The recent advancements in signal acquisition and data mining have revealed the importance of data-driven tools for analyzing signals and images. The availability of large and complex data has also highlighted the need for investigative tools that provide autonomy, noise-robustness, and efficiently utilize data collected from different settings but pertaining to the same phenomenon. State-of-the-art approaches include using tools such as Fourier analysis, wavelets, and Empirical Mode Decomposition for extracting informative features from the data. These features can then be combined with machine learning for clustering, classification, and inference. However, these tools typically require human intervention for feature extraction, and they are sensitive to the input parameters that the user chooses during the laborious but often necessary manual data pre-processing. Therefore, this dissertation was motivated by the need for automatic, adaptive, and noise-robust methods for efficiently leveraging machine learning for studying images as well as time series of dynamical systems. Specifically, this work investigates three application areas: chatter detection in manufacturing processes, image analysis of manufactured surfaces, and tool wear detection during titanium alloys machining. This work's novel investigations are enabled by combining machine learning with methods from Topological Data Analysis (TDA), a relatively recent field of applied topology that encompasses a variety of mature tools for quantifying the shape of data. First, this study experimentally shows for the first time that persistent homology (or persistence) from TDA can be used for chatter classification with accuracies that rival existing detection methods. Further, the efficient use of chatter data sets from different sources is formulated and studied as a transfer learning problem using experimental turning and milling vibration signals. Classification results are shown using comparisons between the TDA pipeline developed in this dissertation and prominent methods for chatter detection. Second, this work describes how to utilize TDA tools for extracting descriptive features from simulated samples generated using different Hurst roughness exponents. The efficiency of the feature extraction is tested by classifying the surfaces according to their roughness level. The resulting accuracies show that TDA can outperform several traditional feature extraction approaches in surface texture analysis. Further, as part of this work, adaptive threshold selection algorithms are developed for Discrete Cosine Transform, and Discrete Wavelet Transform to bypass the need for subjective operator input during surface roughness analysis. Both experimental and synthetic data sets are used to test the effectiveness of these two algorithms. This study also discusses a TDA-based framework that can potentially provide a feasible approach for building an automatic surface finish monitoring system. Finally, this work shows that persistence can be used for tool condition monitoring during titanium alloys machining. Since, in these processes, the cutting tools typically fracture catastrophically before the gradual tool wear reaches the maximum tool life criteria, the industry uses very conservative criteria for replacing the tools. An extensive experiment is described for relating wear markers in various sensor signals to the tool condition at different stages of the tool life. This work shows how, in this setting, TDA provides significant advantages in terms of robustness to noise and alleviating the need for an expert user to extract the informative features. The obtained TDA-based features are compared to existing state-of-the-art featurization tools using feature-level data fusion. The temporal location of the most representative tool condition features is also studied in the signals by considering a variety of window lengths preceding tool wear milestones.
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