Compressive sensing is a sampling method which provides a new approach to efficient signal compression and recovery by exploiting the fact that a sparse signal can be suitably reconstructed from very few measurements. One of the most concerns in compressive sensing is the construction of the sensing matrices. While random sensing matrices have been widely studied, only a few deterministic sensing matrices have been considered. These matrices are highly desirable on structure which allows fast implementation with reduced storage requirements. In this paper, a survey of deterministic sensing matrices for compressive sensing is presented. We introduce a basic problem in compressive sensing and some disadvantage of the random sensing matrices. Some recent results on construction of the deterministic sensing matrices are discussed.

Consider a scenario that

For an

The existence and uniqueness of the solution can be guaranteed as soon as the measurement matrix

There are two common ways to solve these problems. First, we can exactly recover

However, in order to ensure unique and stable reconstruction, the sensing matrix

Recently, several deterministic sensing matrices have been proposed. We can classify them into two categories. First are those matrices which are based on coherence [

The rest of this paper is organized as follows. Section

Recall that

If

In practice, the original signals may be affected by noise, so the recovered signals are not exact, and rather they are almost sparse instead. Hence, some modified criteria were proposed as follows.

Suppose that

A new result on RIC was proposed by Candès as follows.

Given

Several inequalities in terms of RIC have been discovered, such as

Random matrices are easy to construct and ensure high probability reconstruction. However, they also have many drawbacks. First, storing random matrices requires a lot of storage. Second, there is no efficient algorithm verifying RIP condition. So far, it is not a good approach because of its lack of efficiency. The recovery problems may be difficult when the dimension of the signal becomes large, and we have to construct a measurement matrix that satisfies RIP with a small

A discrete chirp of length

Most of the sensing chirp matrices admit a fast reconstruction algorithm which reduces the complexity to

The second-order Reed-Muller code is given as follows:

Denote

An example of binary matrices formed by BCH code is given as follows. Let

In [

An

An

These constructions allow recovery methods for which expected performance is sublinear in

In [

Let

There are several deterministic constructions of sensing matrices via algebraic curves over finite fields called algebraic geometry codes [

In [

A bipartite graph with

A bipartite graph

They constructed a large class of binary and sparse matrices satisfying a different form of the RIP property called RIP-

Consider any

This approach utilizes sparse matrices interpreted as adjacency matrices of sparsity to recover an approximation to the original signal. The new property RIP-

In this paper, various deterministic sensing matrices have been investigated and presented in terms of coherence and RIP. The advantages of these matrices, in addition to their deterministic constructions, are the simplicity in sampling and recovery process as well as small storage requirement. It can be possible to make further improvement in both reconstruction efficiency and accuracy using these deterministic matrices in compressive sensing, particularly when some

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

This research was partly supported by Mid-career Researcher Program through NRF Grant funded by MEST, Korea (No. 2013-030059), and by the MSIP, Korea, in the ICT R&D Program 2013 (KCA-2012-12-911-01-107).

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