Total roto-translational variation

Abstract

We consider curvature depending variational models for image regularization, such as Euler's elastica. These models are known to provide strong priors for the continuity of edges and hence have important applications in shape-and image processing. We consider a lifted convex representation of these models in the roto-translation space: In this space, curvature depending variational energies are represented by means of a convex functional defined on divergence free vector fields. The line energies are then easily extended to any scalar function. It yields a natural generalization of the total variation to the roto-translation space. As our main result, we show that the proposed convex representation is tight for characteristic functions of smooth shapes. We also discuss cases where this representation fails. For numerical solution, we propose a staggered grid discretization based on an averaged Raviart-Thomas finite elements approximation. This discretization is consistent, up to minor details, with the underlying continuous model. The resulting non-smooth convex optimization problem is solved using a first-order primal-dual algorithm. We illustrate the results of our numerical algorithm on various problems from shape-and image processing. vision) since at least the 90s [12,13,16,14,17,15,30]. It is shown already in [12] that the boundary of many sets with cusps can be approximated by sets with smooth boundaries of bounded energy, showing that the relaxation of the Elastica for boundaries of sets is already far from trivial. The study of this lower semi-continuity of course enters the long history of the study of general curvature dependent energies of manifolds and in particular the Willmore energy [77].Quite early, it has been suggested to lift the manifold in a larger space where a variable represents its direction or orientation, by means in particular (for co-dimension one manifolds) of the Gauss map (x, ν(x)), ν(x) being the normal to the manifold at x [5,6]. Such approach allows to study very general curvature energies and has been successfully used for establishing lower-semicontinuity and existence results [6,7,32,31], and in particular to lines energies such as ours in higher codimension [2,1] or in dimension 2 [31]. We must mention also in this class an older approach based on "curvature varifolds" (which is very natural since varifolds are defined on the cross product of spatial and directional variables) which has allowed to show existence results since the 80s [45], see also [50].Interestingly, it was understood much earlier [44] that (cats') vision was functioning in a similar way (eg., using a sort of "Gauss map"), thanks to neurons sensitive to particular directions which were found to be stacked inside the visual cortex into ordered columns, making us sensitive to changes of orientations (and thus curvature intensity). These findings (which might explain some of Kanizsa's experiments) inspired some mathematical models quite early [48], however they were formalized into a consistent geometric inte...

Total Variation and Mean Curvature PDEs on the Space of Positions and Orientations

“…The OS is obtained by convolving the image with a set of rotated wavelets allowing for stable reconstruction [14,4,24]. 2nd row: Vessel-tracking in a 2D image via geodesic PDE-flows in OS that underly TVF: [4,18,8], with n = (cos θ, sin θ) T ∈ S 1 . 3rd row: CED-OS diffusion of a 3D image [24,17] visualized as a field of angular profiles.…”

“…For {a, b} = {1, 1} we have a geometric Mean Curvature Flow (MCF) PDE. For {a, b} = {0, 1} we have a Total Variation Flow (TVF) [8]. For {a, b} = {0, 0} we obtain a linear diffusion for which exact smooth solutions exist [27].…”

“…In the last decade, many PDE-based image analysis techniques for tracking and enhancement of curvilinear structures took advantage of lifting the image data to the homogeneous space M = R d S d−1 of d-dimensional positions and orientations, cf. [14,10,34,6,4,8]. The precise definition of this homogeneous space follows in the next subsection.…”

“…Also for geodesic tracking methods, it helps that crossing line structures are disentangled in the lifted data. Geodesic flows prior to steepest descent also rely on (related [32,5,8]) PDEs on M commuting with roto-translations. They can account for crossings/bifurcations/corners [4,18,8].…”

“…As for PDE-based image denoising and enhancement, total variation flows (TVF) are more popular than nonlinear diffusion flows. Recently, Chambolle & Pock generalized TVF from R 2 to M = R 2 × S 1 , via 'total roto-translation variation' (TV-RT) flows [8] of 2D images. They employ (a)symmetric Finslerian geodesic models on M cf.…”