146 research outputs found
Practical Distance Functions for Path-Planning in Planar Domains
Path planning is an important problem in robotics. One way to plan a path
between two points within a (not necessarily simply-connected) planar
domain , is to define a non-negative distance function on
such that following the (descending) gradient of this
distance function traces such a path. This presents two equally important
challenges: A mathematical challenge -- to define such that has a
single minimum for any fixed (and this is when ), since a local
minimum is in effect a "dead end", A computational challenge -- to define
such that it may be computed efficiently. In this paper, given a description of
, we show how to assign coordinates to each point of and
define a family of distance functions between points using these coordinates,
such that both the mathematical and the computational challenges are met. This
is done using the concepts of \emph{harmonic measure} and
\emph{-divergences}.
In practice, path planning is done on a discrete network defined on a finite
set of \emph{sites} sampled from , so any method that works well on the
continuous domain must be adapted so that it still works well on the discrete
domain. Given a set of sites sampled from , we show how to define a
network connecting these sites such that a \emph{greedy routing} algorithm
(which is the discrete equivalent of continuous gradient descent) based on the
distance function mentioned above is guaranteed to generate a path in the
network between any two such sites. In many cases, this network is close to a
(desirable) planar graph, especially if the set of sites is dense
Secure Data Hiding for Contact Tracing
Contact tracing is an effective tool in controlling the spread of infectious
diseases such as COVID-19. It involves digital monitoring and recording of
physical proximity between people over time with a central and trusted
authority, so that when one user reports infection, it is possible to identify
all other users who have been in close proximity to that person during a
relevant time period in the past and alert them. One way to achieve this
involves recording on the server the locations, e.g. by reading and reporting
the GPS coordinates of a smartphone, of all users over time. Despite its
simplicity, privacy concerns have prevented widespread adoption of this method.
Technology that would enable the "hiding" of data could go a long way towards
alleviating privacy concerns and enable contact tracing at a very large scale.
In this article we describe a general method to hide data. By hiding, we mean
that instead of disclosing a data value x, we would disclose an "encoded"
version of x, namely E(x), where E(x) is easy to compute but very difficult,
from a computational point of view, to invert. We propose a general
construction of such a function E and show that it guarantees perfect recall,
namely, all individuals who have potentially been exposed to infection are
alerted, at the price of an infinitesimal number of false alarms, namely, only
a negligible number of individuals who have not actually been exposed will be
wrongly informed that they have
On affine rigidity
We define the notion of affine rigidity of a hypergraph and prove a variety
of fundamental results for this notion. First, we show that affine rigidity can
be determined by the rank of a specific matrix which implies that affine
rigidity is a generic property of the hypergraph.Then we prove that if a graph
is is -vertex-connected, then it must be "generically neighborhood
affinely rigid" in -dimensional space. This implies that if a graph is
-vertex-connected then any generic framework of its squared graph must
be universally rigid.
Our results, and affine rigidity more generally, have natural applications in
point registration and localization, as well as connections to manifold
learning.Comment: Updated abstrac
Meshing Genus-1 Point Clouds Using Discrete One-Forms
We present an algorithm to mesh point clouds sampled from a closed manifold surface of genus 1. The method relies on a doubly periodic global parameterization of the point cloud to the plane, so no segmentation of the point cloud is required. Based on some recent techniques for parameterizing higher genus meshes, when some mild conditions on the sampling density are satisfied, the algorithm generates a closed toroidal manifold which interpolates the input and is geometrically similar to the sampled surface.Engineering and Applied Science
Path Planning with Divergence-Based Distance Functions
Distance functions between points in a domain are sometimes used to
automatically plan a gradient-descent path towards a given target point in the
domain, avoiding obstacles that may be present. A key requirement from such
distance functions is the absence of spurious local minima, which may foil such
an approach, and this has led to the common use of harmonic potential
functions. Based on the planar Laplace operator, the potential function
guarantees the absence of spurious minima, but is well known to be slow to
numerically compute and prone to numerical precision issues. To alleviate the
first of these problems, we propose a family of novel divergence distances.
These are based on f-divergence of the Poisson kernel of the domain. We define
the divergence distances and compare them to the harmonic potential function
and other related distance functions.
Our first result is theoretical: We show that the family of divergence
distances are equivalent to the harmonic potential function on simply-connected
domains, namely generate paths which are identical to those generated by the
potential function. The proof is based on the concept of conformal invariance.
Our other results are more practical and relate to two special cases of
divergence distances, one based on the Kullback-Leibler divergence and one
based on the total variation divergence. We show that using divergence
distances instead of the potential function and other distances has a
significant computational advantage, as, following a pre-processing stage, they
may be computed up to an order of magnitude faster than the others when taking
advantage of certain sparsity properties of the Poisson kernel. Furthermore,
the computation is "embarrassingly parallel", so may be implemented on a GPU
with up to three orders of magnitude speedup
Recommended from our members
Routing with Guaranteed Delivery on Virtual Coordinates
We propose four simple algorithms for routing on planar graphs using virtual coordinates. These algorithms are superior to existing algorithms in that they are oblivious, work also for non-triangular graphs, and their virtual coordinates are easy to construct.Engineering and Applied Science
- …