121,596 research outputs found
Learning Laplacian Matrix in Smooth Graph Signal Representations
The construction of a meaningful graph plays a crucial role in the success of
many graph-based representations and algorithms for handling structured data,
especially in the emerging field of graph signal processing. However, a
meaningful graph is not always readily available from the data, nor easy to
define depending on the application domain. In particular, it is often
desirable in graph signal processing applications that a graph is chosen such
that the data admit certain regularity or smoothness on the graph. In this
paper, we address the problem of learning graph Laplacians, which is equivalent
to learning graph topologies, such that the input data form graph signals with
smooth variations on the resulting topology. To this end, we adopt a factor
analysis model for the graph signals and impose a Gaussian probabilistic prior
on the latent variables that control these signals. We show that the Gaussian
prior leads to an efficient representation that favors the smoothness property
of the graph signals. We then propose an algorithm for learning graphs that
enforces such property and is based on minimizing the variations of the signals
on the learned graph. Experiments on both synthetic and real world data
demonstrate that the proposed graph learning framework can efficiently infer
meaningful graph topologies from signal observations under the smoothness
prior
Discrete Signal Processing on Graphs: Frequency Analysis
Signals and datasets that arise in physical and engineering applications, as
well as social, genetics, biomolecular, and many other domains, are becoming
increasingly larger and more complex. In contrast to traditional time and image
signals, data in these domains are supported by arbitrary graphs. Signal
processing on graphs extends concepts and techniques from traditional signal
processing to data indexed by generic graphs. This paper studies the concepts
of low and high frequencies on graphs, and low-, high-, and band-pass graph
filters. In traditional signal processing, there concepts are easily defined
because of a natural frequency ordering that has a physical interpretation. For
signals residing on graphs, in general, there is no obvious frequency ordering.
We propose a definition of total variation for graph signals that naturally
leads to a frequency ordering on graphs and defines low-, high-, and band-pass
graph signals and filters. We study the design of graph filters with specified
frequency response, and illustrate our approach with applications to sensor
malfunction detection and data classification
A graph-based mathematical morphology reader
This survey paper aims at providing a "literary" anthology of mathematical
morphology on graphs. It describes in the English language many ideas stemming
from a large number of different papers, hence providing a unified view of an
active and diverse field of research
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