Functional Liftings of Vectorial Variational Problems with Laplacian Regularization

We propose a functional lifting-based convex relaxation of variational problems with Laplacian-based second-order regularization. The approach rests on ideas from the calibration method as well as from sublabel-accurate continuous multilabeling approaches, and makes these approaches amenable for variational problems with vectorial data and higher-order regularization, as is common in image processing applications. We motivate the approach in the function space setting and prove that, in the special case of absolute Laplacian regularization, it encompasses the discretization-first sublabel-accurate continuous multilabeling approach as a special case. We present a mathematical connection between the lifted and original functional and discuss possible interpretations of minimizers in the lifted function space. Finally, we exemplarily apply the proposed approach to 2D image registration problems.Comment: 12 pages, 3 figures; accepted at the conference "Scale Space and Variational Methods" in Hofgeismar, Germany 201

Piecewise smooth reconstruction of normal vector field on digital data

International audienceWe propose a novel method to regularize a normal vector field defined on a digital surface (boundary of a set of voxels). When the digital surface is a digitization of a piecewise smooth manifold, our method localizes sharp features (edges) while regularizing the input normal vector field at the same time. It relies on the optimisation of a variant of the Ambrosio-Tortorelli functional, originally defined for denoising and contour extraction in image processing [AT90]. We reformulate this functional to digital surface processing thanks to discrete calculus operators. Experiments show that the output normal field is very robust to digitization artifacts or noise, and also fairly independent of the sampling resolution. The method allows the user to choose independently the amount of smoothing and the length of the set of discontinuities. Sharp and vanishing features are correctly delineated even on extremely damaged data. Finally, our method can be used to enhance considerably the output of state-of- the-art normal field estimators like Voronoi Covariance Measure [MOG11] or Randomized Hough Transform [BM12]

Building Scene Models by Completing and Hallucinating Depth and Semantics

Building 3D scene models has been a longstanding goal of computer vision. The great progress in depth sensors brings us one step closer to achieving this in a single shot. However, depth sensors still produce imperfect measurements that are sparse and contain holes. While depth completion aims at tackling this issue, it ignores the fact that some regions of the scene are occluded by the foreground objects. Building a scene model would therefore require to hallucinate the depth behind these objects. In contrast with existing methods that either rely on manual input, or focus on the indoor scenario, we introduce a fully-automatic method to jointly complete and hallucinate depth and semantics in challenging outdoor scenes. To this end, we develop a two-layer model representing both the visible information and the hidden one. At the heart of our approach lies a formulation based on the Mumford-Shah functional, for which we derive an effective optimization strategy. Our experiments evidence that our approach can accurately fill the large holes in the input depth maps, segment the different kinds of objects in the scene, and hallucinate the depth and semantics behind the foreground objects

Generalized ordering constraints for multilabel optimization

We propose a novel framework for imposing label ordering constraints in multilabel optimization. In particular, label jumps can be penalized differently depending on the jump direction. In contrast to the recently proposed MRF-based approaches, the proposed method arises from the viewpoint of spatially continuous optimization. It unifies and generalizes previous approaches to label ordering constraints: Firstly, it provides a common solution to three different problems which are otherwise solved by three separate approaches [4, 10, 14]. We provide an exact characterization of the penalization functions expressible with our approach. Secondly, we show that it naturally extends to three and higher dimensions of the image domain. Thirdly, it allows novel applications, such as the convex shape prior. Despite this generality, our model is easily adjustable to various label layouts and is also easy to implement. On a number of experiments we show that it works quite well, producing solutions comparable and superior to those obtained with previous approaches