Topological Signal Processing Research Articles

Topological Signal Processing Over Generalized Cell Complexes

Topological Signal Processing Over Generalized Cell Complexes

Robust crossings detection in noisy signals using topological signal processing

Robust crossings detection in noisy signals using topological signal processing

Persistence of Weighted Ordinal Partition Networks for Dynamic State Detection

One of the most important problems arising in time series analysis is that of bifurcation, or change point detection. That is, given a collection of time series over a varying parameter, when has the structure of the underlying dynamical system changed? For this task, we turn to the field of topological data analysis (TDA), which encodes information about the shape and structure of data. The idea of utilizing tools from TDA for signal processing tasks, known as topological signal processing (TSP), has gained much attention in recent years, largely through a standard pipeline that computes the persistent homology of the point cloud generated by the Takens' embedding. However, this procedure is limited by computation time since the simplicial complex generated in this case is large, but also has a great deal of redundant data. For this reason, we turn to a more recent method for encoding the structure of the attractor, which constructs an ordinal partition network (OPN) representing information about when the dynamical system has passed between certain regions of state space. The result is a weighted graph whose structure encodes information about the underlying attractor. Our previous work began to find ways to package the information of the OPN in a manner that is amenable to TDA; however, that work only used the network structure and did nothing to encode the additional weighting information. In this paper, we take the next step: building a pipeline to analyze the weighted OPN with TDA and showing that this framework provides more resilience to noise or perturbations in the system and improves the accuracy of the dynamic state detection.

Topological inference and correlation of signals with application to electroencephalography in epilepsy

Electroencephalography (EEG) is an important neurophysiological modality for understanding brain functions and disorders. Topological signal processing allows us to capture how local variations in EEG signals affect their global structures. It reveals changes of topological structures across different temporal and spectral scales of signals that may not be observable through the standard signal processing methods. A topological signal processing framework that tracks the evolution of time segments corresponding to electric potential below a horizontal threshold has shown promise in applications to EEG studies of brain disorders. Gradient filtration is a generalization of the framework for extracting topological features in a signal treated as a two-dimensional curve and tracks the evolution of arcs in the curve cut off by a straight line moving in an arbitrary direction. In this study, we set up a statistical inference framework on gradient filtrations and an improved topological correlation measure for EEG signals by correlating features across gradient filtrations in multiple directions. We compare its performance with standard correlation measures in simulation studies and application to intracranial EEG signals recorded in canine with epilepsy.

Super-resolving star clusters with sheaves

This article presents an optimization-based approach for counting and localizing stars within a small cluster, based on photon counts in a focal plane array. The array need not be arranged in any particular way, and relatively small numbers of photons are required to ensure convergence. The stars can be located close to one another, as the location and magnitude errors were found to be low when the separation was larger than 0.2 Rayleigh radii. To ensure generality of our approach, it was constructed as a special case of a general theory built upon topological signal processing using the mathematics of sheaves that is applicable to general source decomposition problems.

Bayesian topological signal processing

<p style='text-indent:20px;'>Topological data analysis encompasses a broad set of techniques that investigate the shape of data. One of the predominant tools in topological data analysis is persistent homology, which is used to create topological summaries of data called persistence diagrams. Persistent homology offers a novel method for signal analysis. Herein, we aid interpretation of the sublevel set persistence diagrams of signals by 1) showing the effect of frequency and instantaneous amplitude on the persistence diagrams for a family of deterministic signals, and 2) providing a general equation for the probability density of persistence diagrams of random signals via a pushforward measure. We also provide a topologically-motivated, efficiently computable statistical descriptor analogous to the power spectral density for signals based on a generalized Bayesian framework for persistence diagrams. This Bayesian descriptor is shown to be competitive with power spectral densities and continuous wavelet transforms at distinguishing signals with different dynamics in a classification problem with autoregressive signals.</p>

Topological signal processing and inference of event-related potential response

Background: Topological signal processing is a novel approach for decoding multiscale features of signals recorded through electroencephalography (EEG) based on topological data analysis (TDA).New method: We establish stability properties of the TDA descriptor persistence landscape (PL) in event-related potential (ERP) across multi-trial EEG signals, state algorithms for computing PL, and propose an exact inference framework on persistence and PLs.Results: We apply the topological signal processing and inference framework to compare ERPs between individuals with post-stroke aphasia and healthy controls under a speech altered auditory feedback (AAF) paradigm. Results show significant PL difference in the ERP response of aphasic individuals and healthy controls over the parietal-occipital and occipital regions with respect to speech onset, and no significant PL difference in any regions with respect to the two pitch-shift stimuli.Comparison with existing methods: In comparison, spatial patterns of difference between aphasic individuals and healthy controls by persistence, local variance, and spectral powers are much more diffuse than the PL patterns. In simulation results, the exact test on persistence and PLs has more robust performance than the baseline tests on local variance and spectral powers.Conclusions: Persistence features provide a more robust EEG marker than local variance, and spectral powers. It could be a potentially powerful tool for comparing electrophysiological correlates in neurological disorders.

