MAP: Medial Axis Based Geometric Routing in Sensor Networks

One of the challenging tasks in the deployment of dense wireless networks (like sensor networks) is in devising a routing scheme for node to node communication. Important consideration includes scalability, routing complexity, the length of the communication paths and the load sharing of the routes. In this paper, we show that a compact and expressive abstraction of network connectivity by the medial axis enables efficient and localized routing. We propose MAP, a Medial Axis based naming and routing Protocol that does not require locations, makes routing decisions locally, and achieves good load balancing. In its preprocessing phase, MAP constructs the medial axis of the sensor field, defined as the set of nodes with at least two closest boundary nodes. The medial axis of the network captures both the complex geometry and non-trivial topology of the sensor field. It can be represented compactly by a graph whose size is comparable with the complexity of the geometric features (e.g., the number of holes). Each node is then given a name related to its position with respect to the medial axis. The routing scheme is derived through local decisions based on the names of the source and destination nodes and guarantees delivery with reasonable and natural routes. We show by both theoretical analysis and simulations that our medial axis based geometric routing scheme is scalable, produces short routes, achieves excellent load balancing, and is very robust to variations in the network model

Skeletal representations of orthogonal shapes

In this paper we present two skeletal representations applied to orthogonal shapes of R^n : the cube axis and a family of skeletal representations provided by the scale cube axis. Orthogonal shapes are a subset of polytopes, where the hyperplanes of the bounding facets are restricted to be axis aligned. Both skeletal representations rely on the L∞ metric and are proven to be homotopically equivalent to its shape. The resulting skeleton is composed of n − 1 dimensional facets. We also provide an efficient and robust algorithm to compute the scale cube axis in the plane and compare the resulting skeleton with other skeletal representations.Postprint (published version

Doctor of Philosophy

dissertationThe medial axis of an object is a shape descriptor that intuitively presents the morphology or structure of the object as well as intrinsic geometric properties of the object’s shape. These properties have made the medial axis a vital ingredient for shape analysis applications, and therefore the computation of which is a fundamental problem in computational geometry. This dissertation presents new methods for accurately computing the 2D medial axis of planar objects bounded by B-spline curves, and the 3D medial axis of objects bounded by B-spline surfaces. The proposed methods for the 3D case are the first techniques that automatically compute the complete medial axis along with its topological structure directly from smooth boundary representations. Our approach is based on the eikonal (grassfire) flow where the boundary is offset along the inward normal direction. As the boundary deforms, different regions start intersecting with each other to create the medial axis. In the generic situation, the (self-) intersection set is born at certain creation-type transition points, then grows and undergoes intermediate transitions at special isolated points, and finally ends at annihilation-type transition points. The intersection set evolves smoothly in between transition points. Our approach first computes and classifies all types of transition points. The medial axis is then computed as a time trace of the evolving intersection set of the boundary using theoretically derived evolution vector fields. This dynamic approach enables accurate tracking of elements of the medial axis as they evolve and thus also enables computation of topological structure of the solution. Accurate computation of geometry and topology of 3D medial axes enables a new graph-theoretic method for shape analysis of objects represented with B-spline surfaces. Structural components are computed via the cycle basis of the graph representing the 1-complex of a 3D medial axis. This enables medial axis based surface segmentation, and structure based surface region selection and modification. We also present a new approach for structural analysis of 3D objects based on scalar functions defined on their surfaces. This approach is enabled by accurate computation of geometry and structure of 2D medial axes of level sets of the scalar functions. Edge curves of the 3D medial axis correspond to a subset of ridges on the bounding surfaces. Ridges are extremal curves of principal curvatures on a surface indicating salient intrinsic features of its shape, and hence are of particular interest as tools for shape analysis. This dissertation presents a new algorithm for accurately extracting all ridges directly from B-spline surfaces. The proposed technique is also extended to accurately extract ridges from isosurfaces of volumetric data using smooth implicit B-spline representations. Accurate ridge curves enable new higher-order methods for surface analysis. We present a new definition of salient regions in order to capture geometrically significant surface regions in the neighborhood of ridges as well as to identify salient segments of ridges

Compensated convexity, multiscale medial axis maps and sharp regularity of the squared-distance function

In this paper we introduce a new stable mathematical model for locating and measuring the medial axis of geometric objects, called the quadratic multiscale medial axis map of scale λ, and provide a sharp regularity result for the squared-distance function to any closed nonempty subset K of Rn. Our results exploit properties of the function Clλ (dist2(・; K)) obtained by applying the quadratic lower compensated convex transform of parameter λ [K. Zhang, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire, 25 (2008), pp. 743–771] to dist2(・; K), the Euclidean squared-distance function to K. Using a quantitative estimate for the tight approximation of dist2(・; K) by Clλ (dist2(・; K)), we prove the C1,1-regularity of dist2(・; K) outside a neighborhood of the closure of the medial axis MK of K, which can be viewed as a weak Lusin-type theorem for dist2(・; K), and give an asymptotic expansion formula for Clλ (dist2(・; K)) in terms of the scaled squared-distance transform to the set and to the convex hull of the set of points that realize the minimum distance to K. The multiscale medial axis map, denoted by Mλ(・; K), is a family of nonnegative functions, parametrized by λ > 0, whose limit as λ→∞exists and is called the multiscale medial axis landscape map, M∞(・; K). We show that M∞(・; K) is strictly positive on the medial axis MK and zero elsewhere. We give conditions that ensure Mλ(・; K) keeps a constant height along the parts of MK generated by two-point subsets with the value of the height dependent on the scale of the distance between the generating points, thus providing a hierarchy of heights (hence, the word “multiscale”) between different parts of MK that enables subsets of MK to be selected by simple thresholding. Asymptotically, further understanding of the multiscale effect is provided by our exact representation of M∞(・; K). Moreover, given a compact subset K of Rn, while it is well known that MK is not Hausdorff stable, we prove that in contrast, Mλ(・; K) is stable under the Hausdorff distance, and deduce implications for the localization of the stable parts of MK. Explicitly calculated prototype examples of medial axis maps are also presented and used to illustrate the theoretical findings

Extracting curve-skeletons from digital shapes using occluding contours

Curve-skeletons are compact and semantically relevant shape descriptors, able to summarize both topology and pose of a wide range of digital objects. Most of the state-of-the-art algorithms for their computation rely on the type of geometric primitives used and sampling frequency. In this paper we introduce a formally sound and intuitive definition of curve-skeleton, then we propose a novel method for skeleton extraction that rely on the visual appearance of the shapes. To achieve this result we inspect the properties of occluding contours, showing how information about the symmetry axes of a 3D shape can be inferred by a small set of its planar projections. The proposed method is fast, insensitive to noise, capable of working with different shape representations, resolution insensitive and easy to implement