IEEE Signal Processing Magazine

Hypercomplex signal and image processing extends upon conventional methods by using hypercomplex numbers in a unified framework for algebra and geometry. The special issue is divided into two parts and is focused on current advances and applications in computational signal and image processing in the hypercomplex domain.

Inference tasks in signal processing are often characterized by the availability of reliable statistical modeling with some missing instance-specific parameters. One conventional approach uses data to estimate these missing parameters and then infers based on the estimated model. Alternatively, data can also be leveraged to directly learn the inference mapping end to end. These approaches for combining partially known statistical models and data in inference are related to the notions of generative and discriminative models used in the machine learning literature [1] , [2] , typically considered in the context of classifiers.

Quaternions are still largely misunderstood and often considered an “exotic” signal representation without much practical utility despite the fact that they have been around the signal and image processing community for more than 30 years now. The main aim of this article is to counter this misconception and to demystify the use of quaternion algebra for solving problems in signal and image processing. To this end, we propose a comprehensive and objective overview of the key aspects of quaternion representations, models, and methods and illustrate our journey through the literature with flagship applications. We conclude this work by an outlook on the remaining challenges and open problems in quaternion signal and image processing.

Deep learning (DL) has been wildly successful in practice, and most of the state-of-the-art machine learning methods are based on neural networks (NNs). Lacking, however, is a rigorous mathematical theory that adequately explains the amazing performance of deep NNs (DNNs). In this article, we present a relatively new mathematical framework that provides the beginning of a deeper understanding of DL. This framework precisely characterizes the functional properties of NNs that are trained to fit to data. The key mathematical tools that support this framework include transform-domain sparse regularization, the Radon transform of computed tomography, and approximation theory, which are all techniques deeply rooted in signal processing.

Compression is essential for efficient storage and transmission of signals. One powerful method for compression is through the application of orthogonal transforms, which convert a group of N data samples into a group of N transform coefficients. In transform coding, the N samples are first transformed, and then the coefficients are individually quantized and entropy coded into binary bits. The transform serves two purposes: one is to compact the energy of the original N samples into coefficients with increasingly smaller variances so that removing smaller coefficients have negligible reconstruction errors, and another is to decorrelate the original samples so that the coefficients can be quantized and entropy coded individually without losing compression performance. 

Twenty-five years ago, the field of computational imaging arguably did not exist, at least not as a standalone arena of research activity and technical development. Of course, the idea of using computation to form images had been around for several decades, largely thanks to the development of medical imaging—such as magnetic resonance imaging (MRI) and X-ray tomography - in the 1970s and synthetic-aperture radar (SAR) even earlier. 

In this article, we summarize the evolution of speech and language processing (SLP) in the past 25 years. We first provide a snapshot of popular research topics and the associated state of the art (SOTA) in various subfields of SLP 25 years ago, and then highlight the shift in research topics over the years. We describe the major breakthroughs in each of the subfields and the main driving forces that led us to the SOTA today. Societal impacts and potential future directions are also discussed.