Guest Editorial: Wavelets in Medical Imaging

I. Introduction Wavelets are the result of collective efforts that recognized common threads between ideas and concepts that had been independently developed and investigated by distinct research communities. They provide a unifying framework for decomposing images, volumes, and time-series data into their elementary constituents across scale. Although a relatively recent construct, wavelets have become a tool of choice for engineers, physicists, and mathematicians, leading to efficient solutions in time and space frequency analysis problems, as well as a multitude of other applications. One of the consequences is that wavelet methods of analysis and representation are presently having a significant impact on the science of medical imaging and the diagnosis of disease and screening protocols. Because of a powerful underlying mathematical theory, they offer exciting opportunities for the design of new multiresolution image processing algorithms, and novel acquisition methods such as wavelet-encoded magnetic resonance imaging (MRI). This special issue of the IEEE Transactions on Medical Imaging focuses on these recent developments and highlights progress that has been accomplished in the areas related to medical imaging. II. Sizing the Wave: Some Facts and Figures Rarely has a mathematical concept generated so much response and enthusiasm within and between the engineering and mathematical research communities at large. To give a rough idea of the phenomenon, we provide a brief chronology. While wavelets have been traced all the way back to Alfred Haar in 1910 [1], for many, the starting point of their modern history coincides with two publications in the late 1980s by S. Mallat [2] and I. Daubechies [3]. These groundbreaking papers established a solid mathematical footing which would both shape and define the field. In a nutshell, S. Mallat identified the important concept of multiresolution analysis which is the corner stone of modern wavelet theory, while I. Daubechies constructed the first orthogonal wavelet bases that were compactly supported. These two contributions count among the most cited papers in the scientific literature (over 1500 SCI citations each). From that point on, the number of contributions relating to wavelet-applications and theory has increased steadily on the order of 9000 journal papers published to date. This trend is likely to continue as suggested by the strong response to the call for papers for this special issue (over 30 submissions). Wavelets have become so popular that distinct communities continue to have conferences and scientific journals entirely devoted to them. There are already historical anecdotes and folklore associated with them; an entertaining account of which can be found in the book of B. Burke Hubbard [4]. Readers who want to dive deeper into the subject have the daunting task of choosing among over 200 books written on wavelets. Our only advice in this regard is: in case of doubt, stick wit