729 research outputs found
Spectrum-Adapted Tight Graph Wavelet and Vertex-Frequency Frames
We consider the problem of designing spectral graph filters for the
construction of dictionaries of atoms that can be used to efficiently represent
signals residing on weighted graphs. While the filters used in previous
spectral graph wavelet constructions are only adapted to the length of the
spectrum, the filters proposed in this paper are adapted to the distribution of
graph Laplacian eigenvalues, and therefore lead to atoms with better
discriminatory power. Our approach is to first characterize a family of systems
of uniformly translated kernels in the graph spectral domain that give rise to
tight frames of atoms generated via generalized translation on the graph. We
then warp the uniform translates with a function that approximates the
cumulative spectral density function of the graph Laplacian eigenvalues. We use
this approach to construct computationally efficient, spectrum-adapted, tight
vertex-frequency and graph wavelet frames. We give numerous examples of the
resulting spectrum-adapted graph filters, and also present an illustrative
example of vertex-frequency analysis using the proposed construction
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
Perfect Reconstruction Two-Channel Filter Banks on Arbitrary Graphs
This paper extends the existing theory of perfect reconstruction two-channel
filter banks from bipartite graphs to non-bipartite graphs. By generalizing the
concept of downsampling/upsampling we establish the frame of two-channel filter
bank on arbitrary connected, undirected and weighted graphs. Then the equations
for perfect reconstruction of the filter banks are presented and solved under
proper conditions. Algorithms for designing orthogonal and biorthogonal banks
are given and two typical orthogonal two-channel filter banks are calculated.
The locality and approximation properties of such filter banks are discussed
theoretically and experimentally.Comment: 33 pages,11 figures. This manuscript has been submitted to
ScienceDirect Applied and Computational Harmonic Analysis (ACHA) on Jan
27,202
Spline-Like Wavelet Filterbanks with Perfect Reconstruction on Arbitrary Graphs
In this work, we propose a class of spline-like wavelet filterbanks for graph
signals. These filterbanks possess the properties of critical sampling and
perfect reconstruction. Besides, the analysis filters are localized in the
graph domain because they are polynomials of the normalized adjacency matrix of
the graph. We generalize the spline-like filters in the literature so that they
have the ability to annihilate signals of some specified frequencies.
Optimization problems are posed for the analysis filters to approximate desired
responses. We conduct some experiments to demonstrate the good locality of the
proposed filters and the good performance of the filterbank in the denoising
task.Comment: 11 pages,12 figures. This work has been submitted to the IEEE for
possible publication. Copyright may be transferred without notice, after
which this version may no longer be accessibl
From spline wavelet to sampling theory on circulant graphs and beyond– conceiving sparsity in graph signal processing
Graph Signal Processing (GSP), as the field concerned with the extension of classical signal processing concepts to the graph domain, is still at the beginning on the path toward providing a generalized theory of signal processing. As such, this thesis aspires to conceive the theory of sparse representations on graphs by traversing the cornerstones of wavelet and sampling theory on graphs.
Beginning with the novel topic of graph spline wavelet theory, we introduce families of spline and e-spline wavelets, and associated filterbanks on circulant graphs, which lever- age an inherent vanishing moment property of circulant graph Laplacian matrices (and their parameterized generalizations), for the reproduction and annihilation of (exponen- tial) polynomial signals. Further, these families are shown to provide a stepping stone to generalized graph wavelet designs with adaptive (annihilation) properties. Circulant graphs, which serve as building blocks, facilitate intuitively equivalent signal processing concepts and operations, such that insights can be leveraged for and extended to more complex scenarios, including arbitrary undirected graphs, time-varying graphs, as well as associated signals with space- and time-variant properties, all the while retaining the focus on inducing sparse representations.
Further, we shift from sparsity-inducing to sparsity-leveraging theory and present a novel sampling and graph coarsening framework for (wavelet-)sparse graph signals, inspired by Finite Rate of Innovation (FRI) theory and directly building upon (graph) spline wavelet theory. At its core, the introduced Graph-FRI-framework states that any K-sparse signal residing on the vertices of a circulant graph can be sampled and perfectly reconstructed from its dimensionality-reduced graph spectral representation of minimum size 2K, while the structure of an associated coarsened graph is simultaneously inferred. Extensions to arbitrary graphs can be enforced via suitable approximation schemes.
Eventually, gained insights are unified in a graph-based image approximation framework which further leverages graph partitioning and re-labelling techniques for a maximally sparse graph wavelet representation.Open Acces
A graph signal processing solution for defective directed graphs
The main purpose of this thesis is to nd a method that allows to systematically adapt GSP
techniques so they can be used on most non-diagonalizable graph operators.
In Chapter 1 we begin by presenting the framework in which GSP is developed, giving
some basic de nitions in the eld of graph theory and in relation with graph signals. We also
present the concept of a Graph Fourier Tranform (GFT), which will be of great importance
in the proposed solution.
Chapter 2 presents the actual motivation of the research: Why the computation of the
GFT is problematic for some directed graphs, and the speci c cases in which this happen. We
will see that the issue can not be assigned to a very speci c graph topography, and therefore
it is important to develop solutions that can be applied to any directed graph.
In Chapter 3 we introduce our proposed new method, which can be used to form, based on
the spectral decomposition of a matrix obtained through its Schur decomposition, a complete
basis of vectors that can be used as a replacement of the previously mentioned Graph Fourier
Transform. The proposed method, the Graph Schur Transform (GST), aims to o er a valid
operator to perform a spectral decomposition of a graph that can be used even in the case of
defective matrices.
Finally, in Chapter 4 we study the main properties of the proposed method and compare
them with the corresponding properties o ered by the Di usion Wavelets design. In the last
section we prove, for a large set of directed graphs, that the GST provides a valid solution for
the proble
A Panorama on Multiscale Geometric Representations, Intertwining Spatial, Directional and Frequency Selectivity
The richness of natural images makes the quest for optimal representations in
image processing and computer vision challenging. The latter observation has
not prevented the design of image representations, which trade off between
efficiency and complexity, while achieving accurate rendering of smooth regions
as well as reproducing faithful contours and textures. The most recent ones,
proposed in the past decade, share an hybrid heritage highlighting the
multiscale and oriented nature of edges and patterns in images. This paper
presents a panorama of the aforementioned literature on decompositions in
multiscale, multi-orientation bases or dictionaries. They typically exhibit
redundancy to improve sparsity in the transformed domain and sometimes its
invariance with respect to simple geometric deformations (translation,
rotation). Oriented multiscale dictionaries extend traditional wavelet
processing and may offer rotation invariance. Highly redundant dictionaries
require specific algorithms to simplify the search for an efficient (sparse)
representation. We also discuss the extension of multiscale geometric
decompositions to non-Euclidean domains such as the sphere or arbitrary meshed
surfaces. The etymology of panorama suggests an overview, based on a choice of
partially overlapping "pictures". We hope that this paper will contribute to
the appreciation and apprehension of a stream of current research directions in
image understanding.Comment: 65 pages, 33 figures, 303 reference
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