43 research outputs found
Multiclass Data Segmentation using Diffuse Interface Methods on Graphs
We present two graph-based algorithms for multiclass segmentation of
high-dimensional data. The algorithms use a diffuse interface model based on
the Ginzburg-Landau functional, related to total variation compressed sensing
and image processing. A multiclass extension is introduced using the Gibbs
simplex, with the functional's double-well potential modified to handle the
multiclass case. The first algorithm minimizes the functional using a convex
splitting numerical scheme. The second algorithm is a uses a graph adaptation
of the classical numerical Merriman-Bence-Osher (MBO) scheme, which alternates
between diffusion and thresholding. We demonstrate the performance of both
algorithms experimentally on synthetic data, grayscale and color images, and
several benchmark data sets such as MNIST, COIL and WebKB. We also make use of
fast numerical solvers for finding the eigenvectors and eigenvalues of the
graph Laplacian, and take advantage of the sparsity of the matrix. Experiments
indicate that the results are competitive with or better than the current
state-of-the-art multiclass segmentation algorithms.Comment: 14 page
Simplified Energy Landscape for Modularity Using Total Variation
Networks capture pairwise interactions between entities and are frequently
used in applications such as social networks, food networks, and protein
interaction networks, to name a few. Communities, cohesive groups of nodes,
often form in these applications, and identifying them gives insight into the
overall organization of the network. One common quality function used to
identify community structure is modularity. In Hu et al. [SIAM J. App. Math.,
73(6), 2013], it was shown that modularity optimization is equivalent to
minimizing a particular nonconvex total variation (TV) based functional over a
discrete domain. They solve this problem, assuming the number of communities is
known, using a Merriman, Bence, Osher (MBO) scheme.
We show that modularity optimization is equivalent to minimizing a convex
TV-based functional over a discrete domain, again, assuming the number of
communities is known. Furthermore, we show that modularity has no convex
relaxation satisfying certain natural conditions. We therefore, find a
manageable non-convex approximation using a Ginzburg Landau functional, which
provably converges to the correct energy in the limit of a certain parameter.
We then derive an MBO algorithm with fewer hand-tuned parameters than in Hu et
al. and which is 7 times faster at solving the associated diffusion equation
due to the fact that the underlying discretization is unconditionally stable.
Our numerical tests include a hyperspectral video whose associated graph has
2.9x10^7 edges, which is roughly 37 times larger than was handled in the paper
of Hu et al.Comment: 25 pages, 3 figures, 3 tables, submitted to SIAM J. App. Mat
Variational methods and its applications to computer vision
Many computer vision applications such as image segmentation can be formulated in a ''variational'' way as energy minimization problems. Unfortunately, the computational task of minimizing these energies is usually difficult as it generally involves non convex functions in a space with thousands of dimensions and often the associated combinatorial problems are NP-hard to solve. Furthermore, they are ill-posed inverse problems and therefore are extremely sensitive to perturbations (e.g. noise). For this reason in order to compute a physically reliable approximation from given noisy data, it is necessary to incorporate into the mathematical model appropriate regularizations that require complex computations.
The main aim of this work is to describe variational segmentation methods that are particularly effective for curvilinear structures. Due to their complex geometry, classical regularization techniques cannot be adopted because they lead to the loss of most of low contrasted details. In contrast, the proposed method not only better preserves curvilinear structures, but also reconnects some parts that may have been disconnected by noise. Moreover, it can be easily extensible to graphs and successfully applied to different types of data such as medical imagery (i.e. vessels, hearth coronaries etc), material samples (i.e. concrete) and satellite signals (i.e. streets, rivers etc.). In particular, we will show results and performances about an implementation targeting new generation of High Performance Computing (HPC) architectures where different types of coprocessors cooperate. The involved dataset consists of approximately 200 images of cracks, captured in three different tunnels by a robotic machine designed for the European ROBO-SPECT project.Open Acces
Stochastic Block Models are a Discrete Surface Tension
Networks, which represent agents and interactions between them, arise in
myriad applications throughout the sciences, engineering, and even the
humanities. To understand large-scale structure in a network, a common task is
to cluster a network's nodes into sets called "communities", such that there
are dense connections within communities but sparse connections between them. A
popular and statistically principled method to perform such clustering is to
use a family of generative models known as stochastic block models (SBMs). In
this paper, we show that maximum likelihood estimation in an SBM is a network
analog of a well-known continuum surface-tension problem that arises from an
application in metallurgy. To illustrate the utility of this relationship, we
implement network analogs of three surface-tension algorithms, with which we
successfully recover planted community structure in synthetic networks and
which yield fascinating insights on empirical networks that we construct from
hyperspectral videos.Comment: to appear in Journal of Nonlinear Scienc