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

Efficient global minimization methods for variational problems in imaging and vision

Energy minimization has become one of the most important paradigms for formulating image processing and computer vision problems in a mathematical language. Energy minimization models have been developed in both the variational and discrete optimization community during the last 20-30 years. Some models have established themselves as fundamentally important and arise over a wide range of applications. One fundamental challenge is the optimization aspect. The most desirable models are often the most difficult to handle from an optimization perspective. Continuous optimization problems may be non-convex and contain many inferior local minima. Discrete optimization problems may be NP-hard, which means algorithms are unlikely to exist which can always compute exact solutions without an unreasonable amount of effort. This thesis contributes with efficient optimization methods which can compute global or close to global solutions to important energy minimization models in imaging and vision. New insights are given in both continuous and combinatorial optimization, as well as a strengthening of the relationships between these fields. One problem that is extensively studied is minimal perimeter partitioning problems with several regions, which arise naturally in e.g. image segmentation applications and is NP-hard in the discrete context. New methods are developed that can often compute global solutions and otherwise very close approximations to global solutions. Experiments show the new methods perform significantly better than earlier variational approaches, like the level set method, and earlier combinatorial optimization approaches. The new algorithms are significantly faster than previous continuous optimization approaches. In the discrete community, max-flow and min-cut (graph cuts) have gained huge popularity because they can efficiently compute global solutions to certain energy minimization models. It is shown that new types of problems can be solved exactly by max-flow and min-cut. Furthermore, variational generalizations of max-flow and min-cut are proposed which bring the global optimization property to the continuous setting, while avoiding grid bias and metrication errors which are major disadvantages of the discrete models. Convex optimization algorithms are derived from the variational max-flow models, which are very efficient and are more parallel friendly than traditional combinatorial algorithms

A Study on Continuous Max-Flow and Min-Cut Approaches

We propose and investigate novel max-flow models in the spatially continuous setting, with or without supervised constraints, under a comparative study of graph based max-flow / min-cut. We show that the continuous max-flow models correspond to their respective continuous min-cut models as primal and dual problems, and the continuous min-cut formulation without supervision constraints regards the well-known Chan-Esedoglu-Nikolova model [15] as a special case. In this respect, basic conceptions and terminologies applied by discrete max-flow / mincut are revisited under a new variational perspective. We prove that the associated nonconvex partitioning problems, unsupervised or supervised, can be solved globally and exactly via the proposed convex continuous max-flow and min-cut models. Moreover, we derive novel fast max-flow based algorithms whose convergence can be guaranteed by standard optimization theories. Experiments on image segmentation, both unsupervised and supervised, show that our continuous max-flow based algorithms outperform previous approaches in terms of efficiency and accuracy

Global Minimization for Continuous Multiphase Partitioning Problems Using a Dual Approach

This paper is devoted to the optimization problem of continuous multi-partitioning, or multi-labeling, which is based on a convex relaxation of the continuous Potts model. In contrast to previous efforts, which are tackling the optimal labeling problem in a direct manner, we first propose a novel dual model and then build up a corresponding dualitybased approach. By analyzing the dual formulation, sufficient conditions are derived which show that the relaxation is often exact, i.e. there exists optimal solutions that are also globally optimal to the original nonconvex Potts model. In order to deal with the nonsmooth dual problem, we develop a smoothing method based on the log-sum exponential function and indicate that such a smoothing approach leads to a novel smoothed primal-dual model and suggests labelings with maximum entropy. Such a smoothing method for the dual model also yields a new thresholding scheme to obtain approximate solutions. An expectation maximization like algorithm is proposed based on the smoothed formulation which is shown to be superior in efficiency compared to earlier approaches from continuous optimization. Numerical experiments also show that our method outperforms several competitive approaches in various aspects, such as lower energies and better visual quality

Efficient Global Minimization Methods for Sparsity and Regularity Promoting Optimization Problems in Imaging and Vision, 2014