Underwater channel estimation exploiting multipath feature morphology.

Real-time accurate channel estimation has been an ongoing challenge because of diverse oceanic events that cause rapid fluctuations of high-energy multipath activity across the delay spread. This work leverages current compressed sensing and sparse optimization techniques with topological signal processing to improve estimation time and localize channel estimation to salient parts of the delay spread. This work improves the estimation time by tracking the channel as a union of overlapping multipath and other scattering events, which are modeled as "feature braids" in the delay-time domain. A channel feature braid may be intuitively visualized as the topologically connected trajectory of a group of channel delay taps, which represent the support of dominant or persistent scattering events, e.g., surface bounce multipath scattering. We present algorithms that harness support-constrained mixed norm optimization techniques to track the evolving support of channel feature braids. We validate our channel feature tracking algorithm independently in experimental field data as well as BELLHOP channel simulations across a diversity of oceanic conditions. This work shows that braiding used in estimation can improve estimation time and track high-energy events that develop within the delay vs time channel representation.

Topological Signal Processing: Making Sense of Data Building on Multiway Relations

Uncovering hidden relations in complex data sets is a key step to making sense of the data, which is a hot topic in our era of data deluge. Graph-based representations are examples of tools able to encode relations in a mathematical structure enabling the uncovering of patterns like clusters and paths. However, graphs only capture pairwise relations encoded in the presence of edges, but there are many forms of interaction that cannot be reduced to pairwise relations. To overcome the limitations of graph-based representations, it is necessary to incorporate multiway relations. In this article, we exploit tools from algebraic topology to handle multiway relations. Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study a topological space, that is, a set of points, along with a set of neighborhoods. More specifically, we illustrate topological signal processing (TSP), a framework encompassing a class of methods for analyzing signals defined over a topological space. Given its generality, TSP incorporates graph signal processing (GSP) as a particular case. After motivating the use of topological and geometrical methods for detecting patterns in the data, we present the signal processing tools based on algebraic topology and then illustrate their advantages with respect to graph-based methodology.

A Method to Differentiate Fatiguing Conditions in Surface Electromyography Signals using Instantaneous Spectral Centroid.

The nonstationarity measure of surface Electromyography (sEMG) signals provide an index for muscle fatigue conditions. In this paper, a new framework has been proposed for the analysis of sEMG signal using Instantaneous Spectral Centroid (ISC). The novelty of the proposed work is use of topological signal processing method to quantify the nonstationarity of sEMG signal. For this, the signals are recorded from the biceps brachii muscles of 25 healthy subjects in isometric contraction. The analytical signals corresponding to nonfatigue and fatigue segments are computed using Hilbert Transform. Further, topological features such as center of gravity (CoG), triangular area function (TAF) and ISC are calculated from the geometrical representation of a transformed signal. The result indicates the increase of TAF in fatigue condition and the significant right shift of CoG in x-axis for 80% of subjects. Importantly, the ISC estimate is decreased by 17% upon fatiguing for 84% of subjects. The obtained results show statistical significance with p < 0.05. It is observed that the shape parameters are varied in accordance with the changes observed in global characteristics of sEMG signals during muscle fatigue. The preliminary results show that the topological features are able to quantify the nonstationarity in sEMG signal. Therefore, the proposed method can be used as a fatigue index for diagnosing various neuromuscular disorders.Clinical Relevance-This method can be used to establish metrics of muscle fatigue for the benefit of physicians especially in the field of fitness, sports, pre and post-surgery surveillance and rehabilitation.

Topological Signal Processing Over Simplicial Complexes

The goal of this paper is to establish the fundamental tools to analyze signals defined over a topological space, i.e. a set of points along with a set of neighborhood relations. This setup does not require the definition of a metric and then it is especially useful to deal with signals defined over non-metric spaces. We focus on signals defined over simplicial complexes. Graph Signal Processing (GSP) represents a special case of Topological Signal Processing (TSP), referring to the situation where the signals are associated only with the vertices of a graph. Even though the theory can be applied to signals of any order, we focus on signals defined over the edges of a graph and show how building a simplicial complex of order two, i.e. including triangles, yields benefits in the analysis of edge signals. After reviewing the basic principles of algebraic topology, we derive a sampling theory for signals of any order and emphasize the interplay between signals of different order. Then we propose a method to infer the topology of a simplicial complex from data. We conclude with applications to real edge signals and to the analysis of discrete vector fields to illustrate the benefits of the proposed methodologies.

