Subdivision Directional Fields

We present a novel linear subdivision scheme for face-based tangent directional fields on triangle meshes. Our subdivision scheme is based on a novel coordinate-free representation of directional fields as halfedge-based scalar quantities, bridging the finite-element representation with discrete exterior calculus. By commuting with differential operators, our subdivision is structure-preserving: it reproduces curl-free fields precisely, and reproduces divergence-free fields in the weak sense. Moreover, our subdivision scheme directly extends to directional fields with several vectors per face by working on the branched covering space. Finally, we demonstrate how our scheme can be applied to directional-field design, advection, and robust earth mover's distance computation, for efficient and robust computation

A Coloring Algorithm for Disambiguating Graph and Map Drawings

Drawings of non-planar graphs always result in edge crossings. When there are many edges crossing at small angles, it is often difficult to follow these edges, because of the multiple visual paths resulted from the crossings that slow down eye movements. In this paper we propose an algorithm that disambiguates the edges with automatic selection of distinctive colors. Our proposed algorithm computes a near optimal color assignment of a dual collision graph, using a novel branch-and-bound procedure applied to a space decomposition of the color gamut. We give examples demonstrating the effectiveness of this approach in clarifying drawings of real world graphs and maps

Mathematical Simulations in Topology and Their Role in Mathematics Education

This thesis presents and discusses several software projects related to the learning of mathematics in general and topological concepts in particular, collecting the results from several publications in this field. It approaches mathematics education by construction of mathematical learning environments, which can be used for the learning of mathematics, as well as by contributing insights gained during the development and use of these learning environments. It should be noted that the presented software environments were not built for the use in schools or other settings, but to provide proofs of concepts and to act as a basis for research into mathematics and its education and communication. The first developed and analyzed environment is Ariadne, a software for the interactive visualization of dots, paths, and homotopies of paths. Ariadne is used as an example of a “mathematical simulation”, capable of supporting argumentation in a way that may be characterized as proving. The software was extended from two to three dimensions, making possible the investigation of two-dimensional manifolds, such as the torus or the sphere, using virtual reality. Another extension, KnotPortal, enables the exploration of three-dimensional manifolds represented as branched covers of knots, after an idea by Bill Thurston to portray these branched covers of knots as knotted portals between worlds. This software was the motivation for and was used in an investigation into embodied mathematics learning, as this virtual reality environment challenges users to determine the structure of the covering by moving their body. Also presented are some unpublished projects that were not completed during the doctorate. This includes work on concept images in topology as well as software for various purposes. One such software was intended for the construction of closed orientable surfaces, while another was focused on the interactive visualization of the uniformization theorem. The thesis concludes with a meta-discussion on the role of design in mathematics education research. While design plays an important role in mathematics education, designing seems to not to be recognized as research in itself, but only as part of theory building or, in most cases, an empirical study. The presented argumentation challenges this view and points out the dangers and obstacles involved

Interactive Design and Optics-Based Visualization of Arbitrary Non-Euclidean Kaleidoscopic Orbifolds

Orbifolds are a modern mathematical concept that arises in the research of hyperbolic geometry with applications in computer graphics and visualization. In this paper, we make use of rooms with mirrors as the visual metaphor for orbifolds. Given any arbitrary two-dimensional kaleidoscopic orbifold, we provide an algorithm to construct a Euclidean, spherical, or hyperbolic polygon to match the orbifold. This polygon is then used to create a room for which the polygon serves as the floor and the ceiling. With our system that implements M\"obius transformations, the user can interactively edit the scene and see the reflections of the edited objects. To correctly visualize non-Euclidean orbifolds, we adapt the rendering algorithms to account for the geodesics in these spaces, which light rays follow. Our interactive orbifold design system allows the user to create arbitrary two-dimensional kaleidoscopic orbifolds. In addition, our mirror-based orbifold visualization approach has the potential of helping our users gain insight on the orbifold, including its orbifold notation as well as its universal cover, which can also be the spherical space and the hyperbolic space.Comment: IEEE VIS 202

Evaluation of Artery Visualizations for Heart Disease Diagnosis

Heart disease is the number one killer in the United States, and finding indicators of the disease at an early stage is critical for treatment and prevention. In this paper we evaluate visualization techniques that enable the diagnosis of coronary artery disease. A key physical quantity of medical interest is endothelial shear stress (ESS). Low ESS has been associated with sites of lesion formation and rapid progression of disease in the coronary arteries. Having effective visualizations of a patient's ESS data is vital for the quick and thorough non-invasive evaluation by a cardiologist. We present a task taxonomy for hemodynamics based on a formative user study with domain experts. Based on the results of this study we developed HemoVis, an interactive visualization application for heart disease diagnosis that uses a novel 2D tree diagram representation of coronary artery trees. We present the results of a formal quantitative user study with domain experts that evaluates the effect of 2D versus 3D artery representations and of color maps on identifying regions of low ESS. We show statistically significant results demonstrating that our 2D visualizations are more accurate and efficient than 3D representations, and that a perceptually appropriate color map leads to fewer diagnostic mistakes than a rainbow color map.Engineering and Applied Science

