ABSTRACT
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 Section 5.2, lapped transforms are introduced. Basic theory and concepts are presented for both orthogonal and nonorthogonal cases. In Section 5.3 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 Section 5.4. Section 5.5 introduces hierarchical lapped transforms (which are constructed by connecting transforms hierarchically in a tree path), briefly introducing time-frequency diagrams and concepts such as the exchange of resolution between time and frequency. Another concept is also introduced in Section 5.5: variable length lapped transforms, which are
also found through hierarchical connection of systems. Practical transforms are then presented. Transforms with symmetric bases including the popular lapped orthogonal transform, its bi-orthogonal, and generalized versions are described in Section 5.6, while fast transforms with variable-length are presented in Section 5.7. The transforms based on cosine modulation are presented in Section 5.8. In order to apply lapped transforms to images, one has to be able to transform signal segments of finite-length. Several methods for doing so are discussed in Section 5.9. Design issues for lapped transforms are discussed in Section 5.10, wherein the emphasis is given to compression applications. In Section 5.11, image compression systems are briefly introduced, including JPEG and other methods based on wavelet transforms. The performance analysis of lapped transforms in image compression is carried in Section 5.12 for different compression systems and several transforms. Finally, the conclusions are presented in Section 5.13.
