Sparse Parallel Algorithms for Recognizing Touch Topology on Curved Interactive Screens

The advent of topological data analysis enabled us to extract topological invariants under object deformations and transformations, such as the number of loops in an image, which was conventionally unachievable with the filters of analog nature. However, the existing algorithms are mostly off-line or non-parallel, and therefore not suitable for interactive applications, such as touch sensors. Therefore, we previously proposed a real-time, distributed algorithm to compute the Euler characteristics as a touch invariant by summing up the Poincare-Hopf index for each pixel, which is fortunately sparse being zero for most pixels. However, the previous algorithm was specialized and restricted to plane screens with triangulated meshes. The explosive growth of the tablet or touch interface technology with super-thin screens or virtual realities is creating a new demand for software, for example, to detect if or not a touch wraps plastic bottles, coffee cup handles, or balls. In this paper, we extended the scope of our previous algorithm from plane to curved screens, including cylinders, toruses, and regular polyhedra, for solving truly topological sensing problems. We demonstrate that our implementation in processing, in the form of solely local logical operations and without any time-consuming iterative operations, returns and updates the correct topological invariants of touches on curved screens in real time.