Skeletonization and Partitioning of Digital Images Using Discrete Morse Theory

Delgado‐Friedrichs, Olaf; Robins, Vanessa; Sheppard, Adrian

doi:10.1109/tpami.2014.2346172

Cited by 86 publications

(81 citation statements)

References 45 publications

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“…The high-level description of our discrete-Morse based skeleton extraction algorithm in Algorithm 1. We note that this is similar to the persistence-guided discrete Morse based graph reconstruction framework introduced in [17] which builds upon [1,16,45] 3 . We thus will only provide a brief description below and for further details we refer the readers to [17] or to Chapter 5 and 6 of [51].…”

Section: Step 2: Graph Skeletonizationmentioning

confidence: 91%

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Topological Skeletonization and Tree-Summarization of Neurons Using Discrete Morse Theory

Wang

Mitra

et al. 2018

Preprint

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Neuroscientific data analysis has classically involved methods for statistical signal and image processing, drawing on linear algebra and stochastic process theory. However, digitized neuroanatomical data sets containing labelled neurons, either individually or in groups labelled by tracer injections, do not fully fit into this classical framework. The tree-like shapes of neurons cannot mathematically be adequately described as points in a vector space (eg, the subtraction of two neuronal shapes is not a meaningful operation). There is therefore a need for new approaches. Methods from computational topology and geometry are naturally suited to the analysis of neuronal shapes. Here we introduce methods from Discrete Morse Theory to extract tree-skeletons of individual neurons from volumetric brain image data, or to summarize collections of neurons labelled by localized anterograde tracer injections. Since individual neurons are topologically trees, it is sensible to summarize the collection of neurons labelled by a localized anterograde tracer injection using a consensus tree-shape. This consensus tree provides a richer information summary than the regional or voxel-based "connectivity matrix" approach that has previously been used in the literature.The algorithmic procedure includes an initial pre-processing step to extract a density field from the raw volumetric image data, followed by initial skeleton extraction from the density field using a discrete version of a 1-(un)stable manifold of the density field. Heuristically, if the density field is regarded as a mountainous landscape, then the 1-(un)stable manifold follows the "mountain ridges" connecting the maxima of the density field. We then simplify this skeletongraph into a tree using a shortest-path approach and methods derived from persistent homology. The advantage of this approach is that it uses global information about the density field and is therefore robust to local fluctuations and non-uniformly distributed input signals. To be able to handle large data sets, we use a divide-and-conquer approach. The resulting software DiMorSC is available on Github [40]. To the best of our knowledge this is currently the only publicly available code for the extraction of the 1-unstable manifold from an arbitrary simplicial complex using the Discrete Morse approach.

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Section: Step 2: Graph Skeletonizationmentioning

confidence: 91%

“…From a computer science perspective, our apporach builds upon prior work on using discrete Morse based methodology for extracting skeletons of images; see e.g, [1,16,42,45,52]. Our software is based on the work of Gyulassy [1,26], of Sousbie [45], and Wang et al [52].…”

Section: Tree Summary Extraction and Simplificationmentioning

confidence: 99%

Topological Skeletonization and Tree-Summarization of Neurons Using Discrete Morse Theory

Wang

Mitra

et al. 2018

Preprint

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“…The area of TDA that relates to the present work is the persistence-guided Discrete Morse theory based framework for reconstructing hidden graphs from observed data. Discrete Morse theory has been utilized to capture hidden structure in 2D or 3D volumetric data [15][16][17] . Extraction of hidden graphs was formulated in 18 , and the framework was simplified and theoretical guarantees provided in 19 .…”

Section: Introductionmentioning

confidence: 99%

Detection and skeletonization of single neurons and tracer injections using topological methods

