687 research outputs found
Sparse Modeling for Image and Vision Processing
In recent years, a large amount of multi-disciplinary research has been
conducted on sparse models and their applications. In statistics and machine
learning, the sparsity principle is used to perform model selection---that is,
automatically selecting a simple model among a large collection of them. In
signal processing, sparse coding consists of representing data with linear
combinations of a few dictionary elements. Subsequently, the corresponding
tools have been widely adopted by several scientific communities such as
neuroscience, bioinformatics, or computer vision. The goal of this monograph is
to offer a self-contained view of sparse modeling for visual recognition and
image processing. More specifically, we focus on applications where the
dictionary is learned and adapted to data, yielding a compact representation
that has been successful in various contexts.Comment: 205 pages, to appear in Foundations and Trends in Computer Graphics
and Visio
On stable reconstructions from nonuniform Fourier measurements
We consider the problem of recovering a compactly-supported function from a
finite collection of pointwise samples of its Fourier transform taking
nonuniformly. First, we show that under suitable conditions on the sampling
frequencies - specifically, their density and bandwidth - it is possible to
recover any such function in a stable and accurate manner in any given
finite-dimensional subspace; in particular, one which is well suited for
approximating . In practice, this is carried out using so-called nonuniform
generalized sampling (NUGS). Second, we consider approximation spaces in one
dimension consisting of compactly supported wavelets. We prove that a linear
scaling of the dimension of the space with the sampling bandwidth is both
necessary and sufficient for stable and accurate recovery. Thus wavelets are up
to constant factors optimal spaces for reconstruction
Wavelets: mathematics and applications
The notion of wavelets is defined. It is briefly described {\it what} are
wavelets, {\it how} to use them, {\it when} we do need them, {\it why} they are
preferred and {\it where} they have been applied. Then one proceeds to the
multiresolution analysis and fast wavelet transform as a standard procedure for
dealing with discrete wavelets. It is shown which specific features of signals
(functions) can be revealed by this analysis, but can not be found by other
methods (e.g., by the Fourier expansion). Finally, some examples of practical
application are given (in particular, to analysis of multiparticle production}.
Rigorous proofs of mathematical statements are omitted, and the reader is
referred to the corresponding literature.Comment: 16 pages, 5 figures, Latex, Phys. Atom. Nuc
Convex and Network Flow Optimization for Structured Sparsity
We consider a class of learning problems regularized by a structured
sparsity-inducing norm defined as the sum of l_2- or l_infinity-norms over
groups of variables. Whereas much effort has been put in developing fast
optimization techniques when the groups are disjoint or embedded in a
hierarchy, we address here the case of general overlapping groups. To this end,
we present two different strategies: On the one hand, we show that the proximal
operator associated with a sum of l_infinity-norms can be computed exactly in
polynomial time by solving a quadratic min-cost flow problem, allowing the use
of accelerated proximal gradient methods. On the other hand, we use proximal
splitting techniques, and address an equivalent formulation with
non-overlapping groups, but in higher dimension and with additional
constraints. We propose efficient and scalable algorithms exploiting these two
strategies, which are significantly faster than alternative approaches. We
illustrate these methods with several problems such as CUR matrix
factorization, multi-task learning of tree-structured dictionaries, background
subtraction in video sequences, image denoising with wavelets, and topographic
dictionary learning of natural image patches.Comment: to appear in the Journal of Machine Learning Research (JMLR
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
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