I enjoy constructing mathematical models and techniques for real-world research problems in engineering and science. Some of my primary research areas are described below; related publications are listed here.
Signal processing on graphs
Data collected in physical and engineering applications, as well as social, biomolecular, commercial, security, and many other domains, are becoming increasingly larger and more complex. When datasets are represented in a numerical form, we can view them as signals and process them with appropriate signal processing techniques. The theory of discrete signal processing on graphs offers a new paradigm for the analysis of high-dimensional numerical data with complex, irregular structure. The framework extends fundamental signal processing concepts, including signals, filters, spectrum, Fourier transform, frequency response, low- and high-pass filtering, from signals residing on regular lattices, which are studied by the classical signal processing theory, to data residing on general graphs. This theory offers a new methodology for formulating and solving data analysis problems, such as data compression, denoising, reconstruction, classification, anomaly detection, and others, as standard signal processing tasks.
Algebraic signal processing
The algebraic signal processing theory is a mathematical framework that generalizes and extends classical signal processing. It abstracts fundamental concepts, such as signals, filters, z-transform, spectrum, frequency response, Fourier transform, and others to signals that are not covered by the traditional signal processing theory. Algebraic signal processing provided a foundation for the discrete signal processing on graphs. It also led to the discovery of numerous fast algorithms for discrete transforms (Fourier, cosine and sine, polynomial) and to the development of a mathematical framework for the discovery of fast algorithms for arbitrary linear transforms.
Design and optimization of fast algorithms
Many areas of engineering and science extensively use linear transforms, such as the discrete Fourier transform, discrete cosine and sine transforms, transforms based on orthogonal polynomials (Hermite, Legendre, Laguerre, and others), and others. Efficient and practical application of these transforms, in particular to large signals and datasets, require highly optimized algorithms and implementations. We have developed a rigorous theoretical framework for the discovery and construction of divide-and-conquer algorithms for general linear transforms. The obtained algorithms can significantly reduce the number of operations required for the computation of transforms, and their structure allows for software- and hardware-based optimizations, such as parallelization and vectorization of code and distributed computation. The framework is formulated using the algebraic signal processing theory.
Biomedical signal processing
Many problems in biomedical data analysis have strong connections to signal processing. Examples include evaluation of electrophysiological signals, classification of biological images, and inverse problems in medical imaging. By building appropriate signal models and optimizing processing algorithms for image and data reconstruction and representation, we aim to improve the quality and performance of medical data analysis.