A standard way to approximate the distance between two vertices p and q in a graph is to compute a shortest path from p to q that goes through one of k sources, which are well-chosen vertices. Precomputing the distance between each of the k sources to all vertices yields an efficient computation of approximate distances between any two vertices. One standard method for choosing k sources is the so-called Farthest Point Sampling (FPS), which starts with a random vertex as the first source, and iteratively selects the farthest vertex from the already selected sources.In this paper, we analyze the stretch factor F FPS of approximate geodesics computed using FPS, which is the maximum, over all pairs of distinct vertices, of their approximated distance over their geodesic distance in the graph. We show that F FPS can be bounded in terms of the minimal value F * of the stretch factor obtained using an optimal placement of k sources as F FPS ≤ 2 r e 2 F * + 2 r e 2 + 8 r e + 1 , where r e is the length ratio of longest edge over the shortest edge in the graph. We further show that the factor r e is not an artefact of the analysis by providing a class of graphs for which F FPS ź 1 2 r e F * .
The FPS algorithm has been used for a variety of isometry-invariant surface processing tasks. The algorithm was first introduced for graph clustering @cite_14 , and later independently developed for 2D images @cite_6 and extended to 3D meshes @cite_9 . Ben @cite_10 and Giard and Macq @cite_7 used this sampling strategy to efficiently compute approximate geodesic distances, Elad and Kimmel @cite_13 and M ' e moli and Sapiro @cite_19 used FPS in the context of shape recognition. @cite_16 and @cite_4 used FPS to efficiently compute point-to-point correspondences between surfaces. While it has been shown experimentally that FPS is a good heuristic for isometry-invariant surface processing tasks @cite_10 @cite_7 @cite_13 @cite_19 @cite_16 @cite_4 , to the best of our knowledge, the worst-case stretch of the geodesics has not been analyzed theoretically.
@cite_13 @cite_14 @cite_4 @cite_7 @cite_9 @cite_6 @cite_19 @cite_16 @cite_10
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We conducted high-frame-rate (HFR) video mosaicing for real-time synthesis of a panoramic image by implementing an improved feature-based video mosaicing algorithm on a field-programmable gate array (FPGA)-based high-speed vision platform. In the implementation of the mosaicing algorithm, feature point extraction was accelerated by implementing a parallel processing circuit module for Harris corner detection in the FPGA on the high-speed vision platform. Feature point correspondence matching can be executed for hundreds of selected feature points in the current frame by searching those in the previous frame in their neighbor ranges, assuming that frame-to-frame image displacement becomes considerably smaller in HFR vision. The system we developed can mosaic 512×512 images at 500 fps as a single synthesized image in real time by stitching the images based on their estimated frame-to-frame changes in displacement and orientation. The results of an experiment conducted, in which an outdoor scene was captured using a hand-held camera-head that was quickly moved by hand, verify the performance of our system. We present a novel method to obtain fine-scale detail in 3D reconstructions generated with low-budget RGB-D cameras or other commodity scanning devices. As the depth data of these sensors is noisy, truncated signed distance fields are typically used to regularize out the noise, which unfortunately leads to over-smoothed results. In our approach, we leverage RGB data to refine these reconstructions through shading cues, as color input is typically of much higher resolution than the depth data. As a result, we obtain reconstructions with high geometric detail, far beyond the depth resolution of the camera itself. Our core contribution is shading-based refinement directly on the implicit surface representation, which is generated from globally-aligned RGB-D images. We formulate the inverse shading problem on the volumetric distance field, and present a novel objective function which jointly optimizes for fine-scale surface geometry and spatially-varying surface reflectance. In order to enable the efficient reconstruction of sub-millimeter detail, we store and process our surface using a sparse voxel hashing scheme which we augment by introducing a grid hierarchy. A tailored GPU-based Gauss-Newton solver enables us to refine large shape models to previously unseen resolution within only a few seconds. We present a robot-pose-registration algorithm, which is entirely based on large planar-surface patches extracted from point clouds sampled from a three-dimensional (3-D) sensor. This approach offers an alternative to the traditional point-to-point iterative-closest-point (ICP) algorithm, its point-to-plane variant, as well as newer grid-based algorithms, such as the 3-D normal distribution transform (NDT). The simpler case of known plane correspondences is tackled first by deriving expressions for least-squares pose estimation considering plane-parameter uncertainty computed during plane extraction. Closed-form expressions for covariances are also derived. To round-off the solution, we present a new algorithm, which is called minimally uncertain maximal consensus (MUMC), to determine the unknown plane correspondences by maximizing geometric consistency by minimizing the uncertainty volume in configuration space. Experimental results from three 3-D sensors, viz., Swiss-Ranger, University of South Florida Odetics Laser Detection and Ranging, and an actuated SICK S300, are given. The first two have low fields of view (FOV) and moderate ranges, while the third has a much bigger FOV and range. Experimental results show that this approach is not only more robust than point- or grid-based approaches in plane-rich environments, but it is also faster, requires significantly less memory, and offers a less-cluttered planar-patches-based visualization. This paper proposes a novel 3D scene interpretation approach for robots in mobile manipulation scenarios using a set of 3D point features (Fast Point Feature Histograms) and probabilistic graphical methods (Conditional Random Fields). Our system uses real time stereo with textured light to obtain dense depth maps in the robot's manipulators working space. For the purposes of manipulation, we want to interpret the planar supporting surfaces of the scene, recognize and segment the object classes into their primitive parts in 6 degrees of freedom (6DOF) so that the robot knows what it is attempting to use and where it may be handled. The scene interpretation algorithm uses a two-layer classification scheme: i) we estimate Fast Point Feature Histograms (FPFH) as local 3D point features to segment the objects of interest into geometric primitives; and ii) we learn and categorize object classes using a novel Global Fast Point Feature Histogram (GFPFH) scheme which uses the previously estimated primitives at each point. To show the validity of our approach, we analyze the proposed system for the problem of recognizing the object class of 20 objects in 500 table settings scenarios. Our algorithm identifies the planar surfaces, decomposes the scene and objects into geometric primitives with 98.27 accuracy and uses the geometric primitives to identify the object's class with an accuracy of 96.69 .