Spectral methods that are based on eigenvectors and eigenvalues of discrete graph Laplacians, such as Diffusion Maps and Laplacian Eigenmaps are often used for manifold learning and non-linear dimensionality reduction. It was previously shown by Belkin and Niyogi [5] that the eigenvectors and eigenvalues of the graph Laplacian converge to the eigenfunctions and eigenvalues of the Laplace-Beltrami operator of the manifold in the limit of infinitely many data points sampled independently from the uniform distribution over the manifold. Recently, we introduced Vector Diffusion Maps and showed that the connection Laplacian of the tangent bundle of the manifold can be approximated from random samples. In this paper, we present a unified framework for approximating other connection Laplacians over the manifold by considering its principle bundle structure. We prove that the eigenvectors and eigenvalues of these Laplacians converge in the limit of infinitely many independent random samples. We generalize the spectral convergence results to the case where the data points are sampled from a non-uniform distribution, and for manifolds with and without boundary.

The multiple fundamental frequency detection problem and the source separation problem from a single-channel signal containing multiple oscillatory components and a nonstationary noise are both challenging tasks. To extract the fetal electrocardiogram (ECG) from a single-lead maternal abdominal ECG, we need to solve both challenges. We propose a novel method to extract the fetal ECG from a single-lead maternal abdominal ECG, without any additional measurement. The algorithm is composed of three components. First, the maternal and fetal heart rates are estimated by the de-shape short time Fourier transform (STFT), which is a recently proposed nonlinear time-frequency analysis technique. The beat tracking technique is the second component which is applied to accurately obtain the maternal and fetal R peaks. The third component consists of establishing the maternal and fetal ECG waveforms by the nonlocal median. The algorithm is tested on two real databases with the annotation provided by experts (adfecgdb database and CinC2013 database) and a simulated database (fecgsym), and provides the state-of-the-art results. We conclude that with the proposed algorithm, the fetal ECG waveform and the fetal heart rate could be accurately obtained from the single-lead maternal abdominal ECG.

Finding appropriate low dimensional representations of high-dimensional multimodal data can be challenging, since each modality embodies unique deformations and interferences. In this paper, we address the problem using manifold learning, where the data from each modality is assumed to lie on some manifold. In this context, the goal is to characterize the relations between the different modalities by studying their underlying manifolds. We propose two new diffusion operators that allow to isolate, enhance and attenuate the hidden components of multi-modal data in a data-driven manner. Based on these new operators, efficient low-dimensional representations can be constructed for such data, which characterize the common structures and the differences between the manifolds underlying the different modalities. The capabilities of the proposed operators are demonstrated on 3D shapes and on a fetal heart rate monitoring application.

This result advocates for an effective and scalable method for recognizing changes in physiological state using non-invasive heart rate monitoring. The CNN model adaptively quantifies IHR fluctuation as well as its location in time and is suitable for differentiating between the wake and sleep stages.

Physiologically, the thoracic (THO) and abdominal (ABD) movement signals, captured using wearable piezo-electric bands, provide information about various types of apnea, including central sleep apnea (CSA) and obstructive sleep apnea (OSA). However, the use of piezo-electric wearables in detecting sleep apnea events has been seldom explored in the literature. This study explored the possibility of identifying sleep apnea events, including OSA and CSA, by solely analyzing one or both the THO and ABD signals. An adaptive non-harmonic model was introduced to model the THO and ABD signals, which allows us to design features for sleep apnea events. To confirm the suitability of the extracted features, a support vector machine was applied to classify three categories - normal and hypopnea, OSA, and CSA. According to a database of 34 subjects, the overall classification accuracies were on average 75.9%±11.7% and 73.8%±4.4%, respectively, based on the cross validation. When the features determined from the THO and ABD signals were combined, the overall classification accuracy became 81.8%±9.4%. These features were applied for designing a state machine for online apnea event detection. Two event-byevent accuracy indices, S and I, were proposed for evaluating the performance of the state machine. For the same database, the S index was 84.01%±9.06%, and the I index was 77.21%±19.01%. The results indicate the considerable potential of applying the proposed algorithm to clinical examinations for both screening and homecare purposes.