Signal Processing 152 (2018) 69–78
Contents lists available at ScienceDirect
Signal Processing
journal homepage: www.elsevier.com/locate/sigpro
A novel multi-dictionary framework with global sensing matrix design
for compressed sensing
Jiajun Ding
a
, Donghai Bao
a
, Qingpei Wang
a
, Xiongxiong He
a , ∗
, Huang Bai
b
, Sheng Li
a
a
College of Information Engineering, Zhejiang University of Technology, Hangzhou, Zhejiang 310023, China
b
School of Information Science and Engineering, Hangzhou Normal University, Hangzhou, Zhejiang 311121, China
a r t i c l e i n f o
Article history:
Received 15 December 2017
Revised 1 May 2018
Accepted 16 May 2018
Available online 24 May 2018
Keywords:
Compressed sensing
Multi-dictionary framework
Global sensing matrix
Image processing
a b s t r a c t
In this paper, a new compressed sensing (CS) system is proposed to reduce computational burden of dic-
tionary learning and improve reconstruction accuracy. The proposed CS system employs a novel frame-
work which contains multiple dictionaries. In multi-dictionary framework, the whole training dataset is
divided into multiple subdatasets for optimizing multiple dictionaries. Dictionary learning process can be
accelerated due to the parallel computation and the reduction of training dataset size. Each dictionary
can get an image (called snapshot) independently with the same measurements in the image reconstruc-
tion process. These snapshots will be fused to be one image with averaging strategy. In order to keep
the measurement size of the proposed CS system same as that of traditional CS system and improve re-
construction accuracy, a new method of designing global sensing matrix for multi-dictionary framework
is also explored. Experiments demonstrate the effectiveness of new framework and the method to de-
sign global sensing matrix. Compared with other CS systems, the proposed CS system shows a superior
performance for real images.
© 2018 Elsevier B.V. All rights reserved.
1.
Introduction
Compressed sensing (CS) has attracted lots of attention since
its introduction [1–4] . At first, CS is a mathematical framework
that recovers sparse signals from the projection of original high-
dimensional signals accurately. That is, a sparse signal vector
x ∈ R
N × 1
can be recovered from the measurement vector y ∈ R
M × 1
( M N ) through a chosen sensing matrix ∈ R
M × N
:
y = x . (1)
Since M N , (1) is an under-determined problem, which may
have more than one solution. Therefore, the sparsity of original sig-
nal x can be an appropriate constraint in the CS system to make
the solution unique. For convince, the L
0
norm ( ·
0
) which is
used to count the number of non-zero entries is employed for
measuring the sparsity of vectors. However, when signal x is dense
in the real world, it should be represented sparsely by an over-
complete dictionary ∈ R
N × L
:
x = θ, (2)
∗
Corresponding author.
E-mail address: hxx@zjut.edu.cn (X. He).
where θ ∈ R
L × 1
is the sparse coefficient of signal x and the sparse
constraint is satisfied by θ. That is, the signal can be K -sparse if
x
0
≤ K or θ
0
≤ K .
There may exist noise or interference in practical. The approx-
imate representation can be obtained by minimizing the error of
sparse representation. Thus, a sparsifying dictionary for x can
be gotten through the formulation:
min
θ,
x − θ
2
2
. (3)
Let the matrix X = [ x
1
, ··· , x
W
] be a set of samples from a class
of signals to be considered [10] . can be obtained through the
formulation:
min
,
X −
2
F
, (4)
where = [ θ
1
, ··· , θ
W
] , is the aggregation of the sparse co-
efficients. The reasonable method to solve (3) or (4) is alterna-
tive iteration, i.e. sparse coding [5–7] and dictionary learning [8–
12]
. Among plenty of practical dictionary learning algorithms, K -
singular value decomposition (K-SVD) [9] is widely used, and or-
thogonal matching pursuit (OMP) is generally employed for sparse
coding [5,6] .
With (1) and (2) , the framework of the CS can be rewritten as:
y = θ = D θ, (5)
https://doi.org/10.1016/j.sigpro.2018.05.012
0165-1684/© 2018 Elsevier B.V. All rights reserved.
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