sparse image representation via combined transforms - Convex ...

30 CHAPTER 3. IMAGE TRANSFORMS AND IMAGE FEATURES
Also let DFT N denote the N-point DFT matrix
DFT N = {e −i 2π N kl } k=0,1,2,... ,N−1; ,
l=0,1,2,... ,N−1
where k is the row index and l is the column index. We have
⃗̂X = DFT N · ⃗X.
Apparently DFT N is an N by N square symmetric matrix. Both X ⃗ and ⃗̂X are N-
dimensional vectors.
It is well known that the matrix DFT N is orthogonal, which is equivalent to saying
that the inverse of DFT N is its complex conjugate transpose. Thus, DFT is an orthogonal
transform and consequently it is an isometric transform. This result is generally attributed
to Parseval.
Since the concise statements and proofs can be found in many textbooks, we skip the
proof.
Why Fourier Transform?
The preponderant reason that Fourier analysis is so powerful in analyzing linear time invariant
system and cyclic-stationary time series is the fact that Fourier series are the eigenvectors
of matrices associated with these transforms. These transforms are ubiquitous in DSP and
time series analysis.
Before we state the key result, let us first introduce some mathematical notation:
Toeplitz matrix, Hankel matrix and circulant matrix.
Suppose A N×N is an N by N real-valued matrix: A = {a ij } 1≤i≤N,1≤j≤N with all a ij ∈ R.
The matrix A is a Toeplitz matrix when the elements on the diagonal, and the rows that
are parallel to the diagonal, are the same, or mathematically a ij = t i−j [74, page 193]. The
following is a Toeplitz matrix:
⎛
T =
⎜
⎝
⎞
t 0 t −1 ... t −(n−2) t −(n−1)
t 1 t 0 ... t −(n−3) t −(n−2)
. . . .. . .
.
t n−2 t n−3 ... t 0 t −1
⎟
⎠
t n−1 t n−2 ... t 1 t 0
N×N