4 Reconstruction of Retinal OCT Images with Sparse Representation
95
their positions are the same, a property which we will exploit in the next subsection
for enhanced compression.
A simple trick that can help us further reduce the total number of nonzero coefficients needed to represent a similar set of patches is to estimate the variance of the
sparse vectors
ˆ
α
t
sim,i
T
t1
. If the variance of the sparse vectors in a set is below
a threshold, we denote this set as “very similar” and then fuse the corresponding sparse vectors into one vector α vs,i . Otherwise, we denote them as “not very
similar”
α
t
nvs,i
T
t1
and keep all the coefficients:
⎧
⎪ ⎨
⎪ ⎩
α vs,i mean
α
t
sim,i
T
t1
, if variance
α
t
sim,i
T
t1
≤ b × ε
α
t
nvs,i
T
t1
α
t
sim,i
T
t1
, if variance
α
t
sim,i
T
t1
> b × ε
,
(4.20)
where b is a constant and the mean is the operation to compute the mean of the
α
t
vs,i
T
t1
.
The “different” nearby patches
x
t
dif,i
T
t1
are independently decomposed on the
sub-dictionaries D
structral
ˆ
h
t
i
that can best fit each of them, which amounts to the problem:
ˆ
α
t
dif,i
T
t1
arg min
α
t
dif
T
t1
α
t
dif,i
0
subject to
t∈{1,...,T }
x
t
dif,i − D
structural
ˆ
h
t
i
α
t
dif,i
2
2
≤ ε.
(4.21)
We solve this problem by applying the OMP algorithm [34] separately on each patch.
Note that the positions and values of the nonzero coefficients in
ˆ
α
t
di f,i
T
t1
might
be varied for reflecting the differences among the nearby patches
x
t
dif,i
T
t1
. The
proposed 3D sparse representation algorithm is summarized in Fig. 4.15.
4.3.3.2 3D Adaptive Encoding, Decoding and Image Reconstruction
To encode the positions and values of the nonzero coefficients representing a set
of nearby patches, we first quantize the sparse vectors using a uniform quantizer
[32]. Then, we utilize an adaptive strategy to preserve the positions and values of the
nonzero coefficients as follows:
For the “very similar” nearby patches, both the positions and values of the nonzero
coefficients are the same and these sparse vectors are already fused into one vector
α vs,i . Thus, only one sequence is required to store the position information and one
sequence is used to preserve the value information, as shown in Fig. 4.16a.
For the “not very similar” nearby patches, the positions of the nonzero coefficients
in
α
t
nvs,i
T
t1
are the same while their values are different. Thus, only one sequence
is needed to store the position information while another T sequences are employed
to preserve the value information, as shown in Fig. 4.16b. For the “different” nearby
95
their positions are the same, a property which we will exploit in the next subsection
for enhanced compression.
A simple trick that can help us further reduce the total number of nonzero coefficients needed to represent a similar set of patches is to estimate the variance of the
sparse vectors
ˆ
α
t
sim,i
T
t1
. If the variance of the sparse vectors in a set is below
a threshold, we denote this set as “very similar” and then fuse the corresponding sparse vectors into one vector α vs,i . Otherwise, we denote them as “not very
similar”
α
t
nvs,i
T
t1
and keep all the coefficients:
⎧
⎪ ⎨
⎪ ⎩
α vs,i mean
α
t
sim,i
T
t1
, if variance
α
t
sim,i
T
t1
≤ b × ε
α
t
nvs,i
T
t1
α
t
sim,i
T
t1
, if variance
α
t
sim,i
T
t1
> b × ε
,
(4.20)
where b is a constant and the mean is the operation to compute the mean of the
α
t
vs,i
T
t1
.
The “different” nearby patches
x
t
dif,i
T
t1
are independently decomposed on the
sub-dictionaries D
structral
ˆ
h
t
i
that can best fit each of them, which amounts to the problem:
ˆ
α
t
dif,i
T
t1
arg min
α
t
dif
T
t1
α
t
dif,i
0
subject to
t∈{1,...,T }
x
t
dif,i − D
structural
ˆ
h
t
i
α
t
dif,i
2
2
≤ ε.
(4.21)
We solve this problem by applying the OMP algorithm [34] separately on each patch.
Note that the positions and values of the nonzero coefficients in
ˆ
α
t
di f,i
T
t1
might
be varied for reflecting the differences among the nearby patches
x
t
dif,i
T
t1
. The
proposed 3D sparse representation algorithm is summarized in Fig. 4.15.
4.3.3.2 3D Adaptive Encoding, Decoding and Image Reconstruction
To encode the positions and values of the nonzero coefficients representing a set
of nearby patches, we first quantize the sparse vectors using a uniform quantizer
[32]. Then, we utilize an adaptive strategy to preserve the positions and values of the
nonzero coefficients as follows:
For the “very similar” nearby patches, both the positions and values of the nonzero
coefficients are the same and these sparse vectors are already fused into one vector
α vs,i . Thus, only one sequence is required to store the position information and one
sequence is used to preserve the value information, as shown in Fig. 4.16a.
For the “not very similar” nearby patches, the positions of the nonzero coefficients
in
α
t
nvs,i
T
t1
are the same while their values are different. Thus, only one sequence
is needed to store the position information while another T sequences are employed
to preserve the value information, as shown in Fig. 4.16b. For the “different” nearby