The main goals of “Efficient Global Minimization Methods for Sparsity and Regularity Promoting Optimization Problems in Imaging and Vision, 2014” are design and analysis of regularity/sparsity promoting energy minimization models in image processing, computer vision and compressed sensing, and design and analysis of efficient numerical methods that can produce global or nearly global solutions of the resulting optimization problems. Imaging and vision are some of the core emerging technologies that are shaping our society. This can partly be explained by progressively better physical sensing devices for acquiring image data. Another important reason is the development of image processing and computer vision software for restoring, analyzing, simplifying and interpreting the image information. Energy minimization has been established as one of the most important paradigms for formulating problems in image processing and computer vision in a mathematical language. The problems can elegantly be formulated as finding the minimal state of some energy potential, which typically encodes the underlying assumption that the image data is regular/sparse, i.e. values at different image pixels are correlated. More recently, it has been realized that the expensive acquisition process can be greatly improved by incorporating the same assumption in an energy minimization framework, a field known as compressed sensing. A major challenge is to solve the resulting optimization problems efficiently. The most desirable models are often the most difficult to handle from an optimization perspective. Continuous optimization problems may be non-convex and contain many inferior local minima. Discrete optimization problems may be NP-hard, which means an algorithm is unlikely to exist which can always compute an exact solution without an unreasonable amount of effort. Even though the underlying problems are NP-hard, computational methods will be constructed that produce global solutions for practical input data, and otherwise close approximations. The data are presented in various articles

A Continuous Max-Flow Approach to Minimal Partitions with Label Cost Prior

This paper investigates a convex relaxation approach for minimum description length (MDL) based image partitioning or labeling, which proposes an energy functional regularized by the spatial smoothness prior joint with a penalty for the total number of appearences or labels, the so-called label cost prior. As common in recent studies of convex relaxation approaches, the total-variation term is applied to encode the spatial regularity of partition boundaries and the auxiliary label cost term is penalized by the sum of convex infinity norms of the labeling functions. We study the proposed such convex MDL based image partition model under a novel continuous flow maximization perspective, where we show that the label cost prior amounts to a relaxation of the flow conservation condition which is crucial to study the classical duality of max-flow and min-cut! To the best of our knowledge, it is new to demonstrate such connections between the relaxation of flow conservation and the penalty of the total number of active appearences. In addition, we show that the proposed continuous max-flow formulation also leads to a fast and reliable max-flow based algorithm to address the challenging convex optimization problem, which significantly outperforms the previous approach by direct convex programming, in terms of speed, computation load and handling large-scale images. Its numerical scheme can by easily implemented and accelerated by the advanced computation framework, e.g. GPU

A Fast Continuous Max-Flow Approach to Non-Convex Multilabeling Problems

This work addresses a class of multilabeling problems over a spatially continuous image domain, where the data fidelity term can be any bounded function, not necessarily convex. Two total variation based regularization terms are considered, the first favoring a linear relationship between the labels and the second independent of the label values (Pott’s model). In the spatially discrete setting, Ishikawa [33] showed that the first of these labeling problems can be solved exactly by standard max-flow and min-cut algorithms over specially designed graphs. We will propose a continuous analogue of Ishikawa’s graph construction [33] by formulating continuous max-flow and min-cut models over a specially designed domain. These max-flow and min-cut models are equivalent under a primal-dual perspective. They can be seen as exact convex relaxations of the original problem and can be used to compute global solutions. Fast continuous max-flow based algorithms are proposed based on the max-flow models whose efficiency and reliability can be validated by both standard optimization theories and experiments. In comparison to previous work [53, 52] on continuous generalization of Ishikawa’s construction, our approach differs in the max-flow dual treatment which leads to the following main advantages: A new theoretical framework which embeds the label order constraints implicitly and naturally results in optimal labeling functions taking values in any predefined finite label set; A more general thresholding theorem which allows to produce a larger set of non-unique solutions to the original problem; Numerical experiments show the new max-flow algorithms converge faster than the fast primal-dual algorithm of [53, 52]. The speedup factor is especially significant at high precisions. In the end, our dual formulation and algorithms are extended to a recently proposed convex relaxation of Pott’s model [50], thereby avoiding expensive iterative computations of projections without closed form solution

Reconstructing Open Surfaces via Graph-Cuts

A novel graph-cuts-basedmethod is proposed for reconstructing open surfaces from unordered point sets. Through a boolean operation on the crust around the data set, the open surface problem is translated to a watertight surface problem within a restricted region. Integrating the variationalmodel, Delaunay-based tetrahedralmesh framework and multi-phase technique, the proposed method can reconstruct open surfaces robustly and effectively. Furthermore, a surface reconstruction method with domain decomposition is presented, which is based on the new open surface reconstruction method. This method can handle more general surfaces, such as non-orientable surfaces. The algorithm is designed in a parallel-friendly way and necessary measures are taken to eliminate cracks at the interface between the subdomains. Numerical examples are included to demonstrate the robustness and effectiveness of the proposed method on watertight, open orientable, open non-orientable surfaces and combinations of such