Digital Signal Processing Reference
In-Depth Information
10
3
0.6
0.5
10
2
0.4
0.3
10
1
0.2
0.1
0
10
0
10
1
4
6
8
10
12
14
16
18
Normalized
norm of the residuals in log scale
norm
10
2
10
1
10
5
10
4
10
3
10
2
10
1
Normalized
norm of the residuals in log scale
Figure 8.6. Comparison of MCA-MOM (solid line) to BP (dashed line) for sparse recovery:
(top left)
1
/
2
curve of both MCA-MOM and BP (dashed line); (top right)
0
/
1
curve of
both MCA-MOM and BP; and (bottom) nonlinear approximation error curve (
0
/
2
) of both
MCA-MOM and BP. See text for details.
wavelets or curvelets, have been proposed (Starck and Murtagh 2006); see also
Chapter 6.
8.5.6 Morphological Component Analysis versus Basis Pursuit
for Sparse Recovery
Performance of BP and MCA-MOM for sparse recovery were compared. BP was
solved with the DR splitting algorithm, Algorithm 26. We define
BG
(
p
1
,
p
2
)asthe
class of signals such that if
y
is drawn from this model, then
y
=
1
α
1
+
2
α
2
, where
α
1
and
α
2
are Bernoulli-Gaussian random vectors whose entries are nonzero with
probabilities
p
1
and
p
2
, respectively. Nonzero coefficients are drawn from
N
,
(0
1).
Signals drawn from this class are compressible in
used in this ex-
periment is taken as the union of the one-dimensional orthogonal wavelet transform
and the discrete cosine transform (DCT). The results are pictured in Fig. 8.6. The
graphs were computed as an average over 50 decompositions of signals belonging to
the model
. The dictionary
p
2
) with random
p
1
and
p
2
.
Figure 8.6 (top right and bottom) show the
BG
(
p
1
,
1
norms of the decomposi-
tion coefficients recovered by MCA (solid line) and BP (dashed line) as a function
0
and
