Set Of Triangles Research Articles

Short-term monitoring of benzene air concentration in an urban area: a preliminary study of application of Kruskal-Wallis non-parametric test to assess pollutant impact on global environment and indoor

In step with the need to develop statistical procedures to manage small-size environmental samples, in this work we have used concentration values of benzene (C6H6), concurrently detected by seven outdoor and indoor monitoring stations over 12 000 minutes, in order to assess the representativeness of collected data and the impact of the pollutant on indoor environment. Clearly, the former issue is strictly connected to sampling-site geometry, which proves critical to correctly retrieving information from analysis of pollutants of sanitary interest. Therefore, according to current criteria for network-planning, single stations have been interpreted as nodes of a set of adjoining triangles; then, a) node pairs have been taken into account in order to estimate pollutant stationarity on triangle sides, as well as b) node triplets, to statistically associate data from air-monitoring with the corresponding territory area, and c) node sextuplets, to assess the impact probability of the outdoor pollutant on indoor environment for each area. Distributions from the various node combinations are all non-Gaussian, in the consequently, Kruskal-Wallis (KW) non-parametric statistics has been exploited to test variability on continuous density function from each pair, triplet and sextuplet. Results from the above-mentioned statistical analysis have shown randomness of site selection, which has not allowed a reliable generalization of monitoring data to the entire selected territory, except for a single "forced" case (70%); most important, they suggest a possible procedure to optimize network design.

Robust and Optimized Algorithms for the Point‐in‐Polygon Inclusion Test without Pre‐processing

Abstract In this work we present new point inclusion algorithms for non‐convex polygons. These algorithms do not perform any pre‐processing or any type of decomposition nor features classification, which makes them especially suitable for deformable or moving polygons. The algorithms are more accurate and robust than others in the sense that they consider the inclusion of the point in the vertices and edges of the polygon, and deal with the special cases correctly. In order to perform this inclusion test efficiently, they use the sign of the barycentric coordinates of the test point with regard to the triangles formed by the edges and an origin that depends on the test point. This set of triangles, which is a special simplicial covering of the polygon, is constructed after a transformation of the polygon that simplifies the calculations involved in the inclusion test. Then, an appropriate ordering of the rejection tests allows us to optimize this method. Our algorithms have been tested for robustness and compared with ray‐crossing methods, showing a significant improvement.

Peeling Meshed Potatoes

We study variants of the potato peeling problem on meshed (triangulated) polygons. Given a polygon with holes, and a triangular mesh that covers its interior (possibly using additional vertices), we want to find a largest-area connected set of triangles of the mesh that is convex, or has some other shape-related property. In particular, we consider (i) convexity, (ii) monotonicity, (iii) bounded backturn, and (iv) bounded total turning angle. The first three problems are solved in polynomial time, whereas the fourth problem is shown to be NP-hard.

FastV: From‐point Visibility Culling on Complex Models

AbstractWe present an efficient technique to compute the potentially visible set (PVS) of triangles in a complex 3D scene from a viewpoint. The algorithm computes a conservative PVS at object space accuracy. Our approach traces a high number of small, volumetric frusta and computes blockers for each frustum using simple intersection tests. In practice, the algorithm can compute the PVS of CAD and scanned models composed of millions of triangles at interactive rates on a multi‐core PC. We also use the visibility algorithm to accurately compute the reflection paths from a point sound source. The resulting sound propagation algorithm is 10–20X faster than prior accurate geometric acoustic methods.

The Use of Precomputed Triangle Clusters for Accelerated Ray Tracing in Dynamic Scenes

AbstractIn this paper we present a hybrid algorithm for building the bounding volume hierarchy (BVH) that is used in accelerating ray tracing of animated models. This algorithm precomputes densely packed clusters of triangles on surfaces. Folowing that, a set of clusters is used to rebuild the BVH in every frame. Our approach utilizes the assumption that groups of connected triangles remain connected throughout the course of the animation. We introduce a novel heuristic to create triangle clusters that are designed for high performance ray tracing. This heuristic combines the density of connectivity, geometric size and the shape of the cluster.Our approach accelerates the BVH builder by an order of magnitude rebuilding only the set of clusters that is much smaller than the original set of triangles. The speed‐up is achieved against a ‘brute‐force’ BVH builder that repartitions all triangles in every frame of animation without using any pre‐clustering. The rendering performance is not affected when a cluster contains a few dozen triangles. We demonstrate the real‐time/interactive ray tracing performance for highly‐dynamic complex models.

