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.