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Topological data analysis (TDA) has proven to be a potent approach for extracting intricate topological structures from complex and high-dimensional data. In this paper, we propose a TDA-based processing pipeline for analyzing multi-channel scalp EEG data. The pipeline starts with extracting both frequency and temporal information from the signals via the Hilbert–Huang Transform. The sequences of instantaneous frequency and instantaneous amplitude across all electrode channels are treated as approximations of curves in the high-dimensional space. TDA features, which represent the local topological structure of the curves, are further extracted and used in the classification models. Three sets of scalp EEG data, including one collected in a lab and two Brain–computer Interface (BCI) competition data, were used to validate the proposed methods, and compare with other state-of-art TDA methods. The proposed TDA-based approach shows superior performance and outperform the winner of the BCI competition. Besides BCI, the proposed method can also be applied to spatial and temporal data in other domains such as computer vision, remote sensing, and medical imaging.