Skeletonization and Partitioning of Digital Images Using Discrete Morse Theory

Cocoon microstructures through the lens of topological persistence

Biological materials display a wide array of functionality, often dictated by complicated microstructures. New geometric and topological strategies allow one to describe the microstructures in a precise and systematic way. This article describes the application of topological persistence and other geometric methods to the microstructural analysis of three-dimensional X-ray micro-computed tomography scans of the Bombyx mori silkworm cocoons. These methods allow conclusions to be drawn about pore space gradients, silk fibre thickness gradients and fibre alignment within the cocoon. The study demonstrates the applicability of these topological and geometric methods to quantify and characterize fibrous materials.

Morse theoretic signal compression and reconstruction on chain complexes

At the intersection of Topological Data Analysis (TDA) and machine learning, the field of cellular signal processing has advanced rapidly in recent years. In this context, each signal on the cells of a complex is processed using the combinatorial Laplacian, and the resultant Hodge decomposition. Meanwhile, discrete Morse theory has been widely used to speed up computations by reducing the size of complexes while preserving their global topological properties. In this paper, we provide an approach to signal compression and reconstruction on chain complexes that leverages the tools of algebraic discrete Morse theory. The main goal is to reduce and reconstruct a based chain complex together with a set of signals on its cells via deformation retracts, preserving as much as possible the global topological structure of both the complex and the signals. We first prove that any deformation retract of real degree-wise finite-dimensional based chain complexes is equivalent to a Morse matching. We will then study how the signal changes under particular types of Morse matchings, showing its reconstruction error is trivial on specific components of the Hodge decomposition. Furthermore, we provide an algorithm to compute Morse matchings with minimal reconstruction error.

A GPU Parallel Algorithm for Computing Morse-Smale Complexes

The Morse-Smale complex is a well studied topological structure that represents the gradient flow behavior between critical points of a scalar function. It supports multi-scale topological analysis and visualization of feature-rich scientific data. Several parallel algorithms have been proposed towards the fast computation of the 3D Morse-Smale complex. Its computation continues to pose significant algorithmic challenges. In particular, the non-trivial structure of the connections between the saddle critical points are not amenable to parallel computation. This paper describes a fine grained parallel algorithm for computing the Morse-Smale complex and a GPU implementation (gmsc). The algorithm first determines the saddle-saddle reachability via a transformation into a sequence of vector operations, and next computes the paths between saddles by transforming it into a sequence of matrix operations. Computational experiments show that the method achieves up to 8.6× speedup over pyms3d and 6× speedup over TTK, the current shared memory implementations. The paper also presents a comprehensive experimental analysis of different steps of the algorithm and reports on their contribution towards runtime performance. Finally, it introduces a CPU based data parallel algorithm for simplifying the Morse-Smale complex via iterative critical point pair cancellation.

Persistence Cycles for Visual Exploration of Persistent Homology

Persistent homology is a fundamental tool in topological data analysis used for the most diverse applications. Information captured by persistent homology is commonly visualized using scatter plots representations. Despite being widely adopted, such a visualization technique limits user understanding and is prone to misinterpretation. This article proposes a new approach for the efficient computation of persistence cycles, a geometric representation of the features captured by persistent homology. We illustrate the importance of rendering persistence cycles when analyzing scalar fields, and we discuss the advantages that our approach provides compared to other techniques in topology-based visualization. We provide an efficient implementation of our approach based on discrete Morse theory, as a new module for the Topology Toolkit. We show that our implementation has comparable performance with respect to state-of-the-art toolboxes while providing a better framework for visually analyzing persistent homology information.

Pore-network extraction using discrete Morse theory: Preserving the topology of the pore space

Pore-scale modeling based on the 3D structural information of porous materials has enormous potential in assessing physical properties beyond the capabilities of laboratory methods. Such capabilities are pricey in terms of computational expenses, and this limits the applicability of the direct simulations to a small volume and requires high-performance computational resources, especially for multiphase flow simulations. The only pore-scale technique capable of dealing with large representative volumes of porous samples is pore-network (PNM) based modeling. The problem of the PNM approach is that 3D pore geometry first needs to be simplified into a graph of pores and throats that conserve topological and geometrical properties of the original 3D image. While significant progress has been achieved in terms of geometry representation, no methodology provides full conservation of the topological features of the pore structure. In this paper we present a pore-network extraction algorithm for binary 3D images based on discrete Morse theory and persistent homology that by design targets topology preservation. In addition to methodological developments, we also clarify the relationship between topological characteristics of constructed Morse chain complex and pore-network elements. We show that the Euler numbers calculated for PNMs based on our methodology coincide with those obtained using the direct topological analysis. The characteristics of the extracted pore network are calculated for several 3D porous binary images and compared with the results of maximum inscribed balls-based and watershed-based approaches as well as a hybrid approach to support our methodology.

