sparse image representation via combined transforms - Convex ...

38 CHAPTER 3. IMAGE TRANSFORMS AND IMAGE FEATURES
3.1.2 Discrete Cosine Transform
Definition
Coefficients of the discrete cosine transform (DCT) are equal to the inner product of a
discrete signal and a discrete cosine function. The discrete cosine function is an equally
spaced sampling of a cosine function. Note that we stay in a discrete setting. Let the
sequence
X = {X[l] :l =0, 1, 2,... ,N − 1}
be the signal sequence, and denote the DCT of X by
Y = {Y [k] :k =0, 1, 2,... ,N − 1}.
Then by definition, each Y [k] is given by
Y [k] =
N−1
∑
l=0
and the element C kl is determined by
C kl = α 1 (k)α 2 (l)
C kl X[l], 0 ≤ k ≤ N − 1,
√
2
N cos ( π(k + δ1 )(l + δ 2 )
N
)
, 0 ≤ k, l ≤ N − 1. (3.11)
For i =1, 2, the choice of function α i (k),k ∈ N is determined by the parameter δ i :
⎧
⎪⎨
⎪⎩
if δ i = 0, then
{
√2 1
, if k =0,
α i (k) =
1, if k =1, 2,... ,N − 1;
if δ i =1/2, then α i (k) ≡ 1.
The purpose of choosing different values for the function α i (k) istomaketheDCT
an orthogonal transform, or equivalently to make the transform C N = {C kl } k=0,1,... ,N−1 an
l=0,1,... ,N−1
orthonormal matrix.
A good reference book about DCT is Rao and Yip [121].
By choosing different values for δ 1 and δ 2 in (3.11), we can define four types of DCT, as
summarized in Table 3.2.