457 research outputs found
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
Network Flow Algorithms for Structured Sparsity
We consider a class of learning problems that involve a structured
sparsity-inducing norm defined as the sum of -norms over groups of
variables. Whereas a lot of effort has been put in developing fast optimization
methods when the groups are disjoint or embedded in a specific hierarchical
structure, we address here the case of general overlapping groups. To this end,
we show that the corresponding optimization problem is related to network flow
optimization. More precisely, the proximal problem associated with the norm we
consider is dual to a quadratic min-cost flow problem. We propose an efficient
procedure which computes its solution exactly in polynomial time. Our algorithm
scales up to millions of variables, and opens up a whole new range of
applications for structured sparse models. We present several experiments on
image and video data, demonstrating the applicability and scalability of our
approach for various problems.Comment: accepted for publication in Adv. Neural Information Processing
Systems, 201
Geometric Surface Processing and Virtual Modeling
In this work we focus on two main topics "Geometric Surface Processing" and "Virtual Modeling". The inspiration and coordination for most of the research work contained in the thesis has been driven by the project New Interactive and Innovative Technologies for CAD (NIIT4CAD), funded by the European Eurostars Programme. NIIT4CAD has the ambitious aim of overcoming the limitations of the traditional approach to surface modeling of current 3D CAD systems by introducing new methodologies and technologies based on subdivision surfaces
in a new virtual modeling framework. These innovations will allow designers and engineers to transform quickly and intuitively an idea of shape in a high-quality geometrical model suited for engineering and manufacturing purposes.
One of the objective of the thesis is indeed the reconstruction and modeling of surfaces, representing arbitrary topology objects, starting from 3D irregular curve networks acquired through an ad-hoc smart-pen device.
The thesis is organized in two main parts: "Geometric Surface Processing" and "Virtual Modeling". During the development of the geometric pipeline in our Virtual Modeling system, we faced many challenges that captured our interest and opened new areas of research and experimentation.
In the first part, we present these theories and some applications to Geometric Surface Processing.
This allowed us to better formalize and give a broader understanding on some of the techniques used in our latest advancements on virtual modeling and surface reconstruction.
The research on both topics led to important results that have been published and presented in articles and conferences of international relevance
Topological approaches for 3D object processing and applications
The great challenge in 3D object processing is to devise computationally efficient algorithms for recovering 3D models contaminated by noise and preserving their geometrical structure. The first problem addressed in this thesis is object denoising formulated in the discrete variational framework. We introduce a 3D mesh denoising method based on kernel density estimation. The proposed approach is able to reduce the over-smoothing effect and effectively remove undesirable noise while preserving prominent geometric features of a 3D mesh such as sharp features and fine details. The feasibility of the approach is demonstrated through extensive experiments. The rest of the thesis is devoted to a joint exploitation of geometry and topology of 3D objects for as parsimonious as possible representation of models and its subsequent application in object modeling, compression, and hashing problems. We introduce a 3D mesh compression technique using the centroidal mesh neighborhood information. The key idea is to apply eigen-decomposition to the mesh umbrella matrix, and then discard the smallest eigenvalues/eigenvectors in order to reduce the dimensionality of the new spectral basis so that most of the energy is concentrated in the low frequency coefficients. We also present a hashing technique for 3D models using spectral graph theory and entropic spanning trees by partitioning a 3D triangle mesh into an ensemble of submeshes, and then applying eigen-decomposition to the Laplace-Beltrami matrix of each sub-mesh, followed by computing the hash value of each sub-mesh. Moreover, we introduce several statistical distributions to analyze the topological properties of 3D objects. These probabilistic distributions provide useful information about the way 3D mesh models are connected. Illustrating experiments with synthetic and real data are provided to demonstrate the feasibility and the much improved performance of the proposed approaches in 3D object compression, hashing, and modeling
Deep Graph-Convolutional Image Denoising
Non-local self-similarity is well-known to be an effective prior for the
image denoising problem. However, little work has been done to incorporate it
in convolutional neural networks, which surpass non-local model-based methods
despite only exploiting local information. In this paper, we propose a novel
end-to-end trainable neural network architecture employing layers based on
graph convolution operations, thereby creating neurons with non-local receptive
fields. The graph convolution operation generalizes the classic convolution to
arbitrary graphs. In this work, the graph is dynamically computed from
similarities among the hidden features of the network, so that the powerful
representation learning capabilities of the network are exploited to uncover
self-similar patterns. We introduce a lightweight Edge-Conditioned Convolution
which addresses vanishing gradient and over-parameterization issues of this
particular graph convolution. Extensive experiments show state-of-the-art
performance with improved qualitative and quantitative results on both
synthetic Gaussian noise and real noise
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
Courbure discrÚte : théorie et applications
International audienceThe present volume contains the proceedings of the 2013 Meeting on discrete curvature, held at CIRM, Luminy, France. The aim of this meeting was to bring together researchers from various backgrounds, ranging from mathematics to computer science, with a focus on both theory and applications. With 27 invited talks and 8 posters, the conference attracted 70 researchers from all over the world. The challenge of finding a common ground on the topic of discrete curvature was met with success, and these proceedings are a testimony of this wor
Deep Graph-Convolutional Image Denoising
3noNon-local self-similarity is well-known to be an effective prior for the image denoising problem. However, little work has been done to incorporate it in convolutional neural networks, which surpass non-local model-based methods despite only exploiting local information. In this paper, we propose a novel end-to-end trainable neural network architecture employing layers based on graph convolution operations, thereby creating neurons with non-local receptive fields. The graph convolution operation generalizes the classic convolution to arbitrary graphs. In this work, the graph is dynamically computed from similarities among the hidden features of the network, so that the powerful representation learning capabilities of the network are exploited to uncover self-similar patterns. We introduce a lightweight Edge-Conditioned Convolution which addresses vanishing gradient and over-parameterization issues of this particular graph convolution. Extensive experiments show state-of-the-art performance with improved qualitative and quantitative results on both synthetic Gaussian noise and real noise.partially_openopenValsesia D.; Fracastoro G.; Magli E.Valsesia, D.; Fracastoro, G.; Magli, E
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