Wang

Magee

Huo

et al. 2020

Preprint

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Neuroscientific data analysis has traditionally relied on linear algebra and stochastic process theory. However, the tree-like shapes of neurons cannot be described easily as points in a vector space (the subtraction of two neuronal shapes is not a meaningful operation), and methods from computational topology are better suited to their analysis. Here we introduce methods from Discrete Morse (DM) Theory to extract the tree-skeletons of individual neurons from volumetric brain image data, and to summarize collections of neurons labelled by tracer injections. Since individual neurons are topologically trees, it is sensible to summarize the collection of neurons using a consensus tree-shape that provides a richer information summary than the traditional regional 'connectivity matrix' approach. The conceptually elegant DM approach lacks handtuned parameters and captures global properties of the data as opposed to previous approaches which are inherently local. For individual skeletonization of sparsely labelled neurons we obtain substantial performance gains over state-of-the-art non-topological methods (over 10% improvements in precision and faster proofreading). The consensus-tree summary of tracer injections incorporates the regional connectivity matrix information, but in addition captures the collective collateral branching patterns of the set of neurons connected to the injection site, and provides a bridge between single-neuron morphology and tracer-injection data. SummaryNeuroscientific data analysis has traditionally involved methods for statistical signal and image processing, drawing on linear algebra and stochastic process theory. However, digitized neuroanatomical datasets containing labelled neurons, either individually or in groups labelled by tracer injections, do not fit into this classical framework. The tree-like shapes of neurons cannot be adequately described as points in a vector space (e.g. the subtraction of two neuronal shapes is not a meaningful operation). There is therefore a need for new approaches, which has become more urgent given the growth in whole-brain datasets with sparsely labelled neurons or tracer injections.Methods from computational topology and geometry are naturally suited to the analysis of neuronal shapes. In this paper we introduce methods from Discrete Morse Theory to extract tree-skeletons of individual neurons from volumetric brain image data, and to summarize collections of neurons labelled by anterograde tracer injections. Since individual neurons are topologically trees, it is sensible to summarize the collection of neurons using a consensus tree-shape. This consensus tree provides a richer information summary than the regional or voxel-based "connectivity matrix" approach that has previously been used in the literature.The algorithmic procedure for single neuron skeletonization includes an initial neurite-detection step to extract a density field from the raw volumetric image data, followed by Dis-crete Morse theoretic skeleton extraction from the density field us...

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“…Persistent homology is a new tool for studying shape in application areas such as digital images [17,18], dynamical systems [19], and high-dimensional data-mining [20]. There are two main strengths of our approach:…”

Section: Introductionmentioning

confidence: 99%

Texture Analysis of Hydrophobic Polycarbonate and Polydimethylsiloxane Surfaces via Persistent Homology

et al. 2017

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Due to recent climate change-triggered, regular dust storms in the Middle East, dust mitigation has become the critical issue for solar energy harvesting devices. One of the methods to minimize and prevent dust adhesion and create self-cleaning abilities is to generate hydrophobic characteristics on surfaces. The purpose of this study is to explore the topological features of hydrophobic surfaces. We use non-standard techniques from topological data analysis to extract morphological features from the AFM images. Our method recovers most of the previous qualitative observations in a robust and quantitative way. Persistence diagrams, which is a summary of topological structures, witness quantitatively that the crystallized polycarbonate (PC) surface possesses spherulites, voids, and fibrils, and the texture height and spherulite concentration increases with the increased immersion period. The approach also shows that the polydimethylsiloxane (PDMS) exactly copied the structures at the PC surface but 80 to 90 percent of the nanofibrils were not copied at PDMS surface. We next extract a feature vector from each persistence diagram to show which experiments hold features with similar variance using principal component analysis (PCA). The K-means clustering algorithm is applied to the matrix of feature vectors to support the PCA result, grouping experiments with similar features.

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Skeletonization and Partitioning of Digital Images Using Discrete Morse Theory

Cited by 86 publications

References 45 publications

Topological Skeletonization and Tree-Summarization of Neurons Using Discrete Morse Theory

Topological Skeletonization and Tree-Summarization of Neurons Using Discrete Morse Theory

Detection and skeletonization of single neurons and tracer injections using topological methods

Texture Analysis of Hydrophobic Polycarbonate and Polydimethylsiloxane Surfaces via Persistent Homology

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