Continuous Markov Random Field Research Articles

Lifting the Convex Conjugate in Lagrangian Relaxations: A Tractable Approach for Continuous Markov Random Fields

Dual decomposition approaches in nonconvex optimization may suffer from a duality gap. This poses a challenge when applying them directly to nonconvex problems such as MAP-inference in a Markov random field (MRF) with continuous state spaces. To eliminate such gaps, this paper considers a reformulation of the original nonconvex task in the space of measures. This infinite-dimensional reformulation is then approximated by a semi-infinite one, which is obtained via a piecewise polynomial discretization in the dual. We provide a geometric intuition behind the primal problem induced by the dual discretization and draw connections to optimization over moment spaces. In contrast to existing discretizations which suffer from a grid bias, we show that a piecewise polynomial discretization better preserves the continuous nature of our problem. Invoking results from optimal transport theory and convex algebraic geometry we reduce the semi-infinite program to a finite one and provide a practical implementation based on semidefinite programming. We show, experimentally and in theory, that the approach successfully reduces the duality gap. To showcase the scalability of our approach, we apply it to the stereo matching problem between two images.

Marginal Inference in Continuous Markov Random Fields Using Mixtures

Exact marginal inference in continuous graphical models is computationally challenging outside of a few special cases. Existing work on approximate inference has focused on approximately computing the messages as part of the loopy belief propagation algorithm either via sampling methods or moment matching relaxations. In this work, we present an alternative family of approximations that, instead of approximating the messages, approximates the beliefs in the continuous Bethe free energy using mixture distributions. We show that these types of approximations can be combined with numerical quadrature to yield algorithms with both theoretical guarantees on the quality of the approximation and significantly better practical performance in a variety of applications that are challenging for current state-of-the-art methods.

HPM-TDP: An efficient hierarchical PatchMatch depth estimation approach using tree dynamic programming

Accurate and efficient estimation of the dense depth information from a pair of stereo images is a key step for many applications such as digital surface model production, 3D reconstruction and visualization, autonomous driving, and robotic navigation. Although great progress has been achieved in stereo matching over the past decade, the matching difficulties in poor and repetitive texture regions remain an issue. Aiming at solving the shortcomings of the current methods, this paper proposes HPM-TDP, which is an efficient hierarchical PatchMatch depth estimation approach that integrates a coarse-to-fine image pyramid strategy with a continuous Markov random field (MRF)-based global energy optimization framework, and minimizes the energy function by combining a hierarchical PatchMatch (HPM) framework and local α-expansion based tree dynamic programming (TDP). Firstly, the coarse-to-fine image pyramid strategy is integrated with the PatchMatch filter algorithm to quickly generate the hierarchical disparity plane prior for initializing each pixel’s disparity plane of the energy function optimization. Secondly, a multi-resolution cost aggregation strategy is adopted to boost the robustness of the matching cost function in the poor and repetitive texture areas. Finally, the HPM framework and local α-expansion based TDP are adopted to solve the non-submodular energy optimization problem, resulting in a globally optimized disparity plane map. Three benchmark datasets—the Middlebury 3.0, KITTI 2015, and Vaihingen datasets—were used to test the performance of HPM-TDP. The comprehensive experimental results demonstrate that HPM-TDP obtains a good performance on all datasets in terms of the (“Out-Noc”, “Avg-Noc”, “Out-All”, “Avg-All”) of (15.45%, 4.16px, 24.26%, 12.14px) and (5.46%, 1.20px, 6.55%, 1.54px) for Middlebury 3.0 and KITTI 2015 training datasets, and the (“Out-All”, “Avg-All”) of (26.32%, 4.04px) for Vaihingen dataset, respectively.

Learning to refine depth for robust stereo estimation

Traditional depth estimation from stereo images is usually formulated as a patch-matching problem, which requires post-processing stages to impose smoothness and handle depth discontinuities and occlusions. While recent deep network approaches directly learn a regressor for the entire disparity map, they still suffer from large errors near the depth discontinuities. In this paper, we propose a novel method to refine the disparity maps generated by deep regression networks. Instead of relying on ad hoc post-processing, we learn a unified deep network model that predicts a confidence map and the disparity gradients from the learned feature representation in regression networks. We integrate the initial disparity estimation, the confidence map and the disparity gradients into a continuous Markov Random Field (MRF) for depth refinement, which is capable of representing rich surface structures. Our disparity MRF model can be solved via efficient global optimization in a closed form. We evaluate our approach on both synthetic and real-world datasets, and the results show it achieves the state-of-art performance and produces more structure-preserving disparity maps with smaller errors in the neighborhood of depth boundaries.

