Set Of Irregular Points Research Articles

Improved stencil selection for meshless finite difference methods in 3D

We introduce a geometric stencil selection algorithm for Laplacian in 3D that significantly improves octant-based selection considered earlier. The goal of the algorithm is to choose a small subset from a set of irregular points surrounding a given point that admits an accurate numerical differentiation formula. The subset serves as an influence set for the numerical approximation of the Laplacian in meshless finite difference methods using either polynomial or kernel-based techniques. Numerical experiments demonstrate a competitive performance of this method in comparison to the finite element method and to other selection methods for solving the Dirichlet problems for the Poisson equation on several STL models. Discretization nodes for these domains are obtained either by 3D triangulations or from Cartesian grids or Halton quasi-random sequences.

Entropy of Irregular Points for Some Dynamical Systems

We derive sufficient conditions for a dynamical systems to have a set of irregular points with full topological entropy. Such conditions are verified for nonuniformly hyperbolic systems such as positive entropy surface diffeomorphisms and rational functions on the Riemann sphere.

The Kellogg property under generalized growth conditions

AbstractWe study minimizers of the Dirichlet φ‐energy integral with generalized Orlicz growth. We prove the Kellogg property, the set of irregular points has zero capacity, and give characterizations of semiregular boundary points. The results are new ever for the special cases double phase and Orlicz growth.

SPU-Net: Self-Supervised Point Cloud Upsampling by Coarse-to-Fine Reconstruction With Self-Projection Optimization.

The task of point cloud upsampling aims to acquire dense and uniform point sets from sparse and irregular point sets. Although significant progress has been made with deep learning models, state-of-the-art methods require ground-truth dense point sets as the supervision, which makes them limited to be trained under synthetic paired training data and not suitable to be under real-scanned sparse data. However, it is expensive and tedious to obtain large numbers of paired sparse-dense point sets as supervision from real-scanned sparse data. To address this problem, we propose a self-supervised point cloud upsampling network, named SPU-Net, to capture the inherent upsampling patterns of points lying on the underlying object surface. Specifically, we propose a coarse-to-fine reconstruction framework, which contains two main components: point feature extraction and point feature expansion, respectively. In the point feature extraction, we integrate the self-attention module with the graph convolution network (GCN) to capture context information inside and among local regions simultaneously. In the point feature expansion, we introduce a hierarchically learnable folding strategy to generate upsampled point sets with learnable 2D grids. Moreover, to further optimize the noisy points in the generated point sets, we propose a novel self-projection optimization associated with uniform and reconstruction terms as a joint loss to facilitate the self-supervised point cloud upsampling. We conduct various experiments on both synthetic and real-scanned datasets, and the results demonstrate that we achieve comparable performances to state-of-the-art supervised methods.

A novel framework for spatio-temporal prediction of environmental data using deep learning

As the role played by statistical and computational sciences in climate and environmental modelling and prediction becomes more important, Machine Learning researchers are becoming more aware of the relevance of their work to help tackle the climate crisis. Indeed, being universal nonlinear function approximation tools, Machine Learning algorithms are efficient in analysing and modelling spatially and temporally variable environmental data. While Deep Learning models have proved to be able to capture spatial, temporal, and spatio-temporal dependencies through their automatic feature representation learning, the problem of the interpolation of continuous spatio-temporal fields measured on a set of irregular points in space is still under-investigated. To fill this gap, we introduce here a framework for spatio-temporal prediction of climate and environmental data using deep learning. Specifically, we show how spatio-temporal processes can be decomposed in terms of a sum of products of temporally referenced basis functions, and of stochastic spatial coefficients which can be spatially modelled and mapped on a regular grid, allowing the reconstruction of the complete spatio-temporal signal. Applications on two case studies based on simulated and real-world data will show the effectiveness of the proposed framework in modelling coherent spatio-temporal fields.

