ZHANG et al.: HYPERSPECTRAL IMAGE RESTORATION USING LOW-RANK MATRIX RECOVERY 4731
Fig. 1. Formulation of the low-rank matrix from an HSI patch.
Differing from the patch matching algorithm used in natural
image/video restoration [18]–[20], which exploits the redun-
dancy in the spatial and temporal domains, the high correlation
of the spectral information is adopted to formulate the low-rank
components here. Fig. 1 shows the matrix representation of a
clean HSI patch. Assuming that we have an original HSI cube
u, denoted by (a) in Fig. 1, the size of u is M × N × B,in
which M represents the width of the image, N stands for the
height of the image, and B is the number of spectral bands.
Select a subcube p
i,j
of size q × q × B centered at the spatial
position (i, j). If we represent the kth band of the subcube
p
i,j
as a vector p
i,j,k
∈ R
q
2
by lexicographically ordering
all its columns, we obtain a 2-D matrix P
i,j
of size q
2
× B
as follows:
P
i,j
=(p
i,j,1
, p
i,j,2
···p
i,j,β
). (5)
Next, we investigate the low-rank property of P
i,j
.
The low-rank property of an HSI can be exploited from
the perspective of a linear spectral mixing model [27], [28].
As known, there exist high correlations among the spectral
signatures (rows of P
i,j
), as each spectral signature can be
represented by a linear combination of a small number of pure
spectral endmembers, which is known as the linear spectral
mixing model. Supposing the upper bound of the number of
pure spectral endmembers for the HSI patch is r, then P
i,j
can be decomposed as P
i,j
= MH
T
, where H ∈ R
B×r
+
is the
matrix whose columns contain pure spectral endmembers and
M ∈ R
q
2
×r
+
is the abundance matrix. As the upper bound value
of the number of endmembers r is usually relatively small,
then the rank of P
i,j
is bounded, i.e., rank(P
i,j
) ≤ r, which
suggests the low-rank property of the matrix P
i,j
.
III. LRMR-B
ASED HSI RESTORATION METHOD
A. HSI Degradation Model
We assume that the HSI is corrupted by four kinds of noise:
Gaussian noise, impulse noise, dead pixels or lines, and stripes.
The noise degradation model of the HSI can be written as
d = u + s + n (6)
where u is the clean HSI, as shown in Fig. 1(a); d is the
observed degradation image; s denotes the mixture of impulse
noise, dead pixels or lines, and stripes; and n stands for the
Gaussian noise. In t his degradation model, the images d, u, s,
and n are of the same size of M × N × B.
B. LRMR-Based HSI Restoration Model
In this paper, we divide the HSI into patches and restore
each patch sequentially, which can effectively preserve the local
details [29], [30]. The size of each patch is q × q × B.Fora
patch centered at the pixel (i, j), the image degradation model
is expressed as follows:
d
i,j
= u
i,j
+ s
i,j
+ n
i,j
(7)
where d
i,j
, u
i,j
, s
i,j
, and n
i,j
are the 3-D subcubes of d, u,
s, and n, respectively. Rearrange d
i,j
, u
i,j
, s
i,j
, and n
i,j
into
2-D matrices according to the process shown in Fig. 1, and
denote them as D
i,j
, L
i,j
, S
i,j
, and N
i,j
, respectively. The
corresponding image degradation model is
D
i,j
= L
i,j
+ S
i,j
+ N
i,j
(8)
where the matrix L
i,j
represents the clean HSI patch, the matrix
S
i,j
is denoted as the mixture of impulse noise, dead lines,
and stripes, and the noise matrix N
i,j
is the representation of
the Gaussian noise. With the low-rank property of the clean
HSI, the matrix L
i,j
is low rank. Clearly, as the impulse noise,
dead pixels or lines, and stripes only corrupt some parts or
some bands of the HSI, therefore, the matrix S
i,j
is sparse. The
Gaussian noise matrix N
i,j
is of a small level and impacts all
the pixels.
With the similarity between the HSI degradation model in
(8) and the LRMR measurement model, it is natural for us to
apply sparse and low-rank matrix decomposition to the noisy
matrix D
i,j
and obtain the clean HSI patch L
i,j
. Thus, the
restoration process for the patch centered at (i, j) of the HSI
is accomplished. After processing all the patches of the HSI,
we synthesize the restored image from these clean patches.
In this paper, the image patches are sampled with overlapping
regions, and most of the pixels are covered by several restored
patches. The value of each pixel in the HSI is then determined
by taking the average of the restored patches at this pixel, which
will suppress the possible artifacts in the neighborhood of the
boundaries of patches.
C. GoDec Algorithm
In recent years, many LRMR optimization algorithms have
been proposed. For example, Lin et al. [31] developed and com-
pared two complementary approaches for solving model (2).
The first is an accelerated proximal gradient algorithm directly
applied to the primal, and the s econd is a gradient algorithm
applied to the dual problem. Yuan and Yang [32] proposed
the alternating direction method for solving problem (2), by
taking full exploitation to the high-level separable structure of
the convex relaxation problem. Recently, Lin et al. [33] applied
the method of ALM to solve this convex program. In [26], the
authors proved that the methods mentioned earlier also fit the
revised model (3).
In this paper, we adopt the GoDec algorithm proposed by
Zhou and Tao in [21] to solve the low-rank HSI patch, which
can remove the Gaussian noise more effectively. Before solving
the optimization problem (4) by the GoDec approach, the upper
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