Large Point Sets Research Articles

Effectively learning spatial indices

Machine learning, especially deep learning, is used increasingly to enable better solutions for data management tasks previously solved by other means, including database indexing. A recent study shows that a neural network can not only learn to predict the disk address of the data value associated with a one-dimensional search key but also outperform B-tree-based indexing, thus promises to speed up a broad range of database queries that rely on B-trees for efficient data access. We consider the problem of learning an index for two-dimensional spatial data. A direct application of a neural network is unattractive because there is no obvious ordering of spatial point data. Instead, we introduce a rank space based ordering technique to establish an ordering of point data and group the points into blocks for index learning. To enable scalability, we propose a recursive strategy that partitions a large point set and learns indices for each partition. Experiments on real and synthetic data sets with more than 100 million points show that our learned indices are highly effective and efficient. Query processing using our indices is more than an order of magnitude faster than the use of R-trees or a recently proposed learned index.

Delaunay triangulation of large-scale datasets using two-level parallelism

Because of the importance of Delaunay Triangulation in science and engineering, researchers have devoted extensive attention to parallelizing this fundamental algorithm. However, generating unstructured meshes for extremely large point sets remains a barrier for scientists working with large scale or high resolution datasets. In our previous paper, we introduced a novel algorithm – Triangulation of Independent Partitions in Parallel (TIPP) which divides the domain into many independent partitions that can be triangulated in parallel. However, using only a single master process introduced a performance bottleneck and inhibited scalability. In this paper, we refine our description of the original TIPP algorithm, and also extend TIPP to employ multiple master processes, distributing computational load across several machines. This new design improves both performance and scalability, and can produce 20 billion triangles using only 10 commodity nodes in under 30 minutes.

Typical differentiability within an exceptionally small set

We verify the existence of a purely unrectifiable set in which the typical Lipschitz function has a large set of differentiability points. The example arises from a construction, due to Csörnyei, Preiss and Tišer, of a universal differentiability set in which a certain Lipschitz function has only a purely unrectifiable set of differentiability points.

A novel algorithm for a continuous and fast 3D projection of points on triangulated surfaces for CAM/CAD/CAE applications

In the present work, a novel algorithm for the continuous projection of point triangles belonging to a triangle mesh is presented. The algorithm uses the normals defined at the vertices of the triangle to perform the projection. The direction of projection is not chosen a priori but depends on the point to be projected: the proposed algorithm, named Fast Continuous Projection method (FCP) lets the projection direction vary continuously on the mesh. Moreover, the direction is coherent with the original surface that is approximated with the triangle mesh. An optimized version of the algorithm is also presented: this uses pre-evaluated matrices to reduce the calculation time. This algorithm can be effectively used when a large set of points has to be projected on a coarse mesh as, for example, to generate the scanning vectors for laser engraving/milling.

Optical flow-based structure-from-motion for the reconstruction of epithelial surfaces

This paper details a novel optical flow-based structure from motion (SfM) approach for the reconstruction of surfaces with few textures using video sequences acquired under strong illumination changes. An original image search and grouping strategy allows to reconstruct each 3D scene point using a large set of 2D homologous points extracted from a reference image and its superimposed images acquired from different viewpoints. A variational optical flow scheme with a descriptor-based data term leads to a robust, accurate and dense homologous point determination between the image pairs. Thus, contrary to classical SfM usable for textured scenes, the proposed dense point cloud reconstruction algorithm requires neither a feature point tracking method nor any multi-view stereo technique. The performance of the proposed SfM approach is assessed on phantoms with known ground truth and on very complex patient data of various medical examinations and image modalities.

