Math 412 - Topology with Applications

Highlights of Data Expedition: • Students explored daily observations of local climate data spanning the past 35 years. • Topological Data Analysis, or TDA for short, provides cutting-edge tools for studying the geometry of data in arbitrarily high dimensions. • Using TDA tools, students discovered intrinsic dynamical features of the data and learned how to quantify periodic phenomenon in a time-series. • Since nature invariably produces noisy data which rarely has exact periodicity, students also considered the theoretical basis of almost-periodicity and even invented and tested new mathematical definitions of almost-periodic functions. Summary The dataset we used for this data expedition comes from the Global Historical Climatology Network. “GHCN (Global Historical Climatology Network)-Daily is an integrated database of daily climate summaries from land surface stations across the globe.” Source: https://www.ncdc.noaa.gov/oa/climate/ghcn-daily/ We focused on the daily maximum and minimum temperatures from January 1, 1980 to April 1, 2015 collected from RDU International Airport. Through a guided series of exercises designed to be performed in Matlab, students explore these time-series, initially by direct visualization and basic statistical techniques. Then students are guided through a special sliding-window construction which transforms a time-series into a high-dimensional geometric curve. These high-dimensional curves can be visualized by projecting down to lower dimensions as in the figure below (Figure 1), however, our focus here was to use persistent homology to directly study the high-dimensional embedding. The shape of these curves has meaningful information but how one describes the “shape” of data depends on which scale the data is being considered. However, choosing the appropriate scale is rarely an obvious choice. Persistent homology overcomes this obstacle by allowing us to quantitatively study geometric features of the data across multiple-scales. Through this data expedition, students are introduced to numerically computing persistent homology using the rips collapse algorithm and interpreting the results. In the specific context of sliding-window constructions, 1-dimensional persistent homology can reveal the nature of periodic structure in the original data. I created a special technique to study how these high-dimensional sliding-window curves form loops in order to quantify the periodicity. Students are guided through this construction and learn how to visualize and interpret this information. Climate data is extremely complex (as anyone who has suffered from a bad weather prediction can attest) and numerous variables play a role in determining our daily weather and temperatures. This complexity coupled with imperfections of measuring devices results in very noisy data. This causes the annual seasonal periodicity to be far from exact. To this end, I have students explore existing theoretical notions of almost-periodicity and test it on the data. They find that some existing definitions are also inadequate in this context. Hence I challenged them to invent new mathematics by proposing and testing their own definition. These students rose to the challenge and suggested a number of creative definitions. While autocorrelation and spectral methods based on Fourier analysis are often used to explore periodicity, the construction here provides an alternative paradigm to quantify periodic structure in almost-periodic signals using tools from topological data analysis.

Refractometry Studies of the Optical Properties of Polymer Films and the Development of Polymer Coated Refractive Index Sensors

Sensors for real-time monitoring of environmental contaminants are essential for protecting ecosystems and human health. Refractive index sensing is a non-selective technique that can be used to measure almost any analyte. Miniaturized refractive index sensors, such as silicon-on-insulator (SOI) microring resonators are one possible platform, but require coatings selective to the analytes of interest. A homemade prism refractometer is reported and used to characterize the interactions between polymer films and liquid or vapour-phase analytes. A camera was used to capture both Fresnel reflection and total internal reflection within the prism. For thin-films (d = 10 μm - 100 μm), interference fringes were also observed. Fourier analysis of the interferogram allowed for simultaneous extraction of the average refractive index and film thickness with accuracies of ∆n = 1-7 ×10-4 and ∆d < 3-5%. The refractive indices of 29 common organic solvents as well as aqueous solutions of sodium chloride, sucrose, ethylene glycol, glycerol, and dimethylsulfoxide were measured at λ = 1550 nm. These measurements will be useful for future calibrations of near-infrared refractive index sensors. A mathematical model is presented, where the concentration of analyte adsorbed in a film can be calculated from the refractive index and thickness changes during uptake. This model can be used with Fickian diffusion models to measure the diffusion coefficients through the bulk film and at the film-substrate interface. The diffusion of water and other organic solvents into SU-8 epoxy was explored using refractometry and the diffusion coefficient of water into SU-8 is presented. Exposure of soft baked SU-8 films to acetone, acetonitrile and methanol resulted in rapid delamination. The diffusion of volatile organic compound (VOC) vapours into polydimethylsiloxane and polydimethyl-co-polydiphenylsiloxane polymers was also studied using refractometry. Diffusion and partition coefficients are reported for several analytes. As a model system, polydimethyl-co-diphenylsiloxane films were coated onto SOI microring resonators. After the development of data acquisition software, coated devices were exposed to VOCs and the refractive index response was assessed. More studies with other polymers are required to test the viability of this platform for environmental sensing applications.

