Optimal Topological Cycles and Their Application in Cardiac Trabeculae Restoration

Wu, Pengxiang; Chen, Chao; Wang, Yusu; Zhang, Shaoting; Yuan, Changhe; Qian, Zhen; Metaxas, Dimitris N.; Axel, Leon

doi:10.1007/978-3-319-59050-9_7

Cited by 31 publications

(33 citation statements)

References 13 publications

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“…Approximately 90% of the classification matches were performed by medical doctors 15 . Moreover, in cardiac image analysis, a study applying TDA to computed tomography images successfully extracted the shape of the trabeculae, the fine muscle columns on the ventricular walls which had been missed by previous methods 45 . These results and ours suggest that TDA is suitable for the image analysis of organs with fine structures.…”

Section: Discussionmentioning

confidence: 99%

Assessment of skin barrier function using skin images with topological data analysis

et al. 2020

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Recent developments of molecular biology have revealed diverse mechanisms of skin diseases, and precision medicine considering these mechanisms requires the frequent objective evaluation of skin phenotypes. Transepidermal water loss (TEWL) is commonly used for evaluating skin barrier function; however, direct measurement of TEWL is time-consuming and is not convenient for daily clinical practice. Here, we propose a new skin barrier assessment method using skin images with topological data analysis (TDA). TDA enabled efficient identification of structural features from a skin image taken by a microscope. These features reflected the regularity of the skin texture. We found a significant correlation between the topological features and TEWL. Moreover, using the features as input, we trained machine-learning models to predict TEWL and obtained good accuracy (R2 = 0.524). Our results suggest that assessment of skin barrier function by topological image analysis is promising.

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Section: Discussionmentioning

confidence: 99%

Assessment of skin barrier function using skin images with topological data analysis

et al. 2020

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“…To make use of persistent homology, one typically computes a persistence diagram (also called barcode) which is a set of intervals with birth and death points. Besides just utilizing the set of intervals, some applications [13,28] need persistence diagrams augmented with representative cycles for the intervals for gaining more insight into the data. These representative cycles, termed as persistent cycles [13], have been studied by Wu et al [28], Obayashi [23], and Dey et al [13] recently from the view-point of optimality.…”

Section: Introductionmentioning

confidence: 99%

Computing Minimal Persistent Cycles: Polynomial and Hard Cases

Dey

Hou

Mandal

2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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Persistent cycles, especially the minimal ones, are useful geometric features functioning as augmentations for the intervals in the purely topological persistence diagrams (also termed as barcodes). In our earlier work, we showed that computing minimal 1-dimensional persistent cycles (persistent 1-cycles) for finite intervals is NP-hard while the same for infinite intervals is polynomially tractable. In this paper, we address this problem for general dimensions with Z 2 coefficients. In addition to proving that it is NP-hard to compute minimal persistent dcycles (d > 1) for both types of intervals given arbitrary simplicial complexes, we identify two interesting cases which are polynomially tractable. These two cases assume the complex to be a certain generalization of manifolds which we term as weak pseudomanifolds. For finite intervals from the d th persistence diagram of a weak (d + 1)-pseudomanifold, we utilize the fact that persistent cycles of such intervals are null-homologous and reduce the problem to a minimal cut problem. Since the same problem for infinite intervals is NP-hard, we further assume the weak (d + 1)-pseudomanifold to be embedded in R d+1 so that the complex has a natural dual graph structure and the problem reduces to a minimal cut problem. Experiments with both algorithms on scientific data indicate that the minimal persistent cycles capture various significant features of the data. ProblemRestrictionMain contributions. We summarize our contributions as follows:• We prove the NP-hardness of PCYC-FIN d and WPCYC-INF d for all d ≥ 2.• We present two polynomial time algorithms for WPCYC-FIN d and WEPCYC-INF d when d ≥ 1, based on the duality of minimal persistent cycles and minimal cuts. Other than the minimal cut computation, steps in both algorithms run in linear or almost linear time.

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“…To address the issue of network threshold selection, a set of brain network analysis methods called graph filtration based on persistent homology (Chen & Edelsbrunner, ; Edelsbrunner & Harer, ; Wu et al, ) has been proposed to measure persistent brain network topology features generated over all possible thresholds (Choi et al, ; Chung, Hanson, Ye, Davidson, & Pollak, ; Lee et al, ; Lee, Kang, Chung, Kim, & Lee, ; Yoo et al, ). It quantifies various persistent topological features at different scales in a coherent manner and avoids the thresholding selection.…”

Section: Introductionmentioning

confidence: 99%

A concise and persistent feature to study brain resting‐state network dynamics: Findings from the Alzheimer's Disease Neuroimaging Initiative

Kuang

Chen

Caselli

et al. 2018

Human Brain Mapping

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Alzheimer’s disease (AD) is the most common type of dementia in the elderly with no effective treatment currently. Recent studies of non-invasive neuroimaging, resting-state functional magnetic resonance imaging (rs-fMRI) with graph theoretical analysis have shown that patients with AD and mild cognitive impairment (MCI) exhibit disrupted topological organization in large-scale brain networks. In previous work, it is a common practice to threshold such networks. However, it is not only difficult to make a principled choice of threshold values, but worse is the discard of potential important information. To address this issue, we propose a threshold-free feature by integrating a prior persistent homology-based topological feature (the zeroth Betti number) and a newly defined connected component aggregation cost feature to model brain networks over all possible scales. We show that the induced topological feature (Integrated Persistent Feature – IPF) follows a monotonically decreasing convergence function and further propose to use its slope as a concise and persistent brain network topological measure. We apply this measure to study rs-fMRI data from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) and compare our approach with five other widely used graph measures across five parcellation schemes ranging from 90 to 1024 region-of-interests (ROIs). The experimental results demonstrate that the proposed network measure shows more statistical power and stronger robustness in group difference studies in that the absolute values of the proposed measure of AD are lower than MCI’s, and much lower than normal control’s, providing empirical evidence for decreased functional integration in AD dementia and MCI.

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Optimal Topological Cycles and Their Application in Cardiac Trabeculae Restoration

Cited by 31 publications

References 13 publications

Assessment of skin barrier function using skin images with topological data analysis

Assessment of skin barrier function using skin images with topological data analysis

Computing Minimal Persistent Cycles: Polynomial and Hard Cases

A concise and persistent feature to study brain resting‐state network dynamics: Findings from the Alzheimer's Disease Neuroimaging Initiative

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