Digital Signal Processing Reference
In-Depth Information
with high-speed raster scanning leads to a dramatic loss in resolution. In Bobin et al.
(2008c), we emphasized the redundancy of raster scan data: 2 consecutive images
are almost the same images up to a small shift
t
t
2
)(
t
1
and
t
2
are the trans-
lations along each direction). Then, jointly compressing/decompressing consecutive
images of the same raster scan has been proposed to alleviate the Herschel/PACS
compression dilemma. The problem consists of recovering a single image
x
from
P
compressed and shifted noisy versions of it:
=
(
t
1
,
x
i
=
S
t
i
(
x
)
+
η
∀
i
∈{
1
,...,
P
}
,
(11.10)
i
where
S
t
i
η
i
models instrumental noise or model imperfections. According to the compressed
sensing paradigm, we observe
is an operator that shifts the original image
x
with a shift
t
i
.Theterm
y
i
=
H
i
x
i
∀
i
∈{
1
,...,
P
}
,
(11.11)
where the sampling matrices (
H
i
)
are such that their union spans
R
N
.Inthis
∈{
1
,...,
P
}
application, each sensing matrix
H
i
takes
m
i
=
measurements such that the
measurement subset extracted by
H
i
is disjoint from the other subsets taken by
H
j
=
i
. Obviously, when there is no shift between consecutive images, this condition
on the measurement subset ensures that the system of linear equations (11.11) is
determined, and hence
x
can be reconstructed uniquely and stably from the mea-
surements (
y
i
)
i
=
1
,...,
P
.
N
/
P
11.7.2 Compressed Sensing Solution
Following the compressed sensing rules, the decoding step amounts to solving the
following optimization problem:
P
)
2
y
i
−
1
P
1
2
min
α
∈R
H
i
S
t
i
(
α
+
λ
α
1
.
(11.12)
T
i
=
1
This problem is very similar to equation (7.4) and can be solved efficiently using
the forward-backward splitting algorithm described in Section 7.3.3.2. Adapted to
equation (11.12), the forward-backward recursion to solve equation (11.12) reads
(
t
)
,
1
∗
S
−
t
i
H
∗
i
y
i
H
i
S
t
i
P
1
P
(
t
+
1)
(
t
)
α
=
SoftThresh
μ
t
λ
α
+
μ
−
α
t
i
=
(11.13)
/
i
|||
2
2
). At each iteration, the sought after image is re-
where
μ
t
∈
(0
,
2
P
H
i
|||
|||
|||
(
t
)
. To accelerate this scheme, and
following our discussion in Section 7.7.2, it was advocated by Bobin et al. (2008c) to
use a sequence of thresholds
(
t
)
as
x
(
t
)
constructed from the coefficients
α
=
α
(
t
)
that decrease with the iteration number toward a
final value that is typically 2-3 times standard deviations of the noise. This allows us
to handle the noise properly.
Two approaches to solve the Herschel/PACS compression problem were com-
pared:
λ
1. The first, called MO6, consists of transmitting the average of
P
=
6 consecu-
tive images.
