My end of term as IEEE Signal Processing Society (SPS) president is fast approaching. It has been an incredible experience that has provided me with so many opportunities to engage with our members around the globe, forge relationships with other IEEE Societies, and meet a diverse range of people that I hope will become active members of our Society in the future. It has been a great privilege to be at the helm of a Society that garners such a high level of worldwide respect and recognition.

Encoding-decoding convolutional neural networks (CNNs) play a central role in data-driven noise reduction and can be found within numerous deep learning algorithms. However, the development of these CNN architectures is often done in an ad hoc fashion and theoretical underpinnings for important design choices are generally lacking. Up to now, there have been different existing relevant works that have striven to explain the internal operation of these CNNs. Still, these ideas are either scattered and/or may require significant expertise to be accessible for a bigger audience.

Joseph Fourier’s methods (and their variants) are omnipresent in audio signal processing. However, it turns out that the underlying ideas took some time to penetrate the field of sound analysis and that different paths were first followed in the period immediately following Fourier’s pioneering work, with or without reference to him. This illustrates the interplay between mathematics and physics as well as the key role played by instrumentation, with notable inventions by outsiders to academia, such as Rudolph Koenig and Édouard-Léon Scott de Martinville.

My three years of service as the editor-in-chief (EIC) of Signal Processing Magazine ( SPM ) are now coming to a close. During the past three years, many of us were deeply affected by serious political, social, and environmental events such as the war in Ukraine; protests for freedom in Iran; coups d’état in Africa; the COVID-19 pandemic; seisms in Turkey, Syria, and Morocco; huge floods in Libya and India; gigantic fires in North America and Southern Europe; and an avalanche of stones in the Alps, to name a few. In such a context, I believe that the IEEE slogan, “Advancing Technology for Humanity,” is incredibly relevant and timely.

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.