Representational Coding of Overt and Covert Orienting of Visuospatial Attention in the Frontoparietal Network

Orienting of visuospatial attention refers to reallocation of attentional focus from one target or location to another and can occur either with (overt) or without (covert) eye movement. Although it has been demonstrated that both types of orienting commonly involve frontal and parietal brain regions as the frontoparietal network (FPN), the underlying representational coding of these two types of orienting remains unclear. In this functional magnetic resonance imaging study, participants performed a task that elicited overt and covert orienting to endogenously or exogenously cued targets with eye-tracking to monitor eye movement. Although the FPN was commonly activated for both overt and covert orienting, multivariate patterns of the activation of voxels in the FPN accurately predicted whether eye movements were involved or not during orienting. These overt- and covert-preferred voxels were topologically distributed as distinct and interlaced clusters in a millimeter scale. Inclusion of the two types of clusters predicted orienting type more accurately than one type of clusters alone. These findings suggest that overt and covert orienting are represented by interdependent functional clusters of neuronal populations in regions of the FPN, which might reflect a generalizable principle in the nervous system for functional organization of closely associated processes.

Semantic segmentation of microscopic neuroanatomical data by combining topological priors with encoder-decoder deep networks

Understanding of neuronal circuitry at cellular resolution within the brain has relied on neuron tracing methods which involve careful observation and interpretation by experienced neuroscientists. With recent developments in imaging and digitization, this approach is no longer feasible with the large scale (terabyte to petabyte range) images. Machine learning based techniques, using deep networks, provide an efficient alternative to the problem. However, these methods rely on very large volumes of annotated images for training and have error rates that are too high for scientific data analysis, and thus requires a significant volume of human-in-the-loop proofreading. Here we introduce a hybrid architecture combining prior structure in the form of topological data analysis methods, based on discrete Morse theory, with the best-in-class deep-net architectures for the neuronal connectivity analysis. We show significant performance gains using our hybrid architecture on detection of topological structure (e.g. connectivity of neuronal processes and local intensity maxima on axons corresponding to synaptic swellings) with precision/recall close to 90% compared with human observers. We have adapted our architecture to a high performance pipeline capable of semantic segmentation of light microscopic whole-brain image data into a hierarchy of neuronal compartments. We expect that the hybrid architecture incorporating discrete Morse techniques into deep nets will generalize to other data domains.

Evolution of local motifs and topological proximity in self-assembled quasi-crystalline phases

Using methods from the field of topological data analysis, we investigate the self-assembly and emergence of three-dimensional quasi-crystalline structures in a single-component colloidal system. Combining molecular dynamics and persistent homology, we analyse the time evolution of persistence diagrams and particular local structural motifs. Our analysis reveals the formation and dissipation of specific particle constellations in these trajectories, and shows that the persistence diagrams are sensitive to nucleation and convergence to a final structure. Identification of local motifs allows quantification of the similarities between the final structures in a topological sense. This analysis reveals a continuous variation with density between crystalline clathrate, quasi-crystalline, and disordered phases quantified by 'topological proximity', a visualization of the Wasserstein distances between persistence diagrams. From a topological perspective, there is a subtle, but direct connection between quasi-crystalline, crystalline and disordered states. Our results demonstrate that topological data analysis provides detailed insights into molecular self-assembly.

Automated labeling of the airway tree in terms of lobes based on deep learning of bifurcation point detection

This paper presents an automatic lobe-based labeling of airway tree method, which can detect the bifurcation points for reconstructing and labeling the airway tree from a computed tomography image. A deep learning-based network structure is designed to identify the four key bifurcation points. Then, based on the detected bifurcation points, the entire airway tree is reconstructed by a new region-growing method. Finally, with the basic airway tree anatomy and topology knowledge, individual branches of the airway tree are classified into different categories in terms of pulmonary lobes. There are several advantages in our method such as the detection of the bifurcation points does not depend on the segmentation of airway tree and only four bifurcation points need to be manually labeled for each sample to prepare the training dataset. The segmentation of airway tree is guided by the detected points, which overcomes the difficulty of manual seed selection of conventional region-growing algorithm. In addition, the bifurcation points can help analyze the tree structure, which provides a basis for effective airway tree labeling. Experimental results show that our method is fast, stable, and the accuracy of our method is 97.85%, which is higher than that of the traditional skeleton-based method. Graphical Abstract The pipeline of our proposed lobe-based airway tree labeling method. Given a raw CT volume, a neural network structure is designed to predict major bifurcation points of airway tree. Based on the detected points, airway tree is reconstructed and labeled in terms of lobes.

