Topological Data Analysis of Human Brain Networks Through Order Statistics

Understanding the topological characteristics of the brain network across a population is central to understanding brain functions. The abstraction of human connectome as a graph has been pivotal in gaining insights on the topological features of the brain network. The development of group-level statistical inference procedures in brain graphs while accounting for the heterogeneity and randomness still remains a difficult task. In this study, we develop a robust statistical framework based on persistent homology using the order statistics for analyzing brain networks. The use of order statistics greatly simplifies the computation of the persistent barcodes. We validate the proposed methods using comprehensive simulation studies and subsequently apply to the resting-state functional magnetic resonance images. We conclude a statistically significant topological difference between the male and female brain networks

Topology Reconstruction of Dynamical Networks via Constrained Lyapunov Equations

The network structure (or topology) of a dynamical network is often unavailable or uncertain. Hence, we consider the problem of network reconstruction. Network reconstruction aims at inferring the topology of a dynamical network using measurements obtained from the network. In this technical note we define the notion of solvability of the network reconstruction problem. Subsequently, we provide necessary and sufficient conditions under which the network reconstruction problem is solvable. Finally, using constrained Lyapunov equations, we establish novel network reconstruction algorithms, applicable to general dynamical networks. We also provide specialized algorithms for specific network dynamics, such as the well-known consensus and adjacency dynamics.Comment: 8 page

New Directions for Contact Integrators

Contact integrators are a family of geometric numerical schemes which guarantee the conservation of the contact structure. In this work we review the construction of both the variational and Hamiltonian versions of these methods. We illustrate some of the advantages of geometric integration in the dissipative setting by focusing on models inspired by recent studies in celestial mechanics and cosmology.Comment: To appear as Chapter 24 in GSI 2021, Springer LNCS 1282

IMAGE UNDERSTANDING OF MOLAR PREGNANCY BASED ON ANOMALIES DETECTION

Cancer occurs when normal cells grow and multiply without normal control. As the cells multiply, they form an area of abnormal cells, known as a tumour. Many tumours exhibit abnormal chromosomal segregation at cell division. These anomalies play an important role in detecting molar pregnancy cancer. Molar pregnancy, also known as hydatidiform mole, can be categorised into partial (PHM) and complete (CHM) mole, persistent gestational trophoblastic and choriocarcinoma. Hydatidiform moles are most commonly found in women under the age of 17 or over the age of 35. Hydatidiform moles can be detected by morphological and histopathological examination. Even experienced pathologists cannot easily classify between complete and partial hydatidiform moles. However, the distinction between complete and partial hydatidiform moles is important in order to recommend the appropriate treatment method. Therefore, research into molar pregnancy image analysis and understanding is critical. The hypothesis of this research project is that an anomaly detection approach to analyse molar pregnancy images can improve image analysis and classification of normal PHM and CHM villi. The primary aim of this research project is to develop a novel method, based on anomaly detection, to identify and classify anomalous villi in molar pregnancy stained images. The novel method is developed to simulate expert pathologists’ approach in diagnosis of anomalous villi. The knowledge and heuristics elicited from two expert pathologists are combined with the morphological domain knowledge of molar pregnancy, to develop a heuristic multi-neural network architecture designed to classify the villi into their appropriated anomalous types. This study confirmed that a single feature cannot give enough discriminative power for villi classification. Whereas expert pathologists consider the size and shape before textural features, this thesis demonstrated that the textural feature has a higher discriminative power than size and shape. The first heuristic-based multi-neural network, which was based on 15 elicited features, achieved an improved average accuracy of 81.2%, compared to the traditional multi-layer perceptron (80.5%); however, the recall of CHM villi class was still low (64.3%). Two further textural features, which were elicited and added to the second heuristic-based multi-neural network, have improved the average accuracy from 81.2% to 86.1% and the recall of CHM villi class from 64.3% to 73.5%. The precision of the multi-neural network II has also increased from 82.7% to 89.5% for normal villi class, from 81.3% to 84.7% for PHM villi class and from 80.8% to 86% for CHM villi class. To support pathologists to visualise the results of the segmentation, a software tool, Hydatidiform Mole Analysis Tool (HYMAT), was developed compiling the morphological and pathological data for each villus analysis

Report / Institute für Physik

In this report the Institutes of Physics of the Universität Leipzig present their scientific activities and major achievements in the year 2003

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described