MCS: Reconstructing and Inferring Topology and Geometry from Point Cloud Data

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Point cloud data (PCD) is pervasive across numerous areas in science and engineering. This project addresses two issues, namely reconstruction in three dimension and inference from PCD in high dimensions. In reconstruction, the problem of approximating a three dimensional shape from its PCD is considered. This problem becomes hopeless from computational view point for shapes sitting in high dimensional spaces. Here, the project focuses on inferring topological and geometric properties of the hidden object from its sampled representation rather than reconstructing it completely.A successful implementation of this project will enable many areas of science and engineering to enhance their scope in modeling, analyzing, prototyping, and visualizing input data with assurance of accuracy. Designing machine parts in automotive industry, creating virtual environments with buildings, simulating cracks and shocks in scientific studies are few examples where new algorithms are needed for reconstructing shapes in presence of non-smoothness. This will be addressed in the reconstruction section of the project. Various scientific and social studies such as the ones in medicine, economics, climate, disease control produce data that presumably sample a hidden parameter space (a manifold). They will benefit from the research on the topology and geometry inference section of the project.