Robin Y. Flatland

SPANNING PROPERTIES OF GRAPHS INDUCED BY DIRECTIONAL ANTENNAS

Let S be a set of points in the plane, such that the unit disk graph with vertex set S is connected. We address the problem of finding orientations and a minimum radius for directional antennas of a fixed cone angle placed at the points of S, such that the induced communication graph G[S] is a hop t-spanner of the unit disk graph for S (meaning that G[S] is strongly connected, and contains a directed path with at most t edges between any pair of points within unit distance). We consider problem instances in which antenna angles are bounded below by 120° and 90°. We show that, in the case of 120° angles, a radius of 5 suffices to establish a hop 4-spanner; and in the case of 90° angles, a radius of 7 suffices to establish a hop 5-spanner.

Geometric constraints and stereo disparity computation

Most stereo techniques compute disparity assuming that it varies slowly along surfaces. We quantify and justify this assumption, using weak assumptions about surface orientation distributions in the world to derive the density of disparity surface orientations. The small disparity change assumption is justified by the orientation densitys heavy bias toward disparity surfaces that are nearly parallel to the image plane. In addition, the bias strengthens with smaller baselines, larger focal lengths, and as surfaces move farther from the cameras. To analyze current stereo techniques, we derive three densities from the first density, those of the disparity gradient magnitude, the directional derivative of disparity, and the difference in disparity between neighboring surface points. The latter may be used in Bayesian algorithms computing dense disparity fields. The directional derivative density and the disparity difference density both show that feature-based algorithms should strongly favor small disparity changes, contrary to several well-known algorithms. Finally, we use our original surface orientation density and the gradient magnitude density to derive two new “surfaces-from-stereo” techniques, techniques combining feature-based matching and surface reconstruction. The first uses the densities to severely restrict the search range for the optimum fit. The second incorporates the surface orientation density into the optimization criteria, producing a Bayesian formulation. Both algorithms are shown to be efficient and effective.

Switching to directional antennas with constant increase in radius and hop distance

For any angle α < 2π, we show that any connected communication graph that is induced by a set P of n transceivers using omni-directional antennas of radius 1, can be replaced by a strongly connected communication graph, in which each transceiver in P is equipped with a directional antenna of angle α and radius rdir, for some constant rdir = rdir(α). Moreover, the new communication graph is a c-spanner of the original graph, for some constant c = c(α), with respect to number of hops.

Unfolding Manhattan Towers

We provide an algorithm for unfolding the surface of any orthogonal polyhedron that falls into a particular shape class we call Manhattan Towers, to a nonoverlapping planar orthogonal polygon. The algorithm cuts along edges of a 4x5x1 refinement of the vertex grid.

Coverage with k-transmitters in the presence of obstacles

For a fixed integer k≥0, a k-transmitter is an omnidirectional wireless transmitter with an infinite broadcast range that is able to penetrate up to k “walls”, represented as line segments in the plane. We develop lower and upper bounds for the number of k-transmitters that are necessary and sufficient to cover a given collection of line segments, polygonal chains and polygons.

Maintaining valid topology with active contours: theory and application

We develop and prove correct an algorithm that enables active contours to correctly represent regions that undergo topology changes as the contours evolve. Using the incremental motion typical of active contours, we introduce the concept of motion regions to determine the new topology. When the topology changes (e.g. by contours intersecting), the motion regions are used to delete and and reconnect the contours to accurately describe the new region. Contour intersections can also occur without topology changes. These are also appropriately handled. The algorithm to perform this task is proved correct in a general framework that makes few assumptions about the contour representation. We describe how this algorithm is applied to a polygonal representation of the contours, and argue that it does not significantly affect execution time. Finally, this polygonal implementation is used in surface extraction and phytoplankton classification.

Realistic Reconfiguration of Crystalline (and Telecube) Robots

In this paper we propose novel algorithms for reconfiguring modular robots that are composed of n atoms. Each atom has the shape of a unit cube and can expand/contract each face by half a unit, as well as attach to or detach from faces of neighboring atoms. For universal reconfiguration, atoms must be arranged in 2×2×2 modules. We respect certain physical constraints: each atom reaches at most unit velocity and (via expansion) can displace at most one other atom. We require that one of the atoms can store a map of the target configuration. Our algorithms involve a total of O(n 2) such atom operations, which are performed in O(n) parallel steps. This improves on previous reconfiguration algorithms, which either use O(n 2) parallel steps [8,10,4] or do not respect the constraints mentioned above [1]. In fact, in the setting considered, our algorithms are optimal, in the sense that certain reconfigurations require Ω(n) parallel steps. A further advantage of our algorithms is that reconfiguration can take place within the union of the source and target configurations.

Unfolding Orthogonal Polyhedra with Quadratic Refinement: The Delta-Unfolding Algorithm

We show that every orthogonal polyhedron homeomorphic to a sphere can be unfolded without overlap while using only polynomially many (orthogonal) cuts. By contrast, the best previous such result used exponentially many cuts. More precisely, given an orthogonal polyhedron with n vertices, the algorithm cuts the polyhedron only where it is met by the grid of coordinate planes passing through the vertices, together with Θ(n2) additional coordinate planes between every two such grid planes.

Efficient constant-velocity reconfiguration of crystalline robots**

In this paper, we propose novel algorithms for reconfiguring modular robots that are composed of n atoms. Each atom has the shape of a unit cube and can expand/contract each face by half a unit, as well as attach to or detach from faces of neighboring atoms. For universal reconfiguration, atoms must be arranged in 2 ?? 2 ?? 2 modules. We respect certain physical constraints: each atom reaches at most constant velocity and can displace at most a constant number of other atoms. We assume that one of the atoms has access to the coordinates of atoms in the target configuration. Our algorithms involve a total of O(n2) atom operations, which are performed in O(n) parallel steps. This improves on previous reconfiguration algorithms, which either use O(n2) parallel steps or do not respect the constraints mentioned above. In fact, in the settings considered, our algorithms are optimal. A further advantage of our algorithms is that reconfiguration can take place within the union of the source and target configuration space, and only requires local communication.

Connecting Polygonizations via Stretches and Twangs

We show that the space of polygonizations of a fixed planar point set S of n points is connected by O(n2) “moves” between simple polygons. Each move is composed of a sequence of atomic moves called “stretches” and “twangs,” which walk between weakly simple “polygonal wraps” of S. These moves show promise to serve as a basis for generating random polygons.