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Agency: GTR | Branch: EPSRC | Program: | Phase: Research Grant | Award Amount: 1.55M | Year: 2013

This proposal sits within a field of great scope, stretching from some of the most fundamental problems in physics, to current practical issues in engineering, to some of the most powerful modern techniques in topology and geometry. Although these topics are all very different, it has become apparent that many of the biggest future developments in each area will require overcoming key research challenges that are remarkably similar. It is these challenges that we will address in this proposed research. At the heart of each of the topics above lie Geometric Partial Differential Equations (PDE). Each of these equations could be perhaps a law of physics, or an equation modelling an industrial process, or more abstractly, a rule under which a geometric object can be processed in order to improve it. Smooth solutions to Geometric PDE have been extremely successful in applications to pure and applied problems, but the equations are generally nonlinear, and it is therefore typical that singularities will occur in solutions. The next generation of applications, with extensive potential impact, require us to transform our understanding of these singularities that develop. We must understand when and why they occur, their structure and stability, and how they encode what the PDE is doing. We must analyse to what extent they break the classical theory of smooth solutions, and what effects this has. These are the main challenges of this proposal, and we have compiled a team to address them with complementary expertise in singularity analysis and experience of applying geometric PDE across subjects such as Mathematical Relativity, Geometric Flows and Minimal Surfaces. In Mathematical Relativity, one sees singularities in solutions of the Einstein equations, first written down by Einstein in 1915 as the fundamental equations of the large-scale universe. Progress in the research challenges we propose will have potentially major impact in some of the most famous open problems in this field such as the Cosmic Censorship Conjectures, and the Black Hole Stability Problem. We also find singularities in the field of Geometric Flows, by which we mean the evolution equations of `parabolic type that are currently being so successful in applications to geometry, topology and engineering, and in modelling phenomena in physics and biology. The most famous application in recent years has been the resolution of the Poincaré conjecture, which was named by the journal `Science as the scientific `Breakthrough of the year, 2006, but is considered by many to be the greatest achievement of mathematics in the past 100 years. The research challenges we propose are central to future applications of these equations, whether we are using them to classify manifolds with a certain curvature condition, or manipulate an image from a medical scanner. Intimately connected with these two subjects is the theory of Minimal Surfaces. These surfaces have been historically used to model soap films, but the general theory has developed into a powerful tool with applications to a wide range of subjects from black holes to topology. In this direction, we are particularly interested in applying progress on the research challenges of this proposal to unravel the connection between the existence of higher-index minimal surfaces and the singularities that occur in flows and variational problems that are designed to find them.

Amorim E.,University of Calgary | Vital Brazil E.,University of Calgary | Mena-Chalco J.,Federal University of ABC | Velho L.,National Institute for Pure and Applied Mathematics | And 3 more authors.
Computers and Graphics (Pergamon) | Year: 2015

Multidimensional projection has become a standard tool for visual analysis of multidimensional data sets, as the 2D representation of multidimensional instances gives an important and informative panorama of the data. Recently, research in this torojection, a recently proposed resampling mechanism that allows users to generate new multidimensional instances by creating reference 2D points in the projection space. Given an m-dimensional data set and its 2D projection, inverse projection transforms a user-defined 2D point into an m-dimensional point by means of a mapping function. In this work, we propose a novel inverse projection technique based on Radial Basis Functions interpolation. Our technique provides a smooth and global mapping from low to high dimensions, in contrast with the former technique (iLAMP) which is local and piecewise continuous. In order to demonstrate the potential of our technique, we use a 3D human-faces data set and a procedure to interactively reconstruct and generate new 3D faces. The results demonstrate the simplicity, robustness and efficiency of our approach to create new face models from a structured data set, a task that would typically require the manipulation of hundreds of parameters. © 2015 Elsevier Ltd. All rights reserved.

Da Silva Pires D.,University of Sao Paulo | Cesar Jr. R.M.,University of Sao Paulo | Velho L.,National Institute for Pure and Applied Mathematics
Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) | Year: 2013

We present an approach for motion estimation from videos captured by depth-sensing cameras. Our method uses the technique of graph matching to find groups of pixels that move to the same direction in subsequent frames. In order to choose the best matching for each patch, we minimize a cost function that accounts for distances on RGB and XYZ spaces. Our application runs at real-time rates for low resolution images and has shown to be a convenient framework to deal with input data generated by the new depth-sensing devices. The results show clearly the advantage obtained in the use of RGB-D images over RGB images. © Springer-Verlag 2013.

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