Often, nowadays, one addresses the public understanding of mathematics and rigor by pointing to important applications and how they underpin a great deal of science and engineering. In this context, multiple resolution methods in image and signal processing, as discussed in depth in this book, are important. Results of such methods are often visual. Results too can often be presented to the layperson in an easily understood way. In addition to those aspects that speak powerfully in favor of the methods presented here, the following is worth noting. Among the most cited articles in statistics and signal processing, one finds works in the general area of what we cover in this book.

The methods discussed in this book are essential underpinnings of data analysis and are of relevance to multimedia data processing and to image, video, and signal processing. The methods discussed here feature very crucially in statistics, in mathematical methods, and in computational techniques.

Domains of application are incredibly wide, including imaging and signal processing in biology, medicine, and the life sciences generally; astronomy, physics, and the natural sciences; seismology and land use studies as indicative subdomains from geology and geography in the earth sciences; materials science, metrology, and other areas of mechanical and civil engineering; image and video compression, analysis, and synthesis for movie and television; and so on.

There is a weakness, though, in regard to well-written available works in this area: the very rigor of the methods also means that the ideas can be very deep. When separated from the means to apply and to experiment with the methods, the theory and underpinnings can require a great deal of background knowledge and diligence, and study too, in order to grasp the essential material.

Our aim in this book is to provide an essential bridge between theoretical background and easily applicable experimentation. We have an additional aim, namely that coverage is as extensive as can be, given the dynamic and broad field with which we are dealing.

Our approach, which is wedded to theory and practice, is based on a great deal of practical engagement across many application areas. Very varied applications are used for illustration and discussion in this book. This is natural, given how ubiquitous the wavelet and other multiresolution transforms have become. These transforms have become essential building blocks for addressing problems across most of data, signal, image, and indeed information handling and processing.