35 research outputs found
Sampling and Reconstruction of Spatial Fields using Mobile Sensors
Spatial sampling is traditionally studied in a static setting where static
sensors scattered around space take measurements of the spatial field at their
locations. In this paper we study the emerging paradigm of sampling and
reconstructing spatial fields using sensors that move through space. We show
that mobile sensing offers some unique advantages over static sensing in
sensing time-invariant bandlimited spatial fields. Since a moving sensor
encounters such a spatial field along its path as a time-domain signal, a
time-domain anti-aliasing filter can be employed prior to sampling the signal
received at the sensor. Such a filtering procedure, when used by a
configuration of sensors moving at constant speeds along equispaced parallel
lines, leads to a complete suppression of spatial aliasing in the direction of
motion of the sensors. We analytically quantify the advantage of using such a
sampling scheme over a static sampling scheme by computing the reduction in
sampling noise due to the filter. We also analyze the effects of non-uniform
sensor speeds on the reconstruction accuracy. Using simulation examples we
demonstrate the advantages of mobile sampling over static sampling in practical
problems.
We extend our analysis to sampling and reconstruction schemes for monitoring
time-varying bandlimited fields using mobile sensors. We demonstrate that in
some situations we require a lower density of sensors when using a mobile
sensing scheme instead of the conventional static sensing scheme. The exact
advantage is quantified for a problem of sampling and reconstructing an audio
field.Comment: Submitted to IEEE Transactions on Signal Processing May 2012; revised
Oct 201
Asymptotically Optimal Matching of Multiple Sequences to Source Distributions and Training Sequences
Consider a finite set of sources, each producing i.i.d. observations that
follow a unique probability distribution on a finite alphabet. We study the
problem of matching a finite set of observed sequences to the set of sources
under the constraint that the observed sequences are produced by distinct
sources. In general, the number of sequences may be different from the
number of sources , and only some of the observed
sequences may be produced by a source from the set of sources of interest. We
consider two versions of the problem -- one in which the probability laws of
the sources are known, and another in which the probability laws of the sources
are unspecified but one training sequence from each of the sources is
available. We show that both these problems can be solved using a sequence of
tests that are allowed to produce "no-match" decisions. The tests ensure
exponential decay of the probabilities of incorrect matching as the sequence
lengths increase, and minimize the "no-match" decisions. Both tests can be
implemented using variants of the minimum weight matching algorithm applied to
a weighted bipartite graph. We also compare the performances obtained by using
these tests with those obtained by using tests that do not take into account
the constraint that the sequences are produced by distinct sources. For the
version of the problem in which the probability laws of the sources are known,
we compute the rejection exponents and error exponents of the tests and show
that tests that make use of the constraint have better exponents than tests
that do not make use of this information.Comment: To appear in IEEE Transactions on Information Theor
Sampling High-Dimensional Bandlimited Fields on Low-Dimensional Manifolds
Consider the task of sampling and reconstructing a bandlimited spatial field
in using moving sensors that take measurements along their path. It is
inexpensive to increase the sampling rate along the paths of the sensors but
more expensive to increase the total distance traveled by the sensors per unit
area, which we call the \emph{path density}. In this paper we introduce the
problem of designing sensor trajectories that are minimal in path density
subject to the condition that the measurements of the field on these
trajectories admit perfect reconstruction of bandlimited fields. We study
various possible designs of sampling trajectories. Generalizing some ideas from
the classical theory of sampling on lattices, we obtain necessary and
sufficient conditions on the trajectories for perfect reconstruction. We show
that a single set of equispaced parallel lines has the lowest path density from
certain restricted classes of trajectories that admit perfect reconstruction.
We then generalize some of our results to higher dimensions. We first obtain
results on designing sampling trajectories in higher dimensional fields.
