103 research outputs found
Submodular relaxation for inference in Markov random fields
In this paper we address the problem of finding the most probable state of a
discrete Markov random field (MRF), also known as the MRF energy minimization
problem. The task is known to be NP-hard in general and its practical
importance motivates numerous approximate algorithms. We propose a submodular
relaxation approach (SMR) based on a Lagrangian relaxation of the initial
problem. Unlike the dual decomposition approach of Komodakis et al., 2011 SMR
does not decompose the graph structure of the initial problem but constructs a
submodular energy that is minimized within the Lagrangian relaxation. Our
approach is applicable to both pairwise and high-order MRFs and allows to take
into account global potentials of certain types. We study theoretical
properties of the proposed approach and evaluate it experimentally.Comment: This paper is accepted for publication in IEEE Transactions on
Pattern Analysis and Machine Intelligenc
A discriminative view of MRF pre-processing algorithms
While Markov Random Fields (MRFs) are widely used in computer vision, they
present a quite challenging inference problem. MRF inference can be accelerated
by pre-processing techniques like Dead End Elimination (DEE) or QPBO-based
approaches which compute the optimal labeling of a subset of variables. These
techniques are guaranteed to never wrongly label a variable but they often
leave a large number of variables unlabeled. We address this shortcoming by
interpreting pre-processing as a classification problem, which allows us to
trade off false positives (i.e., giving a variable an incorrect label) versus
false negatives (i.e., failing to label a variable). We describe an efficient
discriminative rule that finds optimal solutions for a subset of variables. Our
technique provides both per-instance and worst-case guarantees concerning the
quality of the solution. Empirical studies were conducted over several
benchmark datasets. We obtain a speedup factor of 2 to 12 over expansion moves
without preprocessing, and on difficult non-submodular energy functions produce
slightly lower energy.Comment: ICCV 201
Optimization of Markov Random Fields in Computer Vision
A large variety of computer vision tasks can be formulated using
Markov Random Fields (MRF). Except in certain special cases,
optimizing an MRF is intractable, due to a large number of
variables and complex dependencies between them. In this thesis,
we present new algorithms to perform inference in MRFs, that are
either more efficient (in terms of running time and/or memory
usage) or more effective (in terms of solution quality), than the
state-of-the-art methods.
First, we introduce a memory efficient max-flow algorithm for
multi-label submodular MRFs. In fact,
such MRFs have been shown to be optimally solvable using max-flow
based on an encoding of the labels proposed by Ishikawa, in which
each variable is represented by nodes (where
is the number of labels) arranged in a column. However, this
method in general requires edges for each pair of
neighbouring variables. This makes it inapplicable to realistic
problems with many variables and labels, due to excessive memory
requirement. By contrast, our max-flow algorithm stores
values per variable pair, requiring much less storage.
Consequently, our algorithm makes it possible to optimally solve
multi-label submodular problems involving large numbers of
variables and labels on a standard computer.
Next, we present a move-making style algorithm for multi-label
MRFs with robust non-convex priors. In particular, our algorithm
iteratively approximates the original MRF energy with an
appropriately weighted surrogate energy that is easier to
minimize. Furthermore, it guarantees that the original energy
decreases at each iteration. To this end, we consider the
scenario where the weighted surrogate energy is multi-label
submodular (i.e., it can be optimally minimized by max-flow), and
show that our algorithm then lets us handle of a large variety of
non-convex priors.
Finally, we consider the fully connected Conditional Random Field
(dense CRF) with Gaussian pairwise potentials that has proven
popular and effective for multi-class semantic segmentation.
While the energy of a dense CRF can be minimized accurately using
a Linear Programming (LP) relaxation, the state-of-the-art
algorithm is too slow to be useful in practice. To alleviate this
deficiency, we introduce an efficient LP minimization algorithm
for dense CRFs. To this end, we develop a proximal minimization
framework, where the dual of each proximal problem is optimized
via block-coordinate descent. We show that each block of
variables can be optimized in a time linear in the number of
pixels and labels. Consequently, our algorithm enables efficient
and effective optimization of dense CRFs with Gaussian pairwise
potentials.
We evaluated all our algorithms on standard energy minimization
datasets consisting of computer vision problems, such as stereo,
inpainting and semantic segmentation. The experiments at the end
of each chapter provide compelling evidence that all our
approaches are either more efficient or more effective than all
existing baselines
Playing with Duality: An Overview of Recent Primal-Dual Approaches for Solving Large-Scale Optimization Problems
Optimization methods are at the core of many problems in signal/image
processing, computer vision, and machine learning. For a long time, it has been
recognized that looking at the dual of an optimization problem may drastically
simplify its solution. Deriving efficient strategies which jointly brings into
play the primal and the dual problems is however a more recent idea which has
generated many important new contributions in the last years. These novel
developments are grounded on recent advances in convex analysis, discrete
optimization, parallel processing, and non-smooth optimization with emphasis on
sparsity issues. In this paper, we aim at presenting the principles of
primal-dual approaches, while giving an overview of numerical methods which
have been proposed in different contexts. We show the benefits which can be
drawn from primal-dual algorithms both for solving large-scale convex
optimization problems and discrete ones, and we provide various application
examples to illustrate their usefulness
MAP inference via Block-Coordinate Frank-Wolfe Algorithm
We present a new proximal bundle method for Maximum-A-Posteriori (MAP)
inference in structured energy minimization problems. The method optimizes a
Lagrangean relaxation of the original energy minimization problem using a multi
plane block-coordinate Frank-Wolfe method that takes advantage of the specific
structure of the Lagrangean decomposition. We show empirically that our method
outperforms state-of-the-art Lagrangean decomposition based algorithms on some
challenging Markov Random Field, multi-label discrete tomography and graph
matching problems
A Comparative Study of Modern Inference Techniques for Structured Discrete Energy Minimization Problems
International audienceSzeliski et al. published an influential study in 2006 on energy minimization methods for Markov Random Fields (MRF). This study provided valuable insights in choosing the best optimization technique for certain classes of problems. While these insights remain generally useful today, the phenomenal success of random field models means that the kinds of inference problems that have to be solved changed significantly. Specifically , the models today often include higher order interactions, flexible connectivity structures, large label-spaces of different car-dinalities, or learned energy tables. To reflect these changes, we provide a modernized and enlarged study. We present an empirical comparison of more than 27 state-of-the-art optimization techniques on a corpus of 2,453 energy minimization instances from diverse applications in computer vision. To ensure reproducibility, we evaluate all methods in the OpenGM 2 framework and report extensive results regarding runtime and solution quality. Key insights from our study agree with the results of Szeliski et al. for the types of models they studied. However, on new and challenging types of models our findings disagree and suggest that polyhedral methods and integer programming solvers are competitive in terms of runtime and solution quality over a large range of model types
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