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.
Abstract. This paper gives some global and uniform convergence estimates for a class of subspace correction (based on space decomposition) iterative methods applied to some unconstrained convex optimization problems. Some multigrid and domain decomposition methods are also discussed as special examples for solving some nonlinear elliptic boundary value problems.
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