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 hand-tuned 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.
Let f be a continuous function defined on a topological space. As we increase the function value, connected components in the level set of f appear, disappear, merge, or split, and the Reeb graph tracks these changes. The join and split trees of f track the changes of connected components in the sublevel and superlevel set respectively. If the Reeb graph is loop-free, then it is a tree called the contour tree. Given a piecewise linear function f defined on a simplicial complex K, Carr, Snoeyink and Axen proposed a simple and elegant algorithm to compute the contour tree of f by "merging" its join and split trees in near-linear time.
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 hand-tuned 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.
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. 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. 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 skeleton-graph 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.
This essay will scrutinises the role of modern jury in promoting civic participation in governance and how it reconciles with the concept of democracy. It will argue that in theory, it is admitted that the jury regime does fit into a successful exercise in democracy. However, in practice, due to it fails to serve democracy and unfortunately represents an unsuccessful exercise in democracy.
Abstract 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.
In recent years, with the rapid growth in the amount of publicly available Volunteered Geographic Information (VGI) data, automatic map generation from GPS trajectories has attracted great attention. Maps generated from these data can for example complement commercial maps in less developed areas. Two main challenges in the automatic generation of maps from volunteered GPS data are the handling of noise and of non-homogeneous sampling of road segments (for example, roads in downtown area can receive significantly more GPS traces than roads in residential areas). In this paper, we present a novel framework for map reconstruction based on a topological idea: the Morse theory. In particular, the use of Morse theory and topological simplification allows us to handle the issues of both noise and non-homogeneous sampling in an elegant unified framework. Our algorithm is significantly simpler than previous approaches, both conceptually and implementation speaking. Little pre- and post-processing is required, and yet the algorithm can reconstruct robust road-networks from challenging data sets (such as GPS traces for Berlin or Beijing cities) that are comparable or better than the output of previous state-of-the-art approaches. The new algorithm is also orders of magnitude faster than previous approaches on large data sets (for example, the entire processing of the Berlin city data with about 27189 trajectories takes less than one minute).
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