Computational Algebraic Topology in Computer Vision

This thesis presents a novel approach to keypoint detection and description in computer vision, grounded in the principles of algebraic topology. Traditional methods for keypoint detection lack robustness under transformations, while more recent deep learning-based approaches face challenges related to scale dependency. To address these limitations, we introduce a framework based on Morse theory and persistent homology, providing a rigorous mathematical foundation for keypoint detection that is inherently scale-invariant. Our method models keypoints as topological invariants derived from the image data, using a differentiable formulation that aligns with modern optimization techniques. We propose a novel loss function inspired by persistent homology, which ensures keypoint repeatability across variations in scale, viewpoint, and illumination. Empirical experiments on standard benchmarks, such as HPatches, demonstrate that our approach outperforms existing methods in terms of keypoint repeatability and robustness. The implications of this work extend to critical applications in visual localization, SLAM, Structure-from-Motion, and 3D reconstruction. By integrating topological concepts into feature detection, this research paves the way for future advancements in computer vision, offering a new paradigm that merges mathematical rigor with practical performance. Moreover, our application is the first method based solely on a topological objective for feature detection, providing new insights into the use of persistent homology in unsupervised learning. The thesis content also covers the foundation of homology theory, persistent homology, and discrete Morse theory, enabling a comprehensive understanding of the methods. Additionally, it details the Pinnacle camera model and the fundamentals of homography estimation using the keypoints and descriptors paradigm.