Parametric model for the reconstruction and representation of hydrofoils and airfoils

In this paper, we present a geometric parametric model that has been mainly designed to accommodate the need for an appropriate analysis model in the context of IsoGeometric Analysis (IGA) [1], but can be also employed as a universal tool for reconstructing and representing a wide range of different airfoils' and/or hydrofoils' families. This model is an extension of the parametric model employed in optimization of hydrofoils' efficiency with the aid of an isogeometric boundary element method solver (IGA-BEM), presented in [2]. Apart from its apparent use in design evaluation and optimization, the parametric model can be further employed in a wider range of relevant applications. Specifically, in this work we begin by demonstrating its use, as an alternative approach, in efficiently reconstructing airfoil and hydrofoil shapes from large sets of points appearing, mainly, in the context of reverse engineering applications. This is further investigated and the parametric model's performance in accurately representing existing airfoils is extensively benchmarked against results from pertinent methods and approaches followed in the aeronautics and aerospace research community. Finally, we demonstrate the use of the parametric model in a foil shape optimization problem. We conclude this work with a discussion on future enhancements and possible research directions.

Lithium iron phosphate power cell fault detection system based on hybrid intelligent system

Abstract Nowadays, batteries play an important role in a lot of different applications like energy storage, electro-mobility, consumer electronic and so on. All the battery types have a common factor that is their complexity, independently of its nature. Usually, the batteries have an electrochemical nature. Several different test are accomplished to check the batteries performance, and commonly, it is predictable how they work depending of their technology. The present research describes the hybrid intelligent system created to accomplish fault detection over a Lithium Iron Phosphate—LiFePO4 power cell type, commonly used in electro-mobility applications. The approach is based on the cell temperatures behaviour for voltage and current specific values. Taken into account the operating range of a real system based on a LiFePO4 cell, a large set of points of operation have been used to achieve the dataset. The different behaviour zones have been obtained by clustering as a first step. Then, different regression techniques have been used over each cluster. Polynomial regression, artificial neural networks and support vector regression were the combined techniques to develop the hybrid intelligent model proposed. The intelligent system gives very good results over the operating range, detecting all the faults tested during the validation.

On large equilateral point-sets in normed spaces

What is the largest cardinal of a two-by-two equilateral point-set in an n-dimensional Minkowski space? It is conjectured that this cardinal is at least $$n+1$$, and several spaces are tight in this regard, such as the Euclidean space. In this paper, we prove in normed spaces X with the $$(H_1,\dots ,H_{n-1})$$-2-intersection property the existence of $$n+1$$ equilateral point-sets of large diameter inscribed to the unit ball B. This extends the construction of Makeev (J Math Sci (N Y) 140:548–550, 2007) in dimension 4.

Turning Big Data Into Tiny Data: Constant-Size Coresets for $k$-Means, PCA, and Projective Clustering

We develop and analyze a method to reduce the size of a very large set of data points in a high dimensional Euclidean space R d to a small set of weighted points such that the result of a predetermined data analysis task on the reduced set is approximately the same as that for the original point set. For example, computing the first k principal components of the reduced set will return approximately the first k principal components of the original set or computing the centers of a k-means clustering on the reduced set will return an approximation for the original set. Such a reduced set is also known as a coreset. The main new feature of our construction is that the cardinality of the reduced set is independent of the dimension d of the input space and that the sets are mergable. The latter property means that the union of two reduced sets is a reduced set for the union of the two original sets (this property has recently also been called composability, see Indyk et. al., PODS 2014). It allows us to turn our methods into streaming or distributed algorithms using standard approaches. For problems such as k-means and subspace approximation the coreset sizes are also independent of the number of input points. Our method is based on projecting the points on a low dimensional subspace and reducing the cardinality of the points inside this subspace using known methods. The proposed approach works for a wide range of data analysis techniques including k-means clustering, principal component analysis and subspace clustering. The main conceptual contribution is a new coreset definition that allows to charge costs that appear for every solution to an additive constant.

A Preprocessing Technique for Fast Convex Hull Computation

Abstract Computing the convex hull of a set of points is a fundamental issue in many fields, including geometric computing, computer graphics, and computer vision. This problem is computationally challenging, especially when the number of points is past the millions. In this paper, we propose a fast filtering technique that reduces the computational cost for computing a convex hull for a large set of points. The proposed method preprocesses the input set and filters all points inside a four-vertex polygon. The experimental results showed the proposed filtering approach achieved a speedup of up to 77 and 12 times faster than the standard Graham scan and Jarvis march algorithms, respectively.

