Compressed sensing (CS) is a novel genre of sampling theory, which has attracted a large number of attention in different areas including applied mathematics, machine learning, pattern recognition, image processing, and so forth. The sparsity of signal is elementary precondition of compressed sensing. In general, one thinks over the model as follows:
where is an measurement matrix () and
is a vector of measurement errors. The aim is to reconstruct the unknown signalbased on and .
Now we all understand that the minimization method presents an efficient approach for recovery of the sparse signal in numerous scenarios. The minimization problem in this settings is
In the noisefree situation, we get . In the noisy situation, we can put  or , where stands for the conjugate transpose of the matrix . Now it is well known that the problem of sparse signal recovery has been well investigated in the framework of the mutual coherence property introduced in . Let
It has been shown that a sparse signal can been reconstructed by minimization with a small or zero error under some appropriate conditions regarding MIP    . In order to further enhance the reconstruction performance, Yin et al.  has recently proposed the approach (i.e., minimization method) as follows:
Additionally, Yin et al. conducted simulations to show that the method (1.4) behaves better than the method (1.2) in recovering sparse signals. Based on this fact, numerous researches    on the minimization approach have been developed. Besides, for recovering , the researchers   proposed minimization method:
However, in practical applications, there exist signals which have special structure form, where the nonzero coefficients appear in some blocks. Such structural signal we called block spare signal in this paper. Such structural sparse signals commonly arise in all kinds of applications, e.g. foetal electrocardiogram (FECG) , motion segmentation, color image , and reconstruction of multi-band signals  . Without loss of generality, suppose that there exist blocks with block size in . Then, can be expressed as
where represents the th block of . We call a vector block -sparse signal if has at most nonzero blocks, i.e., Therefore, the measurement matrix can also be described as
where and respectively stand for the th column vector and th sub-block matrix of .
In this paper, we propose the following minimization to recover block sparse signal:
In this paper, we study the block mutual coherence conditions for the stable recovery of signals with blocks structure from (1.6) via minimization in noise case. Sufficient conditions for stable signal reconstruction by minimization are established. Moreover, we also gain upper bound estimation of error concerning the recovery of block sparse signal. As far as we know, this is the first block mutual coherence based sufficient condition of stably reconstructing via solving (1.8).
The remainder of the paper is organized as follows. In, Section 2, we present some notations and lemmas that will be used. The main theoretical results and their proofs are given in Section 3. Finally, the conclusion is summarized in Section 4.
In this section, we primarily present several lemmas to prove our main results. Before giving these lemmas, we first of all explain some symbols in this paper.
Notations: denotes block indices, is the complement of in . For any vector , denote to imply that maintains the blocks indexed by of and displaces other blocks by zero. represents the block support of . In addition, we often assume that , where is the solution of (1.8) and is the signal to be recovered.
(block mutual coherence) Given matrix , we define its block mutual coherence as
(, Lemma 3) For any block -sparse vector , we have
Recollect that . Since is a minimizer of (1.8), we get
By the reverse triangular inequality of , we get
Note that is block -sparse and , then
which brings about the result. ∎
3 Main results
With the preparations provided in Section , we establish the main results in this section-block mutual coherence conditions for the stable reconstruction of block -sparse signals. We will reveal that the measurement matrix satisfies the block mutual coherence property with , then every block -sparse signal can be stably reconstructed via the minimization method in presence of noise. We first think about stable reconstruction of block -sparse signals with -error.
We then consider stable reconstructing of block -sparse signals with error in the bounded set .
Let be noisy measurement of a signal with . If the block -sparse signal obeys the block mutual coherence property with , then the solution of (1.8) with fulfills
Proof of Theorem 3.1.
Due to the feasibility of , we get
Notice that . It follows from the facts , for , and (2.2) that
On the other hand, by (2.2), we get
Combining with (3) and , it implies
Then, one can easily check that
Because of the fact , for , we get
where (a) follows from (2.3), and (b) is due to the Cauchy-Schwarz inequality.
The equation above can be adapted as
Therefore, due to , we get
Accordingly, by Quadratic Formula, we get
where (a) is from the fact for any nonnegative constants and , and (b) is because both and are monotonically reducing when .
The above equation can be recast as
Owing to the condition of Theorem 3.1, , thereby,
By utilizing Quadratic Formula, we obtain
where (a) is from the fact that both and are monotonically descending when .
The above equation can be reworded as
From , and when , hence
Proof of Theorem 3.2.
Notice that from the first portion of the proof of Theorem 3.1, we get
Employing the fact , where , we have
which combines with (3.9) and the condition , it leads to
Thus, it is easy to check that
By (3), we get
By (2.3), the fact that and Cauchy-Schwarz inequality, we get
which combines with (3.11), it implies that