A Note on Block-Sparse Signal Recovery with Coherent Tight Frames

This note discusses the recovery of signals from undersampled data in the situation that such signals are nearly block sparse in terms of an overcomplete and coherent tight frameD. By introducing the notion of blockD-restricted isometry property (D-RIP), we establish several sufficient conditions for the proposedmixed l 2 /l 1 -analysis method to guarantee stable recovery of nearly blocksparse signals in terms ofD. One of themain results of this note shows that if themeasurementmatrix satisfies the blockD-RIPwith constants δ k < 0.307, then the signals which are nearly block k-sparse in terms ofD can be stably recovered via mixed l 2 /l 1 -analysis in the presence of noise.


Introduction
Compressed sensing (CS) [1,2] has attracted great interests in a number of fields including information processing, electrical engineering, and statistics.In principle, CS theory states that it is possible to recover an unknown signal from considerably few information if the unknown signal has a sparse or nearly sparse representation in an orthonormal basis.However, a large number of applications in signal and image processing point to problems where signals are not sparse in an orthogonal basis but in an overcomplete and tight frame; see, for example, [3,4] and the reference therein.Examples include the reflected radar and sonar signals (Gabor frames) and the images with curves (curvelet frames).In such contexts, the signal f ∈ R  can be expressed as f = x, where  ∈ R × ( ≥ ) is a matrix whose  columns  1 , . . .,   form a tight frame and x ∈ R  is sparse or nearly sparse.Then one acquires f via the observed few linear measurements y = Φf + z, where Φ is a known  ×  matrix ( < ), y ∈ R  are available measurements, and z ∈ R  is a vector of measurements error.
There are two common ways to recover f based on y and Φ.One natural way is first solving an  1 -minimization problem: to find the sparse transform coefficients x; here B is a bounded set determined by the noise structure and then reconstructing the signal y by a synthesis operation; that is, ŷ = x.This is the so-called  1 -synthesis or synthesis based method [5,6].Since the entries of Φ are correlated when  is highly coherent, Φ may no longer satisfy the standard restricted isometry property (RIP) and the mutual incoherence property (MIP) which are commonly used in the standard CS framework.Therefore, it is not easy to study the theoretical performance of the  1 -synthesis method.
An alternative to  1 -synthesis is the  1 -analysis method, which finds the estimator f directly by solving the following  1 -minimization problem: It has been shown that there is a remarkable difference between the two methods despite their apparent similarity [5].To investigate the theoretical performance of  1 -analysis, Candès et al. [7] introduced the definition of -RIP: a measurement matrix Φ is said to satisfy the restricted isometry property adapted to  (abbreviated -RIP) with constant   if (3) holds for every vector c ∈ R  that is -sparse.Note that it is computationally difficult to verify the -RIP for a given deterministic matrix.But as discussed in [7], the matrices which satisfy the standard RIP requirements will also satisfy the -RIP requirements.It has been shown that  1 -analysis can recover a signal that is (nearly) sparse in terms of  with a small and zero error under various conditions on the -RIP, such as  2 < 0.08 or  7 < 0.6 used in [7],   < 0.307 used in [8], and so forth.
There are a growing number of practical scenarios in which the transform coefficient x is not only sparse, but the nonzero entries of x appear in some fixed blocks as well.Such block sparsity (in terms of ) naturally arises in MR imaging [9], color imaging [10], and source localization [11].Mathematically, such vector can be modeled x ∈ R  over a block index set I = { 1 , . . .,   } as follows: Here x[] denotes the th subblock of x and   is the block size for the th subblock.In this term, a vector x is called block sparse over index set I if x[] is nonzero for at most  indices .Now our objective of this note is to recover an unknown signal f ∈ R  that is nearly block -sparse in terms of  from a collection of  linear measurements y.We suggest the use of the following mixed  2 / 1 -analysis method to recover such signals: In this note, we will consider two types of bounded noise: To study the performance of such method, we will introduce the definition of the block -RIP of a measurement matrix, which is an extension of the block RIP introduced in [12].
Definition 1.Let  be an × matrix whose  columns form a tight frame for R  .An × measurement matrix Φ is said to satisfy the block -RIP over I = { 1 , . . .,   } with constant  |I if holds for every vector c ∈ R  that is block -sparse over I.
Note that we only require the -RIP for block-sparse signals, which are a certain subset of -sparse signals ( is the sum of the  largest values of   ).Thus, the block -RIP constant  |I is typically smaller than the -RIP constant   .With a slight abuse of notation, we still denote   by the block -RIP constant  |I without further specification.It is also computationally difficult to verify the block -RIP for a given deterministic matrix.But using the similar arguments as those in [12], one can easily verify that random matrices with Gaussian, sub-Gaussian entries satisfy the block -RIP with overwhelming probability.With this, we will establish some sufficient conditions for mixed  2 / 1 -analysis to guarantee stable recovery of the nearly block -sparse signal f in terms of .The main contribution of this note is to show that if the measurement matrix Φ has the block -RIP constants   < 0.307, then the signal f can be stably recovered via mixed  2 / 1 -analysis in the noise case.Note that the block -sparse signal f is also a standard -sparse signal.Thus, from [8], f can be stably recovered via  1 -analysis if Φ has the -RIP constants   < 0.307.Since the block -RIP constant   is typically smaller than the -RIP constant   , our obtained condition is not as stringent as that obtained by  1 -analysis.This reveals the advantage of mixed  2 / 1 -analysis over  1analysis in the block-sparse (in terms of ) setting.
In the next section, we give two key lemmas.In Section 3, some new block -RIP conditions for stable recovery of nearly block -sparse signals (in terms of ) via mixed  2 / 1analysis are presented.We conclude this paper with some discussions on the block -RIP bounds in Section 4.

