A study on continuous max-flow and min-cut approaches

Abstract

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

A Continuous Max-Flow Approach to Potts Model

“…By the results of (28), (26) and (27), it is easy to conclude that the maximization of the primal-dual model (20) over flow functions p s , p and q gives its equivalent dual model (22), hence we have In this work, we focus on the case when C i (x) = α, ∀x ∈ Ω and i = 1, . .…”

“…Continuous Max-Flow Model with 2 Labels Before we introduce the continuous max-flow model with n labels, we first introduce the recent study of the continuous max-flow model with 2 labels proposed by the authors [27] which is dual to the continuous s-t cut. This is directly analoguous to the graph-based max-flow and s-t cut: given the continuous image domain Ω, we assume there are two terminals, the source s and the sink t, see figure (a) of Fig.…”

“…Yuan et al [27] proved that such a continuous max-flow formulation (12) is equivalent to the continuous s-t min-cut problem [3,28] as follows:…”

Directed Acyclic Graph Continuous Max-Flow Image Segmentation for Unconstrained Label Orderings

“…The modeling approach is derived from those presented by Yuan et al [41][42] and follows the same format, showing the duality of a max-flow primal formation to this minimization problem through an intermediate primal-dual optimization problem. An augmented Lagrangian framework and a proximal Bregman framework are proposed for minimizing this intermediate representation.…”

“…These have been shown to reduce metrication artifacts associated with graph-cut segmentation. [32,41,42] Primal-dual optimization provided an efficient framework for solving these models iteratively [11,41]. Extendable max-flow models, ones which handle an arbitrary number of labels, analogous to the above have been proposed and are described with more [37] Pock et al [32] Continuous Potts model Primal-dual optimization Global fuzzy Projected gradient descent Delong et al [16] Discrete Sub/supermodular models Single graph cut Global binary (subset of partial orderings) Yuan et al [42] Continuous Potts model Primal-dual optimization Global fuzzy Augmented Langrangian Delong et al [17] Discrete Hierarchical models α-expansion-based Approximate binary (subset of partial orderings) Bae et al [2] Continuous Ishikawa model Primal-dual optimization Global fuzzy & (linear ordering)…”