When the boundary of a familiar object is shown by a series of isolated dots, humans can often recognize the object with ease. This ability can be sustained with addition of distracting dots around the object. However, such capability has not been reproduced algorithmically on computers. We introduce a new algorithm that groups a set of dots into multiple non-disjoint subsets. It connects the dots into a spanning tree using the proximity cue. It then applies the straight polygon transformation to an initial polygon derived from the spanning tree. The straight polygon divides the space into polygons recursively and each polygon can be viewed as grouping of a subset of the dots. The number of polygons generated is O( @math ). We also introduce simple shape selection and recognition algorithms that can be applied to the grouping result. We used both natural and synthetic images to show effectiveness of these algorithms.
When a dot pattern represents a single cluster, the problem is to derive a polygonal representation of the cluster. The problem is often called . A trivial but important representation is a convex hull. proposed a generalization of convex hull called @math -. Given a real number @math , the @math - is the intersection of all closed generalized discs with radius @math that contain all the points in the pattern @cite_20 . If @math , a generalized disk is the complement of a disc of radius @math and if @math , it is a half-plane. The convex hull is a case with @math . Furthermore, @math - is a polygonal representation of the dot pattern derived from the corresponding @math - and can be computed in @math . proposed a representation called @math -, which is simpler and computationally more efficient ( @math ) than the @math - @cite_4 .
@cite_4 @cite_20
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In this paper we present several results on the expected complexity of a convex hull of @math points chosen uniformly and independently from a convex shape. (i) We show that the expected number of vertices of the convex hull of @math points, chosen uniformly and independently from a disk is @math , and @math for the case a convex polygon with @math sides. Those results are well known (see rs-udkhv-63,r-slcdn-70,ps-cgi-85 ), but we believe that the elementary proof given here are simpler and more intuitive. (ii) Let @math be a set of directions in the plane, we define a generalized notion of convexity induced by @math , which extends both rectilinear convexity and standard convexity. We prove that the expected complexity of the @math -convex hull of a set of @math points, chosen uniformly and independently from a disk, is @math , where @math is the largest angle between two consecutive vectors in @math . This result extends the known bounds for the cases of rectilinear and standard convexity. (iii) Let @math be an axis parallel hypercube in @math . We prove that the expected number of points on the boundary of the quadrant hull of a set @math of @math points, chosen uniformly and independently from @math is @math . Quadrant hull of a set of points is an extension of rectilinear convexity to higher dimensions. In particular, this number is larger than the number of maxima in @math , and is also larger than the number of points of @math that are vertices of the convex hull of @math . Those bounds are known bkst-anmsv-78 , but we believe the new proof is simpler. We discuss a new conceptual framework for the convexification of discrete optimization problems, and a general technique for obtaining approximations to the convex hull of the feasible set. The concepts come from disjunctive programming and the key tool is a description of the convex hull of a union of polyhedra in terms of a higher dimensional polyhedron. Although this description was known for several years, only recently was it shown by Jeroslow and Lowe to yield improved representations of discrete optimization problems. We express the feasible set of a discrete optimization problem as the intersection (conjunction) of unions of polyhedra, and define an operation that takes one such expression into another, equivalent one, with fewer conjuncts. We then introduce a class of relaxations based on replacing each conjunct (union of polyhedra) by its convex hull. The strength of the relaxations increases as the number of conjuncts decreases, and the class of relaxations forms a hierarchy that spans the spec... We study the convex hull of @math , the set of @math orthogonal matrices with unit determinant, from the point of view of semidefinite programming. We show that the convex hull of @math is doubly spectrahedral, i.e., both it and its polar have a description as the intersection of a cone of positive semidefinite matrices with an affine subspace. Our spectrahedral representations are explicit and are of minimum size, in the sense that there are no smaller spectrahedral representations of these convex bodies. Finite convex geometries are combinatorial structures. It follows from a recent result of M. Richter and L.G. Rogers that there is an infinite set @math of planar convex polygons such that @math with respect to geometric convex hulls is a locally convex geometry and every finite convex geometry can be represented by restricting the structure of @math to a finite subset in a natural way. An @math is a differentiable convex simple closed curve @math in the plane having an inscribed circle of radius @math and a circumscribed circle of radius @math such that the ratio @math is at least @math . Motivated by Richter and Rogers' result, we construct a set @math such that (1) @math contains all points of the plane as degenerate singleton circles and all of its non-singleton members are differentiable convex simple closed planar curves; (2) @math with respect to the geometric convex hull operator is a locally convex geometry; (3) as opposed to @math , @math is closed with respect to non-degenerate affine transformations; and (4) for every (small) positive @math and for every finite convex geometry, there are continuum many pairwise affine-disjoint finite subsets @math of @math such that each @math consists of almost-circles of accuracy @math and the convex geometry in question is represented by restricting the convex hull operator to @math . The affine-disjointness of @math and @math means that, in addition to @math , even @math is disjoint from @math for every non-degenerate affine transformation @math .