An efficient EM-ICP algorithm for non-linear registration of large 3D point sets

In this paper, we present a new method for non-linear pairwise registration of 3D point sets. In this method, we consider the points of the first set as the draws of a Gaussian mixture model whose centres are the displaced points of the second set. Next we perform a maximum a posteriori estimation of the parameters (which include the unknown transformation) of this model using the expectation–maximisation (EM) algorithm. Compared to other methods using the same “EM-ICP” framework, we propose four key modifications leading to an efficient algorithm allowing for fast registration of large 3D point sets: (1) truncation of the cost function; (2) symmetrisation of the point-to-point correspondences; (3) specification of priors on these correspondences using differential geometry; (4) efficient encoding of deformations using the RKHS theory and the Fourier analysis. We evaluate the added value of these modifications and compare our method to the state-of-the-art CPD algorithm on real and simulated data.

On Almost Empty Monochromatic Triangles and Convex Quadrilaterals in Colored Point Sets

Let S be a set of N points on the plane in general position, colored arbitrarily with c colors ($$c\in {\mathbb {N}}$$). A subset of points is said to be monochromatic if all its points have the same color. In this paper we show that if N is sufficiently large, then S contains a monochromatic triangle with vertices in S and containing at most $$c-3$$ points in its interior $$(c \ge 4)$$. The particular case $$c=4$$ solves the conjecture of Basu, Bhattacharya, and Das in [6]. For the case of two colors, it is still unknown whether every sufficiently large bichromatic point set contains an empty monochromatic convex quadrilateral [11]. In this direction we show that if S is sufficiently large, then it always contains a convex monochromatic quadrilateral with at most $$2c-3$$ points in its interior $$(c \ge 2)$$. We also show a general result on the number of empty convex quadrilaterals with disjoint interiors. Let P be a set of n points in general position. It is straightforward to see that if the elements of P are the vertices of a convex polygon, $$n \ge 4$$, P always has $$\left\lceil \frac{n-3}{2} \right\rceil $$ empty convex quadrilaterals with disjoint interiors. In this paper we prove that any point set P in general position has at least $$\left\lfloor \frac{n-3}{2}\right\rfloor $$ interior disjoint empty convex quadrilaterals. We also give direct consequences of this theorem related to simultaneously flippable edges in triangulations and convex decompositions of point sets. We show that for any point set P there is always a triangulation that has $$\left\lfloor \frac{n-3}{2}\right\rfloor $$ simultaneously flippable edges. We also show that there is always a convex decomposition of P consisting of triangles and convex quadrilaterals with at most $${3n-2h\over 2}$$ elements, where h is the number of points in the convex hull of P.

A hybrid meta-model based global optimization method for expensive problems

The meta-model based global optimization algorithms usually select the new promising points from a large set of points, which are generated using the Latin hypercube design (LHD) and evaluated by the meta-model. Once the poor points are generated by the random number based LHD, the desired results may not be obtained. In this work, a hybrid meta-model based global optimization method (HMGO) is proposed. In this method, three different meta-model, kriging, radial basis functions (RBF) and quadratic function (QF) are used together in the search process. And multiple sets of large points are generated and multiple screening strategy is used for the selection of the new promising points to avoid the poor points. Through test by six benchmark math function with the number of the variables ranging from 10 to 24 and compared with the famous efficient global optimization (EGO), the proposed method shows excellent accuracy, efficiency and robustness. The HMGO method is then applied in a vehicle lightweight design problem with 30 design variables, desired results have been obtained.

