Digital Signal Processing Reference
In-Depth Information
residuals
r
k
and are likely to contain mainly the salient information of
x
k
.This
intuition dictates a coordinate relaxation algorithm that cycles through the com-
ponents at each iteration and applies a thresholding to the marginal residuals. This
is what justifies the steps of the MCA algorithm summarized in Algorithm 30.
Besides coordinate relaxation, another important ingredient of MCA is iter-
ative thresholding with
varying threshold
. Thus MCA can be viewed as a stage-
wise hybridization of matching pursuit (MP) (Mallat and Zhang 1993) with block-
coordinate relaxation (BCR) (Sardy et al. 2000) that attempts to solve equation
(8.18). The adjective
stagewise
is used because MCA exploits the fact that the dictio-
nary is structured (union of transforms), and the atoms enter the solution by groups,
rather than individually, unlike matching pursuit (MP). As such, MCA is a
salient-
to-fine
process in which, at each iteration, the most salient content of each morpho-
logical component is iteratively computed. These estimates are then progressively
refined as the threshold
λ
min
. It is worth noting that in practice,
hard thresholding leads generally to better results than soft thresholding.
λ
evolves toward
Algorithm 30
MCA Decomposition Algorithm
Task:
Signal/image decomposition.
Parameters:
The signal/image
x
, the dictionary
=
[
1
···
K
], number of itera-
tions
N
iter
, stopping threshold
λ
min
, threshold update schedule.
Initialization:
Initial solution
x
(0)
k
=
0
,
∀
k
.
Initial residual
r
(0)
y
.
Initial threshold: let
k
=
=
argmax
k
T
k
y
∞
;set
λ
0
=
max
k
=
k
T
k
y
∞
.
Main iteration:
for
t
1
to
N
iter
do
for
k
=
1
to
K
do
Compute marginal residuals
r
(
t
)
k
=
x
(
t
−
1
k
.
Update
k
th component coefficients by hard (or soft) thresholding
r
(
t
−
1)
=
+
(
t
)
k
α
=
HardThresh
λ
t
−
1
(
T
k
r
(
t
k
).
Update
k
th component
x
(
t
)
k
(
t
)
k
=
k
α
.
−
k
=
1
x
(
t
)
Update the residuals
r
(
t
)
=
y
.
k
Update the threshold
λ
t
according to the given schedule.
λ
t
≤
λ
min
then
stop.
Output:
Morphological components
x
(
N
iter
)
k
if
k
=
1
,
···
,
K
.
8.5.3 Thresholding Strategies
The way the threshold is decreased along the iterations of the MCA algorithm is
paramount in terms of separation quality.
