Curve Reconstruction via the Global Statistics of Natural Curves

Reconstructing the missing parts of a curve has been the subject of much computational research, with applications in image inpainting, object synthesis, etc. Different approaches for solving that problem are typically based on processes that seek visually pleasing or perceptually plausible completions. In this work we focus on reconstructing the underlying physically likely shape by utilizing the global statistics of natural curves. More specifically, we develop a reconstruction model that seeks the mean physical curve for a given inducer configuration. This simple model is both straightforward to compute and it is receptive to diverse additional information, but it requires enough samples for all curve configurations, a practical requirement that limits its effective utilization. To address this practical issue we explore and exploit statistical geometrical properties of natural curves, and in particular, we show that in many cases the mean curve is scale invariant and oftentimes it is extensible. This, in turn, allows to boost the number of examples and thus the robustness of the statistics and its applicability. The reconstruction results are not only more physically plausible but they also lead to important insights on the reconstruction problem, including an elegant explanation why certain inducer configurations are more likely to yield consistent perceptual completions than others.Comment: CVPR versio

Existence, regularity and structure of confined elasticae

We consider the problem of minimizing the bending or elastic energy among Jordan curves confined in a given open set $\Omega$. We prove existence, regularity and some structural properties of minimizers. In particular, when $\Omega$ is convex we show that a minimizer is necessarily a convex curve. We also provide an example of a minimizer with self-intersections

On a nonlinear partial differential algebraic system arising in technical textile industry: Analysis and numerics

In this paper we explore a numerical scheme for a nonlinear fourth order system of partial differential algebraic equations that describes the dynamics of slender inextensible elastica as they arise in the technical textile industry. Applying a semi-discretization in time, the resulting sequence of nonlinear elliptic systems with the algebraic constraint for the local length preservation is reformulated as constrained optimization problems in a Hilbert space setting that admit a solution at each time level. Stability and convergence of the scheme are proved. The numerical realization is based on a finite element discretization in space. The simulation results confirm the analytically predicted properties of the scheme.Comment: Abstract and introduction are partially rewritten. The numerical study in Section 4 is completely rewritte

Existence of planar curves minimizing length and curvature

In this paper we consider the problem of reconstructing a curve that is partially hidden or corrupted by minimizing the functional $\int \sqrt{1+K_\gamma^2} ds$, depending both on length and curvature $K$. We fix starting and ending points as well as initial and final directions. For this functional we discuss the problem of existence of minimizers on various functional spaces. We find non-existence of minimizers in cases in which initial and final directions are considered with orientation. In this case, minimizing sequences of trajectories can converge to curves with angles. We instead prove existence of minimizers for the "time-reparameterized" functional \int \| \dot\gamma(t) \|\sqrt{1+K_\ga^2} dt for all boundary conditions if initial and final directions are considered regardless to orientation. In this case, minimizers can present cusps (at most two) but not angles

Visualizing 3D Euler spirals

This video describes a new type of 3D curves, which gener-alizes the family of 2D Euler spirals. They are defined as the curves having both their curvature and their torsion evolve linearly along the curve. The utility of these spirals for curve completion applications is demonstrated. This video accom-panies the paper presented in [4]

Clothoid-Based Three-Dimensional Curve for Attitude Planning

Interest in flying robots, also known as unmanned aerial vehicles (UAVs), has grown during last years in both military and civil fields [1, 2]. The same happens to autonomous underwater vehicles (AUVs) [3]. These vehicles, UAVs and AUVs, offer a wide variety of possible applications and challenges, such as control, guidance or navigation [2, 3]. In this sense, heading and attitude control in UAVs is very important [4], particularly relevant in airplanes (fixed-wing flying vehicles), because they are strongly non-linear, coupled, and tend to be underactuated systems with non-holonomic constraints. Hence, designing a good attitude controller is a difficult task [5, 6, 7, 8, 9], where stability must be taken into account by the controller [10]. Indeed, if the reference is too demanding for the controller or non-achievable because its dynamics is too fast, the vehicle might become unstable. In order to address this issue, autonomous navigation systems usually include a high-level path planner to generate smooth reference trajectories to be followed by the vehicle using a low-level controller. Usually a set of waypoints is given in GPS coordinates, normally from a map, in order to apply a smooth point-to-point control trajectory [11, 12]

Geometry acquisition and grid generation: Recent experiences with complex aircraft configurations

Important issues involved in working with complex geometries are discussed. Approaches taken to address complex geometry issues in the McDonnell Aircraft Computational Grid System and related geometry processing tools are discussed. The efficiency of acquiring a suitable geometry definition, the need to manipulate the geometry, and the time and skill level required to generate the grid while preserving geometric fidelity are discussed