A simple but efficient voice activity detection algorithm through Hilbert transform and dynamic threshold for speech pathologies

A simple but efficient voice activity detector based on the Hilbert transform and a dynamic threshold is presented to be used on the pre-processing of audio signals -- The algorithm to define the dynamic threshold is a modification of a convex combination found in literature -- This scheme allows the detection of prosodic and silence segments on a speech in presence of non-ideal conditions like a spectral overlapped noise -- The present work shows preliminary results over a database built with some political speech -- The tests were performed adding artificial noise to natural noises over the audio signals, and some algorithms are compared -- Results will be extrapolated to the field of adaptive filtering on monophonic signals and the analysis of speech pathologies on futures works

Automatic dynamic mask extraction for PIV images containing an unsteady interface, bubbles, and a moving structure

When performing Particle Image Velocimetry (PIV) measurements in complex fluid flows with moving interfaces and a two-phase flow, it is necessary to develop a mask to remove non-physical measurements. This is the case when studying, for example, the complex bubble sweep-down phenomenon observed in oceanographic research vessels. Indeed, in such a configuration, the presence of an unsteady free surface, of a solid–liquid interface and of bubbles in the PIV frame, leads to generate numerous laser reflections and therefore spurious velocity vectors. In this note, an image masking process is developed to successively identify the boundaries of the ship and the free surface interface. As the presence of the solid hull surface induces laser reflections, the hull edge contours are simply detected in the first PIV frame and dynamically estimated for consecutive ones. As for the unsteady surface determination, a specific process is implemented like the following: i) the edge detection of the gradient magnitude in the PIV frame, ii) the extraction of the particles by filtering high-intensity large areas related to the bubbles and/or hull reflections, iii) the extraction of the rough region containing these particles and their reflections, iv) the removal of these reflections. The unsteady surface is finally obtained with a fifth-order polynomial interpolation. The resulted free surface is successfully validated from the Fourier analysis and by visualizing selected PIV images containing numerous spurious high intensity areas. This paper demonstrates how this data analysis process leads to PIV images database without reflections and an automatic detection of both the free surface and the rigid body. An application of this new mask is finally detailed, allowing a preliminary analysis of the hydrodynamic flow.

Ambiente para aquisição e processamento de sinal eletromiográfico de superfície

In this work, a platform to the conditioning, digitizing, visualization and recording of the EMG signals was developed. After the acquisition, the analysis can be done by signal processing techniques. The platform consists of two modules witch acquire electromyography (EMG) signals by surface electrodes, limit the interest frequency band, filter the power grid interference and digitalize the signals by the analogue-to- digital converter of the modules microcontroller. Thereby, the data are sent to the computer by the USB interface by the HID specification, displayed in real-time in graphical form and stored in files. As processing resources was implemented the operations of signal absolute value, the determination of effective value (RMS), Fourier analysis, digital filter (IIR) and the adaptive filter. Platform initial tests were performed with signal of lower and upper limbs with the aim to compare the EMG signal laterality. The open platform is intended to educational activities and academic research, allowing the addition of other processing methods that the researcher want to evaluate or other required analysis.

A Numerical study of fluid flow in peristaltic pumps and polymeric elastic pipes

This thesis work deals with a mathematical description of flow in polymeric pipe and in a specific peristaltic pump. This study involves fluid-structure interaction analysis in presence of complex-turbulent flows treated in an arbitrary Lagrangian-Eulerian (ALE) framework. The flow simulations are performed in COMSOL 4.4, as 2D axial symmetric model, and ABAQUS 6.14.1, as 3D model with symmetric boundary conditions. In COMSOL, the fluid and structure problems are coupled by monolithic algorithm, while ABAQUS code links ABAQUS CFD and ABAQUS Standard solvers with single block-iterative partitioned algorithm. For the turbulent features of the flow, the fluid model in both codes is described by RNG k-ϵ. The structural model is described, on the basis of the pipe material, by Elastic models or Hyperelastic Neo-Hookean models with Rayleigh damping properties. In order to describe the pulsatile fluid flow after the pumping process, the available data are often defective for the fluid problem. Engineering measurements are normally able to provide average pressure or velocity at a cross-section. This problem has been analyzed by McDonald's and Womersley's work for average pressure at fixed cross section by Fourier analysis since '50, while nowadays sophisticated techniques including Finite Elements and Finite Volumes exist to study the flow. Finally, we set up peristaltic pipe simulations in ABAQUS code, by using the same model previously tested for the fl uid and the structure.

