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Several CAD applications require a surface model of the modeled object consisting of a mesh of triangular facets. In this paper, a new algorithm for triangulation of trimmed surfaces is presented. The algorithm generates a triangulation that approximates the initial surface within a predefined tolerance. The approximation is conformal, without cracks in edges: a closed polyhedron is obtained in the case of a closed initial surface. The proposed algorithm improves the algorithm presented in (Brunet and Vigo, 1995) because it is based on more precise bounds which take into account the directional behavior of local surface curvature, and the resulting triangulation has a lower number of triangles.

We study biplane graphs drawn on a nite point set S in the plane in general position. This is the family of geometric graphs whose vertex set is S and which can be decomposed into two plane graphs. We show that every su ciently large point set admits a 5-connected biplane graph and that there are arbitrarily large point sets that do not admit any 6- connected biplane graph. Furthermore, we show that every plane graph (other than a wheel or a fan) can be augmented into a 4-connected biplane graph. However, there are arbitrarily large plane graphs that cannot be augmented to a 5-connected biplane graph by adding pairwise noncrossing edges.

Describes a method to code a decimated model of an isosurface on an octree representation while maintaining volume data if it is needed. The proposed technique is based on grouping the marching cubes (MC) patterns into five configurations according the topology and the number of planes of the surface that are contained in a cell. Moreover, the discrete number of planes on which the surface lays is fixed. Starting from a complete volume octree, with the isosurface codified at terminal nodes according to the new configuration, a bottom-up strategy is taken for merging cells. Such a strategy allows one to implicitly represent co-planar faces in the upper octree levels without introducing any error. At the end of this merging process, when it is required, a reconstruction strategy is applied to generate the surface contained in the octree intersected leaves. Some examples with medical data demonstrate that a reduction of up to 50% in the number of polygons can be achieved.

Given a tetrahedral mesh immersed in a voxel model, we present a method to refine the mesh to reduce the discrepancy between interpolated values based on either scheme at arbitrary locations. An advantage of the method presented is that it requires few subdivisions and all decisions are made locally at each tetrahedron. We discuss the algorithm’s performance and applications.

The graph-based geometric constraint solving technique works in two steps. First the geometric problem is translated into a graph whose vertices represent the set of geometric elements and whose edges are the constraints. Then the constraint problem is solved by decomposing the graph into a collection of subgraphs each representing a standard problem which is solved by a dedicated equational solver. In this work we report on an algorithm to decompose biconnected tree-decomposable graphs representing either underor wellconstrained 2D geometric constraint problems. The algorithm recursively first computes a set of fundamental circuits in the graph then splits the graph into a set of subgraphs each sharing exactly three vertices with the fundamental circuit. Practical experiments show that the reported algorithm clearly outperforms the treedecomposition approach based on identifying subgraphs by applying specific decomposition rules.

Consider a set of n pairwise disjoint axis-parallel rectangles in the plane. We call this set the source rectangles S. The aim is to move S to a set of (pairwise disjoint) target rectangles T . A move consists in a horizontal or vertical translation of one rectangle, such that it does not collide with any other rectangle during the move. We study how many moves are needed to transform S into T . We obtain bounds on the number of needed moves for labeled and for unlabeled rectangles, and for rectangles of different and of equal dimensions.

We study the problem of computing correlation between two piecewise-linear bivariate functions defined over a common domain, where the surfaces they define in three dimensions---polyhedral terrains---can be transformed vertically by a linear transformation of the third coordinate (scaling and translation). We present a randomized algorithm that minimizes the maximum vertical distance between the graphs of the two functions, over all linear transformations of one of the terrains, in $O(n^{4/3}\operatorname{polylog}n)$ expected time, where $n$ is the total number of vertices in the graphs of the two functions. We also present approximation algorithms for minimizing the mean distance between the graphs of univariate and bivariate functions. For univariate functions we present a $(1+\varepsilon)$-approximation algorithm that runs in $O(n (1 + \log^2 (1/\varepsilon)))$ expected time for any fixed $\varepsilon >0$. The $(1+\varepsilon)$-approximation algorithm for bivariate functions runs in $O(n/\varepsilon)$ time, for any fixed $\varepsilon >0$, provided the two functions are defined over the same triangulation of their domain.

