Persistence Cycles for Visual Exploration of Persistent Homology

Iuricich, Federico

doi:10.1109/tvcg.2021.3110663

Cited by 6 publications

(4 citation statements)

References 69 publications

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“…All our experiments were run on a Desktop computer mounting a Core i7-8700 3.20 GHz processor and 32 GB of RAM. In the first stage we used a dedicated TTK plugin for computing persistent homology and persistence cycles on each image [15]. This step requires less than 50MB of RAM and less than a second per image.…”

Section: Analysis Pipelinementioning

confidence: 99%

“…Red pixels indicate high-density regions that contain more persistence pairs than the blue pixels. The visualization of the persistence image is also paired with the input digit image, and the persistence cycles [15] (see Figure 2a right), an explicit visualization of 1-cycles originated and destroyed by the filtration. We recall that persistent homology associates an importance value (i.e., persistence) to each feature according to its lifespan in the filtration.…”

Section: Visualization and User Interactionsmentioning

confidence: 99%

“…It also provides two ways to visualize persistent homology other than the persistence diagram: the embedded persistence pairs, which represent each dot of the persistence diagram as a pair of points in the filtration domain, and the persistence curve, which is a line chart plotting the number of pairs at the vary of the persistence values. Recently, a new plugin for TTK has been developed which allows the visualization of persistence cycles [15], an explicit representation of the boundary of each hole, or void originated by the filtration.…”

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confidence: 99%

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TopoEmbedding, a web tool for the interactive analysis of persistent homology

Bao¹,

Liu²,

Iuricich³

2022

Preprint

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Software libraries for Topological Data Analysis (TDA) offer limited support for interactive visualization. Most libraries only allow to visualize topological descriptors (e.g., persistence diagrams), and lose the connection with the original domain of data. This makes it challenging for users to interpret the results of a TDA pipeline in an exploratory context. In this paper, we present TopoEmbedding, a web-based tool that simplifies the interactive visualization and analysis of persistence-based descriptors. TopoEmbedding allows non-experts in TDA to explore similarities and differences found by TDA descriptors with simple yet effective visualization techniques.

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Section: Analysis Pipelinementioning

confidence: 99%

Section: Visualization and User Interactionsmentioning

confidence: 99%

mentioning

confidence: 99%

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TopoEmbedding, a web tool for the interactive analysis of persistent homology

Bao¹,

Liu²,

Iuricich³

2022

Preprint

Self Cite

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“…In the early 2000's, mathematicians introduced an idea that is now called TDA [12][13][14][15], implemented using a notion called persistent homology [16,17]. The idea was to take the concepts of homology from the mathematical fields of algebraic topology and algebraic geometry and apply them to real-world data.…”

Section: Introductionmentioning

confidence: 99%

Topological Structures in the Space of Treatment-Naïve Patients With Chronic Lymphocytic Leukemia

McGee,

Reed,

Coombes

et al. 2024

Preprint

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Objectives Patients are complex and heterogeneous; clinical data sets are complicated by noise, missing data, and the presence of mixed-type data. Using such data sets requires understanding the high-dimensional "space of patients", composed of all measurements that define all relevant phenotypes. The current state-of-the-art settles for defining simple spatial groupings of patients using clustering analyses or dimension reduction. Our goal is to see if topological data analysis (TDA), a relatively new unsupervised technique, is able to obtain a more complete understanding of patient space. Methods TDA is optimized to detect "holes" in data, such as the insides of circles (loops) or the insides of spheres (voids). We apply TDA to a space of 266 previously untreated patients with Chronic Lymphocytic Leukemia (CLL). We use the "daisy" distance metric defined by Kaufman and Rouseeuw to compute distances between clinical records. We describe novel computational and graphical methods to interpret the structures detected by TDA. Results Using TDA, we find clear evidence of the existence of both loops and voids in the CLL data. The most persistent loop and the most persistent void can be interpreted using three dichotomized, prognostically important factors in CLL: _IGHV_ somatic mutation status, beta-2 microglobulin, and Rai stage. Conclusion We applied a cutting-edge analysis tool, TDA, to better define the "space of patients" in CLL clinical data. Patient space turns out to be richer and more complex than current models imply. TDA could become a powerful tool in the biomedical informatician's arsenal for interpreting high-dimensional data. It may provide novel insights into biological processes and improve our understanding of clinical and biological data sets.

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Stable volumes for persistent homology

Obayashi

2023

J Appl. and Comput. Topology

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This paper proposes a stable volume and a stable volume variant, referred to as a stable sub-volume, for more reliable data analysis using persistent homology. In prior research, an optimal cycle and similar ideas have been proposed to identify the homological structure corresponding to each birth-death pair in a persistence diagram. While this is helpful for data analysis using persistent homology, the results are sensitive to noise. The sensitivity affects the reliability and interpretability of the analysis. In this paper, stable volumes and stable sub-volumes are proposed to solve this problem. For a special case, we prove that a stable volume is the robust part of an optimal volume against noise. We implemented stable volumes and sub-volumes on HomCloud, a data analysis software package based on persistent homology, and show examples of stable volumes and sub-volumes.

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Persistence Cycles for Visual Exploration of Persistent Homology

Cited by 6 publications

References 69 publications

TopoEmbedding, a web tool for the interactive analysis of persistent homology

TopoEmbedding, a web tool for the interactive analysis of persistent homology

Topological Structures in the Space of Treatment-Naïve Patients With Chronic Lymphocytic Leukemia

Stable volumes for persistent homology

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