Related papers: A Persistent Homology Design Space for 3D Point Cl…
Popular network models such as the mixed membership and standard stochastic block model are known to exhibit distinct geometric structure when embedded into $\mathbb{R}^{d}$ using spectral methods. The resulting point cloud concentrates…
Recognition of occluded objects in unseen indoor environments is a challenging problem for mobile robots. This work proposes a new slicing-based topological descriptor that captures the 3D shape of object point clouds to address this…
Persistence-based topological optimization deforms a point cloud $X \subset \mathbb{R}^d$ by minimizing objectives of the form $L(X) = \ell(\mathrm{Dgm}(X))$, where $\mathrm{Dgm}(X)$ is a persistence diagram. In practice, optimization is…
The irregularity and permutation invariance of point cloud data pose challenges for effective learning. Conventional methods for addressing this issue involve converting raw point clouds to intermediate representations such as 3D voxel…
Mathematical morphology (MM) is a powerful and widely used framework in image processing. Through set-theoretic and discrete geometric principles, MM operations such as erosion, dilation, opening, and closing effectively manipulate digital…
Persistent homology (PH) is a powerful mathematical method to automatically extract relevant insights from images, such as those obtained by high-resolution imaging devices like electron microscopes or new-generation telescopes. However,…
Point clouds are a set of data points in space to represent the 3D geometry of objects. A fundamental step in the processing is to identify a subset of points to represent the shape. While traditional sampling methods often ignore to…
Recent advancements in Diffusion Transformer (DiT) models have significantly improved 3D point cloud generation. However, existing methods primarily focus on local feature extraction while overlooking global topological information, such as…
While conventional computer vision emphasizes pixel-level and feature-based objectives, medical image analysis of intricate biological structures necessitates explicit representation of their complex topological properties. Despite their…
Characterizing the structural properties of neural networks is crucial yet poorly understood, and there are no well-established similarity measures between networks. In this work, we observe that neural networks can be represented as…
Techniques from computational topology, in particular persistent homology, are becoming increasingly relevant for data analysis. Their stable metrics permit the use of many distance-based data analysis methods, such as multidimensional…
Appropriately representing elements in a database so that queries may be accurately matched is a central task in information retrieval; recently, this has been achieved by embedding the graphical structure of the database into a manifold in…
Topological data analysis (TDA) is gaining prominence across a wide spectrum of machine learning tasks that spans from manifold learning to graph classification. A pivotal technique within TDA is persistent homology (PH), which furnishes an…
This paper investigates the problem of temporally interpolating dynamic 3D point clouds with large non-rigid deformation. We formulate the problem as estimation of point-wise trajectories (i.e., smooth curves) and further reason that…
Persistent homology barcodes and diagrams are a cornerstone of topological data analysis that capture the "shape" of a wide range of complex data structures, such as point clouds, networks, and functions. However, their use in statistical…
Persistent homology is a central methodology in topological data analysis that has been successfully implemented in many fields and is becoming increasingly popular and relevant. The output of persistent homology is a persistence diagram --…
In this paper we propose a rotation-invariant deep network for point clouds analysis. Point-based deep networks are commonly designed to recognize roughly aligned 3D shapes based on point coordinates, but suffer from performance drops with…
Persistent Homology is a powerful tool in Topological Data Analysis (TDA) to capture topological properties of data succinctly at different spatial resolutions. For graphical data, shape, and structure of the neighborhood of individual data…
Point clouds, being the simple and compact representation of surface geometry of 3D objects, have gained increasing popularity with the evolution of deep learning networks for classification and segmentation tasks. Unlike human, teaching…
The Extended Persistent Homology Transform (XPHT) is a topological transform which takes as input a shape embedded in Euclidean space, and to each unit vector assigns the extended persistence module of the height function over that shape…