-diffusion equations.

We present an entropy for the model, compute equilibria, and show that the system
has a formal gradient flow structure. We analyse the time-dependent system for two
species and compute the entropy dissipation and equilibria. Linearisation around
equilibrium is examined. We investigate the system with respect to well-posedness
close to equilibrium, global existence and asymptotic behaviour at different diffusion
scales.

The stationary model coupled with the Poisson equation to compute the potential
(drift) is discussed. We use the model to compute densities as well as electrical
potentials for an ion channel, the L-type calcium selective channel. Existence and
uniqueness of the stationary problem is analysed. We simulate this channel and compare
the numerical results with the commonly used Poisson-Nernst-Planck model,
which does not include volume-exclusion. As a result, we observe that the current in
the model with volume-exclusion lies significantly below the current without volumeexclusion.
This is due to the fact that ions inside the channel are crowded and cannot
move uninterrupted.

Next, we extend the model for particles with different radii. We recover basically
all the above mentioned results for the model. We also simulate the L-type calcium
channel with the model including different radii. As ion channels do not only select
over charge, but also over radii, interesting biological phenomena such as the
Anomalous Mole Fraction Effect can be observed.

Moreover, we examine the Brazil Nut Effect, which is an effect generated at the
mixing of particles with different radii and masses. Mixed particles segregate on
shaking. At equal masses, larger particles go to the top, while smaller particles sink
to the bottom. We investigate in which cases this effect arises in our model and carry
out some simulations on this effect, as well as on the Reverse Brazil Nut Effect: In
case that one species is larger, but also more heavy, this species sinks to the bottom.

Finally, we apply the problem to human motion, more precisely to a typical pedestrian
setup with two directions. We investigate linear stability. Arising instabilities
for pedestrians with opposite walking directions mean the forming of lanes of pedestrians
with the same desired direction. The model is able to reproduce lane formation,
and we are able to predict the number of lanes dependent on the width of the domain
and on the density.

[pdf-file]

##
(Nonlocal) Total Variation in Medical Imaging

Alex Sawatzky, 2011 (Reviewer: Gabriele Steidl, TU Kaiserslautern)

This thesis deals with the reconstruction of images where the measured data are corrupted
by Poisson noise and a specific signal-dependent speckle noise, which occurs e.g
in medical ultrasound images. Since both noise types fundamentally differ from the
frequently studied additive Gaussian noise in mathematical image processing, adapted
variational models are required to handle these types of noise accurately.

The first part of this thesis introduces variational regularization frameworks for inverse
problems with data corrupted by Poisson and ultrasound speckle noise. Due to the
strong nonlinearity of both data delity terms, a forward-backward splitting approach
is used to provide efficient numerical schemes allowing the use of arbitrary convex regularization
energies, in particular singular ones. Moreover, analytical results such as the
well-posedness of the variational problems as well as the positivity preservation and convergence
of the proposed iteration methods are proved. Finally, an iterative extension
of both frameworks is proposed in order to refine the systematic errors of variational
regularization techniques, using inverse scale space methods and Bregman distance iterations.

The second part of this thesis considers the use of the (nonlocal) total variation functional
as regularization energy in both previously developed frameworks. In particular,
a modified version of the projected gradient descent algorithm of Chambolle and an augmented
Lagrangian method are presented to solve the weighted (nonlocal) ROF model
arising in both previously developed frameworks. In the case of the total variation regularization
strategy, analytical results obtained previous in the general context of a convex
regularization functional are carried over to the TV seminorm. In the case of the nonlocal
regularization approach, a continuous framework of nonlocal derivative operators
on directed graphs is introduced. This framework generalizes the nonlocal operators on
undirected graphs in continuous and discrete setting and is consistent to the discrete
local derivative operators.

Finally, the performance of the proposed algorithms is illustrated by 2D and 3D synthetic
and real data reconstructions. To validate the method proposed for inverse problems
with data corupted by Poisson noise, simulated PET data (2D) and real cardiac H2
15O
(2D) and 18F-FDG (3D) PET measurements with low count rates are used. Additionally,
a denoising and reconstruction comparison between TV and nonlocal TV regularization
is presented using 2D synthetic Poisson data. In the case of denoising problems in
medical US imaging, results on real patient data (2D) are illustrated.

[pdf-file]

##
Singular Regularization of Inverse Problems

Martin Benning, 2011 (Reviewer: Elena Resmerita, JKU Linz)

This thesis comments on the use of Bregman distances in the context of singular regularization
schemes for inverse problems. According to previous works the use of Bregman distances in combination
with variational frameworks, based on singular regularization energies, leads to improved
approximations of inverse problems solutions. The Bregman distance has become a powerful tool
for the analysis of these frameworks, and has brought iterative algorithms to life that enhance the
quality of solutions of existing frameworks significantly. However, most works have yet considered
Bregman distances in the context of variational frameworks with quadratic fidelity only.

