^{1, 2}

^{1}

^{3}

^{1, 3}

^{1}

^{2}

^{3}

The paper discusses the relationship between the null space property (NSP) and the

Compressed sensing has been drawing extensive and hot attention as soon as it was proposed since 2006 [

Candès et al. gave the following

We see that the

The null space property (NSP) was introduced when one checked the equivalence of the solutions between the

For any set

The thoughts of the null space have appeared since one researched the best approximation of

Given a matrix

This theorem not only gives an existence condition of the solution of the

As for RIP, the null space property is a necessary and sufficient condition to exactly reconstruct the signal using the

This paper focuses on the different types of null space property on the

In this section, we will discuss the

For any set

Definition

Given a matrix

The proof of Theorem

However, in more realistic scenarios, we can only claim that the vectors are close to sparse vectors but no absolutely sparse ones. In such cases, we would like to recover a vector

For any set

Formula (

Obviously, Definition

For any set

We defer the proof of this theorem to Appendix. Under this theorem, we give the main stability result as follows.

Suppose that a matrix

Take

By Corollary

Suppose that a matrix

Compared with Theorem

Suppose that a matrix

This result can be derived from the following inequality [

Under the condition of Corollary

In realistic situations, it is also inconceivable to measure a signal

For any set

We see that Definition

For any set

The proof of Theorem

Suppose that a matrix

Noting that

When

Using inequality

Inequality (

The rest of this section will focus on another robust null space property, that is, the

Given

Obviously, when

Given

We defer the proof of this theorem to Appendix. In the proof, we will see that the condition

Given

In Corollary

In this section, we will discuss two extensions of the

The techniques above hold for signals which are sparse in the standard coordinate basis or sparse with respect to some other orthonormal basis. However, in practical examples, there are numerous signals which are sparse in an overcomplete dictionary

The

Empirical studies show that the

In this subsection, we only propose the

Given a frame

Fix a dictionary

This null space property is abbreviated to

The following theorem asserts that the null space property of a frame

Fix a dictionary

Combining the proof of Theorem 4.2 in [

On the null space property of the

The conventional compressed sensing only considers the sparsity that the signal

To recover a block sparse signal, similar to the standard

Based on the previous discussion of the performance of the

In this subsection, we will extend the

For any set

Inequality (

Using Definition

Given a matrix

We defer the proof of Theorem

So far, we only extend the standard

In this section, we provide the proofs of Theorems

Given a set

Using the following inequalities, we can easily obtain inequality (

For any set

Conversely, let us assume that the matrix

For any set

The proof of sufficiency is similar to that of Theorem

Let us now assume that the matrix

In order to prove Theorem

Now we give the proof of Theorem

Let us first assume that every block

Conversely, let us assume that the block null space property relative to

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors would like to thank the anonymous reviewers for their insightful comments and valuable suggestions. This work was supported by the National Natural Science Foundation of China (NSFC) under Grant no. 11131006 and by a Marie Curie International Research Staff Exchange Scheme Fellowship within the 7th European Community Framework Programme, via EYE2E (269118) and LIVCODE (295151), and in part by the Science Research Project of Ningxia Higher Education Institutions under Grant no. NGY20140147.