S'ha fet aquest projecte degut a la constant necessitat d'expansió d'algunes empreses de serveis, com les cadenes hoteleres, de restauració o tèxtils. Aquestes empreses solen comprar o construir noves seus amb regularitat. Però per fer-ho necessiten fer molts procediments manuals que costen molt temps i diners. Aquests procediments consten d'estudis de viabilitat i d'estudis de la competència. L'objectiu d'aquest projecte és desenvolupar una eina on donat un mapa d'una ciutat i les seus existents amb un valor d'influència associat, indicant el seu poder d'atracció o la seva importància, mostri la viabilitat de construir una nova seu en cada punt del mapa. Les seus existents que hi haurà en el mapa seran de dos tipus: unes que aporten benefici a la nova seu i que per tant són seus amb influència positiva, com podria ser una catedral, una zona de compres, una zona de negocis. I també hi pot haver seus que treuen benefici, és a dir, amb influència negativa que bàsicament serien seus semblants a la que es vol construir, en el cas de voler construir un hotel, la seva competència seran hotels de característiques semblants

We consider matchings between a set R of red points and a set B of blue points with diametral disks. In other words, for each pair of matched points p ¿ R and q ¿ B, we consider the diametral disk defined by p and q. We prove that for any R and B such that |R| = |B|, there exists a perfect matching such that the diametral disks of the matched point pairs have a common intersection. More precisely, we show that a maximum weight perfect matching has this property

Trabajo presentado en la IEEE/CVF Conference on Computer Vision and Pattern Recognition, celebrada en Salt Lake City (UT, USA), del 18 al 23 de junio de 2018, We present a novel approach for synthesizing photorealistic images of people in arbitrary poses using generative adversarial learning. Given an input image of a person and a desired pose represented by a 2D skeleton, our model renders the image of the same person under the new pose, synthesizing novel views of the parts visible in the input image and hallucinating those that are not seen. This problem has recently been addressed in a supervised manner [16, 35], i.e., during training the ground truth images under the new poses are given to the network. We go beyond these approaches by proposing a fully unsupervised strategy. We tackle this challenging scenario by splitting the problem into two principal subtasks. First, we consider a pose conditioned bidirectional generator that maps back the initially rendered image to the original pose, hence being directly comparable to the input image without the need to resort to any training image. Second, we devise a novel loss function that incorporates content and style terms, and aims at producing images of high perceptual quality. Extensive experiments conducted on the DeepFashion dataset demonstrate that the images rendered by our model are very close in appearance to those obtained by fully supervised approaches., This work is supported in part by a Google Faculty Research Award, by the Spanish Ministry of Science and Innovation under projects HuMoUR TIN2017- 90086-R, ColRobTransp DPI2016-78957 and Mar´ıa de Maeztu Seal of Excellence MDM-2016-0656; and by the EU project AEROARMS ICT-2014-1-644271. We also thank Nvidia for hardware donation under the GPU Grant Program.

The determination of the correct integer number of carrier cycles (integer ambiguity) is the key to high accuracy positioning with carrier phase measurements from Global Navigation Satellite Systems (GNSS). There are a number of current methods for resolving ambiguities including the Least-squares AMBiguity Decorrelation Adjustment (LAMBDA) method, which is a combination of least-squares and a transformation to reduce the search space. The current techniques to determine the level of confidence (integrity) of the resolved ambiguities (i.e. ambiguity validation), usually involve the construction of test statistics, characterisation of their distribution and definition of thresholds. Example tests applied include ratio, F-distribution, t-distribution and Chi-square distribution. However, the assumptions that underpin these tests have weaknesses. These include the application of a fixed threshold for all scenarios, and therefore, not always able to provide an acceptable integrity level in the computed ambiguities. A relatively recent technique referred to as Integer Aperture (IA) based on the ratio test with a large number of simulated samples of float ambiguities requires significant computational resources. This precludes the application of IA in real time. This paper proposes and demonstrates the power of an integrity monitoring technique that is applied at the ambiguity resolution and positioning stages. The technique has the important benefit of facilitating early detection of any potential threat to the position solution, originating in the ambiguity space, while at the same time giving overall protection in the position domain based on the required navigation performance. The proposed method uses the conventional test statistic for ratio testing together with a doubly non-central F distribution to compute the level of confidence (integrity) of the ambiguities. Specifically, this is determined as a function of geometry and the ambiguity residuals from least squares based ambiguity resolution algorithms including LAMBDA. A numerical method is implemented to compute the level of confidence in real time. The results for Precise Point Positioning (PPP) with simulated and real data demonstrate the power and efficiency of the proposed method in monitoring both the integrity of the ambiguity computation and position solution processes. Furthermore, due to the fact that the method only requires information from least squares based ambiguity resolution algorithms, it is easily transferable to conventional Real Time Kinematic (RTK) positioning.

