Computation and visualization of ideal knot shapes

We investigate numerical simulations and visualizations of the problem of tying a knot in a piece of rope. The goal is to use the least possible rope of a fixed, prescribed radius to tie a particular knot, e.g. a trefoil, a figure eight, and so on. The ropelength of the knot, the ratio to be minimized, is its length divided by its radius. An overview of existing algorithms to minimize the ropelength is given. They are based on different discretizations. Our work builds on the biarc discretization, for which we have developed an entire C++ library "libbiarc". The library contains a variety of tools to manipulate curves, knots or links. The biarc discretization is particularly well suited to evaluation of thickness. To compute ideal knot shapes we use simulated annealing software, which is also included in "libbiarc", on a biarc discretization. Simulated annealing is a stochastic optimization algorithm that randomly changes the point or tangent data. In the quest to find appropriate moves for this process we arrived upon a Fourier representation for knots, which allows global changes to the curve in the annealing process. Moreover, with the Fourier representation we can enforce symmetries that a given knot might have. To identify these symmetries we use visualization of simulations where symmetry was not enforced. Visualization of knot shapes and their properties is another important aspect in this work. It ranges from simple graphs of the curvature of a knot, through 2-dimensional plots of certain distance, circle or sphere functions, to 3-dimensional images of contact properties. Specially designed color gradients have been developed to emphasize crucial regions of the plots. We show that the contact set of ideal torus knots is a curve that is ambient isotopic to the knot itself, which is a result first suggested by visualization. A combination of numerics and visualization made us aware of a closed trajectory within the trefoil knot, a 9-billiard. Consequently the symmetries and the billiard make it possible to represent the trefoil with only two curve sub segments. We also anneal and visualize knot shapes in the unit 3-sphere or S3. In particular we present the contact set of a candidate for optimality, whose curved contact chords form Villarceau circles, which in turn span a Clifford torus embedded in the unit 3-sphere. Finally some knots and contact surfaces are constructed as physical 3D models using 3D printers

Advances in Computer Recognition, Image Processing and Communications, Selected Papers from CORES 2021 and IP&C 2021

As almost all human activities have been moved online due to the pandemic, novel robust and efficient approaches and further research have been in higher demand in the field of computer science and telecommunication. Therefore, this (reprint) book contains 13 high-quality papers presenting advancements in theoretical and practical aspects of computer recognition, pattern recognition, image processing and machine learning (shallow and deep), including, in particular, novel implementations of these techniques in the areas of modern telecommunications and cybersecurity

Lunar Rover Motion Planning and Commands

Space exploration is moving forward and one of the topics currently being researched is mining. The objective of this thesis is to design and develop software for the auton- omous navigation of a wheeled rover that is being built for NASA’s Lunabotics Mining Competition. The motion control system is a crucial component of a planetary rover system and its implementation heavily depends on the chassis configuration. The configuration of the rover enables us to use three steering modes: Ackermann, Point- turn and Crab steering. The implementation takes advantages of all the modes and involves algorithms for path planning, path smoothing and path following. In addi- tion, the system offers a feature of automatic steering mode selection. The system can be tuned and controlled by the cross-platform application specifically developed for this purpose. The performance of the implemented system is analyzed by testing in a simulator with a realistic physics engine and 3D visualization capabilities. Our con- ducted tests confirm that the system is sufficient in the framework of the Lunabotics Mining Competition

A Parametrization-Based Surface Reconstruction System for Triangular Mesh Simplification with Application to Large Scale Scenes