Reconfigurable Topological Phases in Two-Dimensional Dielectric Photonic Crystals

The extensive research on photonic topological insulators has opened up an intriguing way to control electromagnetic (EM) waves. In this work, we numerically demonstrate reconfigurable microwave photon analogues of topological insulator (TIs) in a triangular lattice of elliptical cylinders, according to the theory of topological defects. Multiple topological transitions between the trivial and nontrivial photonic phases can be realized by inhomogeneously changing the ellipse orientation, without altering the lattice structure. Topological protection of the edge states and reconfigurable topological one-way propagation at microwave frequencies, are further verified. Our approach provides a new route towards freely steering light propagations in dielectric photonic crystals (PCs), which has potential applications in the areas of topological signal processing and sensing.

Windowed persistent homology: A topological signal processing algorithm applied to clinical obesity data

Overweight and obesity are highly prevalent in the population of the United States, affecting roughly 2/3 of Americans. These diseases, along with their associated conditions, are a major burden on the healthcare industry in terms of both dollars spent and effort expended. Volitional weight loss is attempted by many, but weight regain is common. The ability to predict which patients will lose weight and successfully maintain the loss versus those prone to regain weight would help ease this burden by allowing clinicians the ability to skip treatments likely to be ineffective. In this paper we introduce a new windowed approach to the persistent homology signal processing algorithm that, when paired with a modified, semimetric version of the Hausdorff distance, can differentiate the two groups where other commonly used methods fail. The novel approach is tested on accelerometer data gathered from an ongoing study at the University of Michigan. While most standard approaches to signal processing show no difference between the two groups, windowed persistent homology and the modified Hausdorff semimetric show a clear separation. This has significant implications for clinical decision making and patient care.

The performance of topological classifiers on sonar data

For various reasons, synthetic aperture sonar (SAS) target classification in various clutter contexts is usually done using a data-driven, machine learning approach. Unfortunately, the resulting feature set can be rather inscrutable—what features is it really using? Topological methods are particularly well-aligned with the goal of gaining insight into physical processes, since they highlight symmetries which are driven by these physical processes. For instance, collating multiple image looks of a round object uncovers rotational symmetries in an appropriate feature space derived from the images. The use of topological invariants allows one to infer that the object is round by reasoning about its feature space. The fact that sonar target signatures are (mostly) translation invariant in range can also be deduced from topological invariants. I will describe a principled, foundational analysis of target echo structure through the lens of topological signal processing, and then analyze the performance of this approach as compared to more traditional classification methods.

Reconfigurable Microwave Photonic Topological Insulator

Using full 3D finite element simulation and underlining Hamiltonian models, we demonstrate reconfigurable photonic analogues of topological insulators on a regular lattice of tunable posts in a re-entrant 3D lumped element type system. The tunability allows dynamical {\it in-situ} change of media chirality and other properties via alteration of the same parameter for all posts, and as a result, great flexibility in choice of bulk/edge configurations. Additionally, one way photon transport without an external magnetic field is demonstrated. The ideas are illustrated by using both full finite element simulation as well as simplified harmonic oscillator models. Dynamical reconfigurability of the proposed systems paves the way to a new class of systems that can be employed for random access, topological signal processing and sensing.

A Novel Framework for Pulse Pressure Wave Analysis Using Persistent Homology

Four characteristic points of pulse pressure waves–the systolic peak, the anacrotic notch, the dicrotic notch, and the diastolic foot–are used to estimate various aspects of cardiovascular function, such as heart rate and augmentation index. We propose a novel approach to extracting these characteristic points using a topological signal processing framework. We characterize the topology of the signals using a collection of persistence intervals, which are encapsulated in a persistence diagram. The characteristic points are identified based on their time of occurrence and their distance from the identity line in the persistence diagram. We validate this approach by collecting radial pulse pressure data from twenty-eight participants using a wearable tonometer, and computing the peripheral augmentation index using a traditional derivative-based method and our novel persistence-based method. The augmentation index values computed using the two methods are statistically indistinguishable, suggesting that this representation merits further exploration as a tool for analyzing pulse pressure waves.