Maximal Flat Antichains of Minimum Weight

We study maximal families ${\cal A}$ of subsets of $[n]=\{1,2,\dots,n\}$ such that ${\cal A}$ contains only pairs and triples and $A\not\subseteq B$ for all $\{A,B\}\subseteq{\cal A}$, i.e. ${\cal A}$ is an antichain. For any $n$, all such families ${\cal A}$ of minimum size are determined. This is equivalent to finding all graphs $G=(V,E)$ with $|V|=n$ and with the property that every edge is contained in some triangle and such that $|E|-|T|$ is maximum, where $T$ denotes the set of triangles in $G$. The largest possible value of $|E|-|T|$ turns out to be equal to $\lfloor(n+1)^2/8\rfloor$. Furthermore, if all pairs and triples have weights $w_2$ and $w_3$, respectively, the problem of minimizing the total weight $w({\cal A})$ of ${\cal A}$ is considered. We show that $\min w({\cal A})=(2w_2+w_3)n^2/8+o(n^2)$ for $3/n\leq w_3/w_2=:\lambda=\lambda(n) < 2$. For $\lambda\ge 2$ our problem is equivalent to the (6,3)-problem of Ruzsa and Szemerédi, and by a result of theirs it follows that $\min w({\cal A})=w_2n^2/2+o(n^2)$.

Disjoint Color-Avoiding Triangles

A set of pairwise edge-disjoint triangles of an edge-colored $K_n$ is r-color avoiding if it does not contain r monochromatic triangles, each having a different color. Let $f_r(n)$ be the maximum integer so that in every edge coloring of $K_n$ with r colors, there is a set of $f_r(n)$ pairwise edge-disjoint triangles that is r-color avoiding. We prove that $0.1177n^2(1-o(1))<f_2(n)<0.1424n^2(1+o(1))$. The proof of the lower bound uses probabilistic arguments, fractional relaxation and some packing theorems. We also prove that $f_r(n)/n^2<\frac{1}{6}(1-0.145^{r-1})+o(1)$. In particular, for every r, if n is sufficiently large, there are edge colorings of $K_n$ with r colors so that the removal of any $o(n^2)$ members from any Steiner triple system does not turn it r-color avoiding.

Surface mesh regeneration considering curvatures

This work describes an automatic algorithm for unstructured mesh regeneration on arbitrarily shaped three-dimensional surfaces. The arbitrary surface may be: a triangulated mesh, a set of points, or an analytical surface (such as a collection of NURBS patches). To be generic, the algorithm works directly in Cartesian coordinates, as opposed to generating the mesh in parametric space, which might not be available in all the cases. In addition, the algorithm requires the implementation of three generic functions that abstractly represent the supporting surface. The first, given a point location, returns the desired characteristic size of a triangular element at this position. The second method, given the current edge in the boundary-contraction algorithm, locates the ideal apex point that forms a triangle with this edge. And the third method, given a point in space and a projection direction, returns the closest point on the geometrical supporting surface. This work also describes the implementation of these three methods to re-mesh an existing triangulated mesh that might present regions of high curvature. In this implementation, the only information about the surface geometry is a set of triangles. In order to test the efficiency of the proposed algorithm of surface mesh generation and implementation of the three abstract methods, results of performance and quality of generated triangular element examples are presented.

Variational method for objective analysis of scalar variable and its derivative

In this study real time data have been used to compare the standard and triangle method by performing the objective analysis of mean sea level pressure. In the standard method, derivative fields are obtained from the grid point data using finite difference scheme whereas in the triangle method, a set of non-overlapping triangles are formed from the observations and the scalar and the spatial derivatives are computed directly at the centroid of each of the non-overlapping triangles. These scalars and their derivatives are then mapped to uniform grids by using the standard method. It has been found that objectively analysed scalar field obtained using standard method is superior to the scalar field derived by the triangle method, whereas the derivative fields produced by triangle method are superior to the derivative fields produced using standard method. A variational objective analysis scheme has been developed and an experiment has been carried out with depression case of June (11–15) 2004. It is found that the new scheme (variational) is able to extract the better parts of both triangle and standard methods. The results of this study will be useful in carrying out diagnostic calculations that involve derivative estimates.

