7,069 research outputs found
Laplacian Matrix Learning for Smooth Graph Signal Representation
The construction of a meaningful graph plays a crucial role in the emerging field of signal processing on graphs. In this paper, we address the problem of learning graph Laplacians, which is similar to learning graph topologies, such that the input data form graph signals with smooth variations on the resulting topology. We adopt a factor analysis model for the graph signals and impose a Gaussian probabilistic prior on the latent variables that control these graph signals. We show that the Gaussian prior leads to an efficient representation that favours the smoothness property of the graph signals, and propose an algorithm for learning graphs that enforce such property. Experiments demonstrate that the proposed framework can efficiently infer meaningful graph topologies from only the signal observations
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
Graph Signal Processing: Overview, Challenges and Applications
Research in Graph Signal Processing (GSP) aims to develop tools for
processing data defined on irregular graph domains. In this paper we first
provide an overview of core ideas in GSP and their connection to conventional
digital signal processing. We then summarize recent developments in developing
basic GSP tools, including methods for sampling, filtering or graph learning.
Next, we review progress in several application areas using GSP, including
processing and analysis of sensor network data, biological data, and
applications to image processing and machine learning. We finish by providing a
brief historical perspective to highlight how concepts recently developed in
GSP build on top of prior research in other areas.Comment: To appear, Proceedings of the IEE
Learning parametric dictionaries for graph signals
In sparse signal representation, the choice of a dictionary often involves a
tradeoff between two desirable properties -- the ability to adapt to specific
signal data and a fast implementation of the dictionary. To sparsely represent
signals residing on weighted graphs, an additional design challenge is to
incorporate the intrinsic geometric structure of the irregular data domain into
the atoms of the dictionary. In this work, we propose a parametric dictionary
learning algorithm to design data-adapted, structured dictionaries that
sparsely represent graph signals. In particular, we model graph signals as
combinations of overlapping local patterns. We impose the constraint that each
dictionary is a concatenation of subdictionaries, with each subdictionary being
a polynomial of the graph Laplacian matrix, representing a single pattern
translated to different areas of the graph. The learning algorithm adapts the
patterns to a training set of graph signals. Experimental results on both
synthetic and real datasets demonstrate that the dictionaries learned by the
proposed algorithm are competitive with and often better than unstructured
dictionaries learned by state-of-the-art numerical learning algorithms in terms
of sparse approximation of graph signals. In contrast to the unstructured
dictionaries, however, the dictionaries learned by the proposed algorithm
feature localized atoms and can be implemented in a computationally efficient
manner in signal processing tasks such as compression, denoising, and
classification
Graph learning under sparsity priors
Graph signals offer a very generic and natural representation for data that
lives on networks or irregular structures. The actual data structure is however
often unknown a priori but can sometimes be estimated from the knowledge of the
application domain. If this is not possible, the data structure has to be
inferred from the mere signal observations. This is exactly the problem that we
address in this paper, under the assumption that the graph signals can be
represented as a sparse linear combination of a few atoms of a structured graph
dictionary. The dictionary is constructed on polynomials of the graph
Laplacian, which can sparsely represent a general class of graph signals
composed of localized patterns on the graph. We formulate a graph learning
problem, whose solution provides an ideal fit between the signal observations
and the sparse graph signal model. As the problem is non-convex, we propose to
solve it by alternating between a signal sparse coding and a graph update step.
We provide experimental results that outline the good graph recovery
performance of our method, which generally compares favourably to other recent
network inference algorithms
Graph Spectral Image Processing
Recent advent of graph signal processing (GSP) has spurred intensive studies
of signals that live naturally on irregular data kernels described by graphs
(e.g., social networks, wireless sensor networks). Though a digital image
contains pixels that reside on a regularly sampled 2D grid, if one can design
an appropriate underlying graph connecting pixels with weights that reflect the
image structure, then one can interpret the image (or image patch) as a signal
on a graph, and apply GSP tools for processing and analysis of the signal in
graph spectral domain. In this article, we overview recent graph spectral
techniques in GSP specifically for image / video processing. The topics covered
include image compression, image restoration, image filtering and image
segmentation
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