First order algorithms in variational image processing

Variational methods in imaging are nowadays developing towards a quite universal and flexible tool, allowing for highly successful approaches on tasks like denoising, deblurring, inpainting, segmentation, super-resolution, disparity, and optical flow estimation. The overall structure of such approaches is of the form ${\cal D}(Ku) + \alpha {\cal R} (u) \rightarrow \min_u$ ; where the functional ${\cal D}$ is a data fidelity term also depending on some input data $f$ and measuring the deviation of $Ku$ from such and ${\cal R}$ is a regularization functional. Moreover $K$ is a (often linear) forward operator modeling the dependence of data on an underlying image, and $\alpha$ is a positive regularization parameter. While ${\cal D}$ is often smooth and (strictly) convex, the current practice almost exclusively uses nonsmooth regularization functionals. The majority of successful techniques is using nonsmooth and convex functionals like the total variation and generalizations thereof or $\ell_1$-norms of coefficients arising from scalar products with some frame system. The efficient solution of such variational problems in imaging demands for appropriate algorithms. Taking into account the specific structure as a sum of two very different terms to be minimized, splitting algorithms are a quite canonical choice. Consequently this field has revived the interest in techniques like operator splittings or augmented Lagrangians. Here we shall provide an overview of methods currently developed and recent results as well as some computational studies providing a comparison of different methods and also illustrating their success in applications.Comment: 60 pages, 33 figure

Nonlinear Integer Programming

Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject to integrality requirements for the variables. This chapter is dedicated to this topic. The primary goal is a study of a simple version of general nonlinear integer problems, where all constraints are still linear. Our focus is on the computational complexity of the problem, which varies significantly with the type of nonlinear objective function in combination with the underlying combinatorial structure. Numerous boundary cases of complexity emerge, which sometimes surprisingly lead even to polynomial time algorithms. We also cover recent successful approaches for more general classes of problems. Though no positive theoretical efficiency results are available, nor are they likely to ever be available, these seem to be the currently most successful and interesting approaches for solving practical problems. It is our belief that the study of algorithms motivated by theoretical considerations and those motivated by our desire to solve practical instances should and do inform one another. So it is with this viewpoint that we present the subject, and it is in this direction that we hope to spark further research.Comment: 57 pages. To appear in: M. J\"unger, T. Liebling, D. Naddef, G. Nemhauser, W. Pulleyblank, G. Reinelt, G. Rinaldi, and L. Wolsey (eds.), 50 Years of Integer Programming 1958--2008: The Early Years and State-of-the-Art Surveys, Springer-Verlag, 2009, ISBN 354068274

Numerical and Analytical Methods in Electromagnetics

Like all branches of physics and engineering, electromagnetics relies on mathematical methods for modeling, simulation, and design procedures in all of its aspects (radiation, propagation, scattering, imaging, etc.). Originally, rigorous analytical techniques were the only machinery available to produce any useful results. In the 1960s and 1970s, emphasis was placed on asymptotic techniques, which produced approximations of the fields for very high frequencies when closed-form solutions were not feasible. Later, when computers demonstrated explosive progress, numerical techniques were utilized to develop approximate results of controllable accuracy for arbitrary geometries. In this Special Issue, the most recent advances in the aforementioned approaches are presented to illustrate the state-of-the-art mathematical techniques in electromagnetics

Enhancing 3D Autonomous Navigation Through Obstacle Fields: Homogeneous Localisation and Mapping, with Obstacle-Aware Trajectory Optimisation

Small flying robots have numerous potential applications, from quadrotors for search and rescue, infrastructure inspection and package delivery to free-flying satellites for assistance activities inside a space station. To enable these applications, a key challenge is autonomous navigation in 3D, near obstacles on a power, mass and computation constrained platform. This challenge requires a robot to perform localisation, mapping, dynamics-aware trajectory planning and control. The current state-of-the-art uses separate algorithms for each component. Here, the aim is for a more homogeneous approach in the search for improved efficiencies and capabilities. First, an algorithm is described to perform Simultaneous Localisation And Mapping (SLAM) with physical, 3D map representation that can also be used to represent obstacles for trajectory planning: Non-Uniform Rational B-Spline (NURBS) surfaces. Termed NURBSLAM, this algorithm is shown to combine the typically separate tasks of localisation and obstacle mapping. Second, a trajectory optimisation algorithm is presented that produces dynamically-optimal trajectories with direct consideration of obstacles, providing a middle ground between path planners and trajectory smoothers. Called the Admissible Subspace TRajectory Optimiser (ASTRO), the algorithm can produce trajectories that are easier to track than the state-of-the-art for flight near obstacles, as shown in flight tests with quadrotors. For quadrotors to track trajectories, a critical component is the differential flatness transformation that links position and attitude controllers. Existing singularities in this transformation are analysed, solutions are proposed and are then demonstrated in flight tests. Finally, a combined system of NURBSLAM and ASTRO are brought together and tested against the state-of-the-art in a novel simulation environment to prove the concept that a single 3D representation can be used for localisation, mapping, and planning

