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In this paper we prove results relating to two homotopy relations and four homology theories developed in the topology of digital images. We introduce a new type of homotopy relation for digitally continuous functions which we call "strong…
This paper presents a mathematically rigorous framework for brain-inspired representation learning founded on the interplay between persistent topological structures and cohomological flows. Neural computation is reformulated as the…
We characterize when the level sets of a continuous quasi-monotone functional defined on a suitable convex subset of a normed space can be uniquely represented by a family of bounded continuous functionals. Furthermore, we investigate how…
Morphing is a long-standing problem in vision and computer graphics, requiring a time-dependent warping for feature alignment and a blending for smooth interpolation. Recently, multilayer perceptrons (MLPs) have been explored as implicit…
Persistent homology is a multiscale method for analyzing the shape of sets and functions from point cloud data arising from an unknown distribution supported on those sets. When the size of the sample is large, direct computation of the…
Image-based 3D reconstruction is one of the most important tasks in Computer Vision with many solutions proposed over the last few decades. The objective is to extract metric information i.e. the geometry of scene objects directly from…
Functional connectomes capture brain interactions via synchronized fluctuations in the functional magnetic resonance imaging signal. If measured during rest, they map the intrinsic functional architecture of the brain. With task-driven…
Topological data analysis is becoming increasingly relevant to support the analysis of unstructured data sets. A common assumption in data analysis is that the data set is a sample---not necessarily a uniform one---of some high-dimensional…
A fundamental open question asking whether all real-valued strongly quasiconvex functions defined on $\mathbb R^n$ are necessarily continuous, akin to their convex counterparts, is answered in detail in this paper. Among other things, we…
We consider maps which preserve functions which are built out of the invariants of some simple vector fields. We give a reduction procedure, which can be used to derive commuting maps of the plane, which preserve the same symplectic form…
We present an analytical theory of topologically protected photonic states for the two-dimensional Maxwell equations for a class of continuous periodic dielectric structures, modulated by a domain wall. We further numerically confirm the…
This paper introduces persistent homology, which is a powerful tool to characterize the shape of data using the mathematical concept of topology. We explain the fundamental idea of persistent homology from scratch using some examples. We…
Topological data analysis (TDA) is an active field of mathematics for quantifying shape in complex data. Standard methods in TDA such as persistent homology (PH) are typically focused on the analysis of data consisting of a single entity…
Dimensionality reduction is an integral part of data visualization. It is a process that obtains a structure preserving low-dimensional representation of the high-dimensional data. Two common criteria can be used to achieve a dimensionality…
Persistent homology provides information about the lifetime of homology classes along a filtration of cell complexes. Persistence barcode is a graphical representation of such information. A filtration might be determined by time in a set…
Topological structures in image data, such as connected components and loops, play a crucial role in understanding image content (e.g., biomedical objects). % Despite remarkable successes of numerous image processing methods that rely on…
The rapid evolution of digital image manipulation techniques poses significant challenges for content verification, with models such as stable diffusion and mid-journey producing highly realistic, yet synthetic, images that can deceive…
We characterize the class of image plane transformations which realize rigid camera motions and call these transformations `rigidity preserving'. In particular, 2D translations of pinhole images are not rigidity preserving. Hence, when…
A multivalued projection is an idempotent linear relation with invariant domain. We characterize multivalued projections that are operator ranges (called semiclosed) and provide several formulae of them. Moreover, we study the…
Several topological and analytical notions of continuity and fading memory for causal and time-invariant filters are introduced, and the relations between them are analyzed. A significant generalization of the convolution theorem that…