Data-Driven Shape Analysis and Processing

Data-driven methods play an increasingly important role in discovering geometric, structural, and semantic relationships between 3D shapes in collections, and applying this analysis to support intelligent modeling, editing, and visualization of geometric data. In contrast to traditional approaches, a key feature of data-driven approaches is that they aggregate information from a collection of shapes to improve the analysis and processing of individual shapes. In addition, they are able to learn models that reason about properties and relationships of shapes without relying on hard-coded rules or explicitly programmed instructions. We provide an overview of the main concepts and components of these techniques, and discuss their application to shape classification, segmentation, matching, reconstruction, modeling and exploration, as well as scene analysis and synthesis, through reviewing the literature and relating the existing works with both qualitative and numerical comparisons. We conclude our report with ideas that can inspire future research in data-driven shape analysis and processing.Comment: 10 pages, 19 figure

A Discrete Adapted Hierarchical Basis Solver For Radial Basis Function Interpolation

In this paper we develop a discrete Hierarchical Basis (HB) to efficiently solve the Radial Basis Function (RBF) interpolation problem with variable polynomial order. The HB forms an orthogonal set and is adapted to the kernel seed function and the placement of the interpolation nodes. Moreover, this basis is orthogonal to a set of polynomials up to a given order defined on the interpolating nodes. We are thus able to decouple the RBF interpolation problem for any order of the polynomial interpolation and solve it in two steps: (1) The polynomial orthogonal RBF interpolation problem is efficiently solved in the transformed HB basis with a GMRES iteration and a diagonal, or block SSOR preconditioner. (2) The residual is then projected onto an orthonormal polynomial basis. We apply our approach on several test cases to study its effectiveness, including an application to the Best Linear Unbiased Estimator regression problem

Simulating 3D Radiation Transport, a modern approach to discretisation and an exploration of probabilistic methods

Light, or electromagnetic radiation in general, is a profound and invaluable resource to investigate our physical world. For centuries, it was the only and it still is the main source of information to study the Universe beyond our planet. With high-resolution spectroscopic imaging, we can identify numerous atoms and molecules, and can trace their physical and chemical environments in unprecedented detail. Furthermore, radiation plays an essential role in several physical and chemical processes, ranging from radiative pressure, heating, and cooling, to chemical photo-ionisation and photo-dissociation reactions. As a result, almost all astrophysical simulations require a radiative transfer model. Unfortunately, accurate radiative transfer is very computationally expensive. Therefore, in this thesis, we aim to improve the performance of radiative transfer solvers, with a particular emphasis on line radiative transfer. First, we review the classical work on accelerated lambda iterations and acceleration of convergence, and we propose a simple but effective improvement to the ubiquitously used Ng-acceleration scheme. Next, we present the radiative transfer library, Magritte: a formal solver with a ray-tracer that can handle structured and unstructured meshes as well as smoothed-particle data. To mitigate the computational cost, it is optimised to efficiently utilise multi-node and multi-core parallelism as well as GPU offloading. Furthermore, we demonstrate a heuristic algorithm that can reduce typical input models for radiative transfer by an order of magnitude, without significant loss of accuracy. This strongly suggests the existence of more efficient representations for radiative transfer models. To investigate this, we present a probabilistic numerical method for radiative transfer that naturally allows for uncertainty quantification, providing us with a mathematical framework to study the trade-off between computational speed and accuracy. Although we cannot yet construct optimal representations for radiative transfer problems, we point out several ways in which this method can lead to more rigorous optimisation

Hierarchical interpolative factorization for elliptic operators: differential equations

This paper introduces the hierarchical interpolative factorization for elliptic partial differential equations (HIF-DE) in two (2D) and three dimensions (3D). This factorization takes the form of an approximate generalized LU/LDL decomposition that facilitates the efficient inversion of the discretized operator. HIF-DE is based on the multifrontal method but uses skeletonization on the separator fronts to sparsify the dense frontal matrices and thus reduce the cost. We conjecture that this strategy yields linear complexity in 2D and quasilinear complexity in 3D. Estimated linear complexity in 3D can be achieved by skeletonizing the compressed fronts themselves, which amounts geometrically to a recursive dimensional reduction scheme. Numerical experiments support our claims and further demonstrate the performance of our algorithm as a fast direct solver and preconditioner. MATLAB codes are freely available.Comment: 37 pages, 13 figures, 12 tables; to appear, Comm. Pure Appl. Math. arXiv admin note: substantial text overlap with arXiv:1307.266