118 research outputs found
A Theoretically Guaranteed Quaternion Weighted Schatten p-norm Minimization Method for Color Image Restoration
Inspired by the fact that the matrix formulated by nonlocal similar patches
in a natural image is of low rank, the rank approximation issue have been
extensively investigated over the past decades, among which weighted nuclear
norm minimization (WNNM) and weighted Schatten -norm minimization (WSNM) are
two prevailing methods have shown great superiority in various image
restoration (IR) problems. Due to the physical characteristic of color images,
color image restoration (CIR) is often a much more difficult task than its
grayscale image counterpart. However, when applied to CIR, the traditional
WNNM/WSNM method only processes three color channels individually and fails to
consider their cross-channel correlations. Very recently, a quaternion-based
WNNM approach (QWNNM) has been developed to mitigate this issue, which is
capable of representing the color image as a whole in the quaternion domain and
preserving the inherent correlation among the three color channels. Despite its
empirical success, unfortunately, the convergence behavior of QWNNM has not
been strictly studied yet. In this paper, on the one side, we extend the WSNM
into quaternion domain and correspondingly propose a novel quaternion-based
WSNM model (QWSNM) for tackling the CIR problems. Extensive experiments on two
representative CIR tasks, including color image denoising and deblurring,
demonstrate that the proposed QWSNM method performs favorably against many
state-of-the-art alternatives, in both quantitative and qualitative
evaluations. On the other side, more importantly, we preliminarily provide a
theoretical convergence analysis, that is, by modifying the quaternion
alternating direction method of multipliers (QADMM) through a simple
continuation strategy, we theoretically prove that both the solution sequences
generated by the QWNNM and QWSNM have fixed-point convergence guarantees.Comment: 46 pages, 10 figures; references adde
Approximation of Images via Generalized Higher Order Singular Value Decomposition over Finite-dimensional Commutative Semisimple Algebra
Low-rank approximation of images via singular value decomposition is
well-received in the era of big data. However, singular value decomposition
(SVD) is only for order-two data, i.e., matrices. It is necessary to flatten a
higher order input into a matrix or break it into a series of order-two slices
to tackle higher order data such as multispectral images and videos with the
SVD. Higher order singular value decomposition (HOSVD) extends the SVD and can
approximate higher order data using sums of a few rank-one components. We
consider the problem of generalizing HOSVD over a finite dimensional
commutative algebra. This algebra, referred to as a t-algebra, generalizes the
field of complex numbers. The elements of the algebra, called t-scalars, are
fix-sized arrays of complex numbers. One can generalize matrices and tensors
over t-scalars and then extend many canonical matrix and tensor algorithms,
including HOSVD, to obtain higher-performance versions. The generalization of
HOSVD is called THOSVD. Its performance of approximating multi-way data can be
further improved by an alternating algorithm. THOSVD also unifies a wide range
of principal component analysis algorithms. To exploit the potential of
generalized algorithms using t-scalars for approximating images, we use a pixel
neighborhood strategy to convert each pixel to "deeper-order" t-scalar.
Experiments on publicly available images show that the generalized algorithm
over t-scalars, namely THOSVD, compares favorably with its canonical
counterparts.Comment: 20 pages, several typos corrected, one appendix adde
Quaternion tensor ring decomposition and application for color image inpainting
In recent years, tensor networks have emerged as powerful tools for solving
large-scale optimization problems. One of the most promising tensor networks is
the tensor ring (TR) decomposition, which achieves circular dimensional
permutation invariance in the model through the utilization of the trace
operation and equitable treatment of the latent cores. On the other hand, more
recently, quaternions have gained significant attention and have been widely
utilized in color image processing tasks due to their effectiveness in encoding
color pixels. Therefore, in this paper, we propose the quaternion tensor ring
(QTR) decomposition, which inherits the powerful and generalized representation
abilities of the TR decomposition while leveraging the advantages of
quaternions for color pixel representation. In addition to providing the
definition of QTR decomposition and an algorithm for learning the QTR format,
this paper also proposes a low-rank quaternion tensor completion (LRQTC) model
and its algorithm for color image inpainting based on the QTR decomposition.
Finally, extensive experiments on color image inpainting demonstrate that the
proposed QTLRC method is highly competitive
A structure-preserving one-sided Jacobi method for computing the SVD of a quaternion matrix
Abstract(#br)In this paper, we propose a structure-preserving one-sided cyclic Jacobi method for computing the singular value decomposition of a quaternion matrix. In our method, the columns of the quaternion matrix are orthogonalized in pairs by using a sequence of orthogonal JRS-symplectic Jacobi matrices to its real counterpart. We establish the quadratic convergence of our method specially. We also give some numerical examples to illustrate the effectiveness of the proposed method
The QRD and SVD of matrices over a real algebra
Recent work in the field of signal processing has shown that the singular
value decomposition of a matrix with entries in certain real algebras can be a
powerful tool. In this article we show how to generalise the QR decomposition
and SVD to a wide class of real algebras, including all finite-dimensional
semi-simple algebras, (twisted) group algebras and Clifford algebras. Two
approaches are described for computing the QRD/SVD: one Jacobi method with a
generalised Givens rotation, and one based on the Artin-Wedderburn theorem.Comment: Uses elsarticle.cl
An Invitation to Hypercomplex Phase Retrieval: Theory and Applications
Hypercomplex signal processing (HSP) provides state-of-the-art tools to
handle multidimensional signals by harnessing intrinsic correlation of the
signal dimensions through Clifford algebra. Recently, the hypercomplex
representation of the phase retrieval (PR) problem, wherein a complex-valued
signal is estimated through its intensity-only projections, has attracted
significant interest. The hypercomplex PR (HPR) arises in many optical imaging
and computational sensing applications that usually comprise quaternion and
octonion-valued signals. Analogous to the traditional PR, measurements in HPR
may involve complex, hypercomplex, Fourier, and other sensing matrices. This
set of problems opens opportunities for developing novel HSP tools and
algorithms. This article provides a synopsis of the emerging areas and
applications of HPR with a focus on optical imaging.Comment: 10 pages, 4 figures, 2 table
- …