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 Physics , 2012, Abstract: This paper proposes an extension to the classical 3D variational data assimilation approach by explicitly incorporating as a prior information, the transform-domain sparsity observed in a large class of geophysical signals. In particular, the proposed framework extends the maximum likelihood estimation of the analysis state to the maximum a posteriori estimator, from a Bayesian perspective. The promise of the methodology is demonstrated via application to a 1D synthetic example.
 Physics , 2012, DOI: 10.1002/wrcr.20424 Abstract: Improved estimation of hydrometeorological states from down-sampled observations and background model forecasts in a noisy environment, has been a subject of growing research in the past decades. Here, we introduce a unified framework that ties together the problems of downscaling, data fusion and data assimilation as ill-posed inverse problems. This framework seeks solutions beyond the classic least squares estimation paradigms by imposing proper regularization, which are constraints consistent with the degree of smoothness and probabilistic structure of the underlying state. We review relevant regularization methods in derivative space and extend classic formulations of the aforementioned problems with particular emphasis on hydrologic and atmospheric applications. Informed by the statistical characteristics of the state variable of interest, the central results of the paper suggest that proper regularization can lead to a more accurate and stable recovery of the true state and hence more skillful forecasts. In particular, using the Tikhonov and Huber regularization in the derivative space, the promise of the proposed framework is demonstrated in static downscaling and fusion of synthetic multi-sensor precipitation data, while a data assimilation numerical experiment is presented using the heat equation in a variational setting.
 Statistics , 2011, DOI: 10.1109/TSP.2011.2129515 Abstract: This paper addresses the problem of inferring sparse causal networks modeled by multivariate auto-regressive (MAR) processes. Conditions are derived under which the Group Lasso (gLasso) procedure consistently estimates sparse network structure. The key condition involves a "false connection score." In particular, we show that consistent recovery is possible even when the number of observations of the network is far less than the number of parameters describing the network, provided that the false connection score is less than one. The false connection score is also demonstrated to be a useful metric of recovery in non-asymptotic regimes. The conditions suggest a modified gLasso procedure which tends to improve the false connection score and reduce the chances of reversing the direction of causal influence. Computational experiments and a real network based electrocorticogram (ECoG) simulation study demonstrate the effectiveness of the approach.
 中国物理 B , 2011, Abstract: A new method of constructing a sea level pressure field from satellite microwave scatterometer measurements is presented. It is based on variational assimilation in combination with a regularization method using geostrophic vorticity to construct a sea level pressure field from scatterometer data that are given in this paper, which offers a new idea for the application of scatterometer measurements. Firstly, the geostrophic vorticity from the scatterometer data is computed to construct the observation field, and the vorticity field in an area and the sea level pressure on the borders are assimilated. Secondly, the gradient of sea level pressure (semi-norm) is used as the stable functional to educe the adjoint system, the adjoint boundary condition and the gradient of the cost functional in which a weight parameter is introduced for the harmony of the system and the Tikhonov regularization techniques in inverse problem are used to overcome the ill-posedness of the assimilation. Finally, the iteration method of the sea level pressure field is developed.
 Computer Science , 2013, Abstract: For high-dimensional sparse parameter estimation problems, Log-Sum Penalty (LSP) regularization effectively reduces the sampling sizes in practice. However, it still lacks theoretical analysis to support the experience from previous empirical study. The analysis of this article shows that, like $\ell_0$-regularization, $O(s)$ sampling size is enough for proper LSP, where $s$ is the non-zero components of the true parameter. We also propose an efficient algorithm to solve LSP regularization problem. The solutions given by the proposed algorithm give consistent parameter estimations under less restrictive conditions than $\ell_1$-regularization.
 Mathematics , 2015, Abstract: This paper introduces a novel variational approach for image compression motivated by recent PDE-based approaches combining edge detection and Laplacian inpainting. The essential feature is to encode the image via a sparse vector field, ideally concentrating on a set of measure zero. An equivalent reformulation of the compression approach leads to a variational model resembling the ROF-model for image denoising, hence we further study the properties of the effective regularization functional introduced by the novel approach and discuss similarities to TV and TGV functionals. Moreover we computationally investigate the behaviour of the model with sparse vector fields for compression in particular for high resolution images and give an outlook towards denoising.
 Mathematical Problems in Engineering , 2014, DOI: 10.1155/2014/917040 Abstract: Poissonian image deconvolution is a key issue in various applications, such as astronomical imaging, medical imaging, and electronic microscope imaging. A large amount of literature on this subject is analysis-based methods. These methods assign various forward measurements of the image. Meanwhile, synthesis-based methods are another well-known class of methods. These methods seek a reconstruction of the image. In this paper, we propose an approach that combines analysis with synthesis methods. The method is proposed to address Poissonian image deconvolution problem by minimizing the energy functional, which is composed of a sparse representation prior over a learned dictionary, the data fidelity term, and framelet based analysis prior constraint as the regularization term. The minimization problem can be efficiently solved by the split Bregman technique. Experiments demonstrate that our approach achieves better results than many state-of-the-art methods, in terms of both restoration accuracy and visual perception. 1. Introduction Poissonian image deconvolution appears in various applications such as astronomical imaging [1], medical imaging [2], and electronic microscope imaging [3]. It aims to reconstruct a high quality image from the degraded image . Mathematically, the process of Poissonian image deconvolution can be generally modeled by where denotes the point spread function (PSF), denotes the unknown image to be estimated, denotes the Poisson noise process, denotes the blurred noisy image, and is the length of image vector which is stacked by columns. It is known that Poissonian image deconvolution is a typical ill-posed inverse problem. In general, the solution of (1) is not unique. Prior knowledge of image, including analysis-based and synthesis-based priors, can be used to address this problem. For an overview of the two classes of priors, we refer to [4]. Analysis-based priors are frequently used as regularization term in energy functional where is the regularization constraint term. Two main analysis-based methods have been proposed to solve problem (2): the total variation (TV) [5, 6] based methods and wavelet frames-based methods. TV-based methods have shown good performance on blurred images for discontinuous solution and edge-preserving advantages. Many authors have combined the TV regularization term with variant methods to solve problem (2). For example, Sawatzky et al. combined expectation-maximization algorithm with TV regularization [7] in positron emission tomography. Setzer et al. considered using TV regularization term with split
 中国物理 B , 2011, Abstract: The sea level pressure field can be computed from sea surface winds retrieved from satellite microwave scatterometer measurements, based on variational assimilation in combination with a regularization method given in part I of this paper. First, the validity of the new method is proved with a simulation experiment. Then, a new processing procedure for the sea level pressure retrieval is built by combining the geostrophic wind, which is computed from the scatterometer 10-meter wind using the University of Washington planetary boundary layer model using this method. Finally, the feasibility of the method is proved using an actual case study.
 物理学报 , 2010, Abstract: multiple solution scheme, two dimensional variational assimilation, generalized variational optimization analysis, regularization method
 中国物理 B , 2010, Abstract: multiple solution scheme, two dimensional variational assimilation, generalized variational optimization analysis, regularization method
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