10 research outputs found
Hexagonal Smoothness-Increasing Accuracy-Conserving Filtering
Discontinuous Galerkin (DG) methods are a popular class of numerical techniques to solve partial differential equations due to their higher order of accuracy. However, the inter-element discontinuity of a DG solution hinders its utility in various applications, including visualization and feature extraction. This shortcoming can be alleviated by postprocessing of DG solutions to increase the inter-element smoothness. A class of postprocessing techniques proposed to increase the inter-element smoothness is SIAC filtering. In addition to increasing the inter-element continuity, SIAC filtering also raises the convergence rate from order k+1k+1 to order 2k+12k+1 . Since the introduction of SIAC filtering for univariate hyperbolic equations by Cockburn et al. (Math Comput 72(242):577–606, 2003), many generalizations of SIAC filtering have been proposed. Recently, the idea of dimensionality reduction through rotation has been the focus of studies in which a univariate SIAC kernel has been used to postprocess a two-dimensional DG solution (Docampo-Sánchez et al. in Multi-dimensional filtering: reducing the dimension through rotation, 2016. arXiv preprint arXiv:1610.02317). However, the scope of theoretical development of multidimensional SIAC filters has never gone beyond the usage of tensor product multidimensional B-splines or the reduction of the filter dimension. In this paper, we define a new SIAC filter called hexagonal SIAC (HSIAC) that uses a nonseparable class of two-dimensional spline functions called hex splines. In addition to relaxing the separability assumption, the proposed HSIAC filter provides more symmetry to its tensor-product counterpart. We prove that the superconvergence property holds for a specific class of structured triangular meshes using HSIAC filtering and provide numerical results to demonstrate and validate our theoretical results
A Practical Box Spline Compendium
Box splines provide smooth spline spaces as shifts of a single generating
function on a lattice and so generalize tensor-product splines. Their elegant
theory is laid out in classical papers and a summarizing book. This compendium
aims to succinctly but exhaustively survey symmetric low-degree box splines
with special focus on two and three variables. Tables contrast the lattices,
supports, analytic and reconstruction properties, and list available
implementations and code.Comment: 15 pages, 10 figures, 8 table
Fast cosine transform for FCC lattices
Voxel representation and processing is an important issue in a broad spectrum
of applications. E.g., 3D imaging in biomedical engineering applications, video
game development and volumetric displays are often based on data representation
by voxels. By replacing the standard sampling lattice with a face-centered
lattice one can obtain the same sampling density with less sampling points and
reduce aliasing error, as well. We introduce an analog of the discrete cosine
transform for the facecentered lattice relying on multivariate Chebyshev
polynomials. A fast algorithm for this transform is deduced based on algebraic
signal processing theory and the rich geometry of the special unitary Lie group
of degree four.Comment: Presented at 13th APCA International Conference on Automatic Control
and Soft Computing (CONTROLO 2018); 9 figure
Rectangular Body-centered Cuboid Packing Lattices and Their Possible Applications
We first introduce several sphere packing ways such as simple cubic packing (SC), face-centered cubic packing (FCC), body-centered cubic packing (BCC), and rectangular body-centered cuboid packing (recBCC), where the rectangular body-centered cuboid packing means the packing method based on a rectangular cuboid whose base is square and whose height is times the length of one side of its square base such that the congruent spheres are centered at the 8 vertices and the centroid of the cuboid. The corresponding lattices are denoted as SCL, FCCL, BCCL, and recBCCL, respectively. Then we consider properties of those lattices, and show that FCCL and recBCCL are the same. Finally we point out some possible applications of the recBCC lattices
Lattice-Based High-Dimensional Gaussian Filtering and the Permutohedral Lattice
High-dimensional Gaussian filtering is a popular technique in image processing, geometry processing and computer graphics for smoothing data while preserving important features. For instance, the bilateral filter, cross bilateral filter and non-local means filter fall under the broad umbrella of high-dimensional Gaussian filters. Recent algorithmic advances therein have demonstrated that by relying on a sampled representation of the underlying space, one can obtain speed-ups of orders of magnitude over the naïve approach. The simplest such sampled representation is a lattice, and it has been used successfully in the bilateral grid and the permutohedral lattice algorithms. In this paper, we analyze these lattice-based algorithms, developing a general theory of lattice-based high-dimensional Gaussian filtering. We consider the set of criteria for an optimal lattice for filtering, as it offers a good tradeoff of quality for computational efficiency, and evaluate the existing lattices under the criteria. In particular, we give a rigorous exposition of the properties of the permutohedral lattice and argue that it is the optimal lattice for Gaussian filtering. Lastly, we explore further uses of the permutohedral-lattice-based Gaussian filtering framework, showing that it can be easily adapted to perform mean shift filtering and yield improvement over the traditional approach based on a Cartesian grid.Stanford University (Reed-Hodgson Fellowship)Nokia Research Cente
Smoothness-Increasing Accuracy-Conserving (SIAC) filtering and quasi interpolation: A unified view
Filtering plays a crucial role in postprocessing and analyzing data in scientific and engineering applications. Various application-specific filtering schemes have been proposed based on particular design criteria. In this paper, we focus on establishing the theoretical connection between quasi-interpolation and a class of kernels (based on B-splines) that are specifically designed for the postprocessing of the discontinuous Galerkin (DG) method called Smoothness-Increasing Accuracy-Conserving (SIAC) filtering. SIAC filtering, as the name suggests, aims to increase the smoothness of the DG approximation while conserving the inherent accuracy of the DG solution (superconvergence). Superconvergence properties of SIAC filtering has been studied in the literature. In this paper, we present the theoretical results that establish the connection between SIAC filtering to long-standing concepts in approximation theory such as quasi-interpolation and polynomial reproduction. This connection bridges the gap between the two related disciplines and provides a decisive advancement in designing new filters and mathematical analysis of their properties. In particular, we derive a closed formulation for convolution of SIAC kernels with polynomials. We also compare and contrast cardinal spline functions as an example of filters designed for image processing applications with SIAC filters of the same order, and study their properties