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Persistent homology analysis provides means to capture the connectivity structure of data sets in various dimensions. On the mathematical level, by defining a metric between the objects that persistence attaches to data sets, we can…
Motivated by the problem of dealing with incomplete or imprecise acquisition of data in computer vision and computer graphics, we extend results concerning the stability of persistent homology with respect to function perturbations to…
We propose to compute approximations to general invariant sets in dynamical systems by minimizing the distance between an appropriately selected finite set of points and its image under the dynamics. We demonstrate, through computational…
We introduce combinatorial multivector fields, associate with them multivalued dynamics and study their topological features. Our combinatorial multivector fields generalize combinatorial vector fields of Forman. We define isolated…
Tracking by detection, the dominant approach for online multi-object tracking, alternates between localization and association steps. As a result, it strongly depends on the quality of instantaneous observations, often failing when objects…
We present a black-box method to numerically investigate the linear stability of arbitrary multi-physics problems. While the user just has to enter the system's residual in weak formulation, i.e. by a finite element method, all required…
A fundamental tool in topological data analysis is persistent homology, which allows extraction of information from complex datasets in a robust way. Persistent homology assigns a module over a principal ideal domain to a one-parameter…
It remains a huge challenge to design effective and efficient trackers under complex scenarios, including occlusions, illumination changes and pose variations. To cope with this problem, a promising solution is to integrate the temporal…
Despite the success of many advanced tracking methods in this area, tracking targets with drastic variation of appearance such as deformation, view change and partial occlusion in video sequences is still a challenge in practical…
Multiple objects tracking finds its applications in many high level vision analysis like object behaviour interpretation and gait recognition. In this paper, a feature based method to track the multiple moving objects in surveillance video…
Understanding the decision-making processes of large language models is critical given their widespread applications. To achieve this, we aim to connect a formal mathematical framework - zigzag persistence from topological data analysis -…
Persistence has proved to be a valuable tool to analyze real world data robustly. Several approaches to persistence have been attempted over time, some topological in flavor, based on the vector space-valued homology functor, other…
In this paper, an online adaptive model-free tracker is proposed to track single objects in video sequences to deal with real-world tracking challenges like low-resolution, object deformation, occlusion and motion blur. The novelty lies in…
A method is presented for the distributed computation of persistent homology, based on an extension of the generalized Mayer-Vietoris principle to filtered spaces. Cellular cosheaves and spectral sequences are used to compute global…
Visual object tracking, which is primarily based on visible light image sequences, encounters numerous challenges in complicated scenarios, such as low light conditions, high dynamic ranges, and background clutter. To address these…
The aim of this article is to describe a new perspective on functoriality of persistent homology and explain its intrinsic symmetry that is often overlooked. A data set for us is a finite collection of functions, called measurements, with a…
In many relevant cases -- e.g., in hamiltonian dynamics -- a given vector field can be characterized by means of a variational principle based on a one-form. We discuss how a vector field on a manifold can also be characterized in a similar…
With the advance of fluorescence imaging technologies, recently cell biologists are able to record the movement of protein vesicles within a living cell. Automatic tracking of the movements of these vesicles become key for qualitative…
Feature tracking is the building block of many applications such as visual odometry, augmented reality, and target tracking. Unfortunately, the state-of-the-art vision-based tracking algorithms fail in surgical images due to the challenges…
Multiparameter persistence is a natural extension of the well-known persistent homology, which has attracted a lot of interest. However, there are major theoretical obstacles preventing the full development of this promising theory. In this…