A Compact Linear Programming Relaxation for Binary Sub-modular MRF

We propose a novel compact linear programming (LP) relaxation for binary sub-modular MRF in the context of object segmentation. Our model is obtained by linearizing an $l_1^+$-norm derived from the quadratic programming (QP) form of the MRF energy. The resultant LP model contains significantly fewer variables and constraints compared to the conventional LP relaxation of the MRF energy. In addition, unlike QP which can produce ambiguous labels, our model can be viewed as a quasi-total-variation minimization problem, and it can therefore preserve the discontinuities in the labels. We further establish a relaxation bound between our LP model and the conventional LP model. In the experiments, we demonstrate our method for the task of interactive object segmentation. Our LP model outperforms QP when converting the continuous labels to binary labels using different threshold values on the entire Oxford interactive segmentation dataset. The computational complexity of our LP is of the same order as that of the QP, and it is significantly lower than the conventional LP relaxation

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

Barrier Frank-Wolfe for Marginal Inference

We introduce a globally-convergent algorithm for optimizing the tree-reweighted (TRW) variational objective over the marginal polytope. The algorithm is based on the conditional gradient method (Frank-Wolfe) and moves pseudomarginals within the marginal polytope through repeated maximum a posteriori (MAP) calls. This modular structure enables us to leverage black-box MAP solvers (both exact and approximate) for variational inference, and obtains more accurate results than tree-reweighted algorithms that optimize over the local consistency relaxation. Theoretically, we bound the sub-optimality for the proposed algorithm despite the TRW objective having unbounded gradients at the boundary of the marginal polytope. Empirically, we demonstrate the increased quality of results found by tightening the relaxation over the marginal polytope as well as the spanning tree polytope on synthetic and real-world instances.Comment: 25 pages, 12 figures, To appear in Neural Information Processing Systems (NIPS) 2015, Corrected reference and cleaned up bibliograph

Robust inversion and detection techniques for improved imaging performance

Thesis (Ph.D.)--Boston UniversityIn this thesis we aim to improve the performance of information extraction from imaging systems through three thrusts. First, we develop improved image formation methods for physics-based, complex-valued sensing problems. We propose a regularized inversion method that incorporates prior information about the underlying field into the inversion framework for ultrasound imaging. We use experimental ultrasound data to compute inversion results with the proposed formulation and compare it with conventional inversion techniques to show the robustness of the proposed technique to loss of data. Second, we propose methods that combine inversion and detection in a unified framework to improve imaging performance. This framework is applicable for cases where the underlying field is label-based such that each pixel of the underlying field can only assume values from a discrete, limited set. We consider this unified framework in the context of combinatorial optimization and propose graph-cut based methods that would result in label-based images, thereby eliminating the need for a separate detection step. Finally, we propose a robust method of object detection from microscopic nanoparticle images. In particular, we focus on a portable, low cost interferometric imaging platform and propose robust detection algorithms using tools from computer vision. We model the electromagnetic image formation process and use this model to create an enhanced detection technique. The effectiveness of the proposed technique is demonstrated using manually labeled ground-truth data. In addition, we extend these tools to develop a detection based autofocusing algorithm tailored for the high numerical aperture interferometric microscope

A learning framework for higher-order consistency models in multi-class pixel labeling problems

Recently, higher-order Markov random field (MRF) models have been successfully applied to problems in computer vision, especially scene understanding problems. One successful higher-order MRF model for scene understanding is the consistency model [Kohli and Kumar, 2010; Kohli et al., 2009] and earlier work by Ladicky et al. [2009, 2013] which contain higher-order potentials composed of lower linear envelope functions. In semantic image segmentation problems, which seek to identify the pixels of images with pre-defined labels of objects and backgrounds, this model encourages consistent label assignments over segmented regions of images. However, solving this MRF problem exactly is generally NP-hard; instead, efficient approximate inference algorithms are used. Furthermore, the lower linear envelope functions involve a number of parameters to learn. But, the typical cross-validation used for pairwise MRF models is not a practical method for estimating such a large number of parameters. Nevertheless, few works have proposed efficient learning methods to deal with the large number of parameters in these consistency models. In this thesis, we propose a unified inference and learning framework for the consistency model. We investigate various issues and present solutions for inference and learning with this higher-order MRF model as follows. First, we derive two variants of the consistency model for multi-class pixel labeling tasks. Our model defines an energy function scoring any given label assignments over an image. In order to perform Maximum a posteriori (MAP) inference in this model, we minimize the energy function using move-making algorithms in which the higher-order problems are transformed into tractable pairwise problems. Then, we employ a max-margin framework for learning optimal parameters. This learning framework provides a generalized approach for searching the large parameter space. Second, we propose a novel use of the Gaussian mixture model (GMM) for encoding consistency constraints over a large set of pixels. Here, we use various oversegmentation methods to define coherent regions for the consistency potentials. In general, Mean shift (MS) produces locally coherent regions, and GMM provides globally coherent regions, which do not need to be contiguous. Our model exploits both local and global information together and improves the labeling accuracy on real data sets. Accordingly, we use multiple higher-order terms associated with each over-segmentation method. Our learning framework allows us to deal with the large number of parameters involved with multiple higher-order terms. Next, we explore a dual decomposition (DD) method for our multi-class consistency model. The dual decomposition MRF (DD-MRF) is an alternative method for optimizing the energy function. In dual decomposition, a complex MRF problem is decomposed into many easy subproblems and we optimize the relaxed dual problem using a projected subgradient method. At convergence, we expect a global optimum in the dual space because it is a concave maximization problem. To optimize our higher-order DD-MRF exactly, we propose an exact minimization algorithm for solving the higher-order subproblems. Moreover, the minimization algorithm is much more efficient than graph-cuts. The dual decomposition approach also solves the max-margin learning problem by minimizing the dual losses derived from DD-MRF. Here, our minimization algorithm allows us to optimize the DD learning exactly and efficiently, which in most cases finds better parameters than the previous learning approach. Last, we focus on improving labeling accuracies of our higher-order model by combining mid-level features, which we call region features. The region features help customize the general envelope functions for individual segmented regions. By assigning specified weights to the envelope functions, we can choose subsets of highly likely labels for each segmented region. We train multiple classifiers with region features and aggregate them to increase prediction performance of possible labels for each region. Importantly, introducing these region features does not change the previous inference and learning algorithms

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 $X_i$ is represented by $\ell$ nodes (where $\ell$ is the number of labels) arranged in a column. However, this method in general requires $2\,\ell^2$ 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 $2\,\ell$ 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