sparse image representation via combined transforms - Convex ...

3.1. DCT AND HOMOGENEOUS COMPONENTS 39
Type of DCT δ 1 δ 2 transform matrix
type-I DCT 0 0 CN
I
type-II DCT 0 1/2 CN
II
type-III DCT 1/2 0 CN
III
type-IV DCT 1/2 1/2 CN
IV
Table 3.2: Definitions of four types of DCTs.
For all four types, the DCT matrices are orthogonal (C −1
N
of course they are unitary. Another noteworthy fact is that C II
N
= CT N
). Since they are real,
=(CIII N )−1 .
We can view the four types of DCT as choosing the first index (which usually corresponds
to the frequency variable) and the second index (which usually corresponds to the time
variable) at different places (which can be either integer points or half integer points).
Figure 3.1 depicts this idea.
DCT via DFT
Particularly for the type-I DCT, the transform matrix CN I is connected to the transform
matrix of a 2N-point DFT (DFT 2N ) in the following way:
⎡⎛
[ ]
C
I
N
•
= R
• • ⎢⎜
⎣⎝
1 √
2
1
. ..
1
⎞ ⎛
DFT
⎟ 2N ⎜
⎠ ⎝
1 √
2
1
. ..
1
⎞⎤
,
⎟⎥
⎠⎦
where R is a real operator: R(A) is the real part of matrix A. Symbol “•” represents an
arbitrary N × N matrix.
The other types of DCT, generally speaking, can be implemented via the 8N-point DFT.
We explain the idea for the case of type-IV DCT. For type-II and type-III DCT, a similar
idea should work. Note that in practice, there are better computational schemes.
Let us first give some mathematical notation. Let symbol I N→8N denote an insert
operator: it takes an N-D vector and expand it into an 8N-D vector, so that the subvector
at locations {2, 4, 6,... ,2N} of the 8N-D vector is exactly the N-D vector, and all other