The Topology ToolKit

This system paper presents the Topology ToolKit (TTK), a software platform designed for topological data analysis in scientific visualization. TTK provides a unified, generic, efficient, and robust implementation of key algorithms for the topological analysis of scalar data, including: critical points, integral lines, persistence diagrams, persistence curves, merge trees, contour trees, Morse-Smale complexes, fiber surfaces, continuous scatterplots, Jacobi sets, Reeb spaces, and more. TTK is easily accessible to end users due to a tight integration with ParaView. It is also easily accessible to developers through a variety of bindings (Python, VTK/C++) for fast prototyping or through direct, dependence-free, C++, to ease integration into pre-existing complex systems. While developing TTK, we faced several algorithmic and software engineering challenges, which we document in this paper. In particular, we present an algorithm for the construction of a discrete gradient that complies to the critical points extracted in the piecewise-linear setting. This algorithm guarantees a combinatorial consistency across the topological abstractions supported by TTK, and importantly, a unified implementation of topological data simplification for multi-scale exploration and analysis. We also present a cached triangulation data structure, that supports time efficient and generic traversals, which self-adjusts its memory usage on demand for input simplicial meshes and which implicitly emulates a triangulation for regular grids with no memory overhead. Finally, we describe an original software architecture, which guarantees memory efficient and direct accesses to TTK features, while still allowing for researchers powerful and easy bindings and extensions. TTK is open source (BSD license) and its code, online documentation and video tutorials are available on TTK's website

A survey of partial differential equations in geometric design

YesComputer aided geometric design is an area where the improvement of surface generation techniques is an everlasting demand since faster and more accurate geometric models are required. Traditional methods for generating surfaces were initially mainly based upon interpolation algorithms. Recently, partial differential equations (PDE) were introduced as a valuable tool for geometric modelling since they offer a number of features from which these areas can benefit. This work summarises the uses given to PDE surfaces as a surface generation technique togethe

Stereoscopic Sketchpad: 3D Digital Ink

--Context-- This project looked at the development of a stereoscopic 3D environment in which a user is able to draw freely in all three dimensions. The main focus was on the storage and manipulation of the ‘digital ink’ with which the user draws. For a drawing and sketching package to be effective it must not only have an easy to use user interface, it must be able to handle all input data quickly and efficiently so that the user is able to focus fully on their drawing. --Background-- When it comes to sketching in three dimensions the majority of applications currently available rely on vector based drawing methods. This is primarily because the applications are designed to take a users two dimensional input and transform this into a three dimensional model. Having the sketch represented as vectors makes it simpler for the program to act upon its geometry and thus convert it to a model. There are a number of methods to achieve this aim including Gesture Based Modelling, Reconstruction and Blobby Inflation. Other vector based applications focus on the creation of curves allowing the user to draw within or on existing 3D models. They also allow the user to create wire frame type models. These stroke based applications bring the user closer to traditional sketching rather than the more structured modelling methods detailed. While at present the field is inundated with vector based applications mainly focused upon sketch-based modelling there are significantly less voxel based applications. The majority of these applications focus on the deformation and sculpting of voxmaps, almost the opposite of drawing and sketching, and the creation of three dimensional voxmaps from standard two dimensional pixmaps. How to actually sketch freely within a scene represented by a voxmap has rarely been explored. This comes as a surprise when so many of the standard 2D drawing programs in use today are pixel based. --Method-- As part of this project a simple three dimensional drawing program was designed and implemented using C and C++. This tool is known as Sketch3D and was created using a Model View Controller (MVC) architecture. Due to the modular nature of Sketch3Ds system architecture it is possible to plug a range of different data structures into the program to represent the ink in a variety of ways. A series of data structures have been implemented and were tested for efficiency. These structures were a simple list, a 3D array, and an octree. They have been tested for: the time it takes to insert or remove points from the structure; how easy it is to manipulate points once they are stored; and also how the number of points stored effects the draw and rendering times. One of the key issues brought up by this project was devising a means by which a user is able to draw in three dimensions while using only two dimensional input devices. The method settled upon and implemented involves using the mouse or a digital pen to sketch as one would in a standard 2D drawing package but also linking the up and down keyboard keys to the current depth. This allows the user to move in and out of the scene as they draw. A couple of user interface tools were also developed to assist the user. A 3D cursor was implemented and also a toggle, which when on, highlights all of the points intersecting the depth plane on which the cursor currently resides. These tools allow the user to see exactly where they are drawing in relation to previously drawn lines. --Results-- The tests conducted on the data structures clearly revealed that the octree was the most effective data structure. While not the most efficient in every area, it manages to avoid the major pitfalls of the other structures. The list was extremely quick to render and draw to the screen but suffered severely when it comes to finding and manipulating points already stored. In contrast the three dimensional array was able to erase or manipulate points effectively while the draw time rendered the structure effectively useless, taking huge amounts of time to draw each frame. The focus of this research was on how a 3D sketching package would go about storing and accessing the digital ink. This is just a basis for further research in this area and many issues touched upon in this paper will require a more in depth analysis. The primary area of this future research would be the creation of an effective user interface and the introduction of regular sketching package features such as the saving and loading of images

