Abstract. 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. [19,16]. There has been a vast amount of research on this topic during the last years [8,10]. Other discrete optimization methods include message passing [45,29] and linear programming [33] etc. One main drawback of such graph-based approaches is the grid bias. The interaction potential penalizes some spatial directions more than other, which leads to visible artifacts in computational results. Reducing such metrication errors can be done by considering more neighboring nodes [9,28] or high-order interaction potentials [27,25]. However, this either results in a heavy memory load and high computation cost or amounts to a more complex algorithmic scheme, e.g. QPBO [7,30].Recent studies [15] showed that formulating min-cut in the spatially continuous setting properly avoids metrication bias and leads to fast and global numerical solvers through convex optimization [11]. G. Strang [41,42] was the first to study max-flow and min-cut problems over a continuous domain. Related studies include [2,3], where Appleton et al proposed an edge-based continuous minimal surface approach to segmenting 2D and 3D objects. Chan et al [15] considered image segmentation with two regions in the form

Abstract. We address the continuous problem of assigning multiple (unordered) labels with the minimum perimeter. The corresponding discrete Potts model is typically addressed with a-expansion which can generate metrication artifacts. Existing convex continuous formulations of the Potts model use TV-based functionals directly encoding perimeter costs. Such formulations are analogous to 'min-cut' problems on graphs. We propose a novel convex formulation with a continous 'max-flow' functional. This approach is dual to the standard TV-based formulations of the Potts model. Our continous max-flow approach has significant numerical advantages; it avoids extra computational load in enforcing the simplex constraints and naturally allows parallel computations over different labels. Numerical experiments show competitive performance in terms of quality and significantly reduced number of iterations compared to the previous state of the art convex methods for the continuous Potts model.

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 ob- tain 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.

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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. This work has been funded by the Norwegian Research Council through the eVita project 166075. During my time as a phd student, I feel privileged to have worked with many interesting, skilled and motivated people. First, I want to thank my supervisor Xue-Cheng Tai. I appreciate his enthusiasm and constant encouragement both before and during my phd period, while at the same time leaving me a lot of freedom. I am also grateful for being introduced to several researchers and research groups, which has resulted in fruitful collaborations, and for hosting me for two semesters at Nanyang Technological University in S...