Brambles and independent packings in chordal graphs

An independent packing of triangles is a set of pairwise disjoint triangles, no two of which are joined by an edge. A triangle bramble is a set of triangles, every pair of which intersect or are joined by an edge. More generally, I consider independent packings and brambles of any specified connected graphs, not just triangles. I give a min–max theorem for the maximum number of graphs in an independent packing of any family of connected graphs in a chordal graph, and a dual min–max theorem for the maximum number of graphs in a bramble in a chordal graph.

Weighted Clustering Coefficient for Identifying Modular Formations in Protein-Protein Interaction Networks

This paper describes a novel approach for deriving modules from protein-protein interaction networks, which combines functional information with topological properties of the network. This approach is based on weighted clustering coefficient, which uses weights representing the functional similarities between the proteins. These weights are calculated according to the semantic similarity between the proteins, which is based on their Gene Ontology terms. We recently proposed an algorithm for identification of functional modules, called SWEMODE (Semantic WEights for MODule Elucidation), that identifies dense sub-graphs containing functionally similar proteins. The rational underlying this approach is that each module can be reduced to a set of triangles (protein triplets connected to each other). Here, we propose considering semantic similarity weights of all triangle-forming edges between proteins. We also apply varying semantic similarity thresholds between neighbours of each node that are not neighbours to each other (and hereby do not form a triangle), to derive new potential triangles to include in module-defining procedure. The results show an improvement of pure topological approach, in terms of number of predicted modules that match known complexes.

Accessibility analysis for automatic inspection in CMMs by using bounding volume hierarchies

Accessibility analysis represents one of the most critical tasks in inspection planning. This analysis determines those probe orientations that can touch an inspection point without collision. This paper presents a methodology based on part discretization and the application of space partitioning techniques (kd-tree) in order to reduce the number of intersection tests between probe and part. A STL model has been used for discretizing the inspection part in a set of triangles, which permits the application of the developed system to any type of part, regardless of its shape and its complexity. Likewise, a recursive ray traversal algorithm has been used in order to speed up the traversal of the kd-tree hierarchical structure and to calculate exclusively the intersection of each probe orientation with those part triangles that can potentially interfere with it. In a further step of the analysis, the real geometry of the probe has been considered. Hence, a simplified model has been developed for each probe component (column, head, touch probe, stylus and tip) using different basic geometrical shapes. Finally, collision-free probe orientations are clustered for minimizing the orientation changes during the inspection process. Furthermore, the applied algorithm allows for determining different valid combinations of clusters. The developed system was applied to two example parts in order to prove that this methodology is adequate for the solution of real cases.

Domain Decomposition for Less Regular Subdomains: Overlapping Schwarz in Two Dimensions

In the theory of domain decomposition methods, it is often assumed that each subdomain is the union of a small set of coarse triangles or tetrahedra. In this study, extensions to the existing theory which accommodate subdomains with much less regular shapes are presented; the subdomains are required only to be John domains. Attention is focused on overlapping Schwarz preconditioners for problems in two dimensions with a coarse space component of the preconditioner, which allows for good results even for coefficients which vary considerably. It is shown that the condition number of the domain decomposition method is bounded by $C(1+H/\delta)(1+\log(H/h))^2$, where the constant C is independent of the number of subdomains and possible jumps in coefficients between subdomains. Numerical examples are provided which confirm the theory and demonstrate very good performance of the method for a variety of subregions including those obtained when a mesh partitioner is used for the domain decomposition.

Measures of accuracy for the Index of Production

In recent times the key measure of quality used for the Index of Production (IoP) has been the revisions performance of key aggregates. This is published as a set of revisions triangles alongside the monthly IoP release on the National Statistics website. Additionally, the IoP homepage on the website has a link to a Summary Quality Report which describes other aspects of quality of the series, for example, their timeliness, punctuality and relevance. This article sets out the results of recent further work on another dimension of the quality of these series: the accuracy of the estimates, based on an analysis of their sampling error. Additionally the article introduces an approach to defining quality bands for each series, to allow users to compare the relative quality of different IoP components.

