1,252 research outputs found
Approximation of multi-variable signals and systems : a tensor decomposition approach
Signals that evolve over multiple variables or indices occur in all fields of science and engineering. Measurements of the distribution of temperature across the globe during a certain period of time are an example of such a signal. Multivariable systems describe the evolution of signals over a spatial-temporal domain. The mathematical equations involved in such a description are called a model and this model dictates which values the signals can obtain as a function of time and space. In an industrial production setting, such mathematical models may be used to monitor the process or determine the control action required to reach a certain set-point. Since their evolution is over both space and time, multi-variable systems are described by Partial Differential Equations (PDEs). Generally, it is not the signals or systems themselves one is interested in, but the information they carry. The main numerical tools to extract system trajectories from the PDE description are Finite Element (FE) methods. FE models allow simulation of the model via a discretization scheme. The main problem with FE models is their complexity, which leads to large simulation time, making them not suitable for applications such as on-line monitoring of the process or model-based control design. Model reduction techniques aim to derive lowcomplexity replacement models from complex process models, in the setting of this work, from FE models. The approximations are achieved by projection on lower-dimensional subspaces of the signals and their dynamic laws. This work considers the computation of empirical projection spaces for signals and systems evolving over multi-dimensional domains. Formally, signal approximation may be viewed as a low-rank approximation problem. Whenever the signal under consideration is a function of multiple variables, low-rank approximations can be obtained via multi-linear functionals, tensors. It has been explained in this work that approximation of multi-variable systems also boils down to low-rank approximation problems.The first problem under consideration was that of finding low-rank approximations to tensors. For order-2 tensors, matrices, this problem is well understood. Generalization of these results to higher-order tensors is not straightforward. Finding tensor decompositions that allow suitable approximations after truncation is an active area of research. In this work a concept of rank for tensors, referred to as multi-linear or modal rank, has been considered. A new method has been defined to obtain modal rank decompositions to tensors, referred to as Tensor Singular Value Decomposition (TSVD). Properties of the TSVD that reflect its sparsity structure have been derived and low-rank approximation error bounds have been obtained for certain specific cases. An adaptation of the TSVD method has been proposed that may give better approximation results when not all modal directions are approximated. A numerical algorithm has been presented for the computation of the (dedicated) TSVD, which with a small adaptation can also be used to compute successive rank-one approximation to tensors. Finally, a simulation example has been included which demonstrates the methods proposed in this work and compares them to a well-known existing method. The concepts that were introduced and discussed with regard to signal approximation have been used in a system approximation context.We have considered the well-known model reduction method of Proper Orthogonal Decompositions (POD). We have shown how the basis functions inferred from the TSVD can be used to define projection spaces in POD. This adaptation is both a generalization and a restriction. It is a generalization because it allows POD to be used in a scalable fashion for problems with an arbitrary number of dependent and independent variables. However, it is also a restriction, since the projection spaces require a Cartesian product structure of the domain. The model reduction method that is thus obtained has been demonstrated on a benchmark example from chemical engineering. This application shows that the method is indeed feasible, and that the accuracy is comparable to existing methods for this example. In the final part of the thesis the problem of reconstruction and approximation of multi-dimensional signals was considered. Specifically, the problem of sampling and signal reconstruction for multi-variable signals with non-uniformly distributed sensors on a Cartesian domain has been considered. The central question of this chapter was that of finding a reconstruction of the original signal from its samples. A specific reconstruction map has been examined and conditions for exact reconstruction have been presented. In case that exact reconstruction was not possible, we have derived an expression for the reconstruction error
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
Perfect Reconstruction Two-Channel Wavelet Filter-Banks for Graph Structured Data
In this work we propose the construction of two-channel wavelet filterbanks
for analyzing functions defined on the vertices of any arbitrary finite
weighted undirected graph. These graph based functions are referred to as
graph-signals as we build a framework in which many concepts from the classical
signal processing domain, such as Fourier decomposition, signal filtering and
downsampling can be extended to graph domain. Especially, we observe a spectral
folding phenomenon in bipartite graphs which occurs during downsampling of
these graphs and produces aliasing in graph signals. This property of bipartite
graphs, allows us to design critically sampled two-channel filterbanks, and we
propose quadrature mirror filters (referred to as graph-QMF) for bipartite
graph which cancel aliasing and lead to perfect reconstruction. For arbitrary
graphs we present a bipartite subgraph decomposition which produces an
edge-disjoint collection of bipartite subgraphs. Graph-QMFs are then
constructed on each bipartite subgraph leading to "multi-dimensional" separable
wavelet filterbanks on graphs. Our proposed filterbanks are critically sampled
and we state necessary and sufficient conditions for orthogonality, aliasing
cancellation and perfect reconstruction. The filterbanks are realized by
Chebychev polynomial approximations.Comment: 32 pages double spaced 12 Figures, to appear in IEEE Transactions of
Signal Processin
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
- …