Joan Lasenby on Applications of Geometric Algebra in Engineering

Core Research Objective: A joint project between the Engineering and Architecture departments at Cambridge uses drones to analyze the built environment by prioritizing line-based computer vision over traditional point-cloud methods.

• Line reconstruction is mathematically more complex than point-based structure from motion due to the difficulty of matching and tracking linear features.
• The project leverages Geometric Algebra (GA) to perform vision processing directly on lines, planes, and other geometric primitives.

Mathematical Foundation and History:

• Origins: The framework originates from Hermann Grassmann's outer product (wedge product) and William Clifford's extension combining inner and outer products (Clifford product).
• Evolution: David Hestenes revived the field in the 1960s, renaming it "Geometric Algebra" to distinguish it from the older "Clifford Algebra" notation.
• Key Advantage: GA provides a unified language where scalars, vectors, bivectors (planes), and trivectors (volumes) are treated as objects within a single algebra, allowing for differentiation and multiplication without switching between matrices or tensors.

Why Geometric Algebra vs. Conventional Methods:

• Unification: Unlike vector algebra or tensor analysis, GA unifies classical mechanics, quantum physics, and relativity into one system without requiring dual spaces or complex coordinate transformations.
• Coordinate Independence: GA allows for the writing of coordinate-free expressions, eliminating confusion regarding reference frames, rotation origins, and translation vectors common in traditional computer vision.
• Differentiation: The algebra supports analytic calculus directly on geometric objects (e.g., differentiating with respect to a rotation), which is significantly harder to achieve with matrix-based approaches.
• Conformal Geometric Algebra (CGA): Introduced in 1999, CGA extends 3D space to 5D, allowing points, lines, circles, spheres, and planes to be treated as single algebraic objects that can be rotated, translated, and intersected effortlessly.
• Rotations: GA uses rotors (derived from bivectors) to represent rotations, avoiding the singularities and numerical constraints associated with Euler angles or the redundancy of 3x3 rotation matrices.
• Comparison to Quaternions: While quaternions are effective for 3D rotations, GA generalizes this concept to any dimension and treats bivectors as the fundamental geometric entities of rotation.

Current State of the Field and Adoption:

• Adoption Barriers: Widespread use is hindered by the dominance of established systems (vector calculus, matrices) and the steep learning curve associated with anti-commutative algebra.
• Community Growth: Interest surged after a 2000 Royal Society paper on GA was featured on Hacker News, though skepticism remains among practitioners who view it as an unnecessary departure from proven tools.
• Target Audience: The technology is particularly accessible to younger researchers and engineers without deep backgrounds in advanced differential geometry, as it offers intuitive geometric insight.
• Computational Cost: GA is not computationally faster in raw calculation than optimized matrix libraries; its value lies in developer productivity, code clarity, and the ease of implementing complex algorithms.

Applications in Computer Vision and Engineering:

• Moving Cameras: GA is the preferred method for problems involving moving cameras, drones, and multiple sensor fusion where triangulation and motion extraction are required.
• Limitations in AI/ML: GA is not currently applied to deep learning tasks like image segmentation or object recognition (e.g., identifying trees or roads in self-driving cars), which rely on Bayesian methods and neural networks.
• Future Direction: Research is focused on parameterizing geometric objects (lines, planes) within GA frameworks and learning them directly from data streams, rather than using them for standard machine learning training.
• Cross-Domain Utility: Beyond vision, GA is being applied to thin-shell elasticity, structural engineering, and electromagnetism, where it simplifies complex tensor calculations.

Resources and Accessibility:

• Key Literature:
  • David Hestenes: Spacetime Algebra, Clifford Algebra to Geometric Calculus, New Foundations of Classical Mechanics.
  • Anthony Lasenby & Chris Doran: Geometric Algebra for Physicists.
  • Leo Dorst, Stephen Mann, Daniel Fonteyn: Geometric Algebra for Computer Scientists.
  • Eduardo Bayro Corrochano: Books focused on robotics.
• Software Tools:
  • Traditional implementations exist in MATLAB, C, and Python.
  • Web Accessibility: A community effort led by students (Hugo, Alex Sarsenovich) is developing a browser-based Python implementation to remove installation barriers (NumPy dependencies, OS compatibility), allowing immediate testing of GA concepts.