Thermochemical CO2 splitting performance of perovskite coated porous ceramics

In this paper, we investigate the redox performance of perovskite coated porous ceramics with various architectures. For this purpose, reticulated porous ceramics (RPCs) in three different pore sizes (5, 12, 75 ppi) were fabricated to represent a broad range of structures and pore sizes. The perovskite material is based on lanthanum manganite and was synthesized and doped with Ca and Al through the Pechini method. Using a deep coating method, the surface of RPC substrates was modified by a thin-film coating with a thickness of ∼15 μm. We evaluated the CO₂ conversion performance of the developed materials in a gold-image IR furnace. X-ray micro-computed tomography along with SEM/EDX were utilized in different steps of the work for a thorough study of the bulk and surface features. Results reveal that the intermediate pore size of 12 ppi delivers the maximum perovskite loading with a high degree of coating homogeneity and connectivity while CO₂ conversion tests showed the highest CO yield for 75 ppi. Our results show that the extreme conditions inside the furnace combined with the flow of gaseous phases cause the RPCs to shrink in length up to 23% resulting in the alteration of the pore phase and elimination of small pores reducing the total specific surface area. Further our results reveal an important mechanism resulting in the inhibition of CO₂ conversion where the perovskite coating layer migrates into the matrix of the RPC frame.

A representative volume of LCMA coated porous SiC showing a maximum of 23% shrinkage when subject to high-temperature CO₂ conversion redox reactions. This results in significant structural changes including a reduction in specific surface area.

Rapid Recognition of Field-Grown Wheat Spikes Based on a Superpixel Segmentation Algorithm Using Digital Images

Wheat spike number, which could be rapidly and accurately estimated by the image processing technology, serves as the basis for crop growth monitoring and yield prediction. In this research, simple linear iterative clustering (SLIC) was performed for superpixel segmentation of the digital images of field-grown wheat. Firstly, certain characteristic color parameters were extracted and analyzed from the digital images, and the classifiers with the highest accuracy were chosen for subsequent image classification. Next, the main body of wheat spike was extracted through a series of morphological transformation and estimate was performed for each region. Backbone of the head was extracted, and the number of inflection points of backbone was detected. Then the wheat spike number was determined by combining the estimate of inflection points of backbone and the estimate for each region. Finally, the wheat spike number estimate was verified under four nitrogen fertilizer levels. The results were as follows: (1) Super green value (Eg) and normalized red green index (Dgr) were used as classification features to recognize wheat spikes, soil and leaves; (2) compared with pixel-based image processing, wheat spike recognition effect was much better after superpixel segmentation, as the main body of wheat spike extracted was more clear and morphology more intact; and (3) wheat plants had better growth under high nitrogen fertilizer level, and the accuracy of wheat spike number estimation was also the highest, which was 94.01%. The growth status was the worst under no nitrogen fertilizer application, and the accuracy of wheat spikes number estimation was also the lowest, which was only 80.8%. After excluding the no nitrogen condition, the accuracy of wheat spikes number estimation among mixed samples with more uniform growth status was up to 93.8%, which was an increase by 10.1% than before the exclusion. Wheat spikes number estimate based on superpixel segmentation and color features was a rapid and accurate method that was applicable to the field environment. However, this method was not recommended for use when the growth status of wheat was poor or of high heterogeneity. The findings provided reference for field-grown wheat yield estimate.

Gene Coexpression Network Comparison via Persistent Homology

Persistent homology, a topological data analysis (TDA) method, is applied to microarray data sets. Although there are a few papers referring to TDA methods in microarray analysis, the usage of persistent homology in the comparison of several weighted gene coexpression networks (WGCN) was not employed before to the very best of our knowledge. We calculate the persistent homology of weighted networks constructed from 38 Arabidopsis microarray data sets to test the relevance and the success of this approach in distinguishing the stress factors. We quantify multiscale topological features of each network using persistent homology and apply a hierarchical clustering algorithm to the distance matrix whose entries are pairwise bottleneck distance between the networks. The immunoresponses to different stress factors are distinguishable by our method. The networks of similar immunoresponses are found to be close with respect to bottleneck distance indicating the similar topological features of WGCNs. This computationally efficient technique analyzing networks provides a quick test for advanced studies.

Pore configuration landscape of granular crystallization

Uncovering grain-scale mechanisms that underlie the disorder–order transition in assemblies of dissipative, athermal particles is a fundamental problem with technological relevance. To date, the study of granular crystallization has mainly focussed on the symmetry of crystalline patterns while their emergence and growth from irregular clusters of grains remains largely unexplored. Here crystallization of three-dimensional packings of frictional spheres is studied at the grain-scale using X-ray tomography and persistent homology. The latter produces a map of the topological configurations of grains within static partially crystallized packings. Using numerical simulations, we show that similar maps are measured dynamically during the melting of a perfect crystal. This map encodes new information on the formation process of tetrahedral and octahedral pores, the building blocks of perfect crystals. Four key formation mechanisms of these pores reproduce the main changes of the map during crystallization and provide continuous deformation pathways representative of the crystallization dynamics.

Emergence and growth of crystalline domains in granular media remains under-explored. Here, the authors analyse tomographic snapshots from partially recrystallized packings of spheres using persistent homology and find agreement with proposed transitions based on continuous deformation of octahedral and tetrahedral voids.