22 research outputs found

    An overview of inverse problem regularization using sparsity

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    International audienceSparsity constraints are now very popular to regularize inverse problems. We review several approaches which have been proposed in the last ten years to solve inverse problems such as inpainting, deconvolution or blind source separation. We will focus especially on optimization methods based on iterative thresholding methods to derive the solution

    Neural Gradient Regularizer

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    Owing to its significant success, the prior imposed on gradient maps has consistently been a subject of great interest in the field of image processing. Total variation (TV), one of the most representative regularizers, is known for its ability to capture the sparsity of gradient maps. Nonetheless, TV and its variants often underestimate the gradient maps, leading to the weakening of edges and details whose gradients should not be zero in the original image. Recently, total deep variation (TDV) has been introduced, assuming the sparsity of feature maps, which provides a flexible regularization learned from large-scale datasets for a specific task. However, TDV requires retraining when the image or task changes, limiting its versatility. In this paper, we propose a neural gradient regularizer (NGR) that expresses the gradient map as the output of a neural network. Unlike existing methods, NGR does not rely on the sparsity assumption, thereby avoiding the underestimation of gradient maps. NGR is applicable to various image types and different image processing tasks, functioning in a zero-shot learning fashion, making it a versatile and plug-and-play regularizer. Extensive experimental results demonstrate the superior performance of NGR over state-of-the-art counterparts for a range of different tasks, further validating its effectiveness and versatility

    Image Restoration for Remote Sensing: Overview and Toolbox

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    Remote sensing provides valuable information about objects or areas from a distance in either active (e.g., RADAR and LiDAR) or passive (e.g., multispectral and hyperspectral) modes. The quality of data acquired by remotely sensed imaging sensors (both active and passive) is often degraded by a variety of noise types and artifacts. Image restoration, which is a vibrant field of research in the remote sensing community, is the task of recovering the true unknown image from the degraded observed image. Each imaging sensor induces unique noise types and artifacts into the observed image. This fact has led to the expansion of restoration techniques in different paths according to each sensor type. This review paper brings together the advances of image restoration techniques with particular focuses on synthetic aperture radar and hyperspectral images as the most active sub-fields of image restoration in the remote sensing community. We, therefore, provide a comprehensive, discipline-specific starting point for researchers at different levels (i.e., students, researchers, and senior researchers) willing to investigate the vibrant topic of data restoration by supplying sufficient detail and references. Additionally, this review paper accompanies a toolbox to provide a platform to encourage interested students and researchers in the field to further explore the restoration techniques and fast-forward the community. The toolboxes are provided in https://github.com/ImageRestorationToolbox.Comment: This paper is under review in GRS

    A General Proximal Alternating Minimization Method with Application to Nonconvex Nonsmooth 1D Total Variation Denoising

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    We deal with a class of problems whose objective functions are compositions of nonconvex nonsmooth functions, which has a wide range of applications in signal/image processing. We introduce a new auxiliary variable, and an efficient general proximal alternating minimization algorithm is proposed. This method solves a class of nonconvex nonsmooth problems through alternating minimization. We give a brilliant systematic analysis to guarantee the convergence of the algorithm. Simulation results and the comparison with two other existing algorithms for 1D total variation denoising validate the efficiency of the proposed approach. The algorithm does contribute to the analysis and applications of a wide class of nonconvex nonsmooth problems

