We address the problem of crowd localization, i.e., prediction dots corresponding to people in a crowded scene. Due various challenges, localization method is prone spatial semantic errors, predicting multiple within same person or collapsing cluttered region. propose topological approach targeting these errors. introduce constraint that teaches model reason about arrangement dots. To enforce t...

In mathematical morphology, connected filters based on dynamics are used to filter the extrema of an image. Similarly, persistence is a concept coming from persistent homology and Morse theory that represents stability function. Since these two concepts seem be closely related, in this paper we examine their relationship, prove they equal n-D functions, $$n\ge 1$$ . More exactly, pairing minimu...

Algebraic topology methods have recently played an important role for statistical analysis with complicated geometric structured data. Among them, persistent homology is a well-known tool to extract robust topological features, and outputs as persistence diagrams. Unfortunately, persistence diagrams are point multi-sets which can not be used in machine learning algorithms for vector data. To de...

The theory of local homology modules was initiated by Matlis in 1974. It is a dual version of the theory of local cohomology modules. Mohammadi and Divaani-Aazar (2012) studied the connection between local homology and Gorenstein flat modules by using Gorenstein flat resolutions. In this paper, we introduce generalized local homology modules for complexes and we give several ways for computing ...

Protein 3D structure can be a powerful predictor of function, but it often faces a critical roadblock at the crystallization step. Rv1738, a protein from Mycobacterium tuberculosis that is strongly implicated in the onset of nonreplicating persistence, and thereby latent tuberculosis, resisted extensive attempts at crystallization. Chemical synthesis of the L- and D-enantiomeric forms of Rv1738...

We use persistence modules and their corresponding barcodes to quantitatively distinguish between different fiberwise star-shaped domains in the cotangent bundle of a fixed manifold. The distance two is measured by nonlinear version classical Banach–Mazur distance, called symplectic denoted [Formula: see text]. relevant come from filtered homology are stable with respect Our main focus on space...

This dissertation extends the theory of persistent homology to time varying systems. Most of the previous work has been dedicated to using this powerful tool in topological data analysis to study static point clouds. In particular, given a point cloud, we can construct its persistence diagram. Since the diagram varies continuously as the point cloud varies continuously, we study the space of ti...

We investigate the relative cohomology and relative homology theories of $F$-Gorenstein modules, consider the relations between classical and $F$-Gorenstein (co)homology theories.

Comparison between multidimensional persistent Betti numbers is often based on the multidimensional matching distance. While this metric is rather simple to define and compute by considering a suitable family of filtering functions associated with lines having a positive slope, it has two main drawbacks. First, it forgets the natural link between the homological properties of filtrations associ...