sparse image representation via combined transforms - Convex ...

50 CHAPTER 3. IMAGE TRANSFORMS AND IMAGE FEATURES
Let Σ denote the covariance matrix, and let Σ 1 and Σ 2 denote two matrices that have
special structures that we will specify later.
We will refer to the following table (Table 3.4) in the next Theorem.
Σ 1 (Toeplitz part) Σ 2 (Hankel part) Σ 1 +Σ 2 (covariance matrix)
type-I B · C 1 · B B ·D(C 2 ) · B B · C 1 · B + B ·D(C 2 ) · B
type-II C 1 C 2 C 1 + C 2
type-III
type-IV
B · C 1 ′ · B
C 1 ′ B ·D(C′ 2 ) · B
C 2 ′ B · C′ 1 · B + B ·D(C′ 2 ) · B
C 1 ′ + C′ 2
Table 3.4: Decomposition of covariance matrix that can be diagonalized by different types
of DCT.
Theorem 3.5 (DCT Diagonalization) If a covariance matrix Σ can be diagonalized by
one type of DCT, then it can be decomposed into two matrices, Σ=Σ 1 +Σ 2 ,andthematrices
Σ 1 and Σ 2 have special structures as listed in Table 3.4. For example, if a covariance matrix
Σ can be diagonalized by DCT-I, then
Σ=Σ 1 +Σ 2 = B · C 1 · B + B ·D(C 2 ) · B.
The matrices B,C 1 ,C 1 ′ ,C 2,C 2 ′ , D(C 2) and D(C 2 ′ ) are defined in equations (3.14), (3.15),
(3.16), (3.17), (3.18) and (3.19), respectively.
Moreover, the elements c i and c ′ i
are determined by the following expressions:
c i = 1 N
c ′ i = 1 N
N−1
∑
k=0
N−1
∑
k=0
λ k b 2 k
πki
cos ,i=0, 1,... ,2N − 1,
N
λ k cos π(k + 1 2 )i ,i=0, 1,... ,2N − 1,
N
where the sequence {λ k : k =0, 1,... ,N − 1} is a real-valued sequence.