Sources de variation intra-populationelle de la morphologie des otolithes: asymétrie directionnelle et régime alimentaire

Otoliths are calcified structures located in Osteichthyes’ inner ear that are involved in audition and balance. Their morphology is used as an indicator of various ecological processes or properties. This application requires identifying the endogenous and exogenous factors that act simultaneously as sources of shape variation. This thesis aims at detecting and quantifying the relative contributions of directional asymmetry and diet to otolith shape variation at the intra-population level. Directional asymmetry between left and right otoliths was found in flat-fishes, the blind-side otolith being always longer and larger, whereas it was negligible in round-fishes. However, asymmetry amplitude never exceeded 18 %, which suggests evolutionary canalization of otolith shape symmetry. A correlation between global diet and otolith was detected in 4 species studied in situ. Diet composition contributed more than food amount to morphological variation and affected otolith shape both globally and locally. An experimental study on sea bass (Dicentrarchus larbrax) showed that diet composition in terms of essential polyunsaturated fatty acids at larval stage affects otolith morphogenesis during juvenile stage without impacting on individuals’ somatic growth. This result suggests a direct effect of diet on otolith shape and not an indirect one through the somatic-otolith growth relationship. This effect disappeared at later stages, morphogenetic trajectories converging back to a similar shape, which suggests ontogenetic canalization of otolith shape.

Otolith shape variation provides a marker of stock origin for north Atlantic bluefin tuna (Thunnus thynnus)

Two stocks of bluefin tuna (Thunnus thynnus) inhabit the north Atlantic; the western and eastern stocks spawn in the Gulf of Mexico and the Mediterranean Sea respectively. Trans-Atlantic movements occur outside spawning time whereas natal homing maintains stock structure. Commercial fisheries may exploit a mixed assemblage of both stocks. The incorporation of mixing rates into stock assessment is precluded by uncertainties surrounding stock discrimination. Otolith shape descriptors were used to characterise western and eastern stocks of Atlantic bluefin tuna in the present study and to estimate stock composition in catches of unknown origin. Otolith shape varied with length and between locations and years. Within a restricted size range (200-297-cm fork length (FL)) the two stocks were distinguished with an accuracy of 83%. Bayesian stock mixture analysis indicated that samples from the east Atlantic and Mediterranean were predominantly of eastern origin. The proportion assigned to the eastern stock showed slight spatial variation; however, overlapping 95% credible intervals indicated no significant difference (200-297 cm FL: central Atlantic, 73-100%; Straits of Gibraltar, 73-100%; Morocco, 50-99%; Portugal 64-100%). Otolith shape could be used in combination with other population markers to improve the accuracy of mixing rate estimates for Atlantic bluefin tuna.

Respiratory syncytial virus seasonality in Brazil: implications for the immunisation policy for at-risk populations

Respiratory syncytial virus (RSV) infection is the leading cause of hospitalisation for respiratory diseases among children under 5 years old. The aim of this study was to analyse RSV seasonality in the five distinct regions of Brazil using time series analysis (wavelet and Fourier series) of the following indicators: monthly positivity of the immunofluorescence reaction for RSV identified by virologic surveillance system, and rate of hospitalisations per bronchiolitis and pneumonia due to RSV in children under 5 years old (codes CID-10 J12.1, J20.5, J21.0 and J21.9). A total of 12,501 samples with 11.6% positivity for RSV (95% confidence interval 11 - 12.2), varying between 7.1 and 21.4% in the five Brazilian regions, was analysed. A strong trend for annual cycles with a stable stationary pattern in the five regions was identified through wavelet analysis of the indicators. The timing of RSV activity by Fourier analysis was similar between the two indicators analysed and showed regional differences. This study reinforces the importance of adjusting the immunisation period for high risk population with the monoclonal antibody palivizumab taking into account regional differences in seasonality of RSV.

Pattern recognition methods for EMG prosthetic control

In this work we focus on pattern recognition methods related to EMG upper-limb prosthetic control. After giving a detailed review of the most widely used classification methods, we propose a new classification approach. It comes as a result of comparison in the Fourier analysis between able-bodied and trans-radial amputee subjects. We thus suggest a different classification method which considers each surface electrodes contribute separately, together with five time domain features, obtaining an average classification accuracy equals to 75% on a sample of trans-radial amputees. We propose an automatic feature selection procedure as a minimization problem in order to improve the method and its robustness.

Fourier descriptors for morphological analysis of vectorcardiograms

The use of Fourier shape descriptors for morphological studies of vectorcardio-grams (VCGs) is p resented . The FDs can effectively be used as features for classf-fication of VCGs of different clinical categories . In addition , they provide cli-nically significant qualitative shape information for use by the Cardiologist. The initial result sofanalysisof nwrmal and abnormal VCGs areencouraging.