. In this paper, we present a novel and general framework for color image denoising by integrating a traditional variational approach and learningbased denoisers. Specifically, we incorporate the deep residual convolutional neural network (CNN) prior into the traditional saturation-value total variation (SVTV) based variational model to integrate the advantages of both approaches. Saturation regularization is applied to process the saturation channel of a color image, while CNN based denoiser handles the value channel of a color image. Then, the denoising process is performed in parallel on saturation channel and value channel by using saturation regularization and deep CNN prior respectively. Theoretically, we investigate the properties of the proposed model, and give a detailed discussion based on the mathematical foundation about the solution. Numerically, we formulate an efficient and effective algorithm to solve the proposed model based on the framework of alternating direction method of multipliers (ADMM). Numerical examples are presented to demonstrate that the performance of the proposed framework is competitive in terms of visual quality and some criteria such as peak signal-to-noise ratio (PSNR), structure similarity (SSIM), quaternion structural similarity (QSSIM), and feature similarity (FSIM).
. This paper addresses the inverse problem of simultaneously recovering multiple unknown parameters for semilinear wave equations from boundary measurements. We consider an initial-boundary value problem for a wave equation with a general semilinear term and an internal source. The inverse problem is to determine the nonlinear coefficients (potentials), the source term, and the initial data from the Dirichlet-to-Neumann (DtN) map. Our approach combines higher-order linearization and the construction of geometric optics (GO) solutions. The main results establish that while unique recovery is not always possible, we can precisely characterize the gauge equivalence classes in the solutions to this inverse problem. For a wave equation with a polynomial nonlinearity of degree n, we prove that only the highest-order coefficient can be uniquely determined from the DtN map; the lower-order coefficients and the source can only be recovered up to a specific gauge transformation involving a function psi. Furthermore, we provide sufficient conditions under which unique determination of all parameters is guaranteed. We also extend these results to various specific non-polynomial nonlinearities, demonstrating that the nature of the nonlinearity critically influences whether unique recovery or a gauge symmetry is obtained.
. The classical Calderon problem with partial data is known to be log-log stable in some special cases, but even the uniqueness problem is open in general. We study the partial data stability of an analogous inverse fractional conductivity problem on bounded smooth domains. Using the fractional Liouville reduction, we obtain a log-log stability estimate when the conductivities a priori agree in the measurement set and their difference has compact support. In the case in which the conductivities are assumed to agree a priori in the whole exterior of the domain, we obtain a sharper logarithmic stability estimate.
This article addresses the recovery of sparse signals and low-rank matrices from limited, noisy observations. We introduce unconstrained models, termed the Robust Penalized Dantzig Vector Selector (RPDvS) for sparse signal recovery and the Robust Penalized Dantzig Matrix Selector (RPDmS) for low-rank matrix recovery. Under restricted isometry assumptions on the measurement operators, we establish stable recovery error bounds and oracle inequalities for both models in the presence of Gaussian noise. For the RPDmS model, we derive a tight oracle inequality that guarantees identification of a "nearly perfect" column space and motivates a simple post-processing procedure that yields an improved estimator, thereby bridging a gap in the earlier analysis of the Dantzig matrix selector by Cande`s and Plan (2011). Parallel oracle-type results are obtained for the RPDvS model. Building on these theoretical guarantees, we develop efficient algorithms for solving the RPDvS and RPDmS problems and implement the proposed postprocessing steps. Extensive simulations and empirical studies demonstrate that the new methods are computationally competitive while achieving superior reconstruction accuracy in both sparse signal and low-rank matrix recovery, thereby confirming the sharpness of our error bounds and the effectiveness of the algorithms.
