American Institute of Mathematical Sciences

ISSN:
1930-8337

eISSN:
1930-8345

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Inverse Problems & Imaging

December 2017 , Volume 11 , Issue 6

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2017, 11(6): 917-948 doi: 10.3934/ipi.2017043 +[Abstract](4156) +[HTML](229) +[PDF](1923.8KB)
Abstract:

The limited-angle projection data of an object, in some practical applications of computed tomography (CT), are obtained due to the restriction of scanning condition. In these situations, since the projection data are incomplete, some limited-angle artifacts will be presented near the edges of reconstructed image using some classical reconstruction algorithms, such as filtered backprojection (FBP). The reconstructed image can be fine approximated by sparse coefficients under a proper wavelet tight frame, and the quality of reconstructed image can be improved by an available prior image. To deal with limited-angle CT reconstruction problem, we propose a minimization model that is based on wavelet tight frame and a prior image, and perform this minimization problem efficiently by iteratively minimizing separately. Moreover, we show that each bounded sequence, which is generated by our method, converges to a critical or a stationary point. The experimental results indicate that our algorithm can efficiently suppress artifacts and noise and preserve the edges of reconstructed image, what's more, the introduced prior image will not miss the important information that is not included in the prior image.

2017, 11(6): 949-974 doi: 10.3934/ipi.2017044 +[Abstract](3367) +[HTML](164) +[PDF](4320.1KB)
Abstract:

Restoration of images contaminated by multiplicative noise (also known as speckle noise) is a key issue in coherent image processing. Notice that images under consideration are often highly compressible in certain suitably chosen transform domains. By exploring this intrinsic feature embedded in images, this paper introduces a variational restoration model for multiplicative noise reduction that consists of a term reflecting the observed image and multiplicative noise, a quadratic term measuring the closeness of the underlying image in a transform domain to a sparse vector, and a sparse regularizer for removing multiplicative noise. Being different from popular existing models which focus on pursuing convexity, the proposed sparsity-aware model may be nonconvex depending on the conditions of the parameters of the model for achieving the optimal denoising performance. An algorithm for finding a critical point of the objective function of the model is developed based on coupled fixed-point equations expressed in terms of the proximity operator of functions that appear in the objective function. Convergence analysis of the algorithm is provided. Experimental results are shown to demonstrate that the proposed iterative algorithm is sensitive to some initializations for obtaining the best restoration results. We observe that the proposed method with SAR-BM3D filtering images as initial estimates can remarkably outperform several state-of-art methods in terms of the quality of the restored images.

2017, 11(6): 975-996 doi: 10.3934/ipi.2017045 +[Abstract](2387) +[HTML](203) +[PDF](6720.5KB)
Abstract:

In this paper, we propose a new augmented Lagrangian method for the mean curvature based image denoising model [33]. Different from the previous works in [21,35], this new method only involves two Lagrange multipliers, which significantly reduces the effort of choosing appropriate penalization parameters to ensure the convergence of the iterative process of finding the associated saddle points. With this new algorithm, we demonstrate the features of the model numerically, including the preservation of image contrasts and object corners, as well as its capability of generating smooth patches of image graphs. The data selection property and the role of the spatial mesh size for the model performance are also discussed.

2017, 11(6): 997-1025 doi: 10.3934/ipi.2017046 +[Abstract](1917) +[HTML](142) +[PDF](493.6KB)
Abstract:

We study a variational problem for simultaneous video inpainting and motion estimation. We consider a functional proposed by Lauze and Nielsen [25] and we study, by means of the relaxation method of the Calculus of Variations, a slightly modified version of this functional. The domain of the relaxed functional is constituted of functions of bounded variation and we compute a representation formula of the relaxed functional. The representation formula shows the role of discontinuities of the various functions involved in the variational model. The present study clarifies the variational properties of the functional proposed in [25] for motion compensated video inpainting.

