TABLE OF CONTENTS This important transform (DCT for short) has originated by [Ahmed et al. 74] and has been used and studied extensively since. Because of its importance for data compression, the DCT is treated here in detail. Section 4.6.1 introduces the mathematical expressions for the DCT in one dimension and two dimensions without any theoretical background or justifications. The use of the transform and its advantages for data compression are then demonstrated by several examples. Sections 4.6.2 and 4.6.3 cover the theory of the DCT and discuss its two interpretations as a rotation and as a basis of a vector space. Section 4.6.4 introduces the four DCT types, and Section 8.15.2 discusses the three-dimensional DCT. Section 4.6.5 describes ways to speed up the computation of the DCT, and Section 4.6.7 is a short discussion of the symmetry of the DCT and how it can be exploited for a hardware implementation. Several sections of important background material follow. Section 4.6.8 explains the QR decomposition of matrices. Section 4.6.9 introduces the concept of vector spaces and their bases. Section 4.6.10 shows how the rotation performed by the DCT relates to general rotations in three dimensions. Finally, the discrete sine transform is introduced in Section 4.6.11 together with the reasons that make it unsuitable for data compression.
The DCT in one dimension is given by
where
The input is a set of n data values p t (pixels, audio samples, or other data), and the output...
