What is the book all about?

This is Euclid's definition for “perpendicular”, which is synonymous with the word “orthogonal” used in the title of this book. Although the meaning of this word has been generalized since Euclid's time to describe the relationship between two functions as well as two vectors, as what we will be mostly concerned with in this book, they are essentially no different from two perpendicular straight lines, as discussed by Euclid some 23 centuries ago.

Orthogonality is of important significance not only in geometry and mathematics, but also in science and engineering in general, and in data processing and analysis in particular. This book is about a set of mathematical and computational methods, known collectively as the orthogonal transforms, that enables us to take advantage of the orthogonal axes of the space in which the data reside. As we will see throughout the book, such orthogonality is a much desired property that can keep things untangled and nicely separated for ease of manipulation, and an orthogonal transform can rotate a signal, represented as a vector in a Euclidean space, or more generally Hilbert space, in such a way that the signal components tend to become, approximately or accurately, orthogonal to each other.