Transforms for Image and Video Coding

The purpose of this chapter is to introduce the principles of decorrelating transforms and to demonstrate how these can provide the basis for image compression. The chapter firstly introduces the principles and properties of decorrelating transforms and explains their relationship to principal component analysis and eigen-analysis. It then introduces some simple but useful transforms, such as the discrete Walsh–Hadamard transform, followed by an overview of the optimum Karhunen–Loeve transform (KLT). After discussing the limitations of the KLT, the remainder of the chapter focuses on deriving and characterizing the discrete cosine transform (DCT). Both the 1-D DCT and its 2-D extension are described. Quantization of DCT coefficients is then presented and performance comparisons are provided in the context of the JPEG standard. The chapter concludes with a discussion of DCT implementations and their complexities. Examples are provided throughout.