Digital Signal Processing Reference
In-Depth Information
Algorithm 31
MCA Texture-Cartoon Part Separation
Task:
Texture-piecewise smooth component separation; solve equation (8.21).
Parameters:
The signal/image
y
, the local DCT and curvelet dictionaries
D
and
C
, number of iterations
N
iter
,
γ
and index of the cartoon part, stopping threshold
λ
min
, threshold update schedule.
Initialization:
x
(0)
x
(0)
(0)
D
=
C
=
0,
λ
=
min (
T
D
y
∞
,
T
C
y
∞
).
Main iteration:
for
t
=
1
to
N
iter
do
1.
Texture part: Update of
x
D
,
assuming
x
C
is fixed:
x
(
t
)
x
(
t
)
C
Compute the residual
r
(
t
)
•
=
y
−
D
−
.
(
t
)
D
•
α
=
T
D
(
x
D
+
r
(
t
)
).
Compute the local DCT coefficients
(
t
)
D
=
(
t
)
D
•
Hard (or soft) with threshold
λ
t
:˜
α
HardThresh
λ
t
(
α
).
(
t
)
D
.
2.
Cartoon part: Update of
x
C
,
Update
x
(
t
)
D
•
=
D
˜
α
assuming
x
D
is fixed:
x
(
t
)
x
(
t
)
C
Compute the residual
r
(
t
)
•
=
y
−
C
−
.
(
t
)
C
=
T
C
(
x
C
+
r
(
t
)
).
•
Compute the curvelet coefficients
α
(
t
)
C
=
(
t
)
C
•
Hard (or soft) with threshold
λ
t
:˜
α
HardThresh
λ
t
(
α
).
(
t
)
C
.
3.
TV regularization of
x
C
:
see Algorithm 32.
4. Update the threshold
Update
x
(
t
)
•
C
=
C
˜
α
(
t
+
1)
λ
according to the given schedule (see Sec-
tion 8.5.3).
5.
if
(
t
+
1)
λ
≤
λ
min
then
stop.
End iteration
Output:
Texture and cartoon components
x
(
N
iter
)
D
.
x
(
N
iter
)
C
,
presented is an approximation to the proximity operator of the TV penalty. An al-
ternative exact way of computing it is the duality-based algorithms of Section 7.3.2.3
with the discrete gradient. The exact solution would, however, entail an iterative
algorithm.
Algorithm 32
Approximate TV Regularization by Soft Thresholding Undecimated
Haar Wavelet Coefficients
Parameters:
The image
x
C
and regularization parameter
γ
.
Compute the undecimated Haar wavelet transform coefficients of
x
C
.
Soft threshold the finest-scale coefficients with threshold
γ
.
Reconstruct by inverse undecimated Haar wavelet transform.