TGNet: Geometric Graph CNN on 3-D Point Cloud Segmentation

Recent geometric deep learning works define convolution operations in local regions and have enjoyed remarkable success on non-Euclidean data, including graph and point clouds. However, the high-level geometric correlations between the input and its neighboring coordinates or features are not fully exploited, resulting in suboptimal segmentation performance. In this article, we propose a novel graph convolution architecture, which we term as Taylor Gaussian mixture model (GMM) network (TGNet), to efficiently learn expressive and compositional local geometric features from point clouds. The TGNet is composed of basic geometric units, TGConv, that conduct local convolution on irregular point sets and are parametrized by a family of filters. Specifically, these filters are defined as the products of the local point features and the neighboring geometric features extracted from local coordinates. These geometric features are expressed by Gaussian weighted Taylor kernels. Then, a parametric pooling layer aggregates TGConv features to generate new feature vectors for each point. TGNet employs TGConv on multiscale neighborhoods to extract coarse-to-fine semantic deep features while improving its scale invariance. Additionally, a conditional random field (CRF) is adopted within the output layer to further improve the segmentation results. Using three point cloud data sets, qualitative and quantitative experimental results demonstrate that the proposed method achieves 62.2% average accuracy on ScanNet, 57.8% and 68.17% mean intersection over union (mIoU) on Stanford Large-Scale 3D Indoor Spaces (S3DIS) and Paris-Lille-3D data sets, respectively.

Some remarks on the Dirichlet problem on infinite trees

Abstract We consider the Dirichlet problem on in_nite and locally _nite rooted trees, andwe prove that the set of irregular points for continuous data has zero capacity. We also give some uniqueness results for solutions in Sobolev W1,p of the tree.

On the irregular points for systems with the shadowing property

We prove that when $f$ is a continuous self-map acting on a compact metric space $(X,d)$ that satisfies the shadowing property, then the set of irregular points (i.e., points with divergent Birkhoff averages) has full entropy. Using this fact, we prove that, in the class of $C^{0}$-generic maps on manifolds, we can only observe (in the sense of Lebesgue measure) points with convergent Birkhoff averages. In particular, the time average of atomic measures along orbits of such points converges to some Sinai–Ruelle–Bowen-like measure in the weak$^{\ast }$ topology. Moreover, such points carry zero entropy. In contrast, irregular points are non-observable but carry infinite entropy.

Topological pressure for the completely irregular set of birkhoff averages

It is well-known that for certain dynamical systems (satisfying specification or its variants), the set of irregular points w.r.t. a continuous function $\phi$ (i.e. points with divergent Birkhoff ergodic averages observed by $\phi$ ) either is empty or carries full topological entropy (or pressure, see [6,17,36,37] etc. for example). In this paper we study the set of irregular points w.r.t. a collection $D$ of finite or infinite continuous functions (that is, points with divergent Birkhoff ergodic averages simultaneously observed by all $\phi∈D$ ) and obtain some generalized results. As consequences, these results are suitable for systems such as mixing shifts of finite type, uniformly hyperbolic diffeomorphisms, repellers and $β-$ shifts.

Non-equispaced B-spline wavelets

This paper has three main contributions. The first is the construction of wavelet transforms from B-spline scaling functions defined on a grid of non-equispaced knots. The new construction extends the equispaced, biorthogonal, compactly supported Cohen–Daubechies–Feauveau wavelets. The new construction is based on the factorization of wavelet transforms into lifting steps. The second and third contributions are new insights on how to use these and other wavelets in statistical applications. The second contribution is related to the bias of a wavelet representation. It is investigated how the fine scaling coefficients should be derived from the observations. In the context of equispaced data, it is common practice to simply take the observations as fine scale coefficients. It is argued in this paper that this is not acceptable for non-interpolating wavelets on non-equidistant data. Finally, the third contribution is the study of the variance in a non-orthogonal wavelet transform in a new framework, replacing the numerical condition as a measure for non-orthogonality. By controlling the variances of the reconstruction from the wavelet coefficients, the new framework allows us to design wavelet transforms on irregular point sets with a focus on their use for smoothing or other applications in statistics.

