4 Reconstruction of Retinal OCT Images with Sparse Representation
79
Learn Multiscale
structural dictionary
Multiscale
structural
dictionary
D
A
Sparse representation
with the selected
dictionary
Training image
Noisy image
Dictionary
selection
Selected
subdictionary
Search for
Similar patches
Similar patches for
the processed one
Reconstruction
Denoised image
Sparse
representation
1,1
Mstr
D
K,1
Mstr
D
1,S
Mstr
D
K,S
Mstr
D
Fig. 4.4 The outline of the MSBTD algorithm
After finding the best representative subdictionary, as in work [37], we define the
objective function for the sparse representation of the noisy patch as follows:
( ˆ
α i , ˆ
β i ) arg min
α i ,β i
x i − D
A
i α i
2
2
+ λ 1 α i 0 + λ 2
α i − β i
0
,
(4.5)
where λ 1 and λ 2 are scalar Lagrange multipliers and α i is a sparse coefficient of the
patch x i . β i represents the sparse coefficient of a noiseless patch. The term α i 0 can
utilize the sparsity in a local patch, while the term
α i − β i
0
(via β i ) attempts to
exploit the non-local similarities in the images [38]. Since there is no noiseless patch
available, we adopt the average patch as the noiseless patch. The average patch is
computed by the average of J patches with the highest similarity to the processed
patch x i searched within a window. The problem (4.5) can be solved by an iterative
reweighted algorithm [37]. After obtaining the sparse coefficient ˆ
α i , we can compute
the related denoised patch and the reconstructed image with the dictionary D
A
i . The
outline of the whole denoising process is illustrated in Fig. 4.4.
4.3.1.3 Experimental Results
To verify the effectiveness of the proposed MSBTD method, its performance is
compared with that of four denoising approaches: Tikhonov [15], New SURE [39],
K-SVD [28], and BM3D [40]. In these experiments, the parameters of the proposed
MSBTD method are empirically selected and kept unchanged. Since most structures
in the SDOCT image lie on the horizontal direction, the patch and the search window
sizes are selected to be rectangle of size 3 × 20 and 40 × 60 pixels, respectively.
The number J of similar patches in each searching window is set to 20, while the
cluster number K is set to 70 in the k-means clustering. Before the multiscale learning process, the original image is upsampled two times, each by a factor 1.25 and
downsampled three times, each by a factor of 1.5625 to create the training images
of six scales. The parameters of the iterative reweighted algorithm for solving the
problem (4.5) are set to the default values in [37]. The parameters of the Tikhonov
method [15] are tuned to achieve its best results, while the parameters of the New
79
Learn Multiscale
structural dictionary
Multiscale
structural
dictionary
D
A
Sparse representation
with the selected
dictionary
Training image
Noisy image
Dictionary
selection
Selected
subdictionary
Search for
Similar patches
Similar patches for
the processed one
Reconstruction
Denoised image
Sparse
representation
1,1
Mstr
D
K,1
Mstr
D
1,S
Mstr
D
K,S
Mstr
D
Fig. 4.4 The outline of the MSBTD algorithm
After finding the best representative subdictionary, as in work [37], we define the
objective function for the sparse representation of the noisy patch as follows:
( ˆ
α i , ˆ
β i ) arg min
α i ,β i
x i − D
A
i α i
2
2
+ λ 1 α i 0 + λ 2
α i − β i
0
,
(4.5)
where λ 1 and λ 2 are scalar Lagrange multipliers and α i is a sparse coefficient of the
patch x i . β i represents the sparse coefficient of a noiseless patch. The term α i 0 can
utilize the sparsity in a local patch, while the term
α i − β i
0
(via β i ) attempts to
exploit the non-local similarities in the images [38]. Since there is no noiseless patch
available, we adopt the average patch as the noiseless patch. The average patch is
computed by the average of J patches with the highest similarity to the processed
patch x i searched within a window. The problem (4.5) can be solved by an iterative
reweighted algorithm [37]. After obtaining the sparse coefficient ˆ
α i , we can compute
the related denoised patch and the reconstructed image with the dictionary D
A
i . The
outline of the whole denoising process is illustrated in Fig. 4.4.
4.3.1.3 Experimental Results
To verify the effectiveness of the proposed MSBTD method, its performance is
compared with that of four denoising approaches: Tikhonov [15], New SURE [39],
K-SVD [28], and BM3D [40]. In these experiments, the parameters of the proposed
MSBTD method are empirically selected and kept unchanged. Since most structures
in the SDOCT image lie on the horizontal direction, the patch and the search window
sizes are selected to be rectangle of size 3 × 20 and 40 × 60 pixels, respectively.
The number J of similar patches in each searching window is set to 20, while the
cluster number K is set to 70 in the k-means clustering. Before the multiscale learning process, the original image is upsampled two times, each by a factor 1.25 and
downsampled three times, each by a factor of 1.5625 to create the training images
of six scales. The parameters of the iterative reweighted algorithm for solving the
problem (4.5) are set to the default values in [37]. The parameters of the Tikhonov
method [15] are tuned to achieve its best results, while the parameters of the New