One of the goals of this thesis is to extend analytical results to more general, nonlinear fidelity
terms arising from applications as e.g. medical imaging. Moreover, the concept of Eigenfunctions
of linear operators is transferred to nonlinear operators arising from the optimality conditions of
the variational frameworks.

From a computational perspective, a novel compressed sensing algorithm based on an inverse
scale space formulation is introduced. Furthermore, important concepts related to Bregman distances
are carried over to non-quadratic frameworks arising from the applications of dynamic
Positron Emission Tomography and Bioluminescence Tomography.
[pdf-file]

##
4D Imaging in Tomography and Optical Nanoscopy

Christoph Brune, 2010 (Reviewer: Stanley Osher, UCLA)

This thesis contributes to the field of mathematical image processing and inverse
problems. An inverse problem is a task, where the values of some model parameters
(in our case images) must be computed from observed data. Such problems arise in
a wide variety of applications in sciences and engineering, such as medical imaging,
biophysics, geophysics, remote sensing, ocean acoustic tomography, nondestructive
testing or astronomy.
Here, we mainly consider reconstruction problems with Poisson noise in tomography
and optical nanoscopy. In optical nanoscopy the task is to reconstruct images from
blurred and noisy measurements, whereas e.g. in positron emission tomography the
task is to visualize biochemical and physiological processes of a patient by measurements from outside the body.
In the literature there are several models and algorithms for 3D static image reconstruction. However, standard methods do not incorporate time-dependent information or dynamics, e.g. heart beat or breathing in medical imaging or cell motion in
microscopy. This can lead to de?cient accuracy particularly at object boundaries,
e.g. at cardiac walls in medical imaging

This dissertation contains a treatise on models, analysis and e?cient algorithms to
solve 3D static and 4D time-dependent inverse problems.
In the first part of this thesis the main goal is to present an accurate, robust and fast
3D static reconstruction framework based on total variation for inverse problems with
non-standard noise models. We will provide a detailed analysis including existence,
uniqueness and convergence proofs.

In the second part our main goal is to study different transport and motion models
and to combine them with the ideas of the first part, to build a joint 4D model for
simultaneous reconstruction, total variation regularization and optimal transport.
The fundamental concepts are based on non-standard noise models, sparsity regularization techniques, Bregman distances, splitting techniques and motion estimation. In the course of this thesis, topics of various areas in applied mathematics
and computer science are brought together, e.g. static and time dependent inverse
problems, regularization of ill-posed problems, applied functional analysis, error estimation, convex optimization theory, numerical algorithms, computational science
(parallelization, GPU programming), continuum mechanics, computer vision, motion estimation or optimal transport.
The performance of our main concepts is illustrated by experimental data in tomography and optical nanoscopy
[pdf-file]

##
Mathematical Modelling and Simulation
of Ion Channels

Kattrin Arning, 2009 (Reviewer: Christoph Romanin, JKU Linz)

This work is concerned with the mathematical modelling and simulation of ion channels.
Ion channels are of major interest and form an area of intensive research in the fields of
biophysics and medicine, since they control many vital physiological functions. As certain
aspects with respect to ion channel structure and function are hard or impossible to address
in experimental investigations, mathematical models form a useful completion and alternative
to these studies.

This thesis will mainly deal with two aspects of the channel behaviour, namely ion conduction
and gating.

The first part is dedicated to the description of ion transport across single open channels, focusing
on a macroscopic model composed of a system of coupled nonlinear partial differential
equations (PDEs), known as the Poisson-Nernst-Planck (PNP) system in biological context.
A one-dimensional approximation of this PDE system is derived by introducing additional
potentials to account for the channel protein geometry, and furthermore a computationally
efficient way to include size-exclusion effects, which become important in narrow geometries,
is developed.

Since for most ion channels the structure of the selectivity filter (the region where the specificity
of the ion channel is determined) cannot be resolved in experiments yet, the PNP
model is subsequently used to address questions from the field of inverse problems. It is
investigated if electrophysiological measurements like current-voltage curves can be used to
characterise the underlying channel structure. A special focus is put on the employment of
surrogate models in the identification procedure.

The second part of the thesis deals with the opening and closing of ion channels, a process
known as gating. Different modelling approaches that can be used to simulate the behaviour
of voltage-gated ion channels are presented, and a model of Fokker-Planck type is derived to
describe the gating currents and open probabilities on a macroscopic, i.e. whole cell, level.
This model is then used to analyse certain characteristic features arising in gating current
data, like the existence of a rising phase under certain conditions.
As above in the case of ion conduction, the derived gating model is subsequently employed
to address inverse problems from the field of parameter identification. Macroscopic current
data are used to investigate what can be inferred about the underlying physical system.
[pdf-file]

##
Forward and Inverse Solvers for Electrodiffusion Problems

Marie-Therese Wolfram, 2008 (Reviewer: Christian Schmeiser, University Vienna)

This work is concerned with forward and inverse solvers for electro-diffusion systems.
Electro-diffusion equations serve as a mathematical model for different
phenomena in physics and biology. Classical examples include the transport of
charged particles in semiconductors or biological channels.

