I enjoy constructing and applying mathematical theories and techniques to research problems in engineering and science. Some of my primary research areas are described below; related publications are listed here.
Algebraic signal processing
The algebraic signal processing theory is a mathematical framework that generalizes and extends linear signal processing theory. It abstracts fundamental concepts, such as signals, filters, z-transform, spectrum, frequency response, Fourier transform, and others to signals and datasets that are not covered by the traditional signal processing theory. We have constructed instantiations of this framework, called signal models, for various types of signals and data, including images, medical signals, sensor measurements, web documents, cellphone networks, and others.
Design and optimization of fast algorithms
Most important linear transforms, such as the discrete Fourier transform, discrete cosine and sine transforms, and others, require highly optimized algorithms and implementations. We have developed a general framework that leads to construction of divide-and-conquer algorithms for any linear transform. It provides a general way to re-discover many known algorithms, as well as identifies previously unknown algorithms. Mathematical description of these algorithms allows for hardware-specific optimization, using, for example, parallelization and vectorization techniques.
Signal processing and machine learning
Signal processing theory offers a wide range of tools and methods for signals analysis; however, the range of data it can be applied to is limited. Machine learning techniques can be applied to a wide variety of datasets; but these techniques differ so much, it is hard to build a rigorous framework stemming from a few fundamental concepts and axioms. We attempt to merge the best from both areas in a rigorous framework that builds a complete model for a complex dataset in a signal processing manner and then expresses relevant machine learning techniques through signal processing transforms and constructions defined by the model, such as filters, Fourier transforms, samplers, predictors, and so on.
Medical imaging and signal analysis
Medical imaging, inverse problems, and medical data analysis have strong connections to signal processing. By building appropriate signal models, optimizing processing algorithms for image and data reconstruction and representation, we can improve the quality and performance of existing techniques.
Image processing is an important area of signal processing. Representation, recognition, compression, and transmission of images can be significantly improved by careful construction of image models and application of appropriate techniques in these models.