Computing the shortest path between two points or a set of points on a point cloud representing a manifold is a crucial problem in Computational Science, particularly in Computer Graphics and CAD/CAM. Most methods calculate geodesic paths on a polygonal model of the manifold, while only a few compute them directly on the point cloud. When the point cloud has noise, constructing a triangular mesh is challenging and results in inaccurate geodesic path calculations. This work calculates the geodesic path by minimizing the discrete geodesic curvature of the connecting curve, using an iterative Newton minimization and a projection procedure based on directed point-projection and Gabriel neighborhoods. The proposed algorithm has quadratic convergence and produces accurate results in synthetic and free-form point clouds, outperforming triangle mesh-based methods in noisy point clouds. The paper concludes with a demonstration of geodesic path computation in diverse free-form point-clouds and a case study in template-based digital surface reconstruction. The source code of the proposed method can be found in https://github.com/agalex1974/LibGeodesicOPPGC .

Most blind hyperspectral unmixing methods exploit convex geometry properties of hyperspectral data. The minimum volume simplex analysis (MVSA) is one of such methods, which, as many others, estimates the minimum volume (MV) simplex where the measured vectors live. MVSA was conceived to circumvent the matrix factorization step often implemented by MV-based algorithms and also to cope with outliers, which compromise the results produced by MV algorithms. Inspired by the recently proposed robust MV enclosing simplex (RMVES) algorithm, we herein introduce the robust MVSA (RMVSA), which is a version of MVSA robust to noise. As in RMVES, the robustness is achieved by employing chance constraints, which control the volume of the resulting simplex. RMVSA differs, however, substantially from RMVES in the way optimization is carried out. In this paper, we develop a linearization relaxation of the nonlinear chance constraints, which can greatly lighten the computational complex of chance constraint problems. The effectiveness of RMVSA is illustrated by comparing its performance with the state of the art.

Linear spectral unmixing aims at estimating the number of pure spectral substances, also calledendmembers, their spectral signatures, and their abundance fractions in remotely sensed hyperspectral images. This paper describes a method for unsupervised hyperspectral unmixing called minimum volume simplex analysis (MVSA) and introduces a new computationally efficient implementation. MVSA approaches hyperspectral unmixing by fitting a minimum volume simplex to the hyperspectral data, constraining the abundance fractions to belong to the probability simplex. The resulting optimization problem, which is computationally complex, is solved in this paper by implementing a sequence of quadratically constrained subproblems using the interior point method, which is particularly effective from the computational viewpoint. The proposed implementation (available online: www.lx.it.pt/%7ejun/DemoMVSA.zip) is shown to exhibit state-of-the-art performance not only in terms of unmixing accuracy, particularly in nonpure pixel scenarios, but also in terms of computational performance. Our experiments have been conducted using both synthetic and real data sets. An important assumption of MVSA is that pure pixels may not be present in the hyperspectral data, thus addressing a common situation in real scenarios which are often dominated by highly mixed pixels. In our experiments, we observe that MVSA yields competitive performance when compared with other available algorithms that work under the nonpure pixel regime. Our results also demonstrate that MVSA is well suited to problems involving a high number of endmembers (i.e., complex scenes) and also for problems involving a high number of pixels (i.e., large scenes).

Abstract Modeling 3D objects from sketches is a process that requires several challenging problems including segmentation, recognition and reconstruction. Some of these tasks are harder for humans and some are harder for the machine. At the core of the problem lies the need for semantic understanding of the shape's geometry from the sketch. In this paper we propose a method to model 3D objects from sketches by utilizing humans specifically for semantic tasks that are very simple for humans and extremely difficult for the machine, while utilizing the machine for tasks that are harder for humans. The user assists recognition and segmentation by choosing and placing specific geometric primitives on the relevant parts of the sketch. The machine first snaps the primitive to the sketch by fitting its projection to the sketch lines, and then improves the model globally by inferringgeosemanticconstraints that link the different parts. The fitting occurs in real‐time, allowing the user to be only as precise as needed to have a good starting configuration for this non‐convex optimization problem. We evaluate the accessibility of our approach with a user study.