Empty Pentagons in Point Sets with Collinearities

The convex hull of a planer set of points can be defined as the set of vertices of the smallest convex polygon containing all the points. If S is a planer set of points then convex layers of S can be derived by iteratively computing the convex hull of S and then removing it from S. Computation of the convex layers has widely been studied in the static environment where the set of points is fixed. The theoretical lower bound of computing the convex layers of a fixed set of points is O (n log n). The static convex layers algorithms with the optimal time complexity have already been proposed in the literature. A set of points where the points can be inserted or deleted is called a dynamic set of points. The set of convex layers should be reconstructed from the scratch at each insertion or deletion if a static convex layers algorithm was used to handle a dynamic set of points. Therefore, it takes O (n log n) time to handle an insertion or a deletion even for an optimal static convex layers algorithm. A dynamic convex layers algorithm has been proposed recently that can perform an insertion or a deletion by doing a slight modification to the existing set of convex layers. It takes O (n) time for an insertion or a deletion. It assumes that the set of points does not contain collinear points and that assumption is not valid in practical applications. Furthermore, the notion of tangent used in that approach restricts the extension of the algorithm into higher dimensions. This paper proposes a novel dynamic convex layers algorithm to eliminate the drawbacks in the existing algorithm. Salient feature of this algorithm is that it represents each layer as a set of line segments. The layers are modified upon an insertion or a deletion of a point by adding some new line segments and deleting some existing line segments. The proposed algorithm takes O (n 3 /k 2 ) time for an insertion or a deletion where k is the number of convex layers. A computer implementation is also presented. The proposed algorithm can work with set of points with collinear points and coincident points. The notion used in the algorithm can easily be extended to higher dimensions. Suppose the set of convex layers have already been found for a given set of points. Further, suppose that the layers are close to each other and new points are expected to fall within the region bounded by the outermost layer. This kind of situations widely occurs in practice and then the proposed algorithm takes O (n) time for an inclusion or a deletion.

Point-to-line mappings as Hough transforms

3-Points Convex Hull Matching (3PCHM) for fast and robust point set registration

Advances in sensors and cameras allow current research in convex hull algorithms to focus on defining methods capable of processing a big set of points. Typically, in most of these algorithms, the orientation function needs around five sums and two multiplications. In this paper, we propose SymmetricHull, a novel algorithm that, unlike the related ones, only performs two comparisons per point, discarding points with a low probability of belonging to the convex hull. Our algorithm takes advantage of the symmetric geometry of convex hulls in 2D spaces and relies on the convexity principle to get convex hulls, without needing further calculations. Our experiments show that SymmetricHull achieves good results, in terms of time and number of necessary operations, resulting especially efficient with sets of points between \(10^4\) and \(10^7\). Given that our datasets are organized by quadrants, the features of our algorithm can be summarized as follows: (1) a fast point discard based on known points with a good chance to be part of the convex hull, (2) a lexicographic sort of points with a high probability of belonging to the convex hull, and (3) a simple slope analysis to verify whether a point is within the convex hull or not.

In the paper, we present and prove Castle Theorem of Convex Hull, design and realize the Fast Rampart Searching Algorithm. The algorithm can be treated as the preprocess of Convex Hull calculation of very large planar point set. When calculating the Convex Hull of a very large planar point set, we can use the Fast Rampart Searching Algorithm to get a very small part points as candidate point set, and then we can get the Convex Hull of the whole planar point set from the candidate point set through other algorithms. Keywords-fast rampart searching algorithm; castle theorem; convex hull; computational geometry

Non-rigid registration finds many applications such as photogrammetry, motion tracking, model retrieval, and object recognition. In this paper we propose a novel convex hull aided registration method (CHARM) to match two point sets subject to a non-rigid transformation. First, two convex hulls are extracted from the source and target respectively. Then, all points of the point sets are projected onto the reference plane through each triangular facet of the hulls. From these projections, invariant features are extracted and matched optimally. The matched feature point pairs are mapped back onto the triangular facets of the convex hulls to remove outliers that are outside any relevant triangular facet. The rigid transformation from the source to the target is robustly estimated by the random sample consensus (RANSAC) scheme through minimizing the distance between the matched feature point pairs. Finally, these feature points are utilized as the control points to achieve non-rigid deformation in the form of thin-plate spline of the entire source point set towards the target one. The experimental results based on both synthetic and real data show that the proposed algorithm outperforms several state-of-the-art ones with respect to sampling, rotational angle, and data noise. In addition, the proposed CHARM algorithm also shows higher computational efficiency compared to these methods.

