Many tasks in imaging can be modeled via the minimization of a nonconvex composite function. A popular class of algorithms for solving such problems are majorization-minimization techniques which iteratively approximate the composite nonconvex function by a majorizing function that is easy to minimize. Most techniques, e.g. gradient descent, utilize convex majorizers in order guarantee that the majorizer is easy to minimize. In our work we consider a natural class of nonconvex majorizers for these functions, and show that these majorizers are still sufficient for a provably convergent optimization scheme under quite general assumptions. Numerical results illustrate that by applying this scheme, one can often obtain superior local optima compared to previous majorization-minimization methods, when the nonconvex majorizers are solved to global optimality. Finally, we illustrate the behavior of our algorithm for depth super-resolution from raw time-of-flight data.
More interesting for imaging tasks is the class of functions where the subproblems can be solved by . It was shown in @cite_7 @cite_16 that free discontinuity-type energies, in particular, with @math being continuous in its second argument, and convex and continuous in its third argument, can be optimized globally by the problem into a higher dimensional space where it admits a convex representation. Recent works, e.g. @cite_27 @cite_18 , discuss how to discretize the continuous formulation accurately.
@cite_27 @cite_16 @cite_18 @cite_7
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We propose a novel spatially continuous framework for convex relaxations based on functional lifting. Our method can be interpreted as a sublabel-accurate solution to multilabel problems. We show that previously proposed functional lifting methods optimize an energy which is linear between two labels and hence require (often infinitely) many labels for a faithful approximation. In contrast, the proposed formulation is based on a piecewise convex approximation and therefore needs far fewer labels. In comparison to recent MRF-based approaches, our method is formulated in a spatially continuous setting and shows less grid bias. Moreover, in a local sense, our formulation is the tightest possible convex relaxation. It is easy to implement and allows an efficient primal-dual optimization on GPUs. We show the effectiveness of our approach on several computer vision problems. We describe a convex relaxation for a family of problems of minimal perimeter partitions. The minimization of the relaxed problem can be tackled numerically: we describe an algorithm and show some results. In most cases, our relaxed problem finds a correct numerical approximation of the optimal solution: we give some arguments to explain why it should be so and also discuss some situations where it fails. We propose an algorithmic framework for computing global solutions of variational models with convex regularity terms that permit quite arbitrary data terms. While the minimization of variational problems with convex data and regularity terms is straightforward (using, for example, gradient descent), this is no longer trivial for functionals with nonconvex data terms. Using the theoretical framework of calibrations, the original variational problem can be written as the maximum flux of a particular vector field going through the boundary of the subgraph of the unknown function. Upon relaxation this formulation turns the problem into a convex problem, although in a higher dimension. In order to solve this problem, we propose a fast primal-dual algorithm which significantly outperforms existing algorithms. In experimental results we show the application of our method to outlier filtering of range images and disparity estimation in stereo images using a variety of convex regularity terms.