2017, 11(6): 1027-1046 doi: 10.3934/ipi.2017047 +[Abstract](1677) +[HTML](133) +[PDF](473.4KB)
Abstract:

This work considers the problem of recovering small electromagnetic inhomogeneities in a bounded domain \begin{document}$Ω \subset \mathbb{R}^3$\end{document}, from a single Cauchy data, at a fixed frequency. This problem has been considered by several authors, in particular in [4]. In this paper, we revisit this work with the objective of providing another identification method and establishing stability results from a single Cauchy data and at a fixed frequency. Our approach is based on the asymptotic expansion of the boundary condition derived in [4] and the extension of the direct algebraic algorithm proposed in [1].

2017, 11(6): 1047-1070 doi: 10.3934/ipi.2017048 +[Abstract](2985) +[HTML](220) +[PDF](1338.5KB)
Abstract:

This paper develops two accelerated Bregman Operator Splitting (BOS) algorithms with backtracking for solving regularized large-scale linear inverse problems, where the regularization term may not be smooth. The first algorithm improves the rate of convergence for BOSVS [5] in terms of the smooth component in the objective function by incorporating Nesterov's multi-step acceleration scheme under the assumption that the feasible set is bounded. The second algorithm is capable of dealing with the case where the feasible set is unbounded. Moreover, it allows more aggressive stepsize than that in the first scheme by properly selecting the penalty parameter and jointly updating the acceleration parameter and stepsize. Both algorithms exhibit better practical performance than BOSVS and AADMM [21], while preserve the same accelerated rate of convergence as that for AADMM. The numerical results on total-variation based image reconstruction problems indicate the effectiveness of the proposed algorithms.

2017, 11(6): 1071-1090 doi: 10.3934/ipi.2017049 +[Abstract](2475) +[HTML](1196) +[PDF](2271.0KB)
Abstract:

In this paper, we investigate the relations between the Radon and weighted divergent beam and cone transforms. Novel inversion formulas are derived for the latter two. The weighted cone transform arises, for instance, in image reconstruction from the data obtained by Compton cameras, which have promising applications in various fields, including biomedical and homeland security imaging and gamma ray astronomy. The inversion formulas are applicable for a wide variety of detector geometries in any dimension. The results of numerical implementation of some of the formulas in dimensions two and three are also provided.

2017, 11(6): 1091-1105 doi: 10.3934/ipi.2017050 +[Abstract](2371) +[HTML](162) +[PDF](733.7KB)
Abstract:

In this paper, we introduce a direct method for the inverse scattering problems in a periodic waveguide from near-field scattered data. The direct scattering problem is to simulate the point sources scattered by a sound-soft obstacle embedded in the periodic waveguide, and the aim of the inverse problem is to reconstruct the obstacle from the near-field data measured on line segments outside the obstacle. Firstly, we will approximate the scattered field by some solutions of a series of Dirichlet exterior problems, and then the shape of the obstacle can be deduced directly from the Dirichlet boundary condition. We will also show that the approximation procedure is reasonable as the solutions of the Dirichlet exterior problems are dense in the set of scattered fields. Finally, we will give several examples to show that this method works well for different periodic waveguides.

2017, 11(6): 1107-1119 doi: 10.3934/ipi.2017051 +[Abstract](2009) +[HTML](150) +[PDF](495.0KB)
Abstract:

We extend the applicability of the Generalized Linear Sampling Method (GLSM)[2] and the Factorization Method (FM)[16] to the case of inhomogeneities where the contrast changes sign. Both methods give an exact characterization of the target shapes in terms of the farfield operator (at a fixed frequency) using the coercivity property of a special solution operator. We prove this property assuming that the contrast has a fixed sign in a neighborhood of the inhomogeneities boundary. We treat both isotropic and anisotropic scatterers with possibly different supports for the isotropic and anisotropic parts. We finally validate the methods through some numerical tests in two dimensions.

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