Unsupervised generation of polygonal approximations based on the convex hull

An algorithm for detecting dominant points (corners and trees) in images of 3D objects is presented. The technique chooses ideal points from the basic definitions of both tree and corner points. The algorithm has two subiterations, in the first one the initial candidate dominant points are determined, and in the second subiteration the candidate dominant points are tested. Experiments were performed to show that the proposed algorithm is reliable for 3D object recognition by meaning of representing independent view point images of the same object by a set of dominant points.

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.

A method for polygonal approximation of digital planar curves is proposed. This method is relied on the supression of dominant points. Initially, the method uses the set of all breakpoints as dominant points. The method eliminates dominant points until a final condition is satisfied. The residuary dominant points are the polygonal approximation. This aproximation is compared with other classical algorithms. The experimental results show that this method works well for digital planar curves with features of several sizes.KeywordsComputer visiondigital planar curvespolygonal approximationdominant points

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

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

Abstract. Recently, building outline extraction from point cloud has gained momentum in particular in the context of 3D building modelling based on a data-driven approach, which has also been our motivation. For an accurate building outline extraction from a point cloud, various factors affecting the quality should be considered. In this research, we analysed the influence of point cloud density on the quality of the extracted building outlines. The input data was a classified photogrammetric point cloud, obtained from the dense image matching of images acquired by an optical sensor mounted on the unmanned aerial vehicle (UAV). For outline extraction, we selected two procedures, namely the direct approach and the raster approach. In the direct approach, building outlines are extracted directly from the points that have been classified as buildings. First, a convex hull with the alpha algorithm is estimated, which is further generalised with the Douglas-Peucker algorithm. This is followed by the shape regularisation to ensure perpendicular angles of the outline. In the raster approach, we first rasterised the building points and then extracted the building outlines using the Hough transform. In both approaches, the result is a roof outline in a 2D plane representing the maximum extent of the building above the surface. The building outlines were extracted from point clouds with five different densities. For both approaches, the quality assessment has shown that point cloud density has an impact on the building outline extraction, especially on the completeness of the outlines.

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.

Detecting the Dominant Points by the Curvature-Based Polygonal Approximation

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.

Given a set Σ of spheres in Ed, with d≥3 and d odd, having a fixed number of m distinct radii ρ1,ρ2,...,ρm, we show that the worst-case combinatorial complexity of the convex hull CHd(Σ) of Σ is Θ(Σ{1≤i≠j≤m}ninj⌊ d/2 ⌋), where ni is the number of spheres in Σ with radius ρi. Our bound refines the worst-case upper and lower bounds on the worst-case combinatorial complexity of CHd(Σ) for all odd d≥3.To prove the lower bound, we construct a set of Θ(n1+n2) spheres in Ed, with d≥3 odd, where ni spheres have radius ρi, i=1,2, and ρ_2≠ρ1, such that their convex hull has combinatorial complexity Ω(n1n2⌊ d/2 ⌋+n2n1⌊ d/2 ⌋). Our construction is then generalized to the case where the spheres have m≥3 distinct radii.For the upper bound, we reduce the sphere convex hull problem to the problem of computing the worst-case combinatorial complexity of the convex hull of a set of m d-dimensional convex polytopes lying on m parallel hyperplanes in Ed+1, where d≥3 odd, a problem which is of independent interest. More precisely, we show that the worst-case combinatorial complexity of the convex hull of a set P{1,P2,...,Pm} of m d-dimensional convex polytopes lying on m parallel hyperplanes of Ed+1 is O(Σ1≤i≠j≤mninj⌊ d/2 ⌋), where ni is the number of vertices of Pi. This bound is an improvement over the worst-case bound on the combinatorial complexity of the convex hull of a point set where we impose no restriction on the points' configuration; using the lower bound construction for the sphere convex hull problem, it is also shown to be tight for all odd d≥3.Finally: (1) we briefly discuss how to compute convex hulls of spheres with a fixed number of distinct radii, or convex hulls of a fixed number of polytopes lying on parallel hyperplanes; (2) we show how our tight bounds for the parallel polytope convex hull problem, yield tight bounds on the combinatorial complexity of the Minkowski sum of two convex polytopes in Ed; and (3) we state some open problems and directions for future work.

This paper presents a computer vision framework for detecting dominant boundary-points on an object’s contour through polygonal approximation of the shape without loss of its significant visual-interpretation. The proposed framework attempts to approximate a polygonal representation of the contour with each polygonal-side having a meaningful thickness to handle noisy curvatures with irregular bumps. The vertices of the polygon are extracted through a novel recursive strategy. The merit of such a scheme depends on how closely it can represent the shape with minimal number of vertices as dominant points without losing its inherent visual characteristics. As per our observation, the proposed framework seems to perform reasonably well in approximating the shape of an object with a small number of dominant points on the contour.

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].

Let P_1,...,P_{d+1} be d-dimensional point sets such that the convex hull of each P_i contains the origin. We call the sets P_i color classes, and we think of the points in P_i as having color i. A colorful choice is a set with at most one point of each color. The colorful Caratheodory theorem guarantees the existence of a colorful choice whose convex hull contains the origin. So far, the computational complexity of finding such a colorful choice is unknown. We approach this problem from two directions. First, we consider approximation algorithms: an m-colorful choice is a set that contains at most m points from each color class. We show that for any fixed epsilon > 0, an (epsilon d)-colorful choice containing the origin in its convex hull can be found in polynomial time. This notion of approximation has not been studied before, and it is motivated through the applications of the colorful Caratheodory theorem in the literature. In the second part, we present a natural generalization of the colorful Caratheodory problem: in the Nearest Colorful Polytope problem (NCP), we are given d-dimensional point sets P_1,...,P_n that do not necessarily contain the origin in their convex hulls. The goal is to find a colorful choice whose convex hull minimizes the distance to the origin. We show that computing local optima for the NCP problem is PLS-complete, while computing a global optimum is NP-hard.

We consider a finite set of lattice points and their convex hull. The author previously gave a geometric proof that the sumsets of these lattice points take over the central regions of dilated convex hulls, thus revealing an interesting connection between additive number theory and geometry. In this paper, we will see an algebraic proof of this fact when the convex hull of points is a simplex, exploring the connection between additive number theory and geometry further.