Multilevel Solvers for Unstructured Surface Meshes

Parameterization of unstructured surface meshes is of fundamental importance in many applications of digital geometry processing. Such parameterization approaches give rise to large and exceedingly ill-conditioned systems which are difficult or impossible to solve without the use of sophisticated multilevel preconditioning strategies. Since the underlying meshes are very fine to begin with, such multilevel preconditioners require mesh coarsening to build an appropriate hierarchy. In this paper we consider several strategies for the construction of hierarchies using ideas from mesh simplification algorithms used in the computer graphics literature. We introduce two novel hierarchy construction schemes and demonstrate their superior performance when used in conjunction with a multigrid preconditioner

Discrete spherical means of directional derivatives and Veronese maps

We describe and study geometric properties of discrete circular and spherical means of directional derivatives of functions, as well as discrete approximations of higher order differential operators. For an arbitrary dimension we present a general construction for obtaining discrete spherical means of directional derivatives. The construction is based on using the Minkowski's existence theorem and Veronese maps. Approximating the directional derivatives by appropriate finite differences allows one to obtain finite difference operators with good rotation invariance properties. In particular, we use discrete circular and spherical means to derive discrete approximations of various linear and nonlinear first- and second-order differential operators, including discrete Laplacians. A practical potential of our approach is demonstrated by considering applications to nonlinear filtering of digital images and surface curvature estimation

Analysis of flow and aerodynamic noise behaviour of a simplified high-speed train bogie inside the bogie cavity

Aerodynamic noise becomes significant for high-speed trains but its prediction in an industrial context is difficult. The flow and aerodynamic noise behaviour of a simplified high-speed train bogie at scale 1:10 are studied here through numerical simulations. The bogie is situated in the bogie cavity and cases without and with a fairing are considered, allowing the shielding effect of the bogie fairing on sound generation and radiation to be investigated. A two-stage hybrid method combining computational fluid dynamics and acoustic analogy is applied. The near-field unsteady flow is obtained by solving the unsteady three-dimensional Navier-Stokes equations numerically using delayed detached-eddy simulation and the data are utilized to predict far-field noise signals based on the Ffowcs Williams-Hawkings acoustic analogy. Results show that when the bogie is located inside the bogie cavity, the shear layer developed from the cavity leading edge interacts strongly with the flow separated from the bogie upstream components and the cavity wall. Therefore, a highly turbulent flow is generated within the bogie cavity due to flow impingement and recirculation within the cavity. It is found that, for noise calculated from the bogie surface sources of both cases, the directivity exhibits a lateral dipole pattern with dominant radiation in the axial direction. Compared with the no fairing case, the noise level is about 1 dB higher in the bogie symmetry plane along the axle mid-span for the fairing case where a stronger flow interaction is produced around the bogie central region. Moreover, the noise radiated to the trackside is predicted based on a permeable integration surface close to the bogie and parallel to the carbody side wall. The results show that the bogie fairing is effective in reducing the noise levels in most of the frequency range due to its shielding effect and a noise reduction around 3 dB is achieved for the current model case by mounting a fairing in the bogie area