Towards a Tool for Early Detection and Estimation of Forest Cuttings by Remotely Sensed Data

Knowing the extent and frequency of forest cuttings over large areas is crucial for forest inventories and monitoring. Remote sensing has amply proved its ability to detect land cover changes, particularly in forested areas. Among various strategies, those focusing on mapping using classification approaches of remotely sensed time series are the most frequently used. The main limit of such approaches stems from the difficulty in perfectly and unambiguously classifying each pixel, especially over wide areas. The same procedure is of course simpler if performed over a single pixel. An automated method for identifying forest cuttings over a predefined network of sampling points (IUTI) using multitemporal Sentinel 2 imagery is described. The method employs normalized difference vegetation index (NDVI) growth trajectories to identify the presence of disturbances caused by forest cuttings using a large set of points (i.e., 1580 “forest” points). We applied the method using a total of 51 S2 images extracted from the Google Earth Engine over two years (2016 and 2017) in an area of about 70 km2 in Tuscany, central Italy.

Non-Model Based Expansion from Limited Points to an Augmented Set of Points Using Chebyshev Polynomials

Expansion of measured dynamic response at a limited set of points is of great interest in the study of structural dynamic systems. Measurements at limited points on a structure can be expanded to unmeasured points by using System Equivalent Reduction Expansion Process (SEREP). However, a finite element model is required to obtain the mode shapes of the system. In this work, a non-model based expansion technique is proposed to expand the vibration characteristics at a sparse set of points to a much larger set of points without the use of a finite element model. Shape functions based on orthogonal polynomials can be used to decompose the deformation of vibrating structures into the summation of a set of shape functions with corresponding weighting coefficients. To obtain the vibration characteristics at an augmented set of points, measurements at a sparse set of points can be related to a set of shape functions which have the same mesh resolution as the sparse configuration. The vibration characteristics at an augmented set of points can then be reconstructed from the shape functions of the high resolution and scaled shape function weighting coefficients of the sparse configuration. Only a sparse set of measurements, geometry and coordinates of measured points are needed. In the work studied here, an analytical plate and a Base-Upright (BU) structure are used as samples to study the application of the proposed method. Both analysis and experiment are used to prove the concept.

Quantum interference among heavy NMSSM Higgs bosons

In the Next-to-Minimal Supersymmetric Standard Model (NMSSM), it is possible to have strong mass degeneracies between the new singlet-like scalar and the heavy doublet-like scalar, as well as between the singlet-like and doublet-like pseudoscalar Higgs states. When the difference in the masses of such states is comparable with the sum of their widths, the quantum mechanical interference between their propagators can become significant. We study these effects by taking into account the full Higgs boson propagator matrix in the calculation of the production process of $\tau^+\tau^-$ pairs in gluon fusion at the Large Hadron Collider (LHC). We find that, while these interference effects are sizeable, they are not resolvable in terms of the distributions of differential cross sections, owing to the poor detector resolution of the $\tau^+\tau^-$ invariant mass. They are, however, identifiable via the inclusive cross sections, which are subject to significant variations with respect to the standard approaches, wherein the propagating Higgs bosons are treated independently from one another. We quantify these effects for several representative benchmark points, extracted from a large set of points, obtained by numerical scanning of the NMSSM parameter space, that satisfy the most important experimental constraints currently available.

SymmetricHull: A Convex Hull Algorithm Based on 2D Geometry and Symmetry

Thanks to the advances in hardware (sensors and cameras) the number of points that convex hull algorithms must process is now much greater than when these algorithms were developed. This limitation entails focusing current research in convex hull algorithms on finding approaches capable of processing large sets of points. From the many works proposed in this area, the heuristic by Akl-Toussaint deserves a special mention, with outstanding results in execution times. In this article, we take the basic principles of this heuristic to create a convex hull algorithm, called SymmetricHull, that takes advantage of geometric and symmetric properties of the formed quadrants in 2D spaces. The proposed algorithm achieves better exe-cution times than that heuristic, reducing the basic operations to a series of comparisons between consecutive points in an array lexicographically sorted by its y coordinate. SymmetricHull organizes the set of points by quadrants and, instead of using the orientation function as the main operation, it appends the slope s between the analyzed point and its predecessor as a property of the point structure (without causing numerical errors). Point operations can be summarized as follows: 1) a rapid discard of a point, based on its coordinates and the ones of its neighbors; 2) a simple comparison between the property s of the current point and the one of its predecessor in the sorted array, in order to determine whether the current point is a strong candidate to be a part of the convex hull; 3) the addition of a point to a stack of candidates for the convex hull; and 4) the simple elimination of a point from the stack of candidates by comparing the property s of the current point with its predecessors in the stack. Our experiments show that SymmetricHull optimizations achieve good results, in terms of time and number of operations required (multiplications, sums, pushs and pops), being espe-cially efficient with point sets greater than 10^4.

