正交梯度测量矩阵优化方法
需积分: 13 38 浏览量
2021-04-29
19:48:19
上传
评论
收藏 2.31MB PDF 举报
IET Image Processing
Research Article
Orthogonal gradient measurement matrix
optimisation method
ISSN 1751-9659
Received on 17th August 2017
Revised 6th February 2018
Accepted on 25th April 2018
E-First on 26th June 2018
doi: 10.1049/iet-ipr.2017.0888
www.ietdl.org
Jinfeng Pan
1
, Jin Shen
1
, Mingliang Gao
1
, Liju Yin
1
, Faying Liu
1
, Guofeng Zou
1
1
School of Electrical and Electronic Engineering, Shandong University of Technology, Zibo Shandong, People's Republic of China
E-mail: pjfbysj@163.com
Abstract: The optimisation of measurement matrix that is within the compressive sensing framework is considered in this study.
Based on the fact that an information factor with smaller mutual coherence performs better, the gradient measurement matrix
optimisation method is improved by an orthogonal search direction revision factor. This algorithm updates the approximation of
ideal Gram matrix of information operator and the measurement matrix alternatingly. Using measurement matrix and sparse
basis to represent the Gram matrix, the measurement matrix is optimised by the gradient algorithm, in which an orthogonal
gradient search direction revision factor is proposed and utilised to further improve the performance of measurement matrix.
This orthogonal factor is computed by the Cayley transform of a real skew symmetric matrix that is related to the gradient and
the measurement matrix. Results of several experiments show that compared with the initial random matrix, the optimised
measurement matrix can lead to better signal reconstruction quality.
1 Introduction
Compressive sensing (CS) is an attracting sampling theory for a
signal x∈R
l
that is sparse in its original or some transform domain
Ψ∈R
l×n
(l ≤ n). Such a signal x can be sampled and almost
perfectly recovered by fewer of its linear projections under the
theory of CS. Suppose x = Ψα, then the vector α∈R
n
is sparse, and
the number of its non-zero entries k (k < n) is called its sparsity
level. The CS measurements of x can be given by
y = ΦΨ α = Dα
(1)
where Φ∈R
m×l
is called measurement matrix or sensing matrix [1],
the parameter m is the measurement number (m≪l). Vector y∈R
m
is the CS measurements. Matrix D∈R
m×n
is known as information
operator [2] and it is the product of Φ and Ψ, that is, D = ΦΨ.
The estimation of α in (1) is the solution of the following
minimisation problem:
α
^
= arg min
α
∥ α ∥
0
, s . t . y = ΦΨ α
(2)
where ℓ
0
-norm ||α||
0
refers to the number of non-zero entries in α.
Though ℓ
0
-norm gives solution for problem (2), but it is a non-
polynomial hard problem. Orthogonal matching pursuit (OMP) [3]
and its improvements [4, 5] is one kind of the methods to solve it.
The easily computed minimum ℓ
1
-norm solution such as basis
pursuit (BP) [6] is also effective, because it is equivalent to
problem (2) when Φ and Ψ satisfy the restricted isometry property
(RIP) [7].
Gaussian random matrix [8], whose entries are independently
and identically distributed (i.i.d.) normalised Gaussian random
variables, is known to satisfy RIP with orthogonal basis Ψ. Banded
block Toeplitz matrix and Toeplitz matrix whose entries are drawn
from zero-mean bounded distributions are both satisfying RIP with
sparse basis with high probability [9, 10]. However, generally
whether a measurement matrix Φ satisfies RIP with Ψ or not is
difficult to determine.
Besides the requirement of RIP, mutual coherence is another
method to evaluate the quality of measurement matrix. It is defined
as the maximum absolute value of the normalised inner product
between different columns in information operator D [11]
μ
mx
(D) = max
1 ≤ i, j ≤ n, i ≠ j
d
i
T
d
j
∥ d
i
∥
2
⋅ ∥ d
j
∥
2
(3)
where d
i
is the ith column of D.
Using the Gershgorin circle theorem [12], Duarte and Eldar
[13] proposed the method to connect the RIP with the mutual
coherence and proved that Φ has the (k,δ)-RIP with δ≤(k − 1)μ
mx
.
Mutual coherence and its derivative concepts are widely used for
the assessment of measurement matrix. The incidence matrix of
packing designs is utilised to generate binary matrices with low
coherence in [14]. In that study, four classes of binary deterministic
sparse measurement matrices are designed using packing designs
named Steiner systems, based on coherence. The tensor product is
applied on the Reed-Solomon generator matrix to produce
deterministic measurement matrix that is asymptotically optimal in
the meaning of coherence [15]. Pereira et al. [16] proposed a
method to construct the measurement matrix that is maximally
incoherent to a given orthogonal or biorthogonal basis. If Ψ is an n
× n orthogonal basis, the optimised measurement matrix is
constructed as the product of Ψ and the matrix formed by m
randomly chosen rows of the n × n Hadamard matrix. Unlike
mutual coherence measures, the entry-wise incoherence, the
cumulative coherence defined in [17], evaluates the average
incoherence between Φ and Ψ. Schnass and Vandergheynst [18]
presented the structurally random matrix (SRM) method and
utilised cumulative coherence to measure its performance. SRM
method prerandomises the sensing signal by first scrambling its
sample locations or flipping its sample signs, and then fast-
transforming the randomised samples, before finally subsampling
the resulting transform coefficients to obtain the SRM that is fast
computation and supports block-based processing. In [19], aiming
at minimising the mutual coherence, a measurement matrix
optimisation method is proposed based on the theory of
equiangular tight frame (ETF), because a matrix in ETF has the
minimum coherence that is called Welch bound [20]
μ
E
=
(n − m)
m(n − 1)
(4)
where m and n are the number of rows and columns of the matrix
in ETF, respectively.
IET Image Process., 2018, Vol. 12 Iss. 10, pp. 1773-1779
© The Institution of Engineering and Technology 2018
1773
评论0
最新资源