Grounding the Lexical Semantics of Verbs in Visual Perception using Force Dynamics and Event Logic

This paper presents an implemented system for recognizing the occurrence of events described by simple spatial-motion verbs in short image sequences. The semantics of these verbs is specified with event-logic expressions that describe changes in the state of force-dynamic relations between the participants of the event. An efficient finite representation is introduced for the infinite sets of intervals that occur when describing liquid and semi-liquid events. Additionally, an efficient procedure using this representation is presented for inferring occurrences of compound events, described with event-logic expressions, from occurrences of primitive events. Using force dynamics and event logic to specify the lexical semantics of events allows the system to be more robust than prior systems based on motion profile

Dual Garside structure of braids and free cumulants of products

We count the n-strand braids whose normal decomposition has length at most two in the dual braid monoid B_n+* by reducing the question to a computation of free cumulants for a product of independent variables, for which we establish a general formula

matching, interpolation, and approximation ; a survey

In this survey we consider geometric techniques which have been used to measure the similarity or distance between shapes, as well as to approximate shapes, or interpolate between shapes. Shape is a modality which plays a key role in many disciplines, ranging from computer vision to molecular biology. We focus on algorithmic techniques based on computational geometry that have been developed for shape matching, simplification, and morphing

Partitioning a Polygon Into Small Pieces

We study the problem of partitioning a given simple polygon $P$ into a minimum number of polygonal pieces, each of which has bounded size. We give algorithms for seven notions of `bounded size,' namely that each piece has bounded area, perimeter, straight-line diameter, geodesic diameter, or that each piece must be contained in a unit disk, an axis-aligned unit square or an arbitrarily rotated unit square. A more general version of the area problem has already been studied. Here we are, in addition to $P$, given positive real values $a_1,\ldots,a_k$ such that the sum $\sum_{i=1}^k a_i$ equals the area of $P$. The goal is to partition $P$ into exactly $k$ pieces $Q_1,\ldots,Q_k$ such that the area of $Q_i$ is $a_i$. Such a partition always exists, and an algorithm with running time $O(nk)$ has previously been described, where $n$ is the number of corners of $P$. We give an algorithm with optimal running time $O(n+k)$. For polygons with holes, we get running time $O(n\log n+k)$. For the other problems, it seems out of reach to compute optimal partitions for simple polygons; for most of them, even in extremely restricted cases such as when $P$ is a square. We therefore develop $O(1)$-approximation algorithms for these problems, which means that the number of pieces in the produced partition is at most a constant factor larger than the cardinality of a minimum partition. Existing algorithms do not allow Steiner points, which means that all corners of the produced pieces must also be corners of $P$. This has the disappointing consequence that a partition does often not exist, whereas our algorithms always produce useful partitions. Furthermore, an optimal partition without Steiner points may require $\Omega(n)$ pieces for polygons where a partition consisting of just $2$ pieces exists when Steiner points are allowed.Comment: 32 pages, 24 figure

Categorization generated by prototypes -- an axiomatic approach

We present a model of categorization based on prototypes. A prototype is an image or template of an idealized member of the category. Once a set of prototypes is defined, entities are sorted into categories on the basis of the prototypes they are closest to. We provide a characterization of those categorizations that are generated by prototypes.categorization, prototype, prototype-orineted decision making

Generating approximate region boundaries from heterogeneous spatial information: an evolutionary approach

Spatial information takes different forms in different applications, ranging from accurate coordinates in geographic information systems to the qualitative abstractions that are used in artificial intelligence and spatial cognition. As a result, existing spatial information processing techniques tend to be tailored towards one type of spatial information, and cannot readily be extended to cope with the heterogeneity of spatial information that often arises in practice. In applications such as geographic information retrieval, on the other hand, approximate boundaries of spatial regions need to be constructed, using whatever spatial information that can be obtained. Motivated by this observation, we propose a novel methodology for generating spatial scenarios that are compatible with available knowledge. By suitably discretizing space, this task is translated to a combinatorial optimization problem, which is solved using a hybridization of two well-known meta-heuristics: genetic algorithms and ant colony optimization. What results is a flexible method that can cope with both quantitative and qualitative information, and can easily be adapted to the specific needs of specific applications. Experiments with geographic data demonstrate the potential of the approach

Steinitz Theorems for Orthogonal Polyhedra

We define a simple orthogonal polyhedron to be a three-dimensional polyhedron with the topology of a sphere in which three mutually-perpendicular edges meet at each vertex. By analogy to Steinitz's theorem characterizing the graphs of convex polyhedra, we find graph-theoretic characterizations of three classes of simple orthogonal polyhedra: corner polyhedra, which can be drawn by isometric projection in the plane with only one hidden vertex, xyz polyhedra, in which each axis-parallel line through a vertex contains exactly one other vertex, and arbitrary simple orthogonal polyhedra. In particular, the graphs of xyz polyhedra are exactly the bipartite cubic polyhedral graphs, and every bipartite cubic polyhedral graph with a 4-connected dual graph is the graph of a corner polyhedron. Based on our characterizations we find efficient algorithms for constructing orthogonal polyhedra from their graphs.Comment: 48 pages, 31 figure

Decomposing and packing polygons / Dania el-Khechen.

