The story of wavelets

The theory and applications of wavelets have undoubtedly dominated the journals in all mathematical, engineering and related fields throughout the last decade. Few other theoretical developments in mathematical sciences have enjoyed this much attention and popularity, have been applied to such a diverse field of disciplines, and perhaps, have been so blindly misused. What was the missing piece in the great puzzle of signal processing, and how did the wavelets fill-in this missing piece? How did it all start, what development stages did it go through and what is the state of the art today? Have we reached the saturation, or do we have a long way to go? In this paper, we present three overviews in an attempt to answer these questions. In a historical overview, we look at the genesis of the wavelet theory as we take a short chronological journey through the wavelet times. In a technical overview, we look at the driving forces that played a key role in the development of the theory of wavelets, and try to find out what was so special that brought them to the center stage of scientific journals. In an application overview, we look at some of the most creative conventional and non-conventional applications of wavelets. On the conventional front, we discuss such applications as image compression, speech processing, and solution of partial differential equations. On the unconventional front, we look at various fields of applications including chemistry, neurophysiology, nondestructive evaluation, fractals, and economics. In particular, we discuss analyzing brain signals for the detection of Alzheimer's disease, analyzing ultrasonic weld inspection signals for the detection of cracks in piping of nuclear power plants, and analyzing fluctuations of financial markets. Finally, we look at new, noteworthy and promising developments, such as wavelet networks, and zero crossing representations, as we conclude what appears to be one of the most remarkable success stories of mathematical and engineering sciences: the story of wavelets.