reconstruction of compressed sampling data by wireless
sensors. This group sparsity has been exploited in dif-
ferent areas and also has different names in different
applications, including, for instance, joint sparsity,
29,30
simultaneous sparsity,
31,32
and group least absolute
shrinkage and selection operator (LASSO).
33
In SHM,
Yang and Nagarajaiah
34
proposed a data recovery
method of structural vibration responses resulting from
incomplete data. This method used the data from mul-
tiple sensors to form a low-rank matrix to recover the
missing data. In their approach, however, the group
sparsity of multiple sensors vibration data is not men-
tioned. In this article, using the special structure of the
Fourier basis, we develop an efficient algorithm based
on the augmented Lagrange multiplier (ALM) method.
In this algorithm, only fast Fourier transform (FFT) or
soft shrinkage is involved, making this method effi-
cient. Section ‘‘CS based on group sparse optimization
algorithm’’ gives the details of this algorithm.
CS based on group sparse optimization
algorithm
Principle of the method
Unlike traditional Nyquist uniform sampling, the CS
enables nonuniform low-rate random sampling as
shown in Figure 1. The Nyquist sampling needs to
sense N samples of a signal to avoid information loss
as shown in Figure 1(a); however, CS randomly senses
much fewer M N, as shown in Figure 1(b).
According to the CS theory, if the original signal is
sparse, CS is able to exactly recover it from far fewer
incoherent random measurements than what is
required by the Shannon sampling theorem.
Suppose that K sensors are implemented in the struc-
ture. The acceleration is measured at discrete time
t
j
, j = 1, ..., M by each sensor. Here, we assume the entire
time span of the samples is ½0, T ,andthesampletimeis
distributed uniformly, that is, t
j
= jT =M, j = 1, ..., M.By
collecting all the data together, we get the M 3K matrix,
U as follows
U =
u
11
u
12
u
1K
u
21
u
22
u
2K
.
.
.
.
.
.
.
.
.
.
.
.
u
N1
u
N2
u
NK
2
6
6
6
4
3
7
7
7
5
ð1Þ
where u
mk
is the data measured by the kth sensor at
time t
m
.
Then, the compressed sensing signal matrix, U is
usually an incomplete matrix. Let V = f(m, k) :
U
m, k
is availableg and P
O
: R
M3K
! R
M3K
is the zero
padding operator, that is, Y = P
O
U, as follows
y
m, k
=
u
m, k
, m, kðÞ2O
0, otherwise
ð2Þ
Then, the problem we are facing is how to calculate
the original signal matrix U from the given signal
Y = P
O
U 2 R
M3K
The signal matrix U can be represented as
U = CX ð3Þ
where C is a Fourier matrix
C =
e
i2pt
1
=T
e
i4pt
1
=T
e
i2Mpt
1
=T
e
i2pt
2
=T
e
i4pt
2
=T
e
i2Mpt
2
=T
.
.
.
.
.
.
.
.
.
.
.
.
e
i2pt
M
=T
e
i4pt
M
=T
e
i2Mpt
M
=T
2
6
6
6
4
3
7
7
7
5
ð4Þ
where X 2 C
M3K
are the Fourier coefficients of the
original signal that has only a few nonzero rows. This
representation gives us an alternative way to recover
the signal matrix U, which is so-called group sparsity.
Generally speaking, a matrix X is said to be group
sparse, if the entries of X are classified as many groups
and only a few groups are nonzero. In different appli-
cations, the classification may be different. More
Figure 1. Nyquist sampling and CS sampling: (a) Nyquist samples and (b) CS samples.
Bao et al. 3
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