    Image restoration: Wavelet frame shrinkage, nonlinear evolution PDEs, and beyond

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    In the past few decades, mathematics based approaches have been widely adopted in various image restoration problems; the partial differential equation (PDE) based approach (e.g., the total variation model [L. Rudin, S. Osher, and E. Fatemi, Phys. D, 60 (1992), pp. 259-268] and its generalizations, nonlinear diffusions [P. Perona and J. Malik, IEEE Trans. Pattern Anal. Mach. Intel., 12 (1990), pp. 629-639; F. Catte et al., SIAM J. Numer. Anal., 29 (1992), pp. 182-193], etc.) and wavelet frame based approach are some successful examples. These approaches were developed through different paths and generally provided understanding from different angles of the same problem. As shown in numerical simulations, implementations of the wavelet frame based approach and the PDE based approach quite often end up solving a similar numerical problem with similar numerical behaviors, even though different approaches have advantages in different applications. Since wavelet frame based and PDE based approaches have all been modeling the same types of problems with success, it is natural to ask whether the wavelet frame based approach is fundamentally connected with the PDE based approach when we trace them all the way back to their roots. A fundamental connection of a wavelet frame based approach with a total variation model and its generalizations was established in [J. Cai, B. Dong, S. Osher, and Z. Shen, J. Amer. Math. Soc., 25 (2012), pp. 1033-1089]. This connection gives the wavelet frame based approach a geometric explanation and, at the same time, it equips a PDE based approach with a time frequency analysis. Cai et al. showed that a special type of wavelet frame model using generic wavelet frame systems can be regarded as an approximation of a generic variational model (with the total variation model as a special case) in the discrete setting. A systematic convergence analysis, as the resolution of the image goes to infinity, which is the key step in linking the two approaches, is also given in Cai et al. Motivated by Cai et al. and [Q. Jiang, Appl. Numer. Math., 62 (2012), pp. 51-66], this paper establishes a fundamental connection between the wavelet frame based approach and nonlinear evolution PDEs, provides interpretations and analytical studies of such connections, and proposes new algorithms for image restoration based on the new understandings. Together with the results in [J. Cai et al., J. Amer. Math. Soc., 25 (2012), pp. 1033-1089], we now have a better picture of how the wavelet frame based approach can be used to interpret the general PDE based approach (e.g., the variational models or nonlinear evolution PDEs) and can be used as a new and useful tool in numerical analysis to discretize and solve various variational and PDE models. To be more precise, we shall establish the following: (1) The connections between wavelet frame shrinkage and nonlinear evolution PDEs provide new and inspiring interpretations of both approaches that enable us to derive new PDE models and (better) wavelet frame shrinkage algorithms for image restoration. (2) A generic nonlinear evolution PDE (of parabolic or hyperbolic type) can be approximated by wavelet frame shrinkage with properly chosen wavelet frame systems and carefully designed shrinkage functions. (3) The main idea of this work is beyond the scope of image restoration. Our analysis and discussions indicate that wavelet frame shrinkage is a new way of solving PDEs in general, which will provide a new insight that will enrich the existing theory and applications of numerical PDEs, as well as those of wavelet frames

    Robust Network Topology Inference and Processing of Graph Signals

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    The abundance of large and heterogeneous systems is rendering contemporary data more pervasive, intricate, and with a non-regular structure. With classical techniques facing troubles to deal with the irregular (non-Euclidean) domain where the signals are defined, a popular approach at the heart of graph signal processing (GSP) is to: (i) represent the underlying support via a graph and (ii) exploit the topology of this graph to process the signals at hand. In addition to the irregular structure of the signals, another critical limitation is that the observed data is prone to the presence of perturbations, which, in the context of GSP, may affect not only the observed signals but also the topology of the supporting graph. Ignoring the presence of perturbations, along with the couplings between the errors in the signal and the errors in their support, can drastically hinder estimation performance. While many GSP works have looked at the presence of perturbations in the signals, much fewer have looked at the presence of perturbations in the graph, and almost none at their joint effect. While this is not surprising (GSP is a relatively new field), we expect this to change in the upcoming years. Motivated by the previous discussion, the goal of this thesis is to advance toward a robust GSP paradigm where the algorithms are carefully designed to incorporate the influence of perturbations in the graph signals, the graph support, and both. To do so, we consider different types of perturbations, evaluate their disruptive impact on fundamental GSP tasks, and design robust algorithms to address them.Comment: Dissertatio

    Deep learning methods for solving linear inverse problems: Research directions and paradigms

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    The linear inverse problem is fundamental to the development of various scientific areas. Innumerable attempts have been carried out to solve different variants of the linear inverse problem in different applications. Nowadays, the rapid development of deep learning provides a fresh perspective for solving the linear inverse problem, which has various well-designed network architectures results in state-of-the-art performance in many applications. In this paper, we present a comprehensive survey of the recent progress in the development of deep learning for solving various linear inverse problems. We review how deep learning methods are used in solving different linear inverse problems, and explore the structured neural network architectures that incorporate knowledge used in traditional methods. Furthermore, we identify open challenges and potential future directions along this research line
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