. 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.
Super-resolution aims to recover high-frequency data from low-frequency measurements, an extrapolation problem well known to be unstable. We propose a model-based super-resolution framework (Model-SR) to analyze its stability, bridging the gap between limited theory and empirical success. The key idea is that, to be determined by its low-frequency components, the target signal must be low-dimensional; instead of assuming it lies on a low-dimensional manifold, we assume it is generated from a model with a low-dimensional parameter space, enabling stability analysis directly through the parameters. Within this framework, we recover the signal by solving a nonlinear least square problem and obtain high-frequency components. The resolution-enhancing map is proven to have Lipschitz continuity, with a constant that depends crucially on parameter separation; consequently, measurements from well-separated parameters yield stable reconstructions. This separation can be enforced via sparsity modeling-using the minimal number of parameters-thus highlighting sparsity's role in stability. Moreover, the Lipschitz constant grows with the high-frequency cutoff, ultimately rendering extrapolation ineffective beyond a certain threshold. We apply the theory to three concrete models, give stability estimates, and present numerical experiments. The model-based framework can be extended to problems with similar structures.
In this paper, we study a broad class of structured monotone inclusion problems in real Hilbert spaces. We propose a novel primal-dual splitting algorithm for solving such inclusions, which accommodates multiple monotone operators and cocoercive terms, as well as a composite monotone operator involving the linear map. The algorithm combines forward evaluations for the cocoercive components with backward resolvent steps for the monotone operators and employs a dual update for the linear composition term. It generalizes and unifies several existing methods, while requiring only a single resolvent or operator evaluation per iteration. We prove weak convergence of the iterates under standard assumptions on monotonicity and cocoercivity. Furthermore, we establish strong convergence under a mild regularity condition, such as uniform monotonicity. Numerical experiments on image deblurring and denoising problems demonstrate the efficiency and flexibility of the proposed algorithm.
We study an inverse source problem for a linearized pseudoparabolic equation in a bounded domain, employing data assimilation techniques to recover the spatial-dependent source in the whole domain from local measurements. Since the inversion input data are given only in a portion of the spatial-temporary domain due to the engineering restriction, the problem is inherently ill-posed related to the classical topic of function extensions. Based on the wellposedness of the forward problem, we reformulate this inverse problem as its optimization version that combines the data-fidelity term with a penalty term incorporating the background information about the unknown source. The Euler-Lagrange equation characterizing the optimal solution is derived, and we rigorously establish the unique solvability of this equation. Moreover, using the Lax-Milgram theorem, we derive the convergence rate for the regularizing solution in terms of the noise level of inversion input and the accuracy of the a-prior background for appropriately chosen regularizing parameter. Finally, an iterative algorithm is proposed to implement the reconstruction procedure, together with several numerical examples demonstrating the effectiveness of our proposed approach.
In image denoising problems, nonsmooth and nonconvex high-order models have shown advantages over the first-order ones in detail-preserving and staircase removal. However, most of them suffer from non-coerciveness, which leads to the ill-posedness in many image reconstruction tasks. To address this issue, we incorporate the box-constraint into them by taking the most recent total scaled-gradient variation (TSGV) based model as an example. The box-constraint helps to overcome the ill-posedness by bounding image intensity values, which stabilizes the reconstruction process. The TSGV is a piecewise-linear structure regularization framework, which extends well the classic total variation (TV) regularization suitable for piecewise-constant structure modeling. To solve the proposed highly nonlinear and nonconvex problem, we introduce an inertial accelerated operator-splitting (IOS) algorithm, which combines the inertial extrapolation and operator-splitting methods based on the Lie scheme and the Marchuk-Yanenko discretization. The derived IOS algorithm is computationally efficient, easy to implement, and preserve the stability and convergence properties of the Lie scheme while improving performance through acceleration strategies. Our proposed method can be applied to many image reconstruction problems, like image deblurring, inpainting, zooming, MRI reconstruction and CT reconstruction. Numerical experiments demonstrate that the proposed method consistently achieves superior image reconstruction quality with fewer parameter-tuning requirements and lower computational cost than existing methods across a series of linear image reconstruction applications.
