| 總結: | <p>Geometry plays an important role in our understanding of the world with its uses spanning multiple fields from astronomy and art to more contemporary fields such as computer vision and robotics. However, as geometry is seldom directly observed the fitting of geometric models becomes a crucial task in these fields. This is however not trivial: as data must be clustered according to the parameters of geometric models that must also be estimated, creating a chicken and egg problem. In addition, most data in practice originates from multiple models, whose numbers are also unknown, and contain noise and clutter which makes geometric multi-model fitting more challenging.</p> <p>In this thesis we present a framework capable of swiftly and accurately fitting multiple geometric models to data contaminated with noise and clutter. Unlike previous greedy approaches this framework efficiently searches for a soft assignment of points to models by minimising a global energy that considers the joint classification of data points to geometric models. Additionally, the energy minimisation is performed through a <b>CO</b>nvex <b>R</b>elaxation <b>AL</b>gorithm (CORAL) that unlike other combinatorial approaches can be efficiently parallelised which leads to an energy minimisation that is significantly faster, as the resolution of data increases, while still performing as well or better than the state-of-the-art when evaluated.</p> <p>While geometry and subsequently this CORAL formulation are not restricted to a single field, in this thesis we are particularly interested in how they can be used to improve various systems in contemporary robotics. We investigate this firstly through mapping wherein geometric models are incorporated to improve, in real-time, dense depth map estimation from cameras in regions where there is little texture. We see that when these improved depth maps are fused there is significantly more coverage (10%) as compared to the state-of-the-art over large-scales. Following this we demonstrate how in tandem with deep learning networks understanding of the road for autonomous vehicles can be improved. In this way, the human-effort needed to obtain training data for deep networks can be reduced while still retrieving qualitatively accurate results as seen in road markings and boundary classification. Finally we present a uniform formulation that encodes relationships between different geometric models. We show that by leveraging this formulation in sparse reconstructions of both indoor and outdoor environments with low-cost LiDAR platforms, we can obtain reconstructions that are as accurate, within the laser noise (0.03m), as professional high-fidelity surveys.</p>
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