55 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
Low-Rank Tensor Recovery with Euclidean-Norm-Induced Schatten-p Quasi-Norm Regularization
The nuclear norm and Schatten- quasi-norm of a matrix are popular rank
proxies in low-rank matrix recovery. Unfortunately, computing the nuclear norm
or Schatten- quasi-norm of a tensor is NP-hard, which is a pity for low-rank
tensor completion (LRTC) and tensor robust principal component analysis
(TRPCA). In this paper, we propose a new class of rank regularizers based on
the Euclidean norms of the CP component vectors of a tensor and show that these
regularizers are monotonic transformations of tensor Schatten- quasi-norm.
This connection enables us to minimize the Schatten- quasi-norm in LRTC and
TRPCA implicitly. The methods do not use the singular value decomposition and
hence scale to big tensors. Moreover, the methods are not sensitive to the
choice of initial rank and provide an arbitrarily sharper rank proxy for
low-rank tensor recovery compared to nuclear norm. We provide theoretical
guarantees in terms of recovery error for LRTC and TRPCA, which show relatively
smaller of Schatten- quasi-norm leads to tighter error bounds.
Experiments using LRTC and TRPCA on synthetic data and natural images verify
the effectiveness and superiority of our methods compared to baseline methods
Interpretable Hyperspectral AI: When Non-Convex Modeling meets Hyperspectral Remote Sensing
Hyperspectral imaging, also known as image spectrometry, is a landmark
technique in geoscience and remote sensing (RS). In the past decade, enormous
efforts have been made to process and analyze these hyperspectral (HS) products
mainly by means of seasoned experts. However, with the ever-growing volume of
data, the bulk of costs in manpower and material resources poses new challenges
on reducing the burden of manual labor and improving efficiency. For this
reason, it is, therefore, urgent to develop more intelligent and automatic
approaches for various HS RS applications. Machine learning (ML) tools with
convex optimization have successfully undertaken the tasks of numerous
artificial intelligence (AI)-related applications. However, their ability in
handling complex practical problems remains limited, particularly for HS data,
due to the effects of various spectral variabilities in the process of HS
imaging and the complexity and redundancy of higher dimensional HS signals.
Compared to the convex models, non-convex modeling, which is capable of
characterizing more complex real scenes and providing the model
interpretability technically and theoretically, has been proven to be a
feasible solution to reduce the gap between challenging HS vision tasks and
currently advanced intelligent data processing models
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