Optimality Bounds for a Variational Relaxation of the Image Partitioning Problem

Abstract

We consider a variational convex relaxation of a class of optimal partitioning and multiclass labeling problems, which has recently proven quite successful and can be seen as a continuous analogue of Linear Programming (LP) relaxation methods for finitedimensional problems. While for the latter case several optimality bounds are known, to our knowledge no such bounds exist in the continuous setting. We provide such a bound by analyzing a probabilistic rounding method, showing that it is possible to obtain an integral solution of the original partitioning problem from a solution of the relaxed problem with an a priori upper bound on the objective, ensuring the quality of the result from the viewpoint of optimization. The approach has a natural interpretation as an approximate, multiclass variant of the celebrated coarea formula.

A Convex Approach to Minimal Partitions

“…as previously in (37). The proof of the following result is then identical to the proof Proposition 4.1 in the scalar setting, provided we use in addition to the approximation Theorem 2.1, the convergence (16):…”

“…5 -this happens less frequently. In the second case, the relaxed solution needs to be binarized -see [38] for a discussion of different binarization strategies, and [37] for interesting bounds on the energy of the binarized solution.…”

Total Variation Regularization for Functions with Values in a Manifold

“…By permitting arbitrary probability measures instead, we obtain a convex and therefore more tractable problem, at the cost of allowing "non-binary" solutions u where u (x) is not a point measure at one or multiple points. Finding rounding strategies for generating u from such u without sacrificing too much in terms of optimality can still be a difficult task, we refer to [16] for recent results on the labeling problem. The usefulness of formulation (5), (6) is still slightly limited, as solving the problem numerically requires to discretize the problem, i.e., choosing a finite set of labels J , in which case (5) reduces to the finite labeling problem (3).…”

Lifting Methods for Manifold-Valued Variational Problems

“…These difficulties may be related to the fact that the original problems are NP-hard [23]. As in the discrete labeling setting [36], so-called rounding strategies have been proposed in the continuous case [45,42] that come with an a priori bound for the relative gap between the minimum of the original functional and the value attained at the projected version of a minimizer to the lifted functional. For the manifold-valued case considered here, we are not aware of a similar result yet.…”