Persistent homology is perhaps the most popular and useful tool offered by topological data analysis, with point-cloud data being the most common setup. Its older cousin, the Euler characteristic curve (ECC) is less expressive, but far easier to compute. It is particularly suitable for analyzing imaging data, and is commonly used in fields ranging from astrophysics to biomedical image analysis. These fields are embracing GPU computations to handle increasingly large datasets. We therefore propose an optimized GPU implementation of ECC computation for 2D and 3D grayscale images. The goal of this paper is twofold. First, we offer a practical tool, illustrating its performance with thorough experimentation, but also explain its inherent shortcomings. Second, this simple algorithm serves as a perfect backdrop for highlighting basic GPU programming techniques that make our implementation so efficient, and some common pitfalls we avoided. This is intended as a step towards a wider usage of GPU programming in computational geometry and topology software. We find this is particularly important as geometric and topological tools are used in conjunction with modern, GPU-accelerated machine learning frameworks.

A key challenge in feature correspondence is the difficulty in differentiating true and false matches at a local descriptor level. This forces adoption of strict similarity thresholds that discard many true matches. However, if analyzed at a global level, false matches are usually randomly scattered while true matches tend to be coherent (clustered around a few dominant motions), thus creating a coherence based separability constraint. This paper proposes a non-linear regression technique that can discover such a coherence based separability constraint from highly noisy matches and embed it into a correspondence likelihood model. Once computed, the model can filter the entire set of nearest neighbor matches (which typically contains over 90 percent false matches) for true matches. We integrate our technique into a full feature correspondence system which reliably generates large numbers of good quality correspondences over wide baselines where previous techniques provide few or no matches.