This thesis contributes to three different, yet loosely connected areas. The first part
deals with the adaptation of well-known semiconductor discretization techniques,
namely exponential fitting schemes and stabilized hybrid discontinuous Galerkin
methods, to different biological applications. Numerical discretization techniques
for semiconductor devices are a highly developed field due to the high demand in
semiconductor industry. It was not until the last few years, that researchers got
more and more interested in biological applications. Therefore numerical methods
for these applications have hardly been researched up to now. In particular we are
interested in transport of ions in biological and synthetic channels as well as the
chemotactical movement of cells.

Electro-diffusion equations can also be formulated in the context of optimal transport
problems. In this field we strive to find an optimal transportation map that
carries one density to another while minimizing the transportation cost and energy.
This formulation provides a basis for a new numerical discretization method, which
can be applied to various optimal transportation problems like the porous medium
equations.

The optimal design of semiconductor poses another challenge. The performance
of a semiconductor is mainly determined by its doping profile. Therefore industry
is interested in efficient numerical methods to obtain a desired behavior, like
maximizing the current flow over a contact by changing the doping profile. Here
computational cost is the limiting factor in optimization schemes. By interpreting
the potential V as an optimal design variable we obtain a decoupled scheme
of electro-diffusion type equations. The resulting system can be solved more efficiently
than conventional schemes, enabling the application of this technique to
more sophisticated semiconductor devices, like a MOSFET.
[pdf-file]

##
Shape Variations, Level Set and Phase-field Methods for Perimeter Regularized Geometric Inverse Problems

Benjamin Hackl, 2006 (Reviewer: Karl Kunisch, KFU Graz)

Geometric inverse problems are in general ill-posed problems whose variable is a shape. A
problem is ill-posed if the solution to the problem depend in a stable way on the data, i.e.
small perturbations of the data may result into arbitrarily large deviations in the solutions.
In order to achieve meaningful solutions, one has to regularize the problem. Usually one
considers the problem in a least squares form and e.g. adds a regularization term to the least
squares functional (Tikhonov regularization).

A geometric equivalent to the Tikhonov regularization is the perimeter regularization, where
one adds a multiple of the perimeter (arclength or surface area of the boundary of the shape)
of the shape to the least squares functional. Therefore heavily oscillating shapes or even
micro-structures are avoided. For a cavity detection problem in linear elasticity we prove,
mathematically rigorous, that perimeter regularization is indeed a convergent regularization
method.

Motivated by the asymptotic regularization for inverse problems in function spaces, we
suggest the terminated velocity method as an alternative regularization method. Numerous
numerical experiments with cavity detection in linear elasticity and thermo-elasticity illustrate
the regularizing behavior of this method.

Next we study the level set method for geometric inverse problems, which is supposed to be
able to handle topology changes easily. In the level set method the shape, more precisely the
boundary of the shape, is represented by the zero level set of a function. The shape evolves in
a velocity field, where the velocity field is chosen such that an appropriately chosen objective
function is non-increasing. Usually the velocity field is chosen due to a shape sensitivity
analysis of the objective function (shape derivatives).

In some cases the level set method using shape derivatives to determine the velocity gets
stuck in local minima and does not perform the desired topology change. Therefore we develop
modifications of the level set method to overcome such local minima and force topology
changes.

One modification incorporates the topological derivative into the level set method. For the
topological derivative one considers the variation of the objective functional with respect to
an additional small cavity. Hence the topological derivative indicates whether it is desirable
to perform a topology change or not. Unfortunately the perimeter is not topologically differentiable
and therefore this modification of the level set method applies just to the terminated
velocity method.

Another modification we suggest is based on a topological expansion of the objective function
with respect to the volume and the perimeter of the topology change. The expansion is
similar to a Taylor expansion and allows to provide estimates of the change of the objective
functional to first and second order. Due to the fact that the topological expansion is also an
expansion in the perimeter, it is possible to treat perimeter regularized problems. Following
the construction ideas of steepest descent- and Newton- type methods, it is possible to enforce
reliable topology changes by solving a sub-minimization problem.

As a final solution method for perimeter regularized problems, we discuss the phase-field
method. In the phase-field method one relaxes the perimeter and represents the shape by a
function in a function space. This relaxation allows for classical function space optimization.
For non-linear source reconstruction problem we compare the classical and newly developed
algorithms.
[pdf-file]