Robust Geometry Processing for Big Dirty Data

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Geometry processing--an extension of signal processing--interprets three-dimensional curves and surfaces as signals. Just as audio and image signal data have exploded in availability, geometric data is massively abundant. We encounter geometric data everywhere: depth scanning guides safe and effective self-driving cars, anatomical curves or surfaces enable medical visualizations and soon robotic telesurgery, and 3D printing brings customization of geometric design to the masses.******However, geometry processing has been outpaced by the data.******Unlike with audio and images, we are not seeing the full application of big data analytics and modern machine learning techniques to geometric data. The core roadblock is lack of robustness in our tool chest of geometry processing techniques.******Geometric data is corrupted with noise, ambiguity and inconsistency. Unlike the regular pixel grid of an image, discrete geometric representations are a zoo with trade-offs in terms of efficiency, accuracy and scope.******Conventional geometry processing is a direct application of continuous mathematical concepts. Its assumption of pristine input surfaces is prohibitively strict and applications suffer.******Objectives******Over the next five years, I will bring geometry processing up to speed with modern geometric data. I will pursue three parallel but complementary tracks:******1. Robust algorithms for recovering structure from unstructured, noisy geometric representations;******2. Robust mathematical and algorithmic foundations for solving partial differential equations (PDEs) on real-world geometric data; and******3. Robust user interfaces for direct manipulation and creation of geometric data.******While solutions uncovered along the way will have immediate practical implications, the long-term impact is a multiplication of progress across all tracks. The combination of successes will unlock machine learning to geometric data, just as robust image processing has done for image data.******Scientific Approach******My general scientific approach is to identify and eliminate unnecessarily strict expectations of "cleanliness" in input data. Often this means returning to first mathematical principles and adapting traditional definitions or concepts to accommodate problems witnessed in real-world data.