Related papers: The Shape of an Image: A Study of Mapper on Images
Summarizing topological information from datasets and maps defined on them is a central theme in topological data analysis. \textsf{Mapper}, a tool for such summarization, takes as input both a possibly high dimensional dataset and a map…
Connected operators are filtering tools that act by merging elementary regions of an image. A popular strategy is based on tree-based image representations: for example, one can compute an attribute on each node of the tree and keep only…
The Mapper algorithm, a technique within topological data analysis (TDA), constructs a simplified graphical representation of high-dimensional data to uncover its underlying shape and structural patterns. The algorithm has attracted…
Mapper is an algorithm that summarizes the topological information contained in a dataset and provides an insightful visualization. It takes as input a point cloud which is possibly high-dimensional, a filter function on it and an open…
We propose a new approach, named PolyMapper, to circumvent the conventional pixel-wise segmentation of (aerial) images and predict objects in a vector representation directly. PolyMapper directly extracts the topological map of a city from…
Node-link diagrams are a popular method for representing graphs that capture relationships between individuals, businesses, proteins, and telecommunication endpoints. However, node-link diagrams may fail to convey insights regarding graph…
Shape inference is classically ill-posed, because it involves a map from the (2D) image domain to the (3D) world. Standard approaches regularize this problem by either assuming a prior on lighting and rendering or restricting the domain,…
Mapper and Ball Mapper are Topological Data Analysis tools used for exploring high dimensional point clouds and visualizing scalar-valued functions on those point clouds. Inspired by open questions in knot theory, new features are added to…
A method for creating a forest of model trees to fit samples of a function defined on images is described in several steps: down-sampling the images, determining a tree's hyperplanes, applying convolutions to the hyperplanes to handle small…
Large-scale vector mapping is important for transportation, city planning, and survey and census. We propose GraphMapper, a unified framework for end-to-end vector map extraction from satellite images. Our key idea is a novel unified…
Topological data analysis aims to extract topological quantities from data, which tend to focus on the broader global structure of the data rather than local information. The Mapper method, specifically, generalizes clustering methods to…
Topological data analysis provides a collection of tools to encapsulate and summarize the shape of data. Currently it is mainly restricted to \emph{mapper algorithm} and \emph{persistent homology}. In this paper we introduce new…
Any function can be constructed using a hierarchy of simpler functions through compositions. Such a hierarchy can be characterized by a binary rooted tree. Each node of this tree is associated with a function which takes as inputs two…
Acquiring plausible pathways on high-dimensional structural distributions is beneficial in several domains. For example, in the drug discovery field, a protein conformational pathway, i.e. a highly probable sequence of protein structural…
Classical unsupervised learning methods like clustering and linear dimensionality reduction parametrize large-scale geometry when it is discrete or linear, while more modern methods from manifold learning find low dimensional representation…
Enhancement is an important step in post-processing digital images for personal use, in medical imaging, and for object recognition. Most existing manual techniques rely on region selection, similarity, and/or thresholding for editing,…
The present paper mainly presents, for example, explicit classifications of compact smooth manifolds having non-empty boundaries and simple structures where the dimensions are general. Studies of this type is fundamental and important. They…
Topological correctness is critical for segmentation of tubular structures, which pervade in biomedical images. Existing topological segmentation loss functions are primarily based on the persistent homology of the image. They match the…
Choosing a suitable filtering function for the Mapper algorithm can be difficult due to its arbitrariness and domain-specific requirements. Finding a general filtering function that can be applied across domains is therefore of interest,…
To investigate the topological structure of Morse functions on the projective plane we use the Reeb graphs. We describe it properties and prove that it is a complete topological invariant of simple Morse function on $\mathbb{R} P^2$. We…