Further, interpreting trajectories as 1-dimensional manifolds, we extend some
of our ideas to higher dimensional sampling manifolds. We formulate the problem
of designing -dimensional sampling manifolds for -dimensional
spatial fields that are minimal in \emph{manifold density}, a natural
generalization of the path density. We show that our results on sampling
trajectories for fields in can be generalized to analogous results on
-dimensional sampling manifolds for -dimensional spatial fields.Comment: Submitted to IEEE Transactions on Information Theory, Nov 2011;
revised July 2012; accepted Oct 201
On Minimal Trajectories for Mobile Sampling of Bandlimited Fields
We study the design of sampling trajectories for stable sampling and the
reconstruction of bandlimited spatial fields using mobile sensors. The spectrum
is assumed to be a symmetric convex set. As a performance metric we use the
path density of the set of sampling trajectories that is defined as the total
distance traveled by the moving sensors per unit spatial volume of the spatial
region being monitored. Focussing first on parallel lines, we identify the set
of parallel lines with minimal path density that contains a set of stable
sampling for fields bandlimited to a known set. We then show that the problem
becomes ill-posed when the optimization is performed over all trajectories by
demonstrating a feasible trajectory set with arbitrarily low path density.
However, the problem becomes well-posed if we explicitly specify the stability
margins. We demonstrate this by obtaining a non-trivial lower bound on the path
density of an arbitrary set of trajectories that contain a sampling set with
explicitly specified stability bounds.Comment: 28 pages, 8 figure
Model-fitting in the presence of outliers
We study the problem of parametric model-fitting in a finite alphabet setting. We characterize the weak convergence of the goodness-of-fit statistic with respect to an exponential family when the observations are drawn from some alternate distribution. We then study the effects of outliers on the model-fitting procedure by specializing our results to -contaminated versions of distributions from the exponential family. We characterize the sensitivity of various distributions from the exponential family to outliers, and provide guidelines for choosing thresholds for a goodness-of-fit test that is robust to outliers in the data
Universal and Composite Hypothesis Testing via Mismatched Divergence
For the universal hypothesis testing problem, where the goal is to decide
between the known null hypothesis distribution and some other unknown
distribution, Hoeffding proposed a universal test in the nineteen sixties.
Hoeffding's universal test statistic can be written in terms of
Kullback-Leibler (K-L) divergence between the empirical distribution of the
observations and the null hypothesis distribution. In this paper a modification
of Hoeffding's test is considered based on a relaxation of the K-L divergence
test statistic, referred to as the mismatched divergence. The resulting
mismatched test is shown to be a generalized likelihood-ratio test (GLRT) for
the case where the alternate distribution lies in a parametric family of the
distributions characterized by a finite dimensional parameter, i.e., it is a
solution to the corresponding composite hypothesis testing problem. For certain
choices of the alternate distribution, it is shown that both the Hoeffding test
and the mismatched test have the same asymptotic performance in terms of error
exponents. A consequence of this result is that the GLRT is optimal in
differentiating a particular distribution from others in an exponential family.
It is also shown that the mismatched test has a significant advantage over the
Hoeffding test in terms of finite sample size performance. This advantage is
due to the difference in the asymptotic variances of the two test statistics
under the null hypothesis. In particular, the variance of the K-L divergence
grows linearly with the alphabet size, making the test impractical for
applications involving large alphabet distributions. The variance of the
mismatched divergence on the other hand grows linearly with the dimension of
the parameter space, and can hence be controlled through a prudent choice of
the function class defining the mismatched divergence.Comment: Accepted to IEEE Transactions on Information Theory, July 201
Sampling 2-D Signals on a Union of Lattices that Intersect on a Lattice
This paper presents new sufficient conditions under which a field (or image) can be perfectly reconstructed from its samples on a union of two lattices that share a common coarse lattice. In particular, if samples taken on the first lattice can be used to reconstruct a field bandlimited to some spectral support region, and likewise samples taken on the second lattice can reconstruct a field bandlimited to another spectral support region, then under certain conditions, a field bandlimited to the union of these two spectral regions can be reconstructed from its samples on the union of the two respective lattices. These results generalize a previous perfect reconstruction theorem for Manhattan sampling, where data is taken at high density along evenly spaced rows and columns of a rectangular grid. Additionally, a sufficient condition is given under which the Landau lower bound is achieved