The laser scanner is nowadays widely used to capture the geometry of art, animation maquettes, or large architectural, industrial, and land form models. It thus poses specific problems depending on the model scale. This thesis provides a solution for simplification of triangulated data and for surface reconstruction of large data sets, where feature edges provide an obvious segmentation structure. It also explores a new method for model segmentation, with the goal of applying multiresolution techniques to data sets characterized by curvy areas and the lack of clear demarcation features. The preliminary stage of surface segmentation, which takes as input single or multiple scan data files, generates surface patches which are processed independently. The surface components are mapped onto a two-dimensional domain with boundary constraints, using a novel parametrization weight coefficient. This stage generates valid parameter domain points, which can be fed as arguments to parametric modeling functions or surface approximation schemes. On this domain, our approach explores two types of remeshing. First, we generate points in a regular grid pattern, achieving multiresolution through a flexible grid step, which nevertheless is designed to produce a globally uniform resampling aspect. In this case, for reconstruction, we attempt to solve the open problem of border reconciliation across adjacent domains by retriangulating the border gap between the grid and the fixed irregular border. Alternatively, we straighten the domain borders in the parameter domain and coarsely triangulate the resulting simplified polygons, resampling the base domain triangles in a 1-4 subdivision pattern, achieving multiresolution from the number of subdivision steps. For mesh reconstruction, we use a linear interpolation method based on the original mesh triangles as control points on local planes, using a saved triangle correspondence between the original mesh and the parametric domain. We also use a region-wide approximation method, applied to the parameter grid points, which first generates data-trained control points, and then uses them to obtain the reconstruction values at the resamples. In the grid resampling scheme, due to the border constraints, the reassembly of the segmented, sequentially processed data sets is seamless. In the subdivision scheme, we align adjacent border fragments in the parameter space, and use a region-to-fragment map to achieve the same border reconstruction across two neighboring components. We successfully process data sets up to 1,000,000 points in one pass of our program, and are capable of assembling larger scenes from sequential runs. Our program consists of a single run, without intermediate storage. Where we process large input data files, we fragment the input using a nested application of our segmentation algorithm to reduce the size of the input scenes, and our pipeline reassembles the reconstruction output from multiple data files into a unique view

A novel parallel algorithm for surface editing and its FPGA implementation

A thesis submitted to the University of Bedfordshire in partial fulfilment of the requirements for the degree of Doctor of PhilosophySurface modelling and editing is one of important subjects in computer graphics. Decades of research in computer graphics has been carried out on both low-level, hardware-related algorithms and high-level, abstract software. Success of computer graphics has been seen in many application areas, such as multimedia, visualisation, virtual reality and the Internet. However, the hardware realisation of OpenGL architecture based on FPGA (field programmable gate array) is beyond the scope of most of computer graphics researches. It is an uncultivated research area where the OpenGL pipeline, from hardware through the whole embedded system (ES) up to applications, is implemented in an FPGA chip. This research proposes a hybrid approach to investigating both software and hardware methods. It aims at bridging the gap between methods of software and hardware, and enhancing the overall performance for computer graphics. It consists of four parts, the construction of an FPGA-based ES, Mesa-OpenGL implementation for FPGA-based ESs, parallel processing, and a novel algorithm for surface modelling and editing. The FPGA-based ES is built up. In addition to the Nios II soft processor and DDR SDRAM memory, it consists of the LCD display device, frame buffers, video pipeline, and algorithm-specified module to support the graphics processing. Since there is no implementation of OpenGL ES available for FPGA-based ESs, a specific OpenGL implementation based on Mesa is carried out. Because of the limited FPGA resources, the implementation adopts the fixed-point arithmetic, which can offer faster computing and lower storage than the floating point arithmetic, and the accuracy satisfying the needs of 3D rendering. Moreover, the implementation includes Bézier-spline curve and surface algorithms to support surface modelling and editing. The pipelined parallelism and co-processors are used to accelerate graphics processing in this research. These two parallelism methods extend the traditional computation parallelism in fine-grained parallel tasks in the FPGA-base ESs. The novel algorithm for surface modelling and editing, called Progressive and Mixing Algorithm (PAMA), is proposed and implemented on FPGA-based ES’s. Compared with two main surface editing methods, subdivision and deformation, the PAMA can eliminate the large storage requirement and computing cost of intermediated processes. With four independent shape parameters, the PAMA can be used to model and edit freely the shape of an open or closed surface that keeps globally the zero-order geometric continuity. The PAMA can be applied independently not only FPGA-based ESs but also other platforms. With the parallel processing, small size, and low costs of computing, storage and power, the FPGA-based ES provides an effective hybrid solution to surface modelling and editing