Holes in 2-convex point sets

Let S be a set of n points in the plane in general position (no three points from S are collinear). For a positive integer k, a k-hole in S is a convex polygon with k vertices from S and no points of S in its interior. For a positive integer l, a simple polygon P is l-convex if no straight line intersects the interior of P in more than l connected components. A point set S is l-convex if there exists an l-convex polygonization of S.Considering a typical Erdős–Szekeres-type problem, we show that every 2-convex point set of size n contains an Ω(log⁡n)-hole. In comparison, it is well known that there exist arbitrarily large point sets in general position with no 7-hole. Further, we show that our bound is tight by constructing 2-convex point sets in which every hole has size O(log⁡n).

Gravitational interactions of stars with supermassive black hole binaries – I. Tidal disruption events

Stars approaching supermassive black holes (SMBHs) in the centers of galaxies can be torn apart by strong tidal forces. We study the physics of tidal disruption by a binary SMBH as a function of the binary mass ratio $q = M_2 / M_1$ and separation $a$, exploring a large set of points in the parameter range $q \in [0.01, 1]$ and $a/r_{t1} \in [10, 1000]$. We simulate encounters in which field stars approach the binary from the loss cone on parabolic, low angular momentum orbits. We present the rate of disruption and the orbital properties of the disrupted stars, and examine the fallback dynamics of the post-disruption debris in the "frozen-in" approximation. We conclude by calculating the time-dependent disruption rate over the lifetime of the binary. Throughout, we use a primary mass $M_1 = 10^6 M_\odot$ as our central example. We find that the tidal disruption rate is a factor of $\sim 2 - 7$ times larger than the rate for an isolated BH, and is independent of $q$ for $q \gtrsim 0.2$. In the "frozen-in" model, disruptions from close, nearly equal mass binaries can produce intense tidal fallbacks: for binaries with $q \gtrsim 0.2$ and $a/r_{t1} \sim 100$, roughly $\sim 18 - 40 \%$ of disruptions will have short rise times ($t_\textrm{rise} \sim 1 - 10$ d) and highly super-Eddington peak return rates ($\dot{M}_\textrm{peak} / \dot{M}_\textrm{Edd} \sim 2 \times 10^2 - 3 \times 10^3$).

A fast-prediction surrogate model for large datasets

Surrogate models approximate a function based on a set of training points and can then predict the function at new points. In engineering, kriging is widely used because it is fast to train and is generally more accurate than other types of surrogate models. However, the prediction time of kriging increases with the size of the dataset, and the training can fail if the dataset is too large or poorly spaced, which limits the accuracy that is attainable. We develop a new surrogate modeling technique—regularized minimal-energy tensor-product splines (RMTS)—that is not susceptible to training failure, and whose prediction time does not increase with the number of training points. The improved scalability with the number of training points is due to the use of tensor-product splines, where energy minimization is used to handle under-constrained problems in which there are more spline coefficients than training points. RMTS scales up to four dimensions with 10–15 spline coefficients per dimension, but scaling beyond that requires coarsening of the spline in some of the dimensions because of the computational cost of the energy minimization step. Benchmarking using a suite of one- to four-dimensional problems shows that while kriging is the most accurate option for a small number of training points, RMTS is the best alternative when a large set of data points is available or a low prediction time is desired. The best-case average root-mean-square error for the 4-D problems is close to 1% for RMTS and just under 10% for kriging.