110 research outputs found
Structured sampling and fast reconstruction of smooth graph signals
This work concerns sampling of smooth signals on arbitrary graphs. We first
study a structured sampling strategy for such smooth graph signals that
consists of a random selection of few pre-defined groups of nodes. The number
of groups to sample to stably embed the set of -bandlimited signals is
driven by a quantity called the \emph{group} graph cumulative coherence. For
some optimised sampling distributions, we show that sampling
groups is always sufficient to stably embed the set of -bandlimited signals
but that this number can be smaller -- down to -- depending on the
structure of the groups of nodes. Fast methods to approximate these sampling
distributions are detailed. Second, we consider -bandlimited signals that
are nearly piecewise constant over pre-defined groups of nodes. We show that it
is possible to speed up the reconstruction of such signals by reducing
drastically the dimension of the vectors to reconstruct. When combined with the
proposed structured sampling procedure, we prove that the method provides
stable and accurate reconstruction of the original signal. Finally, we present
numerical experiments that illustrate our theoretical results and, as an
example, show how to combine these methods for interactive object segmentation
in an image using superpixels
Random sampling of bandlimited signals on graphs
We study the problem of sampling k-bandlimited signals on graphs. We propose
two sampling strategies that consist in selecting a small subset of nodes at
random. The first strategy is non-adaptive, i.e., independent of the graph
structure, and its performance depends on a parameter called the graph
coherence. On the contrary, the second strategy is adaptive but yields optimal
results. Indeed, no more than O(k log(k)) measurements are sufficient to ensure
an accurate and stable recovery of all k-bandlimited signals. This second
strategy is based on a careful choice of the sampling distribution, which can
be estimated quickly. Then, we propose a computationally efficient decoder to
reconstruct k-bandlimited signals from their samples. We prove that it yields
accurate reconstructions and that it is also stable to noise. Finally, we
conduct several experiments to test these techniques
Sampling and Reconstruction of Graph Signals via Weak Submodularity and Semidefinite Relaxation
We study the problem of sampling a bandlimited graph signal in the presence
of noise, where the objective is to select a node subset of prescribed
cardinality that minimizes the signal reconstruction mean squared error (MSE).
To that end, we formulate the task at hand as the minimization of MSE subject
to binary constraints, and approximate the resulting NP-hard problem via
semidefinite programming (SDP) relaxation. Moreover, we provide an alternative
formulation based on maximizing a monotone weak submodular function and propose
a randomized-greedy algorithm to find a sub-optimal subset. We then derive a
worst-case performance guarantee on the MSE returned by the randomized greedy
algorithm for general non-stationary graph signals. The efficacy of the
proposed methods is illustrated through numerical simulations on synthetic and
real-world graphs. Notably, the randomized greedy algorithm yields an
order-of-magnitude speedup over state-of-the-art greedy sampling schemes, while
incurring only a marginal MSE performance loss
Graph Vertex Sampling with Arbitrary Graph Signal Hilbert Spaces
Graph vertex sampling set selection aims at selecting a set of ver-tices of a
graph such that the space of graph signals that can be reconstructed exactly
from those samples alone is maximal. In this context, we propose to extend
sampling set selection based on spectral proxies to arbitrary Hilbert spaces of
graph signals. Enabling arbitrary inner product of graph signals allows then to
better account for vertex importance on the graph for a sampling adapted to the
application. We first state how the change of inner product impacts sampling
set selection and reconstruction, and then apply it in the context of geometric
graphs to highlight how choosing an alternative inner product matrix can help
sampling set selection and reconstruction.Comment: Accepted at ICASSP 202
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