Computerized Analysis of Magnetic Resonance Images to Study Cerebral Anatomy in Developing Neonates

The study of cerebral anatomy in developing neonates is of great importance for the understanding of brain development during the early period of life. This dissertation therefore focuses on three challenges in the modelling of cerebral anatomy in neonates during brain development. The methods that have been developed all use Magnetic Resonance Images (MRI) as source data. To facilitate study of vascular development in the neonatal period, a set of image analysis algorithms are developed to automatically extract and model cerebral vessel trees. The whole process consists of cerebral vessel tracking from automatically placed seed points, vessel tree generation, and vasculature registration and matching. These algorithms have been tested on clinical Time-of- Flight (TOF) MR angiographic datasets. To facilitate study of the neonatal cortex a complete cerebral cortex segmentation and reconstruction pipeline has been developed. Segmentation of the neonatal cortex is not effectively done by existing algorithms designed for the adult brain because the contrast between grey and white matter is reversed. This causes pixels containing tissue mixtures to be incorrectly labelled by conventional methods. The neonatal cortical segmentation method that has been developed is based on a novel expectation-maximization (EM) method with explicit correction for mislabelled partial volume voxels. Based on the resulting cortical segmentation, an implicit surface evolution technique is adopted for the reconstruction of the cortex in neonates. The performance of the method is investigated by performing a detailed landmark study. To facilitate study of cortical development, a cortical surface registration algorithm for aligning the cortical surface is developed. The method first inflates extracted cortical surfaces and then performs a non-rigid surface registration using free-form deformations (FFDs) to remove residual alignment. Validation experiments using data labelled by an expert observer demonstrate that the method can capture local changes and follow the growth of specific sulcus

Improved 3D MR Image Acquisition and Processing in Congenital Heart Disease

Congenital heart disease (CHD) is the most common type of birth defect, affecting about 1% of the population. MRI is an essential tool in the assessment of CHD, including diagnosis, intervention planning and follow-up. Three-dimensional MRI can provide particularly rich visualization and information. However, it is often complicated by long scan times, cardiorespiratory motion, injection of contrast agents, and complex and time-consuming postprocessing. This thesis comprises four pieces of work that attempt to respond to some of these challenges. The first piece of work aims to enable fast acquisition of 3D time-resolved cardiac imaging during free breathing. Rapid imaging was achieved using an efficient spiral sequence and a sparse parallel imaging reconstruction. The feasibility of this approach was demonstrated on a population of 10 patients with CHD, and areas of improvement were identified. The second piece of work is an integrated software tool designed to simplify and accelerate the development of machine learning (ML) applications in MRI research. It also exploits the strengths of recently developed ML libraries for efficient MR image reconstruction and processing. The third piece of work aims to reduce contrast dose in contrast-enhanced MR angiography (MRA). This would reduce risks and costs associated with contrast agents. A deep learning-based contrast enhancement technique was developed and shown to improve image quality in real low-dose MRA in a population of 40 children and adults with CHD. The fourth and final piece of work aims to simplify the creation of computational models for hemodynamic assessment of the great arteries. A deep learning technique for 3D segmentation of the aorta and the pulmonary arteries was developed and shown to enable accurate calculation of clinically relevant biomarkers in a population of 10 patients with CHD

Linear programming based models for resilient supply systems design

Ph.DDOCTOR OF PHILOSOPH

LIPIcs, Volume 274, ESA 2023, Complete Volume

LIPIcs, Volume 274, ESA 2023, Complete Volum

Integrality and cutting planes in semidefinite programming approaches for combinatorial optimization

Many real-life decision problems are discrete in nature. To solve such problems as mathematical optimization problems, integrality constraints are commonly incorporated in the model to reflect the choice of finitely many alternatives. At the same time, it is known that semidefinite programming is very suitable for obtaining strong relaxations of combinatorial optimization problems. In this dissertation, we study the interplay between semidefinite programming and integrality, where a special focus is put on the use of cutting-plane methods. Although the notions of integrality and cutting planes are well-studied in linear programming, integer semidefinite programs (ISDPs) are considered only recently. We show that manycombinatorial optimization problems can be modeled as ISDPs. Several theoretical concepts, such as the Chvátal-Gomory closure, total dual integrality and integer Lagrangian duality, are studied for the case of integer semidefinite programming. On the practical side, we introduce an improved branch-and-cut approach for ISDPs and a cutting-plane augmented Lagrangian method for solving semidefinite programs with a large number of cutting planes. Throughout the thesis, we apply our results to a wide range of combinatorial optimization problems, among which the quadratic cycle cover problem, the quadratic traveling salesman problem and the graph partition problem. Our approaches lead to novel, strong and efficient solution strategies for these problems, with the potential to be extended to other problem classes