87 research outputs found
Efficient Relaxations for Dense CRFs with Sparse Higher Order Potentials
Dense conditional random fields (CRFs) have become a popular framework for
modelling several problems in computer vision such as stereo correspondence and
multi-class semantic segmentation. By modelling long-range interactions, dense
CRFs provide a labelling that captures finer detail than their sparse
counterparts. Currently, the state-of-the-art algorithm performs mean-field
inference using a filter-based method but fails to provide a strong theoretical
guarantee on the quality of the solution. A question naturally arises as to
whether it is possible to obtain a maximum a posteriori (MAP) estimate of a
dense CRF using a principled method. Within this paper, we show that this is
indeed possible. We will show that, by using a filter-based method, continuous
relaxations of the MAP problem can be optimised efficiently using
state-of-the-art algorithms. Specifically, we will solve a quadratic
programming (QP) relaxation using the Frank-Wolfe algorithm and a linear
programming (LP) relaxation by developing a proximal minimisation framework. By
exploiting labelling consistency in the higher-order potentials and utilising
the filter-based method, we are able to formulate the above algorithms such
that each iteration has a complexity linear in the number of classes and random
variables. The presented algorithms can be applied to any labelling problem
using a dense CRF with sparse higher-order potentials. In this paper, we use
semantic segmentation as an example application as it demonstrates the ability
of the algorithm to scale to dense CRFs with large dimensions. We perform
experiments on the Pascal dataset to indicate that the presented algorithms are
able to attain lower energies than the mean-field inference method
Worst-case Optimal Submodular Extensions for Marginal Estimation
Submodular extensions of an energy function can be used to efficiently
compute approximate marginals via variational inference. The accuracy of the
marginals depends crucially on the quality of the submodular extension. To
identify the best possible extension, we show an equivalence between the
submodular extensions of the energy and the objective functions of linear
programming (LP) relaxations for the corresponding MAP estimation problem. This
allows us to (i) establish the worst-case optimality of the submodular
extension for Potts model used in the literature; (ii) identify the worst-case
optimal submodular extension for the more general class of metric labeling; and
(iii) efficiently compute the marginals for the widely used dense CRF model
with the help of a recently proposed Gaussian filtering method. Using synthetic
and real data, we show that our approach provides comparable upper bounds on
the log-partition function to those obtained using tree-reweighted message
passing (TRW) in cases where the latter is computationally feasible.
Importantly, unlike TRW, our approach provides the first practical algorithm to
compute an upper bound on the dense CRF model.Comment: Accepted to AISTATS 201
A Study of Lagrangean Decompositions and Dual Ascent Solvers for Graph Matching
We study the quadratic assignment problem, in computer vision also known as
graph matching. Two leading solvers for this problem optimize the Lagrange
decomposition duals with sub-gradient and dual ascent (also known as message
passing) updates. We explore s direction further and propose several additional
Lagrangean relaxations of the graph matching problem along with corresponding
algorithms, which are all based on a common dual ascent framework. Our
extensive empirical evaluation gives several theoretical insights and suggests
a new state-of-the-art any-time solver for the considered problem. Our
improvement over state-of-the-art is particularly visible on a new dataset with
large-scale sparse problem instances containing more than 500 graph nodes each.Comment: Added acknowledgment
A study of lagrangean decompositions and dual ascent solvers for graph matching
We study the quadratic assignment problem, in computer vision also known as graph matching. Two leading solvers for this problem optimize the Lagrange decomposition duals with sub-gradient and dual ascent (also known as message passing) updates. We explore this direction further and propose several additional Lagrangean relaxations of the graph matching problem along with corresponding algorithms, which are all based on a common dual ascent framework. Our extensive empirical evaluation gives several theoretical insights and suggests a new state-of-the-art anytime solver for the considered problem. Our improvement over state-of-the-art is particularly visible on a new dataset with large-scale sparse problem instances containing more than 500 graph nodes each
Higher-order inference in conditional random fields using submodular functions
Higher-order and dense conditional random fields (CRFs) are expressive graphical
models which have been very successful in low-level computer vision applications
such as semantic segmentation, and stereo matching. These models are able to
capture long-range interactions and higher-order image statistics much better
than pairwise CRFs. This expressive power comes at a price though - inference
problems in these models are computationally very demanding. This is a
particular challenge in computer vision, where fast inference is important and
the problem involves millions of pixels.
In this thesis, we look at how submodular functions can help us designing
efficient inference methods for higher-order and dense CRFs. Submodular
functions are special discrete functions that have important properties from
an optimisation perspective, and are closely related to convex functions. We
use submodularity in a two-fold manner: (a) to design efficient MAP inference
algorithm for a robust higher-order model that generalises the widely-used
truncated convex models, and (b) to glean insights into a recently proposed
variational inference algorithm which give us a principled approach for applying
it efficiently to higher-order and dense CRFs
Efficient Linear Programming for Dense CRFs
The fully connected conditional random field (CRF) with Gaussian pairwise
potentials 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 efficiently optimized. Specifically, for one block, the problem
decomposes into significantly smaller subproblems, each of which is defined
over a single pixel. For the other block, the problem is optimized via
conditional gradient descent. This has two advantages: 1) the conditional
gradient can be computed in a time linear in the number of pixels and labels;
and 2) the optimal step size can be computed analytically. Our experiments on
standard datasets provide compelling evidence that our approach outperforms all
existing baselines including the previous LP based approach for dense CRFs.Comment: 24 pages, 10 figures and 4 table
A Projected Gradient Descent Method for CRF Inference allowing End-To-End Training of Arbitrary Pairwise Potentials
Are we using the right potential functions in the Conditional Random Field
models that are popular in the Vision community? Semantic segmentation and
other pixel-level labelling tasks have made significant progress recently due
to the deep learning paradigm. However, most state-of-the-art structured
prediction methods also include a random field model with a hand-crafted
Gaussian potential to model spatial priors, label consistencies and
feature-based image conditioning.
In this paper, we challenge this view by developing a new inference and
learning framework which can learn pairwise CRF potentials restricted only by
their dependence on the image pixel values and the size of the support. Both
standard spatial and high-dimensional bilateral kernels are considered. Our
framework is based on the observation that CRF inference can be achieved via
projected gradient descent and consequently, can easily be integrated in deep
neural networks to allow for end-to-end training. It is empirically
demonstrated that such learned potentials can improve segmentation accuracy and
that certain label class interactions are indeed better modelled by a
non-Gaussian potential. In addition, we compare our inference method to the
commonly used mean-field algorithm. Our framework is evaluated on several
public benchmarks for semantic segmentation with improved performance compared
to previous state-of-the-art CNN+CRF models.Comment: Presented at EMMCVPR 2017 conferenc
- …