A patch‐growing approach to 3D model segmentation using a shortest‐path labeling technique

This paper presents a new segmentation algorithm for 3D models based on a series of patch growing and merging processes using a shortest‐path labeling technique. This algorithm starts with the patch growing process, which groups triangles into homogeneous surfaces by shortest‐path labeling. Each resulting small patch is then merged to the patch that is the nearest to it in terms of geometric similarity and spatial proximity. Finally, 3D models are separated into sets of polygonal patches, each of them consisting of a set of similar triangles in terms of normal differences. One problem with patch growing is its inherent dependency on the selection of starting triangles, which can be fine tuned iteratively. Experimental results demonstrate the good performance of the new method in terms of better segmentation and less sensitivity to noise, as well as in terms of computational efficiency.

Description and visualization of graminaceous plants with an organ-based 3D architectural model, exemplified for spring barley (Hordeum vulgare L.)

Within the framework of functional-structural plant models (FSPMs) this paper presents a structural (architectural) plant model that describes the morphology of spring barley (Hordeum vulgare L.) plants. A set of equations is introduced and implemented in self-written Matlab® computer programs to calculate the surface shape of vegetative and generative plant organs. Organ surfaces are approximated as geometrical primitives, visualized as triangulated surface meshes. The output of the model is a set of triangles that can be associated with geometrical (e.g. area), topological (e.g. main stem) and physiological (e.g. chlorophyll content) attributes, depending on the measurement techniques applied. This information is a prerequisite for functional (process) models to compute, e.g., the radiation field or gas exchange of the respective canopy.

The finite-element method, Part I: R. L. Courant [Historical Corner

This paper focused on finite elements. Also contains a brief biography of Richard L. Courant, the father of finite elements. The finite-element method, in its presently accepted forms, can be credited to no lesser a person than Richard L. Courant. Using piecewise-linear approximants on a set of triangles, which he called elements, he dashed off a few two-dimensional examples - and the finite-element method was born.

Ray tracing deformable scenes using dynamic bounding volume hierarchies

The most significant deficiency of most of today's interactive ray tracers is that they are restricted to static walkthroughs. This restriction is due to the static nature of the acceleration structures used. While the best reported frame rates for static geometric models have been achieved using carefully constructed kd-trees, this article shows that bounding volume hierarchies (BVHs) can be used to efficiently ray trace large static models.More importantly, the BVH can be used to ray trace deformable models (sets of triangles whose positions change over time) with little loss of performance. A variety of efficiency techniques are used to achieve this performance, but three algorithmic changes to the typical BVH algorithm are mainly responsible. First, the BVH is built using a variant of the surface area heuristic conventionally used to build kd-trees. Second, the topology of the BVH is not changed over time so that only the bounding volumes need to be refit from frame-to-frame. Third, and most importantly, packets of rays are traced together through the BVH using a novel integrated packet-frustum traversal scheme. This traversal scheme elegantly combines the advantages of both packet traversal and frustum traversal and allows for rapid hierarchy descent for packets that hit bounding volumes as well as rapid exits for packets that miss. A BVH-based ray tracing system using these techniques is shown to achieve performance for deformable models comparable to that previously available only for static models.

Affine invariance contour descriptor based on iso-area normalisation

In pattern recognition, by observing a contour extracted from a shape, the position of the points of this contour can be normalised by using a set of triangles with the same area whose vertices consist of the centroid of the contour and a pair of successive points on the contour. Contour characterisation can be obtained by lowpass filtering and observing the resulting changes in the areas of the triangles. It is proved that, for a filtered contour, the areas of the triangles are linear under affine transforms. The experimental results show that the approach is resistant to affine transformations and noise.

Feature - On the roots of digital signal processing - Part II

Part I of this article dealt with the mathematical methodology of antiquity which was essentially of a discrete character whereby a circle was deemed to be a multi-sided polygon whose perimeter or area could be more easily deduced by considering the multi-sided polygon to be made up of a finite set of elemental triangles. Then the emergence of numerical methods during the 1700s as a tool for interpolating numerical data was explored. Part II deals with certain spectacular mathematical discoveries made in France just before, during, and after the French Revolution by Laplace, Fourier, Poisson, and Laurent, which form the foundation of modern spectral analysis. Then the contributions of Nyquist and Shannon to the sampling theorem are examined. The article also reviews the work of Babbage from the perspective of a DSP practitioner and examines the historical circumstances that eventually led to the invention of digital computers and their application as general-purpose inexpensive components in DSP systems. The article concludes with a summary of some of the innovations of the sixties that led to the emergence of what we now call digital signal processing.