A Homologically Persistent Skeleton is a fast and robust descriptor of interest points in 2D images

2D images often contain irregular salient features and interest points with non-integer coordinates. Our skeletonization problem for such a noisy sparse cloud is to summarize the topology of a given 2D cloud across all scales in the form of a graph, which can be used for combining local features into a more powerful object-wide descriptor. We extend a classical Minimum Spanning Tree of a cloud to a Homologically Persistent Skeleton, which is scale-and-rotation invariant and depends only on the cloud without extra parameters. This graph (1) is computable in time O(nlogn) for any n points in the plane; (2) has the minimum total length among all graphs that span a 2D cloud at any scale and also have most persistent 1-dimensional cycles; (3) is geometrically stable for noisy samples around planar graphs

A one-dimensional Homologically Persistent Skeleton of an unstructured point cloud in any metric space

Real data are often given as a noisy unstructured point cloud, which is hard to visualize. The important problem is to represent topological structures hidden in a cloud by using skeletons with cycles. All past skeletonization methods require extra parameters such as a scale or a noise bound. We define a homologically persistent skeleton, which depends only on a cloud of points and contains optimal subgraphs representing 1-dimensional cycles in the cloud across all scales. The full skeleton is a universal structure encoding topological persistence of cycles directly on the cloud. Hence a 1-dimensional shape of a cloud can be now easily predicted by visualizing our skeleton instead of guessing a scale for the original unstructured cloud. We derive more subgraphs to reconstruct provably close approximations to an unknown graph given only by a noisy sample in any metric space. For a cloud of n points in the plane, the full skeleton and all its important subgraphs can be computed in time O(n log n)

A fast approximate skeleton with guarantees for any cloud of points in a Euclidean space

The tree reconstruction problem is to find an embedded straight-line tree that approximates a given cloud of unorganized points in $\mathbb{R}^m$ up to a certain error. A practical solution to this problem will accelerate a discovery of new colloidal products with desired physical properties such as viscosity. We define the Approximate Skeleton of any finite point cloud $C$ in a Euclidean space with theoretical guarantees. The Approximate Skeleton ASk$(C)$ always belongs to a given offset of $C$, i.e. the maximum distance from $C$ to ASk$(C)$ can be a given maximum error. The number of vertices in the Approximate Skeleton is close to the minimum number in an optimal tree by factor 2. The new Approximate Skeleton of any unorganized point cloud $C$ is computed in a near linear time in the number of points in $C$. Finally, the Approximate Skeleton outperforms past skeletonization algorithms on the size and accuracy of reconstruction for a large dataset of real micelles and random clouds

Geometric and Topological Methods for Applications to Materials and Data Skeletonisation

Crystal Structure Prediction (CSP) aims to speed up functional materials discovery by using supercomputers to predict whether an input molecule can form stable crystal struc- tures with desirable properties. The process produces large datasets where each entry is a simulated arrangement of copies of the input molecule to form a crystal. However, these datasets have little structure themselves, and it is the aim of this thesis to contribute towards simplifying and analysing such datasets. Crystals are unbounded collections of atoms or molecules, extending infinitely in the space they lie within. As such, rigorously quantifying the geometric similarity of crystal structures, and even just identifying identical structures, is a challenging problem. To solve it, we seek a continuous, complete, isometry classification of crystals. Consequently, by modelling crystals as periodic point sets, we introduce the density fingerprint, which is invariant under isometries, Lipschitz continuous, and complete for an open and dense space of crystal structures. Such a classification will be able to identify and remove near- duplicates from these large CSP datasets, and potentially even guide future searches. We describe how this fingerprint can be computed using periodic higher Voronoi zones. This geometric concept of concentric regions around a fixed centre characterises relative positions of points from the centre in a periodic point set. We present an algorithm to compute these zones in addition to proving key structural properties. We later discuss research into skeletonisation algorithms, proving theoretical guarantees of the homological persistent skeleton (HoPeS), subsequently formulating and performing an experimental comparison of HoPeS with other relevant algorithms. Such algorithms, if effectively used, can be applied to large datasets including those produced by CSP to reveal the shape of the data, helping to highlight regions of interest and branches that merit further study

Topological Data Analysis and Machine Learning for Recognizing Atmospheric River Patterns in Large Climate Datasets

Abstract. Identifying weather patterns that frequently lead to extreme weather events is a crucial first step in understanding how they may vary under different climate change scenarios. Here we propose an automated method for recognizing atmospheric rivers (ARs) in climate data using topological data analysis and machine learning. The method provides useful information about topological features (shape characteristics) and statistics of ARs. We illustrate this method by applying it to outputs of 5 version 5.1 of the Community Atmosphere Model (CAM5.1) and reanalysis product of the second Modern-Era Retrospective Analysis for Research &amp;amp; Applications (MERRA-2). An advantage of the proposed method is that it is threshold-free. Hence this method may be useful in evaluating model biases in calculating AR statistics. Further, the method can be applied to different climate scenarios without tuning since it does not rely on threshold conditions. We show that the method is suitable for rapidly analyzing large amounts of climate model and reanalysis output data. </jats:p

Topological data analysis and machine learning for recognizing atmospheric river patterns in large climate datasets

Identifying weather patterns that frequently lead to extreme weather events is a crucial first step in understanding how they may vary under different climate change scenarios. Here, we propose an automated method for recognizing atmospheric rivers (ARs) in climate data using topological data analysis and machine learning. The method provides useful information about topological features (shape characteristics) and statistics of ARs. We illustrate this method by applying it to outputs of version 5.1 of the Community Atmosphere Model version 5.1 (CAM5.1) and the reanalysis product of the second Modern-Era Retrospective Analysis for Research and Applications (MERRA-2). An advantage of the proposed method is that it is threshold-free – there is no need to determine any threshold criteria for the detection method – when the spatial resolution of the climate model changes. Hence, this method may be useful in evaluating model biases in calculating AR statistics. Further, the method can be applied to different climate scenarios without tuning since it does not rely on threshold conditions. We show that the method is suitable for rapidly analyzing large amounts of climate model and reanalysis output data.</p

Topological Data Analysis and Machine Learning for Recognizing Atmospheric River Patterns in Large Climate Datasets

Abstract. Identifying weather patterns that frequently lead to extreme weather events is a crucial first step in understanding how they may vary under different climate change scenarios. Here we propose an automated method for recognizing atmospheric rivers (ARs) in climate data using topological data analysis and machine learning. The method provides useful information about topological features (shape characteristics) and statistics of ARs. We illustrate this method by applying it to outputs of 5 version 5.1 of the Community Atmosphere Model (CAM5.1) and reanalysis product of the second Modern-Era Retrospective Analysis for Research &amp;amp; Applications (MERRA-2). An advantage of the proposed method is that it is threshold-free. Hence this method may be useful in evaluating model biases in calculating AR statistics. Further, the method can be applied to different climate scenarios without tuning since it does not rely on threshold conditions. We show that the method is suitable for rapidly analyzing large amounts of climate model and reanalysis output data. </jats:p