A simple method to generate a 2-D binary grid pattern, which allows for absolute and accurate self-location in a finite planar region, is proposed. The pattern encodes position information in a local way so that reading a small number of its black or white pixels at any place provides sufficient data from which the location can be decoded both efficiently and robustly.

Border gateway protocol (BGP) is the de facto standard used for interdomain routing. Since packet forwarding may not be possible until stable routes are learned, it is not only critical for BGP to converge but it is important that the convergence be rapid. The distributed and asynchronous nature of BGP in conjunction with local policies makes it difficult to analyze with respect to convergence behavior. We present a novel model which, to our knowledge, is the first one to permit analysis of convergence in the aggregate (i.e., over all message exchange orders between routers regarding route advertisements), rather than worst case behavior. We introduce the notion of probabilistic safety as requiring the probability of convergence to be 1. We provide a necessary and sufficient condition characterizing probabilistic safety that shows that probabilistic safety accommodates BGP configurations whose potential divergence stems solely from pathological message sequences. More generally, we show how to compute for any BGP configuration its probability of convergence. For probabilistically safe configurations, we present procedures for computing their expected time to converge as well as the probability distribution on their convergence times. The ability to compute these quantitative characteristics makes our work "constructive" and provides the basis for further understanding and deriving procedures for optimizing network characteristics. Finally, we simulate several network examples and verify the consistency between our analysis and the simulations

Inspired by the abilities of both the praying mantis and the pigeon to judge distance by use of motion-based visually mediated odometry, we create miniature models for depth estimation that are similar to the head movements of these animals. We develop mathematical models of the praying mantis and pigeon visual behavior and describe our implementation and experimental environment. We investigate structure from motion problems when images are taken from a camera whose focal point is translating the first case is reminiscent of a praying mantis peering its head left and right, apparently to obtain depth perception, hence the moniker "mantis head camera." In the second case this motion is reminiscent of a pigeon bobbing its head back and forth, also apparently to obtain depth perception, hence the moniker " pigeon head camera." We present the performance of the mantis head camera and pigeon head camera models and provide experimental results of the algorithms. We provide the comparison of the definitiveness of the results obtained by both models. The precision of our mathematical model and its implementation is consistent with the experimental facts obtained from various biological experiments.

Lossy transform compression of images is very successful and widespread. The JPEG standard uses the discrete cosine transform on blocks of the image and a bit allocation process that takes advantage of the uneven energy distribution in the transform domain. For most images 10:1 compression ratios can be achieved with no visible degradations. Suppose however that multiple versions of the compressed image exist in a distributed environment such as the Internet, and several of them could be made available upon request. The classical approach would provide no improvement in the image quality if more than one version of the compressed image became available. In this paper we propose a method, based on multiple description scalar quantization, that yields decompressed image quality improving with the number of compressed versions available.

We investigate how a given fixed number of points should be located in space so that the pose of a camera viewing them from unknown locations can be estimated with the greatest accuracy. We show that optimum solutions are obtained when the points form concentric complete regular polyhedra. For the case of optimal configurations, we provide a worst-case error analysis and use it to analyze the effects of weak perspective approximation to true perspective viewing. Comprehensive computer simulations validate the theoretical results.

Several unconventional ideas for viewer/camera pose estimation are discussed. The methods proposed so far advocate the use of advanced image processing for identification and precise location of calibration objects in the images acquired and base pose recovery on the identification of the viewing dependent deformations of these objects. We propose to more fully exploit the freedom in the design of "space fiducials" or calibration objects showing that we can build objects whose images directly encode, in easily identifiable gray-level/color or temporal patterns, the pose of their viewer. We also show how to construct high-precision fiducials, which can determine a viewing direction quite accurately when it is known to lie within a relatively narrow range.

We discuss a new type of holographic image representations that have advantages in a "distributed" world. We call these representations holographic. Arbitrary portions of a holographic representation enable reconstruction of the whole image, with distortions that decrease gradually with the increase in the size of the portions available. Holographic representations enable progressive refinement in image communication or retrieval tasks, with no restrictions on the order in which the data fragments (sections of the representation) are accessed or become available.

Real cubic algebraic surfaces may be described by either implicit or parametric equations. One particularly useful representation is the rational parametrization, where the three spatial coordinates are given by rational functions of two parameters. These parametrizations take on different forms for different classes of cubic surfaces. Classification of real cubic algebraic surfaces into five families for the nonsingular case is based on the configuration of 27 lines on them. We provide a method of extracting all these lines by constructing and solving a polynomial of degree 27. Simple roots of this polynomial correspond to real lines on the surface, and real skew lines are used to form rational parametrizations for three of these families. Complex conjugate skew lines are used to parametrize surfaces from the fourth family. The parametrizations for these four families involve quotients of polynomials of degree no higher than four. Each of these parametrizations covers the whole surface except for a few points, lines, or conic sections. The parametrization for the fifth family, as noted previously in the literature, requires a square root. We also analyze the image of the derived rational parametrization for both real and complex parameter values, together with “base” points where the parametrizations are ill-defined.

We discuss holographic image representations. Arbitrary portions of a holographic representation enable reconstruction of the whole image, with distortions that decrease gradually with the increase of the size of the portions available. Holographic representations enable progressive refinement in image communication or retrieval tasks, with no restrictions on the order in which the data fragments (sections of the representation) are accessed or become available.

Many important problems in image processing and computer vision can be formulated as the solution of a system of simultaneous polynomial equations. Crucial issues include the uniqueness of solution and the number of solutions (if not unique), and how to find numerically all the solutions. The goal of this paper is to introduce to engineers and scientists some mathematical tools from algebraic geometry which are very useful in resolving these issues. Three-dimensional motion/structure estimation is used as the context. However, these tools should also be helpful in other areas including surface intersection in computer-aided design, and inverse position problems in kinematics/robotics. The tools to be described are Bezout numbers, Grobner bases, homotopy methods, and a powerful theorem which states that under rather general conditions one can draw general conclusions on the number of solutions of a polynomial system from a single numerical example.

We show that there is, in general, a unique solution for the relative orientation of three cameras simultaneously photographing four feature points on a fixed object. However, multiple solutions are possible, in rare cases, even when the four feature points are not coplanar.>

We present a purely geometrical proof that under the weak perspective model, the 3D pose of a 3-point configuration is determined uniquely up to a reflection by its 2D projection.

A planar shape distorted by a projective viewing transformation can be recognized under partial occlusion if an invariant description of its boundary is available. Research in this area has provided a theory for invariant boundary descriptions based on an interplay of differential, local, and global invariants. Differential invariants require high-order derivatives. The use of global invariants and point matches on the distorting transformations enables one to reduce the order. Trade-offs between the highest order derivatives required and the quantity of additional information constraining the distorting viewing transformations are made explicit.>