Comparing Features of Three-Dimensional Object Models Using Registration Based on Surface Curvature Signatures

This dissertation presents a technique for comparing local shape properties for similar three-dimensional objects represented by meshes. Our novel shape representation, the curvature map, describes shape as a function of surface curvature in the region around a point. A multi-pass approach is applied to the curvature map to detect features at different scales. The feature detection step does not require user input or parameter tuning. We use features ordered by strength, the similarity of pairs of features, and pruning based on geometric consistency to efficiently determine key corresponding locations on the objects. For genus zero objects, the corresponding locations are used to generate a consistent spherical parameterization that defines the point-to-point correspondence used for the final shape comparison

Methods for Real-time Visualization and Interaction with Landforms

This thesis presents methods to enrich data modeling and analysis in the geoscience domain with a particular focus on geomorphological applications. First, a short overview of the relevant characteristics of the used remote sensing data and basics of its processing and visualization are provided. Then, two new methods for the visualization of vector-based maps on digital elevation models (DEMs) are presented. The first method uses a texture-based approach that generates a texture from the input maps at runtime taking into account the current viewpoint. In contrast to that, the second method utilizes the stencil buffer to create a mask in image space that is then used to render the map on top of the DEM. A particular challenge in this context is posed by the view-dependent level-of-detail representation of the terrain geometry. After suitable visualization methods for vector-based maps have been investigated, two landform mapping tools for the interactive generation of such maps are presented. The user can carry out the mapping directly on the textured digital elevation model and thus benefit from the 3D visualization of the relief. Additionally, semi-automatic image segmentation techniques are applied in order to reduce the amount of user interaction required and thus make the mapping process more efficient and convenient. The challenge in the adaption of the methods lies in the transfer of the algorithms to the quadtree representation of the data and in the application of out-of-core and hierarchical methods to ensure interactive performance. Although high-resolution remote sensing data are often available today, their effective resolution at steep slopes is rather low due to the oblique acquisition angle. For this reason, remote sensing data are suitable to only a limited extent for visualization as well as landform mapping purposes. To provide an easy way to supply additional imagery, an algorithm for registering uncalibrated photos to a textured digital elevation model is presented. A particular challenge in registering the images is posed by large variations in the photos concerning resolution, lighting conditions, seasonal changes, etc. The registered photos can be used to increase the visual quality of the textured DEM, in particular at steep slopes. To this end, a method is presented that combines several georegistered photos to textures for the DEM. The difficulty in this compositing process is to create a consistent appearance and avoid visible seams between the photos. In addition to that, the photos also provide valuable means to improve landform mapping. To this end, an extension of the landform mapping methods is presented that allows the utilization of the registered photos during mapping. This way, a detailed and exact mapping becomes feasible even at steep slopes

Neural function approximation on graphs: shape modelling, graph discrimination & compression

Graphs serve as a versatile mathematical abstraction of real-world phenomena in numerous scientific disciplines. This thesis is part of the Geometric Deep Learning subject area, a family of learning paradigms, that capitalise on the increasing volume of non-Euclidean data so as to solve real-world tasks in a data-driven manner. In particular, we focus on the topic of graph function approximation using neural networks, which lies at the heart of many relevant methods. In the first part of the thesis, we contribute to the understanding and design of Graph Neural Networks (GNNs). Initially, we investigate the problem of learning on signals supported on a fixed graph. We show that treating graph signals as general graph spaces is restrictive and conventional GNNs have limited expressivity. Instead, we expose a more enlightening perspective by drawing parallels between graph signals and signals on Euclidean grids, such as images and audio. Accordingly, we propose a permutation-sensitive GNN based on an operator analogous to shifts in grids and instantiate it on 3D meshes for shape modelling (Spiral Convolutions). Following, we focus on learning on general graph spaces and in particular on functions that are invariant to graph isomorphism. We identify a fundamental trade-off between invariance, expressivity and computational complexity, which we address with a symmetry-breaking mechanism based on substructure encodings (Graph Substructure Networks). Substructures are shown to be a powerful tool that provably improves expressivity while controlling computational complexity, and a useful inductive bias in network science and chemistry. In the second part of the thesis, we discuss the problem of graph compression, where we analyse the information-theoretic principles and the connections with graph generative models. We show that another inevitable trade-off surfaces, now between computational complexity and compression quality, due to graph isomorphism. We propose a substructure-based dictionary coder - Partition and Code (PnC) - with theoretical guarantees that can be adapted to different graph distributions by estimating its parameters from observations. Additionally, contrary to the majority of neural compressors, PnC is parameter and sample efficient and is therefore of wide practical relevance. Finally, within this framework, substructures are further illustrated as a decisive archetype for learning problems on graph spaces.Open Acces

13th Annual Review of Progress in Applied Computational Electromagnetics at the Naval Postgraduate School, Monterey, CA, March 17-21, 1997, Conference Proceedings Volumes I & II

Includes Volumes 1 &

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