. This paper establishes uniqueness results of inverse elastic scattering problem with phaseless near-field data in periodic structures in R2 and periodic/biperiodic structures in R3. We use a superposition of two point sources in each periodic unit with different positions as the incident field, and measure the phaseless near-field data on a line parallel to x1-axis in 2D, or on a plane parallel to (x1, x2)-plane in 3D. We first calculate the explicit formula of quasi-periodic/quasi-biperiodic Green's functions of Lame system in R3. Then, to establish the uniqueness results, the reciprocity relations for point sources, scattered fields, and total fields are derived. Finally, with the help of Rayleigh's expansion, the uniqueness results are proved. The quasiperiodic/quasi-biperiodic Green's functions of Lame system in R3, the reciprocity relations, and Rayleigh's expansion in R3 are novel results as important by-products in the proof process.
. Nonrigid image registration aims at finding a displacement field densely aligning a moving image and a fixed one, which is a fundamental and challenging task in the field of medical image analysis. Variational models are widely used for nonrigid image registration, which consists of a data term and a regularization term. A common assumption of nonrigid medical image registration of the same modality is that the aligned pixels of the moving and fixed images have the same intensity values, which implies that the difference of registered moving and fixed images is highly sparse. However, existing data terms like sum of squared difference (SSD) and sum of absolute difference (SAD), (i.e., an L2 norm and an L1 norm, respectively) are not optimal choices for modeling sparsity. As to the regularization term, existing models usually assume the displacement field to be smooth, and hence the registration transformation is smooth or diffeomorphic, whereas for medical images, this assumption is not valid in many situations due to complex anatomy structures and possible lesions in images, especially along the edges of different organs and tissues. In this paper, we employ the L0 norm as the data term to capture the sparsity of difference of registered images and regularize the displacement fields as functions of bounded generalized deformation (BGD), which allow both smooth and discontinuous displacement fields. To our knowledge, this is the first nonrigid image registration model that directly takes sparsity of the data term into account. We then introduce a log-sum type sparse approximation of the L0-norm data term, which turns the NP-hard L0 problem into a novel nonconvex and nonsmooth variational registration problem, called the log-BGD model. We prove the existence of its global solutions. We also derive a fast primal-dual iteratively reweighting (PDIR) algorithm for solving the model and prove its convergence. Numerical examples show the effectiveness of both the proposed log-BGD model and the derived algorithm.
Solving inverse problems Ax = y is central to a variety of practically important fields such as medical imaging, remote sensing, and nondestructive testing. The most successful and theoretically best-understood method is convex variational regularization, where approximate but stable solutions are defined as minimizers of parallel to A(& centerdot;)-y delta parallel to 2/2 + alpha R(& centerdot;), with R a regularization functional. Recent methods such as deep equilibrium models and plug-and-play approaches, however, go beyond variational regularization. Motivated by these innovations, we introduce implicit non-variational (INV) regularization, where approximate solutions are defined as solutions of A & lowast;(Ax-y delta) + alpha R(x) = 0 for some regularization operator R. When the regularization operator is the gradient of a functional, INV reduces to classical variational regularization. However, in methods like DEQ and PnP, R is not a gradient field, and the existing theoretical foundation remains incomplete. To address this, we establish stability and convergence results in this broader setting, including convergence rates and stability estimates measured via a absolute Bregman distance.
A single-particle cryo-electron microscopy (cryo-EM) measurement, called a micrograph, consists of multiple two-dimensional tomographic projections of a three-dimensional (3-D) molecular structure at unknown locations, taken under unknown viewing directions. All existing cryo-EM algorithmic pipelines first locate and extract the projection images, and then reconstruct the structure from the extracted images. However, if the molecular structure is small, the signal-to-noise ratio (SNR) of the data is very low, making it challenging to accurately detect projection images within the micrograph. Consequently, all standard techniques fail in low-SNR regimes. To recover molecular structures from measurements of low SNR, and in particular small molecular structures, we devise an approximate expectation-maximization algorithm to estimate the 3-D structure directly from the micrograph, bypassing the need to locate the projection images. We corroborate our computational scheme with numerical experiments and present successful structure recoveries from simulated noisy measurements.