In this thesis, we study three different problems in the field of computational geometry: the partitioning of a simple polygon into two congruent components, the partitioning of squares and rectangles into equal area components while minimizing the perimeter of the cuts, and the packing of the maximum number of squares in an orthogonal polygon. To solve the first problem, we present three polynomial time algorithms which given a simple polygon P partitions it, if possible, into two congruent and possibly nonsimple components P 1 and P 2 : an O ( n 2 log n ) time algorithm for properly congruent components and an O ( n 3 ) time algorithm for mirror congruent components. In our analysis of the second problem, we experimentally find new bounds on the optimal partitions of squares and rectangles into equal area components. The visualization of the best determined solutions allows us to conjecture some characteristics of a class of optimal solutions. Finally, for the third problem, we present three linear time algorithms for packing the maximum number of unit squares in three subclasses of orthogonal polygons: the staircase polygons, the pyramids and Manhattan skyline polygons. We also study a special case of the problem where the given orthogonal polygon has vertices with integer coordinates and the squares to pack are (2 {604} 2) squares. We model the latter problem with a binary integer program and we develop a system that produces and visualizes optimal solutions. The observation of such solutions aided us in proving some characteristics of a class of optimal solutions

Spatial reasoning with RCC8 and connectedness constraints in Euclidean spaces

The language RCC8 is a widely-studied formalism for describing topological arrangements of spatial regions. The variables of this language range over the collection of non-empty, regular closed sets of n-dimensional Euclidean space, here denoted RC+(R^n), and its non-logical primitives allow us to specify how the interiors, exteriors and boundaries of these sets intersect. The key question is the satisfiability problem: given a finite set of atomic RCC8-constraints in m variables, determine whether there exists an m-tuple of elements of RC+(R^n) satisfying them. These problems are known to coincide for all n ≥ 1, so that RCC8-satisfiability is independent of dimension. This common satisfiability problem is NLogSpace-complete. Unfortunately, RCC8 lacks the means to say that a spatial region comprises a ‘single piece’, and the present article investigates what happens when this facility is added. We consider two extensions of RCC8: RCC8c, in which we can state that a region is connected, and RCC8c0, in which we can instead state that a region has a connected interior. The satisfiability problems for both these languages are easily seen to depend on the dimension n, for n ≤ 3. Furthermore, in the case of RCC8c0, we show that there exist finite sets of constraints that are satisfiable over RC+(R^2), but only by ‘wild’ regions having no possible physical meaning. This prompts us to consider interpretations over the more restrictive domain of non-empty, regular closed, polyhedral sets, RCP+(R^n). We show that (a) the satisfiability problems for RCC8c (equivalently, RCC8c0) over RC+(R) and RCP+(R) are distinct and both NP-complete; (b) the satisfiability problems for RCC8c over RC+(R^2) and RCP+(R^2) are identical and NP-complete; (c) the satisfiability problems for RCC8c0 over RC+(R^2) and RCP+(R^2) are distinct, and the latter is NP-complete. Decidability of the satisfiability problem for RCC8c0 over RC+(R^2) is open. For n ≥ 3, RCC8c and RCC8c0 are not interestingly different from RCC8. We finish by answering the following question: given that a set of RCC8c- or RCC8c0-constraints is satisfiable over RC+(R^n) or RCP+(R^n), how complex is the simplest satisfying assignment? In particular, we exhibit, for both languages, a sequence of constraints Φ_n, satisfiable over RCP+(R^2), such that the size of Φ_n grows polynomially in n, while the smallest configuration of polygons satisfying Φ_n cuts the plane into a number of pieces that grows exponentially. We further show that, over RC+(R^2), RCC8c again requires exponentially large satisfying diagrams, while RCC8c0 can force regions in satisfying configurations to have infinitely many components

Discrete Geometry

The workshop on Discrete Geometry was attended by 53 participants, many of them young researchers. In 13 survey talks an overview of recent developments in Discrete Geometry was given. These talks were supplemented by 16 shorter talks in the afternoon, an open problem session and two special sessions. Mathematics Subject Classification (2000): 52Cxx. Abstract regular polytopes: recent developments. (Peter McMullen) Counting crossing-free configurations in the plane. (Micha Sharir) Geometry in additive combinatorics. (József Solymosi) Rigid components: geometric problems, combinatorial solutions. (Ileana Streinu) • Forbidden patterns. (János Pach) • Projected polytopes, Gale diagrams, and polyhedral surfaces. (Günter M. Ziegler) • What is known about unit cubes? (Chuanming Zong) There were 16 shorter talks in the afternoon, an open problem session chaired by Jesús De Loera, and two special sessions: on geometric transversal theory (organized by Eli Goodman) and on a new release of the geometric software Cinderella (Jürgen Richter-Gebert). On the one hand, the contributions witnessed the progress the field provided in recent years, on the other hand, they also showed how many basic (and seemingly simple) questions are still far from being resolved. The program left enough time to use the stimulating atmosphere of the Oberwolfach facilities for fruitful interaction between the participants