Algorithms for processing brain scans and algorithms for processing general clouds of data points usually are treated as unrelated solutions to unrelated problems. Superficially, we can apply tactile and spatial intuition to processing shapes like brain surfaces, while noisy and abstract data may be more understandable through mathematical and statistical formality. Hence, three-dimensional shapes are treated in the geometry processing and vision literatures, while data analysis and pattern extraction are categorized as topics for machine learning and related disciplines.
Shape processing and data processing, however, use remarkably similar geometric language to state objectives and procedures. Three-dimensional objects and datasets both contain distinctive feature points connected by regions of varying curvature. Both encode intrinsic notions of proximity and distance along a domain rather than through the surrounding volume. And, both can be edited plausibly by stretching and bending motions.
Merging our understanding of these and other forms of input suggests our group’s long-term goal: To establish the theory and practice of geometric data processing, considered in the following two senses:
- The processing of geometric data
- Abstract data processing using geometric techniques
With these related but distinct applications in mind, our group widens the scope of “geometric data analysis” from a specialized branch of statistics to a broad field encapsulating the mathematical theory, algorithms, and computations of shape processing applied to abstract datasets and scans of physical objects alike. More generally, our group aims to become a center for applied geometry research at MIT and within the broader computational community.
Our research spans from foundational tools laying out rigorous groundwork for a theory of geometric data to exploration of applications across a broad variety of disciplines. A few themes in our recent work are highlighted below.
Optimization: Our group has developed state-of-the-art numerical algorithms for geometrically-structured problems, achieving scalability and stability needed for real-world applications in computer graphics, computer-aided design (CAD), vision, finite element (FEM) analysis, and operations research. Our recent contributions include efficient numerical methods for surface parameterization, a basic element of the 3D modeling toolbox, and surface-to-surface mapping, used in graphics and imaging to find relationships between different shapes or scans. An additional focus of our group is development of algorithms for optimal transportation problems, phrased in terms of matching supply to demand over a network or map.
Shape analysis: We have proposed fine-grained tools for understanding the structure of geometric data, from 3D scans in computer vision to animated characters in graphics and even abstract high-dimensional clouds of data points. Taking inspiration from differential geometry and machine learning alike, our methods have revealed geometric features and relationships suitable for transferring textures, organizing shape collections, interpolating between shapes, efficiently performing Monte Carlo sampling, and propagating labels for semi-supervised learning.
Collaboration: Our group draws from many application areas, including graphic design, machine vision, manufacturing, and physical simulation. We seek interaction with and feedback from industrial partners and a worldwide network of collaborators.