Terrain modeling is a fast growing field with many applications such as computer graphics, resource management, Earth and environmental sciences, civil and military engineering, surveying and photogrammetry and games programming. One of the most widely used terrain model is the Digital Elevation Model (DEM). A DEM is a simple regularly spaced grid of elevation points that represent the continuous variation of relief over space. DEMs require simple storage and are compatible with satellite data. However, they do not easily account for overhangs. In this work we report on the Grounded Heightmap Tree, a new data structure for terrain representation built as a generalization of the DEM. The new data structure allows to naturally represent terrain overhangs. We illustrate the performance of the Grounded Heightmap Tree when applied to represent terrains that undergo big changes.

In this report an algorithm for the satisfaction of geometric constraints of distance in 2D models is proposed. It is based on the step-by-step relaxation of each constraint and, under well-bounded conditions, it converges to the desired solution. En aquest report es presenta un algorisme de satisfacció de restriccions geomètriques de distancia basat en la relaxació restricció a restricció en models 2D on només hi hagi restriccions de distància. En primer lloc es formalitzen una serie de definicions bàsiques per després presentar l'algorisme de satisfacció i la seva convergència a partir de l'evolució de la norma euclidea associada a les restriccions d'un model.

Este trabajo consiste en la creación de un plug-in para QGIS (un editor de geometrías). Consiste en proveer a usuarios de herramientas de simplificación de geometrías con restricciones especiales. En el trabajo se explica cómo funcionan y en que consisten las simplificaciones y las restricciones.

We propose a 3D mesh curving method that converts a straight-sided mesh to an optimal-quality curved high-order mesh that interpolates a CAD boundary representation. The main application of this method is the generation of discrete approximations of curved domains that are valid for simulation analysis with unstructured high-order methods. We devise the method as follows. First, the boundary of a straight-sided high-order mesh is curved to match the curves and surfaces of a CAD model. Second, the method minimizes the volume mesh distortion with respect to the coordinates of the inner nodes and the parametric coordinates of the curve and surface nodes. The proposed minimization features untangling capabilities and therefore, it repairs the invalid elements that may arise from the initial curving step. Compared with other mesh curving methods, the only goal of the proposed residual system is to minimize the volume mesh distortion. Furthermore, it is less constrained since the boundary nodes are free to slide on the CAD curves and surfaces. Hence, the proposed method is well suited to generate curved high-order meshes of optimal quality from CAD models that contain thin parts or high-curvature entities. To illustrate these capabilities, we generate several curved high-order meshes from CAD models with the implementation detailed in this work. Specifically, we detail a node-by-node non-linear iterative solver that minimizes the proposed objective function in a block Gauss–Seidel manner. This research was partially supported by CONACYT-SENER (‘‘Fondo Sectorial CONACYT SENER HIDROCARBUROS’’, grant contract 163723) and by the Spanish Ministerio de Economía y Competitividad under grant CTM2014-55014-C3-3-R.The work of the second author was partially supported by the Boeing Co. and US Air Force Office of Scientific Research and European Commis- sion through the Boeing–MIT Alliance and Computational Math Program and Marie Skłodowska-Curie Actions (HiPerMeGaFlows project),respectively.

The complexity of virtual environments has grown spectacularly over the recent years, mainly thanks to the use of the currently cheap high performance graphics cards. As the graphics cards improve the performance and the geometry complexity grows, many of the objects present in the scene only project to a few pixels on the screen. This represents a waste in computing effort for the transforming and clipping of maybe a lot of polygons that could be substituted by a simple point or a small set of points. Recently, efficient rendering algorithms for point models have been proposed. However, little attention has been focused on building a point-based modeler, using the advantages that such a representation can provide. In this paper we present a modeler that can generate 3D geometry from an image, completely built on points. It takes as input an image and creates a point-based representation from it. Then, a set of operators allow to modify the geometry in order to produce 3D geometry from the image. With our system it is possible to generate in short time complex geometries that would be difficult to model with a polygon-based modeler.

Geometric constraint solving is a growing field which plays a paramount role in industrial applications and that is deeply rooted in automated deduction in geometry. In this work we report on an algorithm to solve geometric constraint-based problems by decomposing biconnected graphs. The algorithm is based on recursively splitting the graph through sets with three vertices located on fundamental circuits of the graph. Preliminary practical experiments suggest that the algorithm runtime is at worst quadratic with the total number of vertices in the graph.