Digital Signal Processing Reference
In-Depth Information
Figure 8.5. Illustration of the image decomposition problem with sparse representation.
separation. MCA is capable of creating atomic sparse representations containing, as
a by-product, a decoupling of the signal content.
As an illustrative example, let our image contain lines and Gaussians (as in
Fig. 8.5 (bottom)). Then the ridgelet transform would be very effective at sparsely
representing the global lines and poor for Gaussians. On the other hand, wavelets
would be very good at sparsifying the Gaussians and clearly worse than ridgelets for
lines.
8.5.2 The Morphological Component Analysis Algorithm
Starck et al. (2004b) and Starck et al. (2005a) proposed estimating the components
(
x
k
)
1
≤
k
≤
K
by solving the following constrained optimization problem:
K
K
y
k
2
≤
σ,
p
p
min
α
1
,...,α
K
1
α
s
.
t
.
−
1
α
(8.18)
k
k
k
=
k
=
p
p
is sparsity promoting (the most interesting regime is for 0
α
≤
≤
where
p
1) and
is typically chosen as a constant times
√
N
σ
σ
ε
. The constraint in this optimization
problem accounts for the presence of noise and model imperfection. If there is no
noise and the linear superposition model is exact (
σ
=
0), an equality constraint is
substituted for the inequality constraint.
Problem (8.18) is not easy to solve in general, especially for
p
<
1 (for
p
=
0,
it is even nonpolynomial hard). Nonetheless, if all component coefficients
α
l
but
the
k
th are fixed, then a solution can be achieved through hard thresholding (for
p
=
0) or soft thresholding (for
p
=
1) the coefficients of the marginal residuals
−
l
=
k
l
α
l
in
r
k
=
y
k
. The other components are relieved of these marginal
