6,563 research outputs found
Signals on Graphs: Uncertainty Principle and Sampling
In many applications, the observations can be represented as a signal defined
over the vertices of a graph. The analysis of such signals requires the
extension of standard signal processing tools. In this work, first, we provide
a class of graph signals that are maximally concentrated on the graph domain
and on its dual. Then, building on this framework, we derive an uncertainty
principle for graph signals and illustrate the conditions for the recovery of
band-limited signals from a subset of samples. We show an interesting link
between uncertainty principle and sampling and propose alternative signal
recovery algorithms, including a generalization to frame-based reconstruction
methods. After showing that the performance of signal recovery algorithms is
significantly affected by the location of samples, we suggest and compare a few
alternative sampling strategies. Finally, we provide the conditions for perfect
recovery of a useful signal corrupted by sparse noise, showing that this
problem is also intrinsically related to vertex-frequency localization
properties.Comment: This article is the revised version submitted to the IEEE
Transactions on Signal Processing on May, 2016; first revision was submitted
on January, 2016; original manuscript was submitted on July, 2015. The work
includes 16 pages, 8 figure
Sampling and Reconstruction of Signals in a Reproducing Kernel Subspace of
In this paper, we consider sampling and reconstruction of signals in a
reproducing kernel subspace of L^p(\Rd), 1\le p\le \infty, associated with an
idempotent integral operator whose kernel has certain off-diagonal decay and
regularity. The space of -integrable non-uniform splines and the
shift-invariant spaces generated by finitely many localized functions are our
model examples of such reproducing kernel subspaces of L^p(\Rd). We show that
a signal in such reproducing kernel subspaces can be reconstructed in a stable
way from its samples taken on a relatively-separated set with sufficiently
small gap. We also study the exponential convergence, consistency, and the
asymptotic pointwise error estimate of the iterative approximation-projection
algorithm and the iterative frame algorithm for reconstructing a signal in
those reproducing kernel spaces from its samples with sufficiently small gap
Surface Networks
We study data-driven representations for three-dimensional triangle meshes,
which are one of the prevalent objects used to represent 3D geometry. Recent
works have developed models that exploit the intrinsic geometry of manifolds
and graphs, namely the Graph Neural Networks (GNNs) and its spectral variants,
which learn from the local metric tensor via the Laplacian operator. Despite
offering excellent sample complexity and built-in invariances, intrinsic
geometry alone is invariant to isometric deformations, making it unsuitable for
many applications. To overcome this limitation, we propose several upgrades to
GNNs to leverage extrinsic differential geometry properties of
three-dimensional surfaces, increasing its modeling power.
In particular, we propose to exploit the Dirac operator, whose spectrum
detects principal curvature directions --- this is in stark contrast with the
classical Laplace operator, which directly measures mean curvature. We coin the
resulting models \emph{Surface Networks (SN)}. We prove that these models
define shape representations that are stable to deformation and to
discretization, and we demonstrate the efficiency and versatility of SNs on two
challenging tasks: temporal prediction of mesh deformations under non-linear
dynamics and generative models using a variational autoencoder framework with
encoders/decoders given by SNs
Uncertainty Principles and Balian-Low type Theorems in Principal Shift-Invariant Spaces
In this paper, we consider the time-frequency localization of the generator
of a principal shift-invariant space on the real line which has additional
shift-invariance. We prove that if a principal shift-invariant space on the
real line is translation-invariant then any of its orthonormal (or Riesz)
generators is non-integrable. However, for any , there exist principal
shift-invariant spaces on the real line that are also \nZ-invariant with an
integrable orthonormal (or a Riesz) generator , but satisfies
for any
and its Fourier transform cannot decay as fast as
for any . Examples are constructed to demonstrate that the above decay
properties for the orthormal generator in the time domain and in the frequency
domain are optimal
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