Preliminaries
We begin with introducing the definition of block -restricted orthogonality constant, which is a natural extension of -restricted orthogonality constant (-ROC) introduced in [ for all c and c  such that c and c  are block -sparse and block   -sparse over block index set I, respectively.
For convenience, in the remainder of this note, we use  ,  , instead of  ,  |I , to represent the block -ROC whenever the confusion is not caused.Similar to the -RIP and the -ROC, the following monotone properties can be easily observed: From the parallelogram and the definition of block -RIP, it is easy to see that for all c and c  such that c and c  are block -sparse and block   -sparse.
In order to derive the main results of this note, we will introduce two useful inequalities related to  ,  and   .Lemma 3.For all nonnegative integers ,   ≤ , one has Proof.Since  is a tight frame, we have ‖c‖ 2 ≤ ‖c‖ 2 for any vector c.Then using (9) and the definition of  ,  , we can directly get the results.
Lemma 4. For any  ≥ 1 and positive integers ,   such that   is an integer, one has Proof.The proof is similar to the procedure of the proof of Lemma 1 in [13] and the proof of Lemma 2.6 in [8] which yields the result.
Remark 5. Note that Lemma 4 is in essence the square-root lifting inequality first introduced by Cai et al. in [13].It is obvious that when  =  and   = 1 for all  = 1, . . ., , Lemma 4 degenerates to Lemma 1 in [13].

Recovery via Mixed 𝑙 2 /𝑙 1 Analysis
In this section, we establish several block -RIP conditions for stable recovery of nearly block-sparse signals (in terms of ) via mixed  2 / 1 analysis approach (5).In the sequel, we denote by x [] the vector consisting of the  largest blocks over (ii)  ∞ noise: Proof.The proof will follow the ideas from [8,14].Let f = f + h be a solution of ( 5), where f is the original signal.Write Let  = {1, 2, . . ., } and let Ω be the block index set over the  blocks with largest  2 norm of  * f.We represent  as a concatenation of columnblocks [] of size  ×   : where   is the matrix  restricted to the column-blocks indexed by .We use  *  to denote (  ) * .Partition {1, 2, . . ., } into the following sets: Since f is a minimizer of ( 5), then we have This, along with (20), gives Similar to the consequence of -RIP (see [8]), we have where, for the second inequality, we have used (24).Let B = {z : ‖z‖ 2 ≤ }.Then, by the feasibility of f (also see [8]), we have Plugging (28) into the previously mentioned inequality and by a direct calculation, we get which yields (13).
Remark 11.When no noise is present and f is exactly block- sparse (in terms of ), f recovers f exactly.

Discussion
This note has presented some conditions of the measurement matrix Φ for the mixed  2 / 1 -analysis method to stably recover signals which are nearly block sparse in terms of a coherent tight frame .To the best of our knowledge, this is the first theoretical characterization of the proposed mixed  2 / 1analysis method in the compressed sensing framework.In a recent paper [15], Cai and Zhang established a sharp RIP condition for the standard sparse recovery via  1 minimization.They showed that in the noiseless case the RIP condition   < 1/3 is sharp for exact recovery of -sparse signals via  1 minimization.One may then argue whether the block -RIP condition   < 1/3 is also sharp for mixed  2 / 1analysis.Moreover, Davies and Gribonval [16] showed that it is impossible to recover certain standard -sparse signals via  1 minimization under RIP condition  2 < 2/ √ 2. It is interesting to investigate whether 2/ √ 2 is also an upper bound on the block -RIP constant  2 for mixed  2 / 1analysis.
with some modifications.Let c and c  be block -sparse and block  sparse, respectively.Without loss of generality, we assume that the indices of nonzero blocks of c  is {1, . . .,   }.For  = 1, . . .,   , let c   ∈ R  be a vector such that c   keeps the ( mod (  ))th, (( +   − 1) mod (  ))th nonzero blocks of c  and replaces other blocks by zero.By the Cauchy-Schwarz inequality, we have      ⟨Φc, Φc  ⟩ − ⟨c, c  ⟩ = √ ,  ‖c‖ 2      c      2 , Different choices of  1 and  2 can result in different sufficient conditions in Theorem 6.For instance, we can list some of them in Table1.Since the condition  2 < 4/9 implies that   + 1.25 , < 1, we get a sufficient condition that involves (5) that is,  2 < 4/9 is a sufficient recovery condition for mixed  2 / 1 -analysis(5).Remark 8. Though we have only considered bounded noise in Theorem 6, the conclusion, however, can be applied directly to Gaussian noise.