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.

本文提出了一种基于函数提升的新型空间连续凸松弛框架，旨在解决多标签问题，与之前提出的基于函数提升的方法相比，本方法基于分段凸近似，因此需要更少的标签；与最近的基于 MRF 的方法相比，本方法在一个空间连续设置中进行，并且显示较少的栅格偏差；此外，本文的公式在局部意义上是可能得到的最紧凑的凸松弛，易于实现并允许在 GPU 上进行高效原始 - 对偶优化。本方法在几个计算机视觉问题上的效果证明了其有效性。