Dinkelbach NCUT: An Efficient Framework for Solving Normalized Cuts Problems with Priors and Convex Constraints

Abstract

In this paper, we propose a novel framework, called Dinkelbach NCUT (DNCUT), which efficiently solves the normalized graph cut (NCUT) problem under general, convex constraints, as well as, under given priors on the nodes of the graph. Current NCUT methods use generalized eigen-decomposition, which poses computational issues especially for large graphs, and can only handle linear equality constraints. By using an augmented graph and the iterative Dinkelbach method for fractional programming (FP), we formulate the DNCUT framework to efficiently solve the NCUT problem under general convex constraints and given data priors. In this framework, the initial problem is converted into a sequence of simpler sub-problems (i.e. convex, quadratic programs (QP's) subject to convex constraints). The complexity of finding a global solution for each sub-problem depends on the complexity of the constraints, the convexity of the cost function, and the chosen initialization. However, we derive an initialization, which guarantees that each sub-problem is a convex QP that can be solved by available convex programming techniques. We apply this framework to the special case of linear constraints, where the solution is obtained by solving a sequence of sparse linear systems using the conjugate gradient method. We validate DNCUT by performing binary segmentation on real images both with and without linear/nonlinear constraints, as well as, multi-class segmentation. When possi-B. Ghanem ( ) · N. Ahuja ble, we compare DNCUT to other NCUT methods, in terms of segmentation performance and computational efficiency. Even though the new formulation is applied to the problem of spectral graph-based, low-level image segmentation, it can be directly applied to other applications (e.g. clustering).

Semantic content-based image retrieval: A comprehensive study

“…By optimizing these cost functions of graph cut, it becomes possible to get the desirable segmentation. Some of representative techniques and recent variants are: minimum cut [45,46] and normalized cut [27,[47][48][49][50]. c) Graph cut on Markov random field (MRF): The MRF theory has been introduced as a consistent approach for modeling contextual information such as image pixels and visual features.…”

3D Aware Correction and Completion of Depth Maps in Piecewise Planar Scenes

“…Since some pixels inÎ d are already assigned to fitted planes with non-background labels, we use a version of graph cuts (popularly known as interactive graph cuts) to guarantee that the labels of these pixels, after optimization, remain the same. Other optimization methods could be used to solve the normalized version of this problem [25].…”

Graph Transduction as a Noncooperative Game

“…Moreover, unlike the proposed approach, hard labeling constraints cannot be embedded into the normalized cuts framework in an explicit way, such that partial grouping constraints could be enforced by introducing extra linear equality constraints (Eriksson, Olsson, & Kahl, 2007;Xu, Li, & Schuurmans, 2009;Yu & Shi, 2004). The framework suggested recently in Ghanem and Ahuja (2010) is an exception, but it is inherently a two-class clustering approach and requires a recursive strategy to solve multiclass problems.…”