475 research outputs found
A Splitting Augmented Lagrangian Method for Low Multilinear-Rank Tensor Recovery
This paper studies a recovery task of finding a low multilinear-rank tensor
that fulfills some linear constraints in the general settings, which has many
applications in computer vision and graphics. This problem is named as the low
multilinear-rank tensor recovery problem. The variable splitting technique and
convex relaxation technique are used to transform this problem into a tractable
constrained optimization problem. Considering the favorable structure of the
problem, we develop a splitting augmented Lagrangian method to solve the
resulting problem. The proposed algorithm is easily implemented and its
convergence can be proved under some conditions. Some preliminary numerical
results on randomly generated and real completion problems show that the
proposed algorithm is very effective and robust for tackling the low
multilinear-rank tensor completion problem
Iterative Singular Tube Hard Thresholding Algorithms for Tensor Completion
Due to the explosive growth of large-scale data sets, tensors have been a
vital tool to analyze and process high-dimensional data. Different from the
matrix case, tensor decomposition has been defined in various formats, which
can be further used to define the best low-rank approximation of a tensor to
significantly reduce the dimensionality for signal compression and recovery. In
this paper, we consider the low-rank tensor completion problem. We propose a
novel class of iterative singular tube hard thresholding algorithms for tensor
completion based on the low-tubal-rank tensor approximation, including basic,
accelerated deterministic and stochastic versions. Convergence guarantees are
provided along with the special case when the measurements are linear.
Numerical experiments on tensor compressive sensing and color image inpainting
are conducted to demonstrate convergence and computational efficiency in
practice
A Non-Local Structure Tensor Based Approach for Multicomponent Image Recovery Problems
Non-Local Total Variation (NLTV) has emerged as a useful tool in variational
methods for image recovery problems. In this paper, we extend the NLTV-based
regularization to multicomponent images by taking advantage of the Structure
Tensor (ST) resulting from the gradient of a multicomponent image. The proposed
approach allows us to penalize the non-local variations, jointly for the
different components, through various matrix norms with .
To facilitate the choice of the hyper-parameters, we adopt a constrained convex
optimization approach in which we minimize the data fidelity term subject to a
constraint involving the ST-NLTV regularization. The resulting convex
optimization problem is solved with a novel epigraphical projection method.
This formulation can be efficiently implemented thanks to the flexibility
offered by recent primal-dual proximal algorithms. Experiments are carried out
for multispectral and hyperspectral images. The results demonstrate the
interest of introducing a non-local structure tensor regularization and show
that the proposed approach leads to significant improvements in terms of
convergence speed over current state-of-the-art methods
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
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