The decomposition of binary images using rectangular blocks of foreground pixels as primitives is considered. Based on this type of decomposition, a fast method for evaluating the Hough transform is introduced. A complexity analysis of the proposed block Hough transform algorithm sets constraints on the complexity of algorithms used for block decomposition, so that the total decomposition and H...

In this chapter, several advanced detection algorithms for Track-Before-Detect (TBD) procedures using the Hough Transform (HT) are proposed and studied. The detection algorithms are based on the scheme described in (Carlson et al., 1994) to use the Hough transform for simultaneous target detection and trajectory estimation. The concept described in (Carlson et al., 1994) accepts that a target m...

In this paper we characterize the convex hull of feasible points for a disjunctive program, a class of problems which subsumes pure and mixed integer programs and many other nonconvex programming problems. Two representations are given for the convex hull of feasible points. each of which provides linear programming equivalents of the disjunctive program. The first one involves a number of new ...

Recently there has been growing interest among psychologists in human performance on the Euclidean Traveling Salesperson problem (E-TSP). A debate has been initiated on what strategy people use in solving visually presented E-TSP instances. The most prominent hypothesis is the convex-hull hypothesis, originally proposed by MacGregor and Ormerod (1996). We argue that, in the literature so far, t...

Given a set Σ of spheres in E, with d ≥ 3 and d odd, having a constant number of m distinct radii ρ1, ρ2, . . . , ρm, we show that the worst-case combinatorial complexity of the convex hull of Σ is Θ( ∑ 1≤i6=j≤m nin ⌊ d 2 ⌋ j ), where ni is the number of spheres in Σ with radius ρi. To prove the lower bound, we construct a set of Θ(n1+n2) spheres in E , with d ≥ 3 odd, where ni spheres have rad...

This is a survey on tropical polytopes from the combinatorial point of view and with a focus on algorithms. Tropical convexity is interesting because it relates a number of combinatorial concepts including ordinary convexity, monomial ideals, subdivisions of products of simplices, matroid theory, finite metric spaces, and the tropical Grassmannians. The relationship between these topics is expl...

The “convex hull problem” is a catch-all phrase for computing various descriptions of a polytope that is either specified as the convex hull of a finite point set in R or as the intersection of a finite number of halfspaces. We first define the various problems and discuss their mutual relationships (Section 26.1). We discuss the very special case of the irredundancy problem in Section 26.2. We...

Problem of defining convexity of a digital region is considered. Definition of DL− (digital line) convexity is proposed, and it is shown to be stronger than the other two definitions, T−(triangle) convexity and L−(line) convexity. In attempt to connect the convexity of digital sets, to the fact that digital set can be a fuzzy set, the notion of convexity of the membership function is introduced...

Consider the classification task of assigning a test object to one of two or more possible groups, or classes. An intuitive way to proceed is to assign the object to that class, to which the distance is minimal. As a distance measure to a class, we propose here to use the distance to the convex hull of that class. Hence the name Nearest Convex Hull (NCH) classification for the method. Convex-hu...