Discrete stochastic approximations of the Mumford-Shah functional

We propose a $\Gamma$-convergent discrete approximation of the Mumford-Shah functional. The discrete functionals act on functions defined on stationary stochastic lattices and take into account general finite differences through a non-convex potential. In this setting the geometry of the lattice strongly influences the anisotropy of the limit functional. Thus we can use statistically isotropic lattices and stochastic homogenization techniques to approximate the vectorial Mumford-Shah functional in any dimension.Comment: 47 pages, reorganized versio

Chirality transitions in frustrated S2S^{2}-valued spin systems

We study the discrete-to-continuum limit of the helical XY $S^{2}$-spin system on the lattice $\mathbb{Z}^{2}$. We scale the interaction parameters in order to reduce the model to a spin chain in the vicinity of the Landau-Lifschitz point and we prove that at the same energy scaling under which the $S^{1}$-model presents scalar chirality transitions, the cost of every vectorial chirality transition is now zero. In addition we show that if the energy of the system is modified penalizing the distance of the $S^{2}$ field from a finite number of copies of $S^{1}$, it is still possible to prove the emergence of nontrivial (possibly trace dependent) chirality transitions

Real-time on-board obstacle avoidance for UAVs based on embedded stereo vision

In order to improve usability and safety, modern unmanned aerial vehicles (UAVs) are equipped with sensors to monitor the environment, such as laser-scanners and cameras. One important aspect in this monitoring process is to detect obstacles in the flight path in order to avoid collisions. Since a large number of consumer UAVs suffer from tight weight and power constraints, our work focuses on obstacle avoidance based on a lightweight stereo camera setup. We use disparity maps, which are computed from the camera images, to locate obstacles and to automatically steer the UAV around them. For disparity map computation we optimize the well-known semi-global matching (SGM) approach for the deployment on an embedded FPGA. The disparity maps are then converted into simpler representations, the so called U-/V-Maps, which are used for obstacle detection. Obstacle avoidance is based on a reactive approach which finds the shortest path around the obstacles as soon as they have a critical distance to the UAV. One of the fundamental goals of our work was the reduction of development costs by closing the gap between application development and hardware optimization. Hence, we aimed at using high-level synthesis (HLS) for porting our algorithms, which are written in C/C++, to the embedded FPGA. We evaluated our implementation of the disparity estimation on the KITTI Stereo 2015 benchmark. The integrity of the overall realtime reactive obstacle avoidance algorithm has been evaluated by using Hardware-in-the-Loop testing in conjunction with two flight simulators.Comment: Accepted in the International Archives of the Photogrammetry, Remote Sensing and Spatial Information Scienc

New homogenization results for convex integral functionals and their Euler-Lagrange equations

We study stochastic homogenization for convex integral functionals u\mapsto \int_D W(\omega,\tfrac{x}\varepsilon,\nabla u)\,\mathrm{d}x,\quad\mbox{where}\quad u:D\subset \mathbb{R}^d\to\mathbb{R}^m, defined on Sobolev spaces. Assuming only stochastic integrability of the map $\omega\mapsto W(\omega,0,\xi)$, we prove homogenization results under two different sets of assumptions, namely $\bullet_1\quad$ $W$ satisfies superlinear growth quantified by the stochastic integrability of the Fenchel conjugate $W^*(\cdot,0,\xi)$ and a mild monotonicity condition that ensures that the functional does not increase too much by componentwise truncation of $u$, $\bullet_2\quad$ $W$ is $p$-coercive in the sense $|\xi|^p\leq W(\omega,x,\xi)$ for some $p>d-1$. Condition $\bullet_2$ directly improves upon earlier results, where $p$-coercivity with $p>d$ is assumed and $\bullet_1$ provides an alternative condition under very weak coercivity assumptions and additional structure conditions on the integrand. We also study the corresponding Euler-Lagrange equations in the setting of Sobolev-Orlicz spaces. In particular, if $W(\omega,x,\xi)$ is comparable to $W(\omega,x,-\xi)$ in a suitable sense, we show that the homogenized integrand is differentiable.Comment: 43 page

Loss of strong ellipticity through homogenization in 2D linear elasticity: A phase diagram

Since the seminal contribution of Geymonat, M\"uller, and Triantafyllidis, it is known that strong ellipticity is not necessarily conserved through periodic homogenization in linear elasticity. This phenomenon is related to microscopic buckling of composite materials. Consider a mixture of two isotropic phases which leads to loss of strong ellipticity when arranged in a laminate manner, as considered by Guti\'errez and by Briane and Francfort. In this contribution we prove that the laminate structure is essentially the only microstructure which leads to such a loss of strong ellipticity. We perform a more general analysis in the stationary, ergodic setting.Comment: 31 pages, 2 figures, slightly changed the presentation of the main result

Random finite-difference discretizations of the Ambrosio-Tortorelli functional with optimal mesh size

We propose and analyze a finite-difference discretization of the Ambrosio-Tortorelli functional. It is known that if the discretization is made with respect to an underlying periodic lattice of spacing $\delta$, the discretized functionals $\Gamma$-converge to the Mumford-Shah functional only if $\delta\ll\varepsilon$, $\varepsilon$ being the elliptic approximation parameter of the Ambrosio-Tortorelli functional. Discretizing with respect to stationary, ergodic and isotropic random lattices we prove this $\Gamma$-convergence result also for $\delta\sim\varepsilon$, a regime at which the discretization with respect to a periodic lattice converges instead to an anisotropic version of the Mumford-Shah functional.Comment: 36 pages, 6 figures. Added some numerical example