Dynamic Visual Abstraction of Soccer Movement

Trajectory-based visualization of coordinated movement data within a bounded area, such as player and ball movement within a soccer pitch, can easily result in visual crossings, overplotting, and clutter. Trajectory abstraction can help to cope with these issues, but it is a challenging problem to select the right level of abstraction (LoA) for a given data set and analysis task. We present a novel dynamic approach that combines trajectory simplification and clustering techniques with the goal to support interpretation and understanding of movement patterns. Our technique provides smooth transitions between different abstraction types that can be computed dynamically and on-the-fly. This enables the analyst to effectively navigate and explore the space of possible abstractions in large trajectory data sets. Additionally, we provide a proof of concept for supporting the analyst in determining the LoA semi-automatically with a recommender system. Our approach is illustrated and evaluated by case studies, quantitative measures, and expert feedback. We further demonstrate that it allows analysts to solve a variety of analysis tasks in the domain of soccer

From cognitive landscapes to digital hyperscapes

The widespread diffusion of e-Learning in organizations has encouraged the discovery of more effective ways for conveying digital information to learners, for instance, via the commonly called Learning Management Systems (LMS). A problem that we have identified is that cognitive variables and pedagogical processes are rarely taken into consideration and sometimes are confused with the mere use by learners of “diversified” hypermedia resources. Within the context of widespread dissemination of multimedia content that has followed the emergence of massive information resources, we discuss the need for more powerful and effective learner-centered tools capable of handling all kinds of design configurations and learning objects.(undefined

Cheminformatics Tools to Explore the Chemical Space of Peptides and Natural Products

Cheminformatics facilitates the analysis, storage, and collection of large quantities of chemical data, such as molecular structures and molecules' properties and biological activity, and it has revolutionized medicinal chemistry for small molecules. However, its application to larger molecules is still underrepresented. This thesis work attempts to fill this gap and extend the cheminformatics approach towards large molecules and peptides. This thesis is divided into two parts. The first part presents the implementation and application of two new molecular descriptors: macromolecule extended atom pair fingerprint (MXFP) and MinHashed atom pair fingerprint of radius 2 (MAP4). MXFP is an atom pair fingerprint suitable for large molecules, and here, it is used to explore the chemical space of non-Lipinski molecules within the widely used PubChem and ChEMBL databases. MAP4 is a MinHashed hybrid of substructure and atom pair fingerprints suitable for encoding small and large molecules. MAP4 is first benchmarked against commonly used atom pairs and substructure fingerprints, and then it is used to investigate the chemical space of microbial and plants natural products with the aid of machine learning and chemical space mapping. The second part of the thesis focuses on peptides, and it is introduced by a review chapter on approaches to discover novel peptide structures and describing the known peptide chemical space. Then, a genetic algorithm that uses MXFP in its fitness function is described and challenged to generate peptide analogs of peptidic or non-peptidic queries. Finally, supervised and unsupervised machine learning is used to generate novel antimicrobial and non-hemolytic peptide sequences

Visualization of Tensor Fields in Mechanics

Tensors are used to describe complex physical processes in many applications. Examples include the distribution of stresses in technical materials, acting forces during seismic events, or remodeling of biological tissues. While tensors encode such complex information mathematically precisely, the semantic interpretation of a tensor is challenging. Visualization can be beneficial here and is frequently used by domain experts. Typical strategies include the use of glyphs, color plots, lines, and isosurfaces. However, data complexity is nowadays accompanied by the sheer amount of data produced by large-scale simulations and adds another level of obstruction between user and data. Given the limitations of traditional methods, and the extra cognitive effort of simple methods, more advanced tensor field visualization approaches have been the focus of this work. This survey aims to provide an overview of recent research results with a strong application-oriented focus, targeting applications based on continuum mechanics, namely the fields of structural, bio-, and geomechanics. As such, the survey is complementing and extending previously published surveys. Its utility is twofold: (i) It serves as basis for the visualization community to get an overview of recent visualization techniques. (ii) It emphasizes and explains the necessity for further research for visualizations in this context