4 Reconstruction of Retinal OCT Images with Sparse Representation
75
x i Dα i
(4.1)
where the α i ∈ R
z×1 is the sparse coefficient vector for the patch x i and the dictionary
D consists of z atoms {d j }
z
j1 .
For a denoising problem, the sparse model assumes that the clean OCT image
patch can be well decomposed on a few atoms selected from the dictionary, whereas
the noise cannot be represented by the dictionary. Therefore, the sparsity based
denoising model can be formulated as follows [28]
ˆ
α i arg min
α i
α i 0 subject tox i − Dα i
2
2 ≤ ε,
(4.2)
where α i 0 is the 0 -norm counting the number of non-zero coefficients in α i and
ε q(Cσ ) is the error tolerance. C is a constant, and σ is the standard deviation
of noise in the input patch x i , which can be estimated by the approach in [33]. The
problem (4.2) can be rewritten by considering the sparsity level constraint,
ˆ
α i arg min
α i
x i − Dα i
2
2 subject to α i 0 ≤ T,
(4.3)
where T is the sparsity level, representing the maximum number of non-zero coefficients in α i . In both (4.2) and (4.3), two fundamental problems need to be addressed:
(1) design the dictionary D to best represent the patch x i and (2) obtain the sparse
coefficient α i . For the first problem, machine learning based methods (e.g., K-SVD
[25] and recursive least squares dictionary learning algorithm [32]) are popularly
utilized to learn the dictionary D from a large number of training samples similar to
the test image. For the second problem, the greedy pursuit (e.g., orthogonal matching
pursuit (OMP) [34]) can be used to obtain an approximate solution. After obtaining
the dictionary D and sparse coefficient ˆ
α i of each patch, we can use D ˆ
α i to reconstruct the related patch and all the reconstructed patches are returned to their original
positions to generate the denoised image.
For the interpolation problem, we first denote an original high resolution image
as Y H ∈ R
N ×M , the decimation operator as S, and the corresponding low resolution
image as Y L SY H ∈ R
(N / S)×(M / S) . Given the observed low resolution image Y L ,
the objective of the image interpolation is to obtain the estimated high resolution
ˆ
Y H , such that ˆ
Y H ≈ Y H . In [29], Yang et al., extended the above sparse model
to interpolation problem by jointly learning two dictionaries D L and D H in the
low resolution feature space χ L and high resolution feature space χ H . This method
assumes that the sparse coefficient of low resolution image patch x L ∈ χ L on D L is
the same as that of the high resolution image patch x H ∈ χ H with respect to D H .
Therefore, given the observed x L , we can seek its sparse coefficient and reconstruct
the high resolution image patch ˆ
x H as well as the corresponding image ˆ
Y H with D H .
For the compression problem, we firstly subtract the mean of each image patch
X i and denote the patch as X
s
i . Then, we solve the (4.2) or (4.3) to obtain the sparse
coefficient α
s
i of each image patch X
s
i . The obtained α
s
i is very sparse which means
that only a very small number of non-zero coefficients exist in α
s
i . Then, the com-
75
x i Dα i
(4.1)
where the α i ∈ R
z×1 is the sparse coefficient vector for the patch x i and the dictionary
D consists of z atoms {d j }
z
j1 .
For a denoising problem, the sparse model assumes that the clean OCT image
patch can be well decomposed on a few atoms selected from the dictionary, whereas
the noise cannot be represented by the dictionary. Therefore, the sparsity based
denoising model can be formulated as follows [28]
ˆ
α i arg min
α i
α i 0 subject tox i − Dα i
2
2 ≤ ε,
(4.2)
where α i 0 is the 0 -norm counting the number of non-zero coefficients in α i and
ε q(Cσ ) is the error tolerance. C is a constant, and σ is the standard deviation
of noise in the input patch x i , which can be estimated by the approach in [33]. The
problem (4.2) can be rewritten by considering the sparsity level constraint,
ˆ
α i arg min
α i
x i − Dα i
2
2 subject to α i 0 ≤ T,
(4.3)
where T is the sparsity level, representing the maximum number of non-zero coefficients in α i . In both (4.2) and (4.3), two fundamental problems need to be addressed:
(1) design the dictionary D to best represent the patch x i and (2) obtain the sparse
coefficient α i . For the first problem, machine learning based methods (e.g., K-SVD
[25] and recursive least squares dictionary learning algorithm [32]) are popularly
utilized to learn the dictionary D from a large number of training samples similar to
the test image. For the second problem, the greedy pursuit (e.g., orthogonal matching
pursuit (OMP) [34]) can be used to obtain an approximate solution. After obtaining
the dictionary D and sparse coefficient ˆ
α i of each patch, we can use D ˆ
α i to reconstruct the related patch and all the reconstructed patches are returned to their original
positions to generate the denoised image.
For the interpolation problem, we first denote an original high resolution image
as Y H ∈ R
N ×M , the decimation operator as S, and the corresponding low resolution
image as Y L SY H ∈ R
(N / S)×(M / S) . Given the observed low resolution image Y L ,
the objective of the image interpolation is to obtain the estimated high resolution
ˆ
Y H , such that ˆ
Y H ≈ Y H . In [29], Yang et al., extended the above sparse model
to interpolation problem by jointly learning two dictionaries D L and D H in the
low resolution feature space χ L and high resolution feature space χ H . This method
assumes that the sparse coefficient of low resolution image patch x L ∈ χ L on D L is
the same as that of the high resolution image patch x H ∈ χ H with respect to D H .
Therefore, given the observed x L , we can seek its sparse coefficient and reconstruct
the high resolution image patch ˆ
x H as well as the corresponding image ˆ
Y H with D H .
For the compression problem, we firstly subtract the mean of each image patch
X i and denote the patch as X
s
i . Then, we solve the (4.2) or (4.3) to obtain the sparse
coefficient α
s
i of each image patch X
s
i . The obtained α
s
i is very sparse which means
that only a very small number of non-zero coefficients exist in α
s
i . Then, the com-