Sharif Digital Repository

We introduce the concept of local spanners for planar point sets with respect to a family of regions, and prove the existence of local spanners of small size for some families. For a geometric graph G on a point set P and a region R belonging to a family R, we define G∩ R to be the part of the graph G that is inside R (or is induced by R). A local t-spanner w.r.t R is a geometric graph G on P such that for any region R∈ R, the graph G∩ R is a t-spanner for K(P) ∩ R, where K(P) is the complete geometric graph on P. For any set P of n points and any constant ε> 0 , we prove that P admits local (1 + ε) -spanners of sizes O(nlog 6n) and O(nlog n) w.r.t axis-parallel squares and vertical slabs,... 

We introduce the concept of local spanners for planar point sets with respect to a family of regions, and prove the existence of local spanners of small size for some families. For a geometric graph G on a point set P and a region R belonging to a family R, we define G∩ R to be the part of the graph G that is inside R (or is induced by R). A local t-spanner w.r.t R is a geometric graph G on P such that for any region R∈ R, the graph G∩ R is a t-spanner for K(P) ∩ R, where K(P) is the complete geometric graph on P. For any set P of n points and any constant ε> 0 , we prove that P admits local (1 + ε) -spanners of sizes O(nlog 6n) and O(nlog n) w.r.t axis-parallel squares and vertical slabs,... 

Give a triangulation of a set of points on the plane, dilation of any two points is defined as the ratio between the length of the shortest path of these points and their Euclidean distance. Minimum dilation triangulation is a triangulation in which the maximum dilation between any pair of the points is minimized. We give upper bounds on the dilation of the minimum dilation triangulation for two kinds of point sets: An upper bound of nsin⁡(π/n)/2 for a centrally symmetric convex point set containing n points, and an upper bound of 1.19098 for a set of points on the boundary of a semicircle. © 2018 Elsevier B.V  

Given a point set S and a polygonal curve P in Rd, we study the problem of finding a polygonal curve through S, which has minimum Fréchet distance to P. We present an efficient algorithm to solve the decision version of this problem in O(nk2) time, where n and k represent the sizes of P and S, respectively. A curve minimizing the Fréchet distance can be computed in O(nk2 log(nk)) time. As a by-product, we improve the map matching algorithm of Alt et al. by an O(log k) factor for the case when the map is a complete grap  

Dilation of a set of points on the plane is the lowest possible dilation of a plane spanner on the point set. We show that dilation of vertices of any regular polygon is less than 1.4482. We introduce a method for constructing a triangulation of a regular polygon and prove this bound on its dilation. The upper bound on dilation is shown using mathematical proofs and experimental results. The new upper bound improves the previously known bound of 1.48454. © 2019 Elsevier B.V  

Point Set Registration (PSR) of anatomic parts is used in different fields such as Patient Specific Implant (PSI) design in Computer Assisted Surgery (CAS) procedure. We designed a modified rigid PSR method based on student's-t mixture model. The proposed method is compared with Coherent Point Drift (CPD) registration method. Higher convergence speed and the lower error value are the advantages of the suggested algorithm in compare with CPD. In our method, the number of iterations decreases by about 69%, and the final error improvement was about 7% in comparison with CPD. The robustness of the proposed algorithm makes it beneficial to be used in the procedure of designing PSI in both...