Solving Non-parametric Inverse Problem in Continuous Markov Random Field Using Loopy Belief Propagation

In this paper, we address the inverse problem, or the statistical machine learning problem, in Markov random fields with a non-parametric pair-wise energy function with continuous variables. The inverse problem is formulated by maximum likelihood estimation. The exact treatment of maximum likelihood estimation is intractable because of two problems: (1) it includes the evaluation of the partition function and (2) it is formulated in the form of functional optimization. We avoid Problem (1) by using Bethe approximation. Bethe approximation is an approximation technique equivalent to the loopy belief propagation. Problem (2) can be solved by using orthonormal function expansion. Orthonormal function expansion can reduce a functional optimization problem to a function optimization problem. Our method can provide an analytic form of the solution of the inverse problem within the framework of Bethe approximation.

FoxPSL: A Fast, Optimized and eXtended PSL implementation

Abstract In this paper, we describe foxPSL , a fast, optimized and extended implementation of Probabilistic Soft Logic (PSL) based on the distributed graph processing framework Signal/Collect . PSL is one of the leading formalisms of statistical relational learning, a recently developed field of machine learning that aims at representing both uncertainty and rich relational structures, usually by combining logical representations with probabilistic graphical models. PSL can be seen as both a probabilistic logic and a template language for hinge-loss Markov Random Fields, a type of continuous Markov Random fields (MRF) in which Maximum a Posteriori inference is very efficient, since it can be formulated as a constrained convex minimization problem, as opposed to a discrete optimization problem for standard MRFs. From the logical perspective, a key feature of PSL is the capability to represent soft truth values, allowing the expression of complex domain knowledge, like degrees of truth, in parallel with uncertainty. foxPSL supports the full PSL pipeline from problem definition to a distributed solver that implements the Alternating Direction Method of Multipliers (ADMM) consensus optimization. It provides a Domain Specific Language that extends standard PSL with a class system and existential quantifiers, allowing for efficient grounding. Moreover, it implements a series of configurable optimizations, like optimized grounding of constraints and lazy inference, that improve grounding and inference time. We perform an extensive evaluation, comparing the performance of foxPSL to a state-of-the-art implementation of ADMM consensus optimization in GraphLab, and show an improvement in both inference time and solution quality. Moreover, we evaluate the impact of the optimizations on the execution time and discuss the trade-offs related to each optimization.

A random cluster algorithm for some continuous Markov random fields

Based on a random cluster representation, the Swendsen–Wang algorithm for the Ising and Potts distributions is extended to a class of continuous Markov random fields. The algorithm can be described briefly as follows. A given configuration is decomposed into clusters. Probabilities for flipping the values of the random variables in each cluster are calculated. According to these probabilities, values of all the random variables in each cluster will be either updated or kept unchanged and this is done independently across the clusters. A new configuration is then obtained. We will show through a simulation study that, like the Swendsen–Wang algorithm in the case of Ising and Potts distributions, the cluster algorithm here also outperforms the Gibbs sampler in beating the critical slowing down for some strongly correlated Markov random fields.

Adaptive partial volume classification of MRI data

Tomographic biomedical images are commonly affected by an imaging artefact known as the partial volume (PV) effect. The PV effect produces voxels composed of a mixture of tissues in anatomical magnetic resonance imaging (MRI) data resulting in a continuity of these tissue classes. Anatomical MRI data typically consist of a number of contiguous regions of tissues or even contiguous regions of PV voxels. Furthermore discontinuities exist between the boundaries of these contiguous image regions. The work presented here probabilistically models the PV effect using spatial regularization in the form of continuous Markov random fields (MRFs) to classify anatomical MRI brain data, simulated and real. A unique approach is used to adaptively control the amount of spatial regularization imposed by the MRF. Spatially derived image gradient magnitude is used to identify the discontinuities between image regions of contiguous tissue voxels and PV voxels, imposing variable amounts of regularization determined by simulation. Markov chain Monte Carlo (MCMC) is used to simulate the posterior distribution of the probabilistic image model. Promising quantitative results are presented for PV classification of simulated and real MRI data of the human brain.

Spatial Bayesian Variable Selection With Application to Functional Magnetic Resonance Imaging

We propose a procedure to undertake Bayesian variable selection and model averaging for a series of regressions located on a lattice. For those regressors that are in common in the regressions, we consider using an Ising prior to smooth spatially the indicator variables representing whether or not the variable is zero or nonzero in each regression. This smooths spatially the probabilities that each independent variable is nonzero in each regression and indirectly smooths spatially the regression coefficients. We discuss how single-site sampling schemes can be used to evaluate the joint posterior distribution. The approach is applied to the problem of functional magnetic resonance imaging in medical statistics, where massive datasets arise that require prompt processing. Here the Ising prior with a three-dimensional neighborhood structure is used to smooth spatially activation maps from regression models of blood oxygenation. The Ising prior also has the advantage of allowing incorporation of anatomic prior information through the external field. Using a visual experiment, we show how a single-site sampling scheme can provide rapid evaluation of the posterior activation maps and activation amplitudes. The approach is shown to result in maps that are superior to those produced by a recent Bayesian approach using a continuous Markov random field for the activation amplitude.