Traditional geometric algorithms are often presented as if input imprecision does not exist, even though it is often unavoidable. We argue that in some cases, it may be desirable for geometric algorithms to treat this imprecision as an explicit component of the input, and to reflect this imprecision in the output. Starting with three problems from computational geometry whose inputs are planar point sets (Voronoi diagrams, convex hulls, and smallest bounding discs), we recast these as problems where each input point’s location is imprecise, but known to lie within a particular region of uncertainty. Where algorithms to solve each of the original problems produce a single geometric object as output, the algorithms that we present typically produce either guaranteed or possible output objects. A guaranteed object represents qualities that can be guaranteed for every point set that is consistent with the uncertain regions, and a possible object represents qualities that exist for at least one such point set. By dealing with input imprecision explicitly, these guaranteed and possible objects can represent a more accurate view of what can be reliably inferred from the input data.

Mixed Integer Linear Programming (MILP) is a very common method of modelling differential and linear bounds. The Convex Hull (CH) modelling, introduced by Sun et al. (Eprint 2013/Asiacrypt 2014), is a popular method in this regard, which can convert the conditions corresponding to a small (4-bit) SBox to MILP constraints efficiently. Our analysis shows, there are SBoxes for which the CH modelling can yield incorrect modelling. The problem arises from the observation that although the CH is generated for a certain set of points, there can be points outside this set which also satisfy all the inequalities of the CH. As apparently no variant of the CH modelling can circumvent this problem, we propose a new modelling for differential and linear bounds. Our modelling makes use of every points of interest individually. Additionally, we also explore the possibility of using redundant constraints, such that the run time for an MILP solver can be reduced while keeping the optimal result unchanged. With our experiments on round-reduced GIFT-128, we show it is possible to reduce the run time a few folds using a suitable choice of redundant constraints. We also present the optimal linear bounds for 11- and 12-rounds of GIFT-128, extending from the best-known result of 10-rounds.

In this paper, we present a robust point set registration method based on the cross ratio invariance of 4 collinear points, which is able to deal with point set registration problem under the most general linear transformation, projective transformation. On the basis of all combinations of 4 collinear points extracted by Hough transform, meaningful correspondences are identified by combining the cross ratio invariance of 4 collinear points and Randomized RANSAC. At the end, the underlying projective transformation matrix was estimate in a least square sense. It has been shown in the simulation experiments that the proposed approach remains robust in high level of degradations, including 45% outliers and 40% overlap ratio. Experiments with Oxford corridor sequence and ZuBuD wide baseline image database proved its usefulness in real application.

This paper is concerned with a recursive algorithm as a preprocessing step to find the convex hull of [Formula: see text] random points uniformly distributed in the plane. For such a set of points, it is shown that eliminating all but [Formula: see text] of points can derive the same convex hull as the input set. Finally it will be shown that the running time of the algorithm is [Formula: see text].

We show that the triangulations of a finite point set form a flip graph that can be embedded isometrically into a hypercube, if and only if the point set has no empty convex pentagon. Point sets of this type include convex subsets of lattices, points on two lines, and several other infinite families. As a consequence, flip distance in such point sets can be computed efficiently.

An interior point of a finite point set is a point of the set that is not on the boundary of the convex hull of the set. For any integer k ≥ 1, let g(k) be the smallest integer such that every set of points in the plane, no three collinear, containing at least g(k) interior points has a subset of points containing exactly k interior points. Similarly, for any integer k ≥ 3, let h(k) be the smallest integer such that every set of points in the plane, no three collinear, containing at least h(k) interior points has a subset of points containing exactly k or k+1 interior points. In this note, we show that g(k) ≥ 3k-1 for k ≥ 3. We also show that h(k) ≥ 2k+1 for 5 ≤ k ≤ 8, and h(k) ≥ 3k-7 for k ≥ 8.KeywordsLower BoundConvex HullConvex SubsetInterior PointSmall IntegerThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Determination of minimum number of sensors and their locations for an automated facility: An algorithmic approach

Unsupervised generation of polygonal approximations based on the convex hull

This paper presents an algorithm to determine whether a point-set composed of n-points is in the polyhedron or not-which is called the convex hull method. With this method, it is not necessary to judge whether each point in a point set within a polyhedron. By constructing three-dimensional convex hull of the remain target points, by means of collision detection of spatial objects to judge the adjacency relations of the polyhedron and the three-dimensional convex hull of the point set, and then to judge whether the vertices of three-dimensional convex hull are within the polyhedron, or the vertices of polyhedron are within the three-dimensional convex hull, thus which points are spots within the polyhedron can be determined. The algorithm can be applied not only to a general polyhedron, but also to concave polyhedron. The efficiency of checking whether the three-dimensional point set is within the polyhedron can be greatly improved by this approach, which is an important support for the realization of real-time spatial analysis algorithm for three-dimensional Geographic Information System (3D GIS). The experimental results show that the algorithm is simple, reliable and highly efficient.