Point-to-line mappings as Hough transforms

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.

Abstract. An empty pentagon in a point set P in the plane is a set of five points in P in strictly convex position with no other point of P in their convex hull. We prove that every finite set of at least 328`2 points in the plane contains an empty pentagon or ` collinear points. This is optimal up to a constant factor since the ( ` − 1) × ( ` − 1) grid contains no empty pentagon and no ` collinear points. The previous best known bound was doubly exponential. 1.

But he opened out the hinges, Pushed and pulled the joints and hinges, Till it looked all squares and oblongs Like a complicated figure In the Second Book of Euclid. C. L. Dodgson After discussing some further properties of triangles and quadrangles (or quadrilaterals), we shall approach the domain of projective geometry (and even trespass a bit). A systematic development of that fascinating subject must be left for another book, but four of its most basic theorems are justifiably mentioned here because they can be proved by the methods of Euclid; in fact, three of the four are so old that no other methods were available at the time of their discovery. All these theorems deal either with collinearity (certain sets of points lying on a line) or concurrence (certain sets of lines passing through a point). The spirit of projective geometry begins to emerge as soon as we notice that, for many purposes, parallel lines behave like concurrent lines. Quadrangles; Varignon's theorem A polygon may be defined as consisting of a number of points (called vertices ) and an equal number of line segments (called sides ), namely a cyclically ordered set of points in a plane, with no three successive points collinear, together with the line segments joining consecutive pairs of the points. In other words, a polygon is a closed broken Sine lying in a plane.

Modal correlation approaches for general second-order systems: Matching mode pairs and an application to Campbell diagrams

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

We prove that the integral points are potentially Zariski dense in the complement of a reduced effective singular anticanonical divisor in a smooth del Pezzo surface, with the exception of ${{\mathbb {P}}}^{2}$ minus three concurrent lines (for which potential density does not hold). This answers positively a question raised by Hassett and Tschinkel and, combined with previous results, completes the proof of the potential density of integral points for complements of anticanonical divisors in smooth del Pezzo surfaces. We then classify the complements that are simply connected and for these we prove that the set of integral points is potentially not thin, as predicted by a conjecture of Corvaja and Zannier.

The paper presents a method for surface reconstruction from large unorganized and noisy point sets without any normal or orientation information. Firstly, the outliers will be selected and deleted, and acquire new point sets being less noisy with improved method of fuzzy c-means clustering. Secondly, we compute the normal of the new point sets through PCA analysis. Then we interpolate single-level and multi-level the new points acquired by clustering method by compactly supported radial basis function (CSRBF). The experiments show that the outliers will be selected well and the noisy point sets will be smoothed. The interpolation points are renewed by clustering method, and implicit reconstruction by CSRBF may provide a solution for surface reconstruction from noisy and incomplete data. In our algorithm, we represent the point sets by an octree-based subdivision of the bounding box to reduce computation complexity

We present a method for performing efficient barycentric interpolation for large grain boundary octonion point sets which reside on the surface of a hypersphere. This method includes removal of degenerate dimensions via singular value decomposition (SVD) transformations and linear projections, determination of intersecting facets via nearest neighbor (NN) searches, and interpolation. This method is useful for hyperspherical point sets for applications such as grain boundaries structure-property models, robotics, and specialized neural networks. We provide a case study of the method applied to the 7-sphere. We provide 1-sphere and 2-sphere visualizations to illustrate important aspects of these dimension reduction and interpolation methods. A MATLAB implementation is available at github.com/sgbaird-5dof/interp.•Barycentric interpolation is combined with hypersphere facet intersections, dimensionality reduction, and linear projections to reduce computational complexity without loss of information•A max nearest neighbor threshold is used in conjunction with facet intersection determination to reduce computational runtime.

Abstract. Erdös asked whether every large enough set of points in general position in the plane contains six points that form a convex 6-gon without any points from the set in its interior. In this note we show how a set of 29 points was found that contains no empty convex 6-gon. To this end a fast incremental algorithm for finding such 6-gons was designed and implemented and a heuristic search approach was used to find promising sets. The experiments also led to two observations that might be useful in proving that large sets always contain an empty convex 6-gon.

We study biplane graphs drawn on a finite 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 sufficiently 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.

Given an input point cloud P in ℜ3, this paper proposes a novel algorithm to identify surface neighbors of each point p∈P respecting the underlying surface S and then to construct a piecewise linear surface for P. The algorithm utilizes the simple k-nearest neighborhood in constructing local surfaces. It makes use of two concepts: a local convexity criterion to extract a set of surface neighbors for each point, and a global projection test to determine an order for the reconstruction. Our algorithm not only produces a topologically correct surface for well-sampled point sets, but also adapts well to handle under-sampled point sets. Furthermore, the computational cost of the algorithm increases almost linearly in the size of the point cloud. It, thus, scales well to deal with large input point sets.

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.

Typical differentiability within an exceptionally small set

A molecular dynamics investigation of kinetic electron emission from silver surfaces under varying angle of projectile impact

In densely populated cities like Hong Kong, GPS alone is not sufficient in providing rich location based information services. In this work, we provide a mobile location search service by allowing users shooting a short video clip about the surrounding buildings and the Cloud will return a tagged image with a summary of the location and services available. The key technical challenges are the robustness of the SIFT based object matching in video sequences, and the computational complexity associated with the large scale of the repository. We solved this by spatio-temporal pruning of SIFT points in the video repository, and PCA projection and indexing of SIFT points in Cloud. Simulation results demonstrated the robustness of the PCAed and indexed SIFT points in building identification and overall effectiveness of the proposed solution on a small scale trial. Tagging and integration with large scale video repositories like YouTube, Tudou are underway with more interesting applications.