A Fast Continuous Max-Flow Approach to Non-convex Multi-labeling Problems

Abstract

Abstract. This work addresses a class of multilabeling problems over a spatially continuous image domain, where the data fidelity term can be any bounded function, not necessarily convex. Two total variation based regularization terms are considered, the first favoring a linear relationship between the labels and the second independent of the label values (Pott's model). In the spatially discrete setting, Ishikawa [33] showed that the first of these labeling problems can be solved exactly by standard max-flow and min-cut algorithms over specially designed graphs. We will propose a continuous analogue of Ishikawa's graph construction [33] by formulating continuous max-flow and min-cut models over a specially designed domain. These max-flow and min-cut models are equivalent under a primal-dual perspective. They can be seen as exact convex relaxations of the original problem and can be used to compute global solutions. Fast continuous max-flow based algorithms are proposed based on the max-flow models whose efficiency and reliability can be validated by both standard optimization theories and experiments. In comparison to previous work [53,52] on continuous generalization of Ishikawa's construction, our approach differs in the max-flow dual treatment which leads to the following main advantages: A new theoretical framework which embeds the label order constraints implicitly and naturally results in optimal labeling functions taking values in any predefined finite label set; A more general thresholding theorem which, under some conditions, allows to produce a larger set of non-unique solutions to the original problem; Numerical experiments show the new max-flow algorithms converge faster than the fast primal-dual algorithm of [53,52]. The speedup factor is especially significant at high precisions. In the end, our dual formulation and algorithms are extended to a recently proposed convex relaxation of Pott's model [50], thereby avoiding expensive iterative computations of projections without closed form solution.Key words. image processing and segmentation, continuous max-flow / min-cut, optimization AMS subject classifications. . . . Introduction.Many problems in image processing and computer vision can be modeled as energy minimization problems. In image restoration, such minimization problems may be defined over a set of functions which indicate the gray value of the restored image at each pixel. In image segmentation, the minimization problem can be defined over a set of partitions of the image domain. More generally, such problems can be formulated in terms of a labeling function. Examples include image denoising [41,57] where gray-scale values are directly regarded as labels, image segmentation [13, 2, 14] for which each label represents a region, two-view stereo reconstruction [40,41] where discrete-valued depths are used as labels, multi-view reconstruction [45] where inside and outside are simply indicated by two labels (see [49] for a good reference to more applications).Such optimization problems can be a...

Submodular functions: from discrete to continuous domains

“…This is done usually by a reduction to the problem of minimizing a submodular function on a ring family [48]. Moreover, particular examples, such as certain cuts with ordered labels [22,43,3] lead to min-cut/max-flow reformulations with efficient algorithms. Finally, another special case corresponds to functions defined as sums of local functions, where it is known that the usual linear programming relaxations are tight for submodular functions (see [57,59] and references therein).…”

“…Infinite sets. While finite sets already lead to interesting applications in computer vision [22,43,3], functions defined on products of sub-intervals of R are particularly interesting. Indeed, when twice-differentiable, a function is submodular if and only if all cross-second-order derivatives are non-positive, i.e., for all x ∈ X:…”

Total Variation Regularization for Functions with Values in a Manifold

“…Inspired by Ishikawa's graph-theoretic solution to spatially discrete multi-label optimization [12], Chambolle et al [3], Zach et al [22], and Lellmann et al [15,17] proposed relaxations of the labeling problem on continuous domains that allow to find good -and often globally optimalsolutions using convex optimization, see also [20,1]. While these approaches do consider a continuous domain, the set of feasible labels remains a finite discrete set.…”

Efficient Vessel Feature Detection for Endoscopic Image Analysis

“…Compare with those methods, the methods proposed in this paper have the advantage that they do not rely on any image segmentation techniques. Therefore, the proposed method do not need to solve optimization problems required by many image segmentation methods, such as [34]. Inspired by FAST feature point detector [13], we propose to place a circle centered at each candidate point on the ridgeness image and examine the ridgeness value and intensity of each point along the circle to determine whether it is a branching point or not.…”