Digital Signal Processing Reference
In-Depth Information
A typical choice is
γ
t
=
(
N
iter
−
t
)
/
(
N
iter
−
1), which satisfies the requirements as
N
iter
→+∞
.
Using a proof similar to Zhang et al. (2008b, Theorem 3), it can be shown that
if
k
corresponds to a tight frame, then instead of directly solving equation (8.10),
one can solve its smoothed strictly convex version by applying equation (8.11) with
asmall
0
+
, every limit of the solution to the smoothed problem
is a solution to equation (8.10). We also point out that in practice, the recursion
(8.11) applies equally well even with
. Formally, as
→
=
0. For
=
0, from the subdifferential of
the
in equation (8.11) can be
implemented in practice as a soft thresholding with a threshold
1
norm (see equation (7.8)), the operator
I
−
γ
t
∇
F
γ
t
.
Now we have all ingredients for the combined denoising algorithm summarized
in Algorithm 29. In our experiments, we observed that step (c) of the inner loop can
be discarded without loss of quality, and the benefit is computational savings. Hence
the computational cost of this algorithm can be reduced to
O
(2
N
iter
k
=
1
V
k
),
where
V
k
is the computational cost of each transform.
Algorithm 29
Combined Denoising Algorithm
Task:
Denoising using several transforms.
Parameters:
The data
y
, the dictionaries
1
,...,
K
(with atoms assumed nor-
malized to a unit norm), the number of iterations
N
iter
.
Initialization:
Initialize
1 and the solution
x
(0)
γ
0
=
=
0. Estimate the noise stan-
dard deviation
σ
ε
in
y
(e.g., see equation (6.9)), and set
e
k
=
σ
ε
/
2
,
∀
k
.
for
k
=
1
to
K
do
Compute the transform coefficients
β
k
=
T
k
y
; deduce the multiresolution
support
M
k
.
for
t
=
0
to
N
iter
−
1
do
x
(
t
)
.
2.
for
k
1.
z
=
=
1 to
K
do
a. Compute the transform coefficients
T
k
z
.
b. From equation (8.12), compute the projection
α
k
=
ζ
k
[
i
]
= P
k
(
α
k
)[
i
]if
=
α
k
[
i
] otherwise.
c. Project onto the column span of
T
k
:
i
∈M
k
, and
ζ
k
[
i
]
T
k
T
k
(
ζ
k
←
ζ
k
).
d. Apply soft thresholding with threshold
λ
to
ζ
k
:
˜
ζ
k
).
e. Reconstruct and project onto the positive orthant:
ζ
k
=
SoftThresh
γ
t
(
= P
+
(
T
k
˜
z
ζ
k
).
3. Update the image:
x
(
t
+
1)
=
z
.
4.
γ
t
=
−
−
/
(
N
iter
t
1)
N
iter
.
+
1
Output:
x
(
N
iter
)
, the denoised image.
