Lapped Transforms for Image Compression

This chapter covers the basic aspects of lapped transforms and their applications to image compression. It is a subject that has been extensively studied mainly because lapped transforms are closely related to lter banks, wavelets, and time-frequency transformations. Some of these topics are also covered in other chapters in this book. In any case it is certainly impractical to reference all the contributions in the eld. Therefore, the presentation will be more focused rather than general. We refer the reader to excellent texts such as [62],[26], [65],[54] for a more detailed treatment of lter banks. For the rest of this introductory section we will cover the basic notation, give a brief history of lapped transforms and introduce block-based transforms. We will describe the principles of a block transform and its corresponding transform matrix along with its factorization. We will also introduce multi-input multi-output systems and relate them to block transforms. In Sec. 0.2, lapped transforms are introduced. Basic theory and concepts are presented for both orthogonal and non-orthogonal cases. In Sec. 0.3 lapped transforms are related to multi-input multi-output discrete systems with memory laying the theoretical basis for the understanding of the factorization of a lapped transform. Such a factorization is then presented in Sec. 0.4. Section 0.5 is an introduction to hierarchical lapped transforms, (which are constructed by connecting transforms hierarchically in a tree path) brie y introducing time-frequency diagrams, and concepts such as the exchange of resolution between time and frequency. Another concept is also introduced in Sec. 0.5: variable length lapped transforms, which are also found through hierarchi-