. We consider Inverse Synthetic Aperture Radar (ISAR) imaging for a rotating scatterer. We begin by employing a linearised scattering model for the case of a rotating scatterer with a known centre of rotation, on a horizontal target plane, being imaged by a point-like transceiver. This model is described using a scattering operator, F, which maps the object's reflectivity function to a scattered wavefield. The image is obtained by backprojecting the scattered field. The resulting image is the output of the combination, F & lowast;F, applied to the reflectivity function that we wish to image/recover. Utilising microlocal analysis, particularly the wavefront relation of F, we examine the nature of fictitious artifacts that may arise in the reconstructed image. This analysis is used to suggest several experimental setups, which we demonstrate numerically, whereby the artifacts can be guaranteed not to interfere with regions selected for imaging. We also explore the scenario when the centre of rotation is not known and what can be done to form an accurate image in this situation. Following this, we consider scatterers rotating about a known axis being imaged by static linear and planar transceiver arrays, respectively. By using a linear or planar transceiver, we are able to extend the analysis so that the scatterer is no longer constrained to a plane, and we again show how to avoid artifacts. We investigate conditions under which the restriction of F & lowast;F to the scene to be imaged is a pseudodifferential operator. When these conditions are satisfied, no artifacts appear in the reconstructed image. Therefore, this paper provides valuable theoretical insights into the location of artifacts which appear in the imaging of rotating objects and presents a strategy for the design of data acquisition geometries that avoid such artifacts.
. We present a novel variational framework for curve reconstruction in inverse problems, formulated through vector field optimization. The problem is defined in a space of vector-valued measures absolutely continuous with respect to the Lebesgue measure, with Sobolev-type densities. Within this setting, we establish an equivalence, in terms of minimizers, between the proposed functional and a previously defined one within the space of divergence-measure fields. This formulation enables the use of standard convex optimization algorithms for numerical implementation. Furthermore, we propose an acquisition model specifically adapted to blurred curves in microscopy images, leading to promising numerical results.
In this article, we consider an inverse problem of hyperspectral image unmixing. Although significant progress has been made in the current unmixing studies, there are still issues regarding insufficient utilization of spatial information and high computational complexity. Therefore, we propose a novel unmixing method, named sparse and total variation (TV) based deep alternating neural network (STV-ANet), that incorporates sparse and TV regularization terms within a deep alternating neural network framework. In particular, an innovative alternating algorithm is first developed to solve the unmixing problem based on sparse and TV regularization, which requires only a few auxiliary variables. Then this alternating algorithm is unrolled within a deep neural network, integrating the prior knowledge provided by the regularization terms with the data features extracted by the network. The proposed algorithm improves the interpretability of the deep neural network, while the network liberates the algorithm from the adjustment of hyperparameters. Experiments on both synthetic and real-world datasets demonstrate the superior unmixing performance of our STV-ANet over state-of-the-art methods, particularly in the suppression of noise and the recovery accuracy of abundance maps.
. The inverse conductivity problem, also known as Calderon's problem, is a notoriously challenging problem. To combat the ill-posedness of the nonlinear inverse problem, strong regularization is required, and traditional variational methods are computationally expensive and highly sensitive to incorrect modeling of the forward problem. However, complex geometrical optics (CGO)-based methods are an attractive alternative due to their near real-time speed and their robustness to domain modeling errors. In this paper, we present the first numerical implementation of the nonlinear "tB method" on simulated and experimental electrode data and compare tB reconstructions to simplifications from the t0 and texp approximations. The computational cost of the tB method is much higher than that of the t0 and texp methods, without significant improvement in spatial resolution. Reconstruction quality is, however, improved across all three methods with the number of electrodes is increased.
Interface optimal design problems, fundamental to diverse scientific and engineering applications, present computational challenges due to their inherent complex geometries and physical constraints. This paper introduces a novel framework integrating Weak Adversarial Networks (WAN) and the Augmented Lagrangian Method (ALM) to efficiently solve physics constrained interface optimal design problems modeled in the phase field based representation. We reformulate these problems as minimax problems via ALM and represent the primal and dual variables using distinct deep neural networks. These networks are then trained through an adversarial process, enabling the method to naturally handle complex interface geometries and physical constraints. The proposed approach demonstrates superior constraint enforcement through learned Lagrange multipliers, significantly reduces sensitivity to constraint scaling and penalty hyperparameters compared to penalty-based methods, and an enhanced ability to escape local minimizers during optimization. Moreover, empirical study indicates that the tanh3 activation function exhibits enhanced robustness during practical hyperparameter tuning while also improving solution accuracy. Extensive numerical experiments, including Ginzburg-Landau energy minimization with mass constraint, optimal partition problems with L2 norm preservation (demonstrated up to 5 dimensions), and fluid-solid topology optimization with partial differential equation constraints-all featuring physical constraints-validate the capability, robustness, and high-dimensional extensibility of the method.
