Sparse kronecker pascal measurement matrices for compressive imaging
Jiang et al. Journal of the European Optical
Society-Rapid Publications
Sparse kronecker pascal measurement matrices for compressive imaging
Yilin Jiang 0
Qi Tong 0
Haiyan Wang 0
Qingbo Ji 0
0 College of Information and Communication Engineering, Harbin Engineering University , Harbin 150001 , China
Background: The construction of measurement matrix becomes a focus in compressed sensing (CS) theory. Although random matrices have been theoretically and practically shown to reconstruct signals, it is still necessary to study the more promising deterministic measurement matrix. Methods: In this paper, a new method to construct a simple and efficient deterministic measurement matrix, sparse kronecker pascal (SKP) measurement matrix, is proposed, which is based on the kronecker product and the pascal matrix. Results: Simulation results show that the reconstruction performance of the SKP measurement matrices is superior to that of the random Gaussian measurement matrices and random Bernoulli measurement matrices. Conclusions: The SKP measurement matrix can be applied to reconstruct high-dimensional signals such as natural images. And the reconstruction performance of the SKP measurement matrix with a proper pascal matrix outperforms the random measurement matrices.
Compressed sensing; Deterministic measurement matrix; Kronecker product; Pascal matrix
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Background
Compressed sensing (CS) theory is a novel sampling
scheme, which indicates that a sparse signal can be
recovered from much fewer samples than conventional
method [1, 2]. The sampling and the compression
procedure are completed by the linear projection in CS. In
matrix notation, it can be expressed as
where x ∈ ℝN is the original signal, Φ is an M × N(M ≪ N)
measurement matrix, y ∈ ℝM is the measurement vector. x
is said to be K-sparse if ‖x‖0 ≤ K. CS theory asserts that if
the measurement matrix Φ satisfies some conditions, the
signal x can be recovered from measurements y without
distortion.
The emergence of CS provides a new inspiration for
optical imaging. Actually most of the nature images are
compressible in terms of some sparsity basis, such as
Discrete cosine transform (DCT) and Discrete wavelet
transform (DWT). The compressibility of the real-word
images shows the potential for optical compressive
imaging. In the past few years, CS technique has made
great progress in many research fields, which include
terahertz compressive imaging [3], spectral imaging [4],
single pixel imaging [5] and infrared imaging [6]. Some
optical imaging applications have been implemented in
specific physical experiments.
Measurement matrix construction is a crucial problem
in CS. The measurements obtained by measurement
matrix are related to whether the signal can be accurately
reconstructed. If there is enough information within the
measurements, the signal can be recovered with high
probability. Random measurement matrices are proved to
have the merit of universality but suffer from several
shortcomings. Firstly, random measurement matrices
consume lots of storage resources. Secondly, there is no
feasible algorithm to verify whether the random matrix
satisfies the requirement as a measurement matrix [7, 8].
The research on deterministic sampling can be tracked
back to the binary matrices via polynomials over finite
field [9]. The deterministic measurement matrix has the
superiority in physical implementation and the advantage
of saving storage space. Therefore, many researches on
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the deterministic measurement matrix construction have
been carried out. Lu introduced a construction of ternary
matrices with small coherence [10]. Yao presented a novel
simple and efficient measurement matrix named
incoherence rotated chaotic matrix [11]. Huang proposed a
symmetric Toeplitz measurement matrix [12]. Zhao
introduced a deterministic complex measurement matrix to
sample the signals in the single pixel imaging [13].
In this paper, we propose a new construction method
of deterministic measurement matrix, termed sparse
kronecker pascal (SKP) measurement matrix. The SKP
measurement matrix combines the properties of the
kronecker product and the pascal matrix. It is suitable
for the reconstruction of natural images, which are
usually high-dimensional signals. Simulations and analyses
confirm that the SKP measurement matrices can
reconstruct the natural images with a better performance.
Methods
The SKP measurement matrix construction
In mathematics, the kronecker product is an operation
on two matrices of arbitrary size resulting in a block
matrix [14].
(...truncated)