Implementation aspects of sequential Gaussian simulation on irregular points

The increasing use of unstructured grids for reservoir modeling motivates the development of geostatistical techniques to populate them with properties such as facies proportions, porosity and permeability. Unstructured grids are often populated by upscaling high-resolution regular grid models, but the size of the regular grid becomes unreasonably large to ensure that there is sufficient resolution for small unstructured grid elements. The properties could be modeled directly on the unstructured grid, which leads to an irregular configuration of points in the three-dimensional reservoir volume. Current implementations of Gaussian simulation for geostatistics are for regular grids. This paper addresses important implementation details involved in adapting sequential Gaussian simulation to populate irregular point configurations including general storage and computation issues, generating random paths for improved long range variogram reproduction, and search strategies including the superblock search and the k-dimensional tree. An efficient algorithm for computing the variogram of very large irregular point sets is developed for model checking.

REFINEMENT INDEPENDENT WAVELETS FOR USE IN ADAPTIVE MULTIRESOLUTION SCHEMES

This paper constructs a class of semi-orthogonal and bi-orthogonal wavelet transforms on possibly irregular point sets with the property that the scaling coefficients are independent from the order of refinement. That means that scaling coefficients at a given scale can be constructed with the configuration at that scale only. This property is of particular interest when the refinement operation is data dependent, leading to adaptive multiresolution analyses. Moreover, the proposed class of wavelet transforms are constructed using a sequence of just two lifting steps, one of which contains a linear interpolating prediction operator. This operator easily allows extensions towards directional offsets from predictions, leading to an edge-adaptive nonlinear multiscale decomposition.

SPLASH: An Interactive Visualisation Tool for Smoothed Particle Hydrodynamics Simulations

AbstractThis paper presents SPLASH, a publicly available interactive visualisation tool for Smoothed Particle Hydrodynamics (SPH) simulations. Visualisation of SPH data is more complicated than for grid-based codes because the data are defined on a set of irregular points and therefore requires a mapping procedure to a two dimensional pixel array. This means that, in practise, many authors simply produce particle plots which offer a rather crude representation of the simulation output. Here we describe the techniques and algorithms which are utilised in SPLASH in order to provide the user with a fast, interactive and meaningful visualisation of one, two and three dimensional SPH results.

Wavelets on irregular point sets

In this article we review techniques for building and analysing wavelets on irregular point sets in one and two dimensions. We discuss current results both on the practical and theoretical side. In particular, we focus on subdivision schemes and commutation rules. Several examples are included.

One-dimensional polyhedral irregular sets of homomorphisms of 3-manifolds

Examples are given to show that there exist homeomorphisms of open 3 3 -manifolds whose sets of irregular points are wildly embedded one-dimensional polyhedra. The main result of the paper is that a one-dimensional polyhedral set of irregular points can fail to be locally tame on, at most, a discrete subset of the set of points of order greater than one. Necessary and sufficient conditions are given so that the set of irregular points is locally tame at each point.

On the Irregular Sets of a Transformation Group

We assume throughout that (X, T, π) is a transformation group [2], where X is a topological space which is always assumed to be regular and Hausdorff. We call a point x ∊ X regular under T if for any open set U in X and any subset G of T such that , there exists an open set V containing x, such that VG ⊆ U [7]. Let R(X) denote the interior of the set of all the regular points of X under T, and I(X) the set of irregular points of X under T, that is the set of points which are not regular under T.

On the homotopy type of irregular sets

If M M is an open connected manifold and h h is a homeomorphism of M M onto itself such that h h is positively regular on M M and the set of irregular points, Irr ⁡ ( h ) \operatorname {Irr} (h) , is a nonseparating compactum, then it is shown that Irr ⁡ ( h ) \operatorname {Irr} (h) is a strong deformation retract of M M .

On Almost Regular Homeomorphisms

Let (X, d) be a metric space with metric d, and h be a homeomorphism of X onto itself. Any point y in X is called a regular point (2) under A if for any given ϵ > 0 there exists a δ > 0 such that d(x, y) < δ implies that d(hn(x), hn(y)) < ϵ for all integers n, where hn is the composition of h or h-1 with itself |n| times, depending upon whether n is positive or negative, and h0 is the identity on X. If y is not regular under h, then y is called irregular. We shall denote the set of regular points by R(h) and the set of irregular points by I(h).

A new proof of the fundamental theorem of Kellogg-Evans on the set of irregular points in the Dirichlet problem

A new proof of the fundamental theorem of Kellogg-Evans on the set of irregular points in the Dirichlet problem