76740872构建块稀疏测量矩阵基于正交向量的方法

The Method for Constructing Block

Sparse Measurement Matrix

Based on Orthogonal Vectors

Ruizhen Zhao1,2,Zhou Qin1,2,and Jinhui Tang3

1Institute of Information Science,Beijing Jiaotong University,

Beijing100044,China

2Key Laboratory of Advanced Information Science and Network

Technology of Beijing,Beijing100044,China

3School of Computer Science,Nanjing University of Science and Technology,

Nanjing210094,China

pressive sensing is a new way of information processing

which recover the original signal through acquiring much fewer measure-

ments with a measurement matrix.The measurement matrix has an im-

portant effect in signal sampling and reconstruction algorithm.However,

there are two main problems in currently existing matrices:the difficulty

of hardware implementation and high computation complexity.In this

paper,we proposed a class of highly sparse and deterministic scram-

bled block measurement matrices based on orthogonal vectors(SBOV).

It could improve sensing efficiency and reduce computation complexity.

Those matrices constructed by the proposed method only need very lit-

tle memory space and they could be easily implemented in hardware due

to their simple entries.Some experiments show the better imaging per-

formance comparable to scrambled block Hadamard matrix(SBH)and

dense partial Hadamard matrix.SBOV matrices are simpler and sparser

than SBH matrix.

Keywords:compressive sensing,measurement matrix,orthogonal vec-

tors,block and sparse matrix.

1Introduction

In recent years,a great of attention has been paid to the new method of data acquisition compressive sensing(CS)[1],[2].The idea of CS is that signal sam-pling and data compression are implemented at the same time.Sparse signal or compressive signal can be recovered by few measurements.Due to the large amount of data in image and video signal,CS is usually used in imaging appli-cations when the measurement is very costly.Some image processing algorithms This work is supported by National Natural Science Foundation of China(61073079, 61103059)and the Fundamental Research Funds for the Central Universities(2011 JBM216).

W.Lin et al.(Eds.):PCM2012,LNCS7674,pp.872–879,2012.

c Springer-Verlag Berlin Heidelberg2012

The Method for Constructing Block Sparse Measurement Matrix873 based on CS are applied in many areas.For example,Fourier transform ma-trix is applied in MRI[3]and binary random matrix in single pixel camera[4]. In order to solve the algorithm problem of high dimension,Gan proposed the method based on block technology[5].It can improve computation speed when using block measurement matrix.Despite the above mentioned works,there still exists a huge gap between the CS theory and applications.Therefore,how to construct simple and efficient measurement matrices is an important problem.

The measurement matrix plays an important role in measurement acquir-ing and signal reconstruction and it is one of the most important parts in CS. In order to meet practical requirements,the measurement matrix are generally expected to have the following properties:(1)The strong incoherence between measurement and sparse basis;(2)the number of measurements for perfect recon-struction is close to the theo-retical bound;(3)fast sampling and reconstruction;

(4)low memory space and simple entries,easily hardware implement.

At present,there are many measurement matrices in theory,but few of them can satisfy the above four conditions.Thefirst family is dense random matrices, e.g.Gaussian i.i.d matrix[1].They can offer strong incoherence and optimal number of measurement.But they are impractical for image applications due to huge memory space and high computational complexity.The second family is deterministic structured matrices such as Toeplitz matrix[6].They havefixed structure and simple elements,which are easily implemented for hardware.But in order to obtain good performance,they need more measurements and smaller sparsity,which leads to high computational complexity.The third family is ma-trices generated from orthogonal matrices such as partial Hadamard matrix[7] and partial Fourier matrix.Their reconstruction effect and computational com-plexity are better.However they are still hard with hardware im-plementation due to the huge memory space.The fourth family is random sparse matrices, e.g.binary random sparse matrix[8]and the very sparse matrix[9].Due to random elements,they still need much memory space.Thefifth family is struc-tured sparse matrices,e.g.block polynomial deterministic matrix and scrambled block Hadamard(SBH)matrix[10],[11].They are sparse and structured but the size of its block is not too small.There are tens of nonzero in each column. Its computational efficiency and memory space are still a little high.

In order to overcome the disadvantages of above measurement matrices,we pro-posed a way of constructing simple and efficient measurement matrix:scram-bled block sparse structure based on orthogonal vectors(SBOV).The rows of a matrix which are orthogonal could improve reconstruction effect[12].Wefirstly constructed some low dimension orthogonal vectors with entries1or-1.Each vector is regarded as a block and a sub-matrix is constructed with the diago-nal block.Then we merged some sub-matrices.Finally the measurement matrix was constructed through randomly permuting the columns.Those matrices are highly sparse and easily implemented for hardware due to the advantages of low memory space,simple entries and fast computation.The most important ad-vantage is that those matrices could get better reconstruction results than SBH matrix.