Neural 3D Morphable Models: Spiral Convolutional Networks for 3D Shape Representation Learning and Generation

Generative models for 3D geometric data arise in many important applications in 3D computer vision and graphics. In this paper, we focus on 3D deformable shapes that share a common topological structure, such as human faces and bodies. Morphable Models and their variants, despite their linear formulation, have been widely used for shape representation, while most of the recently proposed nonlinear approaches resort to intermediate representations, such as 3D voxel grids or 2D views. In this work, we introduce a novel graph convolutional operator, acting directly on the 3D mesh, that explicitly models the inductive bias of the fixed underlying graph. This is achieved by enforcing consistent local orderings of the vertices of the graph, through the spiral operator, thus breaking the permutation invariance property that is adopted by all the prior work on Graph Neural Networks. Our operator comes by construction with desirable properties (anisotropic, topology-aware, lightweight, easy-to-optimise), and by using it as a building block for traditional deep generative architectures, we demonstrate state-of-the-art results on a variety of 3D shape datasets compared to the linear Morphable Model and other graph convolutional operators.Comment: to appear at ICCV 201

3D mesh processing using GAMer 2 to enable reaction-diffusion simulations in realistic cellular geometries

Recent advances in electron microscopy have enabled the imaging of single cells in 3D at nanometer length scale resolutions. An uncharted frontier for in silico biology is the ability to simulate cellular processes using these observed geometries. Enabling such simulations requires watertight meshing of electron micrograph images into 3D volume meshes, which can then form the basis of computer simulations of such processes using numerical techniques such as the Finite Element Method. In this paper, we describe the use of our recently rewritten mesh processing software, GAMer 2, to bridge the gap between poorly conditioned meshes generated from segmented micrographs and boundary marked tetrahedral meshes which are compatible with simulation. We demonstrate the application of a workflow using GAMer 2 to a series of electron micrographs of neuronal dendrite morphology explored at three different length scales and show that the resulting meshes are suitable for finite element simulations. This work is an important step towards making physical simulations of biological processes in realistic geometries routine. Innovations in algorithms to reconstruct and simulate cellular length scale phenomena based on emerging structural data will enable realistic physical models and advance discovery at the interface of geometry and cellular processes. We posit that a new frontier at the intersection of computational technologies and single cell biology is now open.Comment: 39 pages, 14 figures. High resolution figures and supplemental movies available upon reques

From 4D medical images (CT, MRI, and Ultrasound) to 4D structured mesh models of the left ventricular endocardium for patient-specific simulations

With cardiovascular disease (CVD) remaining the primary cause of death worldwide, early detection of CVDs becomes essential. The intracardiac flow is an important component of ventricular function, motion kinetics, wash-out of ventricular chambers, and ventricular energetics. Coupling between Computational Fluid Dynamics (CFD) simulations and medical images can play a fundamental role in terms of patient-specific diagnostic tools. From a technical perspective, CFD simulations with moving boundaries could easily lead to negative volumes errors and the sudden failure of the simulation. The generation of high-quality 4D meshes (3D in space + time) with 1-to-l vertex becomes essential to perform a CFD simulation with moving boundaries. In this context, we developed a semiautomatic morphing tool able to create 4D high-quality structured meshes starting from a segmented 4D dataset. To prove the versatility and efficiency, the method was tested on three different 4D datasets (Ultrasound, MRI, and CT) by evaluating the quality and accuracy of the resulting 4D meshes. Furthermore, an estimation of some physiological quantities is accomplished for the 4D CT reconstruction. Future research will aim at extending the region of interest, further automation of the meshing algorithm, and generating structured hexahedral mesh models both for the blood and myocardial volume

High order direct Arbitrary-Lagrangian-Eulerian schemes on moving Voronoi meshes with topology changes

We present a new family of very high order accurate direct Arbitrary-Lagrangian-Eulerian (ALE) Finite Volume (FV) and Discontinuous Galerkin (DG) schemes for the solution of nonlinear hyperbolic PDE systems on moving 2D Voronoi meshes that are regenerated at each time step and which explicitly allow topology changes in time. The Voronoi tessellations are obtained from a set of generator points that move with the local fluid velocity. We employ an AREPO-type approach, which rapidly rebuilds a new high quality mesh rearranging the element shapes and neighbors in order to guarantee a robust mesh evolution even for vortex flows and very long simulation times. The old and new Voronoi elements associated to the same generator are connected to construct closed space--time control volumes, whose bottom and top faces may be polygons with a different number of sides. We also incorporate degenerate space--time sliver elements, needed to fill the space--time holes that arise because of topology changes. The final ALE FV-DG scheme is obtained by a redesign of the fully discrete direct ALE schemes of Boscheri and Dumbser, extended here to moving Voronoi meshes and space--time sliver elements. Our new numerical scheme is based on the integration over arbitrary shaped closed space--time control volumes combined with a fully-discrete space--time conservation formulation of the governing PDE system. In this way the discrete solution is conservative and satisfies the GCL by construction. Numerical convergence studies as well as a large set of benchmarks for hydrodynamics and magnetohydrodynamics (MHD) demonstrate the accuracy and robustness of the proposed method. Our numerical results clearly show that the new combination of very high order schemes with regenerated meshes with topology changes lead to substantial improvements compared to direct ALE methods on conforming meshes