74
L. Fang and S. Li
Denoising, interpolation and compression are well-known reconstruction problems in the image processing field [6]. During the past decades, various models have
been proposed to reconstruct high quality OCT images for many applications [3,
4, 7–14]. Classical reconstruction methods often design a smoothness priori based
model (e.g., anisotropic filtering, Tikhonov filtering [15], and total variation [7]), and
reconstruct the image in spatial domain. Some recent approaches transform the input
image into another domain (e.g., using the discrete cosine transform (DCT) [16],
wavelet transformation [17], and curvelet transformation [18]). Although the transform based methods can provide a better reconstruction performance compared with
the spatial domain methods, the transform based methods (e.g. DCT and wavelet)
are often built on a fixed mathematical model and may have limited adaptability [19]
for representing structures in ocular OCT volumes.
Recently, motivated by the sparse coding mechanism of mammalian vision system
[20], the sparse representation theory has been demonstrated to be a very powerful
tool for numerous image processing applications [2, 5, 21–24]. The sparse representation can decompose the input image as a linear combination of basis functions
(also called as the atoms) selected from the dictionary. The dictionary atoms can be
trained from a number of sampled images similar to the input image [25], and so
can be more adaptive for representing the input image. Several very recent works
have also applied the sparse representation to OCT image reconstruction problems
[2, 5, 9, 11–13, 26, 27]. Unlike the 2-D natural image, the 3-D OCT image has more
complex spatial-temporal structures. For example, the 3-D OCT image has many
types and scales of pathology structures (e.g., different layers and drusen) in the
spatial domain, while still has high correlations in the temporal domain. Therefore,
according to the special structures of 3-D OCT image, this chapter will introduce
three sparse representation models and apply them to the OCT image denoising,
interpolation and compression.
The rest of this chapter is organized as follows. Section 4.2 briefly reviews the
traditional sparse reconstruction model and how to utilize it for the reconstruction
problems. Considering the OCT image structures, we introduce three our proposed
sparsity based methods and apply them for the OCT image denoising, interpolation
and compression problems in Sect. 4.3. Section 4.4 concludes this chapter.
4.2 Sparse Representation for Image Reconstruction
Given an input 2-D image of size N × M, most sparse representation methods first
divide this image into ϒ overlapping (for image denoising and interpolation [28,
29]) or non-overlapping (for image compression [30–32]) patches X i ∈ R
n×m
, i
1, 2 . . . , ϒ, n < N and m < M. Here, i represents a particular patch corresponding
to the lateral and axial position of its center in a 2D image. The vector form of each
patch X i is represented as x i ∈ R
q×1 (q n×m), obtained by lexicographic ordering.
The sparse representation can represent the input patch x i as a linear combination of
a few atoms selected from a dictionary (D ∈ R
q×z
, q < z), as follows
L. Fang and S. Li
Denoising, interpolation and compression are well-known reconstruction problems in the image processing field [6]. During the past decades, various models have
been proposed to reconstruct high quality OCT images for many applications [3,
4, 7–14]. Classical reconstruction methods often design a smoothness priori based
model (e.g., anisotropic filtering, Tikhonov filtering [15], and total variation [7]), and
reconstruct the image in spatial domain. Some recent approaches transform the input
image into another domain (e.g., using the discrete cosine transform (DCT) [16],
wavelet transformation [17], and curvelet transformation [18]). Although the transform based methods can provide a better reconstruction performance compared with
the spatial domain methods, the transform based methods (e.g. DCT and wavelet)
are often built on a fixed mathematical model and may have limited adaptability [19]
for representing structures in ocular OCT volumes.
Recently, motivated by the sparse coding mechanism of mammalian vision system
[20], the sparse representation theory has been demonstrated to be a very powerful
tool for numerous image processing applications [2, 5, 21–24]. The sparse representation can decompose the input image as a linear combination of basis functions
(also called as the atoms) selected from the dictionary. The dictionary atoms can be
trained from a number of sampled images similar to the input image [25], and so
can be more adaptive for representing the input image. Several very recent works
have also applied the sparse representation to OCT image reconstruction problems
[2, 5, 9, 11–13, 26, 27]. Unlike the 2-D natural image, the 3-D OCT image has more
complex spatial-temporal structures. For example, the 3-D OCT image has many
types and scales of pathology structures (e.g., different layers and drusen) in the
spatial domain, while still has high correlations in the temporal domain. Therefore,
according to the special structures of 3-D OCT image, this chapter will introduce
three sparse representation models and apply them to the OCT image denoising,
interpolation and compression.
The rest of this chapter is organized as follows. Section 4.2 briefly reviews the
traditional sparse reconstruction model and how to utilize it for the reconstruction
problems. Considering the OCT image structures, we introduce three our proposed
sparsity based methods and apply them for the OCT image denoising, interpolation
and compression problems in Sect. 4.3. Section 4.4 concludes this chapter.
4.2 Sparse Representation for Image Reconstruction
Given an input 2-D image of size N × M, most sparse representation methods first
divide this image into ϒ overlapping (for image denoising and interpolation [28,
29]) or non-overlapping (for image compression [30–32]) patches X i ∈ R
n×m
, i
1, 2 . . . , ϒ, n < N and m < M. Here, i represents a particular patch corresponding
to the lateral and axial position of its center in a 2D image. The vector form of each
patch X i is represented as x i ∈ R
q×1 (q n×m), obtained by lexicographic ordering.
The sparse representation can represent the input patch x i as a linear combination of
a few atoms selected from a dictionary (D ∈ R
q×z
, q < z), as follows