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3100 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 26, NO. 7, JULY 2017
Fig. 1. Comparison of grouping methods with intensity and manifold-structure based measurements for the same exemplar patch. (a) Left: Patches with
similar intensities are searched in a search window by intensity measurement. Right: Patches with similar manifold structures are searched in a search window
by manifold structure measurement. (b) Top: Patch group constructed by the patches (in the searching window of the left image of (a)) with similar intensities.
Bottom: Patch group constructed by the patches(in the searching window of the right image of (a)) with similar manifold structures.
Fig. 2. Comparison of intensity and manifold-structure based measurements in 1-dimensional vectorized patches. (a) Vectorized patches in the group
constructed by intensity measurement. (b) Measured by manifold structure measurement, the patches in (a) are distinguished more precisely.
elaborates the proposed method called structure-based low-
rank model with graph nuclear norm regularization (SLG).
Experimental results on AWGN and IN removal are reported
in Section IV. And in Section V, we summarize this paper.
II. G
ROUPING-BASED SPARSE REPRESENTATION
For utilizing the local and nonlocal priors, the low-rank and
sparse representation models [3]–[5], [7], [17], [18], [29], [32],
[36], [37] usually represent an image in an overlapped patch
space. The patch-based sparse coding methods [3], [7], [29],
[32], [36], [37] generally consider the overlapped patches of
size
√
n ×
√
n pixels, ordered lexicographically as column
vectors x
h
= R
h
x, h = 1, 2, ···, m,whereR
h
represents the
matrix extracting at the position h in an image x and m is the
number of patches extracted from the image. Each patch can be
sparsely represented as x
h
≈ Uα
h
by given a dictionary U ∈
R
n×P
, P > n and the sparse code α
h
. The image denoising
task can be formulated as the following minimization problem,
(
U, α
)
= arg min
α
h
m
h
y
h
− Uα
h
2
2
+ τ
m
h
α
h
1
(1)
where y
h
is the noisy version of x
h
,andτ is a regularization
parameter. The grouped sparse representation methods [3]–[5],
[17], [18], [36]–[37] group the patches according to intensity
similarity and stack the similar patches into a data matrix
Y =
y
1
, y
2
, ···, y
m
∈ R
n×m
, where the column
vectors y
h
, h = 1 ···m are patch vectors with noise.
We reconstruct the image ˆx by using the sparse coding set
{
α
h
}
m
h=1
as follows,
ˆx ≈ Uα
m
h=1
R
T
h
R
h
−1
m
h=1
R
T
h
Uα
h
. (2)
According to grouping-based sparse coding [38], the local and
nonlocal priors of the grouped patch data matrix are character-
ized simultaneously by solving the adaptive dictionary learning
problem. Dong et al. [3] applied eigenvalue decomposition
to the sparse coefficient matrix A,whereA = V
T
.Then
the sparse coding problem is reformulated as a low rank
approximation problem with a nuclear norm regularization,
(
U, A
)
= arg min
U,A
Y −UA
2
F
+ τ
A
∗
, (3)
where A = [α
1
, ···, α
m
] is the coefficient matrix, α
h
=
a
h,1
;a
h,2
;···;a
h,n
denotes the h
th
column of A,and
nuclear norm
A
∗
=
i
ω
i
(
A
)
represents the sum of the
singular values. The authors applied a spatially adaptive itera-
tive singular-value thresholding algorithm (SAIST) [3] to solve
this low rank approximation problem, i.e., Eq. (3). SAIST and
other grouping-based methods search best matched patches
as one group according to intensity similarity. In the next
section, we will discuss the shortcoming of these intensity-
based grouping method.