Incremental Visual-Inertial 3D Mesh Generation with Structural Regularities

Visual-Inertial Odometry (VIO) algorithms typically rely on a point cloud representation of the scene that does not model the topology of the environment. A 3D mesh instead offers a richer, yet lightweight, model. Nevertheless, building a 3D mesh out of the sparse and noisy 3D landmarks triangulated by a VIO algorithm often results in a mesh that does not fit the real scene. In order to regularize the mesh, previous approaches decouple state estimation from the 3D mesh regularization step, and either limit the 3D mesh to the current frame or let the mesh grow indefinitely. We propose instead to tightly couple mesh regularization and state estimation by detecting and enforcing structural regularities in a novel factor-graph formulation. We also propose to incrementally build the mesh by restricting its extent to the time-horizon of the VIO optimization; the resulting 3D mesh covers a larger portion of the scene than a per-frame approach while its memory usage and computational complexity remain bounded. We show that our approach successfully regularizes the mesh, while improving localization accuracy, when structural regularities are present, and remains operational in scenes without regularities.Comment: 7 pages, 5 figures, ICRA accepte

3D mesh refinement procedure using the bisection and rivara algorithms with mesh quality assessment

Mesh refinement procedures for the solution of three dimensional problems are described. The computational\ud domain is represented by an assembly of tetrahedral elements and the mesh refinement is acheived by the bisection\ud and Rivara methods using an explicit mesh density function coupled with an automatic 3D mesh generator.\ud A couple of benchmark examples is used to compare the performance of both refinement methods in terms of mesh\ud and size qualities, number of generated elements and CPU time consume

Adaptive mesh and geodesically sliced Schwarzschild spacetime in 3+1 dimensions

We present first results obtained with a 3+1 dimensional adaptive mesh code in numerical general relativity. The adaptive mesh is used in conjunction with a standard ADM code for the evolution of a dynamically sliced Schwarzschild spacetime (geodesic slicing). We argue that adaptive mesh is particularly natural in the context of general relativity, where apart from adaptive mesh refinement for numerical efficiency one may want to use the built in flexibility to do numerical relativity on coordinate patches.Comment: 21 pages, LaTeX, 7 figures included with eps

Rich: Open Source Hydrodynamic Simulation on a Moving Voronoi Mesh

We present here RICH, a state of the art 2D hydrodynamic code based on Godunov's method, on an unstructured moving mesh (the acronym stands for Racah Institute Computational Hydrodynamics). This code is largely based on the code AREPO. It differs from AREPO in the interpolation and time advancement scheme as well as a novel parallelization scheme based on Voronoi tessellation. Using our code we study the pros and cons of a moving mesh (in comparison to a static mesh). We also compare its accuracy to other codes. Specifically, we show that our implementation of external sources and time advancement scheme is more accurate and robust than AREPO's, when the mesh is allowed to move. We performed a parameter study of the cell rounding mechanism (Llyod iterations) and it effects. We find that in most cases a moving mesh gives better results than a static mesh, but it is not universally true. In the case where matter moves in one way, and a sound wave is traveling in the other way (such that relative to the grid the wave is not moving) a static mesh gives better results than a moving mesh. Moreover, we show that Voronoi based moving mesh schemes suffer from an error, that is resolution independent, due to inconsistencies between the flux calculation and change in the area of a cell. Our code is publicly available as open source and designed in an object oriented, user friendly way that facilitates incorporation of new algorithms and physical processes

Distributions of several infinite families of mesh patterns

Br\"and\'en and Claesson introduced mesh patterns to provide explicit expansions for certain permutation statistics as linear combinations of (classical) permutation patterns. The first systematic study of avoidance of mesh patterns was conducted by Hilmarsson et al., while the first systematic study of the distribution of mesh patterns was conducted by the first two authors. In this paper, we provide far-reaching generalizations for 8 known distribution results and 5 known avoidance results related to mesh patterns by giving distribution or avoidance formulas for certain infinite families of mesh patterns in terms of distribution or avoidance formulas for smaller patterns. Moreover, as a corollary to a general result, we find the distribution of one more mesh pattern of length 2.Comment: 27 page

Efficient index handling of multidimensional periodic boundary conditions

An efficient method is described to handle mesh indexes in multidimensional problems like numerical integration of partial differential equations, lattice model simulations, and determination of atomic neighbor lists. By creating an extended mesh, beyond the periodic unit cell, the stride in memory between equivalent pairs of mesh points is independent of their position within the cell. This allows to contract the mesh indexes of all dimensions into a single index, avoiding modulo and other implicit index operations.Comment: 2 pages, 0 figure

A fast and robust patient specific Finite Element mesh registration technique: application to 60 clinical cases

Finite Element mesh generation remains an important issue for patient specific biomechanical modeling. While some techniques make automatic mesh generation possible, in most cases, manual mesh generation is preferred for better control over the sub-domain representation, element type, layout and refinement that it provides. Yet, this option is time consuming and not suited for intraoperative situations where model generation and computation time is critical. To overcome this problem we propose a fast and automatic mesh generation technique based on the elastic registration of a generic mesh to the specific target organ in conjunction with element regularity and quality correction. This Mesh-Match-and-Repair (MMRep) approach combines control over the mesh structure along with fast and robust meshing capabilities, even in situations where only partial organ geometry is available. The technique was successfully tested on a database of 5 pre-operatively acquired complete femora CT scans, 5 femoral heads partially digitized at intraoperative stage, and 50 CT volumes of patients' heads. The MMRep algorithm succeeded in all 60 cases, yielding for each patient a hex-dominant, Atlas based, Finite Element mesh with submillimetric surface representation accuracy, directly exploitable within a commercial FE software