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28 CHAPTER 3. IMAGE TRANSFORMS AND IMAGE FEATURES
3.1 DCT and Homogeneous Components
We start with the Fourier transform (FT). The reason is that the discrete cosine transform
(DCT) is actually a real version of the discrete Fourier transform (DFT). We answer three
key questions about the DFT:
1. (Definition) What is the discrete Fourier transform?
2. (Motivation) Why do we need the discrete Fourier transform?
3. (Fast algorithms) How can it be computed quickly?
These three items comprise the content of Section 3.1.1.
discrete cosine transform (DCT). It has two ingredients:
Section 3.1.2 switches to the
1. definition,
2. fast algorithms.
For fast algorithms, the theory follows two lines: (1) An easy way to implement fast DCT is
to utilize fast DFT; the fast DFT is also called the fast Fourier transform (FFT). (2) We can
usually do better by considering the sparse matrix factorization of the DCT matrix itself.
(This gives a second method for finding fast DCT algorithms.) The algorithms from the
second idea are computationally faster than those from the first idea. We summarize recent
advances for fast DCT in Section 3.1.2. Section 3.1.3 gives the definitions of Discrete Sine
Transform (DST). Section 3.1.4 provides a framework for homogeneous images, and explains
when a transform is good at processing homogeneous images. The key idea is that the
DCT almost diagonalizes the covariance matrix of certain Gaussian Markov random fields.
Section 3.1.4 shows that the 2-D DCT is a good transform for images with homogeneous
components.
3.1.1 Discrete Fourier Transform
Definition
In order to introduce the DFT, we need to first introduce the continuous Fourier transform.
From the book by Y. Meyer [107, Page 14], we learn that the idea of continuous Fourier
transform dates to 1807, the year Joseph Fourier asserted that any 2π-periodic function
in L 2 , which is a functional space made by functions whose square integral is finite, is