Quaternion block sparse representation for signal recovery and classification
Introduction
The past decades have witnessed the great advancement of sparse signal recovery and its widespread applications in signal processing [1], [2], [3] and pattern recognition [4], [5]. Given a sampling matrix (m < n), the problem of sparse signal recovery is to recover the sparse signal from its compressed measurement vectorwhere denotes the noise vector. In many applications such as multiband signals reconstruction [6], face recognition [5] and measuring gene expression levels [7], the nonzero elements of sparse signals appear in a few blocks and exhibit the block sparse structure. Many recent studies [5], [8], [9] have shown that by leveraging such block structure of the sparse is beneficial to improving the recovery performance.
In recent years, a number of block sparse recovery (BSR) methods have been proposed to further promote signal recovery performance. The first category of BSR methods attempt to revise the greedy algorithms originally designed for sparse recovery by incorporating the block structure information into the algorithm. In 2010, Eldar et al. [10] propose a block version of the OMP (Orthogonal Matching Pursuit) algorithm [11], termed BOMP for block sparse signal recovery. Recently, the block version of StOMP (Stagewise OMP) [12] and CoSaMP (Compressive Sampling Matching Pursuit) [13] have also been proposed and analyzed in [14] and [15], respectively. Unlike the greedy algorithms above, another category of BSR methods are based on convex optimization using the mixed ℓq/ℓ1 norm [16] where q > 1. A common choice of q is . The works [10], [16], [17] study theoretical conditions under which the ℓq/ℓ1 norm-based BSR method can recover the ground-truth block sparse signal. For noisy data, Yuan and Lin [18] study the group Lasso model, referred to as GLasso in short, which is a generalization of the conventional Lasso model [19]. The works [5], [20] provide both theoretical and empirical evidence to show the superiority of GLasso over the standard Lasso. Some other recent works on BSR can be found in the references [9], [21], [22], [23].
To the best of our knowledge, the existing algorithms on block sparse signal recovery only deal with the real-valued or complex-valued signals. As for quaternion-valued signals, since the multiplication of quaternions is noncommunicative, these algorithms cannot be applied to the recovery of quaternion-valued signals. As a natural extension of the real and complex number system, quaternion has been widely used in a variety of applications, such as vector-sensor array signal processing [24], color image restoration [25] and color face recognition [26]. In recent years, a number of works [25], [26], [27] have shown that there is strong correlation among the different color channels of a color image. Specifically, Xu et al. [25] generalize the OMP algorithm to the quaternion domain and develop the QOMP (Quaternion Orthogonal Matching Pursuit) algorithm with application to color image restoration. To capitalize on the correlation information among different color channels, in our previous work [26] we propose a quaternion sparse representation method called QLasso with application to color face recognition. However, both QOMP and QLasso fail to consider the block structure of the quaternion sparse signals.
The contributions of this work are summarized below.
- 1.
We propose a quaternion block sparse representation (QBSR) method for the recovery of quaternion block sparse signals. To the best of the authors’ knowledge, this work is the first attempt to develop a block sparse signal recovery method in the quaternion space in the open literature.
- 2.
We develop a QBSR based classification (QBSRC) method for quaternion signal classification. Compared with existing real-valued representation based classifiers, QBSRC can take full advantage of the correlation information of multiple channels of a quaternion signal, such as the multiple color channels of a color image.
- 3.
We establish the theoretical guarantees of QBSRC for classification. Specifically, we prove that QBSRC is guaranteed to succeed in classifying any new test sample under appropriate conditions on the data distribution.
Advantages of QBSR. Compared with the real and complex methods which are only designed for real-valued or complex-valued data such as gray-scale images, QBSR can be directly applied for quaternion-valued data such as color images [25], [28], [29] and vector-sensor array signals [24]. The advantages of QBSR lie in the following two aspects.
- •
It broadens the range of applications of block sparse representation methods for more kinds of data, such as color images and vector-sensor array signals.
- •
For quaternion-valued data such as color images, the real and complex methods can only deal the multiple channels of the data separately and independently. Thus, they ignore the structural correlation information among the multiple channels. In contrast, QBSR processes the multiple channels of quaternion-valued data in a holistic manner and can preserve correlation information among them to achieve better performance.
Many recent studies [25], [28], [29] have shown that compared to real and complex numbers, quaternions are better adapted to color images by encoding the multiple color channels holistically in many applications, such as color image super-resolution [25], segmentation [28] and quality assessment [29].
The rest of this paper is organized as follows. In Section 2.2, we propose the quaternion block sparse representation (QBSR) method for the recovery of quaternion block sparse signals. Section 3 devises a QBSR based classifier and provides the theoretical analysis. Section 4 presents the experimental results on synthetic and real data. Finally, Section 5 summarizes this work.
In this work, scalars, vectors and matrices are represented using italic letters (e.g., x), boldface lowercase letters (e.g., x), and boldface capital letters (e.g., X), respectively. To distinguish real-valued and quaternion-valued variables, we add a dot above the variable to indicate a quaternion variable. For instance, denote a quaternion scalar, vector and matrix, respectively. For each vector and an index set denotes a subvector of x containing entries indexed by . Table 1 summarizes the key notations and acronyms used in this paper.
Section snippets
Quaternion block sparse representation (QBSR)
In this section, we describe the proposed QBSR (Quaternion Block Sparse Representation) method as well as the optimization algorithm. This section is arranged as follows. Firstly, we introduce some basic concepts of quaternion algebra. Secondly, we formulate the problem of recovering quaternion block sparse signals and build the QBSR model. Finally, we devise the optimization algorithm for QBSR within the ADMM (Alternating Direction Method of Multipliers) framework [30].
Quaternion block sparse representation based classification (QBSRC)
In this section, we leverage the QBSR method to devise a classifier referred to as QBSRC for quaternion signal classification. Assume that we have a collection of n quaternion training samples from K classes. For ease of presentation, we arrange them as columns of a quaternion matrix . Let be the submatrix of containing its columns corresponding to class k for . Given any new test sample the goal of classification is to correctly assign
Experiments
In this section, we evaluate the performance of the proposed QBSR method for the recovery of quaternion block sparse signals and the proposed QBSRC method for color face recognition, respectively.
Conclusion
In this paper, we propose a quaternion block sparse representation (QBSR) method for quaternion-valued signal recovery. In addition, we also leverage QBSR to devise a novel classifier termed as QBSRC for quaternion data classification, such as color face recognition. The theoretical guarantees of QBSRC have also been established. Experiments on both synthetic and real-world data validate the efficacy of the proposed method for signal recovery and classification in quaternion space.
CRediT authorship contribution statement
Cuiming Zou: Conceptualization, Methodology, Writing - original draft. Kit Ian Kou: Supervision, Project administration. Yulong Wang: Software, Validation. Yuan Yan Tang: Writing - review & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
This work was supported by the National Natural Science Foundation of China under Grant Nos. 61806027, 61702057, 61672114 and 11771130. This work was also partly supported by The Science and Technology Development Fund, Macau SAR (File no. FDCT/085/2018/A2) and University of Macau (File no. MYRG2019-00039-FST).
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