sparse image representation via combined transforms - Convex ...

3.1. DCT AND HOMOGENEOUS COMPONENTS 29
a superposition of sinusoid functions. Nowadays, Fourier analysis has become the most
powerful tool in signal processing. Every researcher in science and engineering should know
it.
The Fourier transform of a continuous function X(t), by definition, is
̂X(ω) = 1
2π
∫ +∞
−∞
X(t)e −i2πωt dt,
where i 2 = −1. The following is the inversion formula:
X(t) =
∫ +∞
−∞
̂X(ω)e i2πωt dω.
Since digital signal processing (DSP) has become the mainstream in signal processing,
and only discrete transforms can be implemented in the digital domain, it is more interesting
to consider the discrete version of Fourier transform.
For a finite sequence X[0],X[1],... ,X[N − 1], where N is a given integer, the discrete
Fourier transform is defined as
and consequently
̂X[l] = √ 1 N−1
∑
X[k]e −i 2π N kl , 0 ≤ l ≤ N − 1; (3.1)
N
k=0
X[k] = √ 1 N−1
∑
N
l=0
̂X[l]e i 2π N kl , 0 ≤ k ≤ N − 1.
The discrete Fourier transform can be considered as a matrix-vector multiplication.
Let X ⃗ and ⃗̂X denote the vectors made by the original signal and its Fourier transform
respectively:
⎛
⎞
X[0]
X[1]
⃗X =
⎜
⎝ . ⎟
⎠
X[N − 1]
N×1
⎛
and ⃗̂X = ⎜
⎝
⎞
̂X[0]
̂X[1]
. ⎟
⎠
̂X[N − 1]
N×1
.