Digital Signal Processing Reference
In-Depth Information
superposing atoms:
x
=
α
. Analysis and synthesis are different linear operations.
In the overcomplete case,
is not invertible, and the reconstruction is not unique
(see also Section 8.2 for further details).
1.1.4 Best Dictionary
Obviously, the best dictionary is the one that leads to the sparsest representation.
Hence we could imagine having a huge dictionary (i.e.,
T
N
), but we would be
faced with a prohibitive computation time cost for calculating the
coefficients.
Therefore there is a trade-off between the complexity of our analysis (i.e., the size of
the dictionary) and computation time. Some specific dictionaries have the advantage
of having fast operators and are very good candidates for analyzing the data. The
Fourier dictionary is certainly the most well known, but many others have been
proposed in the literature such as wavelets (Mallat 2008), ridgelets (Candes and
Donoho 1999), curvelets (Cand es and Donoho 2002; Candes et al. 2006a; Starck
et al. 2002), bandlets (Le Pennec and Mallat 2005), and contourlets (Do and Vetterli
2005), to name but a few candidates. We will present some of these in the chapters
to follow and show how to use them for many inverse problems such as denoising
or deconvolution.
α
1.2 FROM FOURIER TO WAVELETS
The Fourier transform is well suited only to the study of stationary signals, in which
all frequencies have an infinite coherence time, or, otherwise expressed, the signal's
statistical properties do not change over time. Fourier analysis is based on global
information that is not adequate for the study of compact or local patterns.
As is well known, Fourier analysis uses basis functions consisting of sine and co-
sine functions. Their frequency content is time-independent. Hence the description
of the signal provided by Fourier analysis is purely in the frequency domain. Music
or the voice, however, imparts information in both the time and the frequency do-
mains. The windowed Fourier transform and the wavelet transform aim at an anal-
ysis of both time and frequency. A short, informal introduction to these different
methods can be found in the work of Bentley and McDonnell (1994), and further
material is covered by Chui (1992), Cohen (2003), and Mallat (2008).
For nonstationary analysis, a windowed Fourier transform (short-time Fourier
transform, STFT) can be used. Gabor (1946) introduced a local Fourier analysis,
taking into account a sliding Gaussian window. Such approaches provide tools for
investigating time and frequency. Stationarity is assumed within the window. The
smaller the window size, the better is the time resolution; however, the smaller the
window size, also, the more the number of discrete frequencies that can be repre-
sented in the frequency domain will be reduced, and therefore the more weakened
will be the discrimination potential among frequencies. The choice of window thus
leads to an uncertainty trade-off.
The STFT transform, for a continuous-time signal
s
(
t
), a window
g
around time
point
τ
, and frequency
ω
,is
+∞
)
e
−
j
ω
t
dt
STFT(
τ,ω
)
=
s
(
t
)
g
(
t
−
τ
.
(1.2)
−∞
