A Continuous Max-Flow Approach to Potts Model

Abstract

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.

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

“…The continuous Potts model [32,42] was the first continuous max-flow model which allowed for the incorporation of more labels than the standard foreground/ background and minimizes the segmentation energy:…”

“…Equation (14) can be optimized iteratively by holding each variable fixed except one which is used to maximize (in the case of q L (x), p L (x), and p S (x)) or minimize (in the case of u L (x)) the objective function accordingly similar to previous augmented Lagrangian approaches to continuous max-flow models [2,3,34,41,42]. Therefore, the solution steps for the augmented Lagrangian approach are:…”

“…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)…”

“…Contributions: This article describes a continuous maxflow framework that allows for labels to be organized as a set of continuous spaces linked as a directed acyclic graph (DAG), which is a strict generalization of previous, more constrained algorithms [3,34,42]. This article also derives two solution algorithms for this framework derived from primal-dual optimization: the first based on augmented Lagrangian optimization, and the second on proximal Bregman projections.…”

Curvature Regularity for Multi-label Problems - Standard and Customized Linear Programming

“…Here, we make use of prior work [14,32] on writing the (length-based) Potts model as a very compact integer linear program. Approaches derived from a continuous formulation of the Potts model include [21,31].…”