Michael Q. Rieck

Association schemes based on isotropic subspaces, Part 1

The subspaces of a given dimension in a finite classical polar space form the points of an association scheme. When the dimension is zero, this is the scheme of the collinearity graph of the space. At the other extreme, when the dimension is maximal, it is the scheme of the corresponding dual polar graph. These extreme cases have been thoroughly studied. In this article, the general case is examined and a detailed computation of the intersection numbers of these association schemes is initiated.

On Shortest Path Routing Schemes for Wireless Ad Hoc Networks

In this paper, we propose two new distributed algorithms for producing sets of nodes that can be used to form backbones of an ad hoc wireless network. Our focus is on producing small sets that are d-hop connected and d-hop dominating and have a desirable ‘shortest path property’. These algorithms produce sets that are considerably smaller than those produced by an algorithm previously introduced by the authors. One of these two new algorithms has constant-time complexity.

A Fundamentally New View of the Perspective Three-Point Pose Problem

The Perspective Three-Point Pose Problem (P3P) is an old and basic problem in the area of camera tracking. While methods for solving it have been largely successful, they are subject to erratic behavior near the so-called “danger cylinder.” Another difficulty with most of these methods is the need to select the physically correct solution from among various mathematical solutions. This article presents a new framework from which to study P3P for non-collinear control points, particularly near the danger cylinder. A multivariate Newton-Raphson method to approximately solve P3P is introduced. Using the new framework, this is then enhanced by adding special procedures for handling the problematic behavior near the danger cylinder. It produces a point on the cylinder, a compromise between two nearly equal mathematical solutions, only one of which is the camera’s actual position. The compromise diminishes the risk of accidentally converging to the other nearby solution. However, it does impose the need, upon receding from the danger cylinder vicinity, to make a selection between two possible approximate solution points. Traditional algebraic methods depend on correctly selecting from up to four points, each time the camera position is recomputed. In the new iterative method, selecting between just two points is only occasionally required. Simulations demonstrate that a considerable improvement results from using this revised method instead of the basic Newton-Raphson method.

An Algorithm for Finding Repeated Solutions to the General Perspective Three-Point Pose Problem

In the Perspective 3-Point Pose Problem (P3P), the transformation that converts the triple of (unknown) camera-to-control-point distances, into the triple of (known) angle cosines between the projection lines, is generally locally invertible. However, this fails to be the case when the camera’s focal point (center of perspective) is on the danger cylinder. This situation corresponds to a double solution to P3P, and presents extra difficulties in solving P3P.An extensive analysis of the danger cylinder setup leads to the introduction of a special rational function that proves to be quite useful in solving P3P in the danger cylinder case. This involves some rather long algebraic expressions that are best manipulated using mathematical software. Ultimately, some fairly simple formulas emerge that serve as a basis for an algorithm, called the Double Solution Algorithm (DSA). Experimental results comparing DSA with Grunert’s quartic polynomial method demonstrate that DSA often has substantially greater accuracy. This is particularly so when the camera is relatively far from the control points, even if it is not very close to the danger cylinder.

Various Distributed Shortest Path Routing Strategies for Wireless Ad Hoc Networks

In this paper, we describe and compare several distributed greedy algorithms that produce sets of nodes that can be used to form a virtual backbone for an ad hoc wireless network. The backbone produced is always a d-hop dominating, d-hop connected set and has a desirable “shortest path property”. The perfomance of these algorithms for various parameters are compared.

An energy-efficient heterogeneous dual routing scheme for mobile ad hoc and sensor networks

New energy-efficient routing algorithms are introduced, based on a generalisation of the k-SPR sets from earlier work by the authors. This generalisation provides a means for the automatic avoidance of certain nodes and links when messages are routed. Sensor networks are modelled as connected graphs with vertex costs and edge costs. In addition, a two-tiered routing system in introduced. The low level routing is used for local routing within k hops, and is essentially (local) link-state routing. The high level routing depends on the routers from a k-SPR set, to manage this global routing.

Hierarchical routing in sensor networks using k -dominating sets

For a connected graph, representing a sensor network, distributed algorithms for the Set Covering Problem can be employed to construct reasonably small subsets of the nodes, called k-SPR sets. Such a set can serve as a virtual backbone to facilitate shortest path routing, as introduced in [4] and [14]. When employed in a hierarchical fashion, together with a hybrid (partly proactive, partly reactive) strategy, the k-SPR set methods become highly scalable, resulting in guaranteed minimal path routing, with comparatively little overhead.

Hierarchical routing in ad hoc networks using k-dominating sets

In this article, the notion of a k-SPR set from previous work is extended to the context of an edge-weighted graph. Under a reasonable assumption, such a set is still k-dominating, and k-hop connected. When a decreasing sequence of such sets is used, together with a hybrid route discovery strategy (partly proactive, partly reactive), the result is a highly scalable and efficient, minimal path routing protocol.

Constructing special k-dominating sets using variations on the greedy algorithm

This paper focuses on the efficient selection of a special type of subset of network nodes, which we call a k-SPR set, for the purpose of coordinating the routing of messages through a network. Such a set is a special k-hop-connected k-dominating set that has an additional property that promotes the regular occurrence of routers in all directions. The distributed algorithms introduced here for obtaining a k-SPR set require that each node broadcast at most three messages to its k-hop neighbors. These transmissions can be made asynchronously. The time required to send these messages and the sizes of the resulting sets are compared by means of data collected from simulations. The main contribution is the adaptation of some variations of the distributed greedy algorithms to the problem of generating a small k-SPR set. These variations are much faster than the standard distributed greedy algorithm. Yet, when used with a sensible choice for a certain parameter, our empirical evidence strongly suggests that the resulting set size will generally be very close to the set size for the standard greedy algorithms.

Related Solutions to the Perspective Three-Point Pose Problem

The perspective three-point pose problem involves solving Grunert’s system of quadratic equations for the distances from the center of perspective to the three control points, typically resulting in multiple mathematical solutions. Relationships between the corresponding possible camera positions in space have only rarely been studied. Several efforts have been made though to understand the number of solution points using various assumptions. In this article, the number of solutions is determined in the limiting case, where the center of perspective is far from the plane containing the control points, as compared with its distance to the danger cylinder. Moreover, concise formulas are given for the other solutions based on a knowledge of one of the solutions. It turns out that the projection onto the control points plane of the various solution points lies at the intersection points of two rectangular hyperbolas. A certain deltoid curve also plays a crucial role.