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Inverse Problems and Imaging

April 2022 , Volume 16 , Issue 2

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Variational source conditions for inverse Robin and flux problems by partial measurements
De-Han Chen, Daijun Jiang, Irwin Yousept and Jun Zou
2022, 16(2): 283-304 doi: 10.3934/ipi.2021050 +[Abstract](1132) +[HTML](372) +[PDF](590.56KB)
Abstract:

This work is devoted to the convergence analysis of the Tikhonov regularization for the inverse Robin and flux problems.Both inverse problems aim at recovering a respective physical quantity on an inaccessible part of the boundary through some measurement on a partial accessible boundary.The convergence and convergence rate in the desirable L2-norm are derived based on two new logarithmic type stabilities (only in some weak norms,e.g.,the negative Sobolev norms), which enable us to construct and rigorously verify the required variational source conditions.

 

Addendum: The abstract is added since it was missing when the article was published online. To reflect this addition an Addendum is published in this same IPI 16-2 April 2022 regular issue. We apologize for any inconvenience this may cause.

Inverse problems for the fractional Laplace equation with lower order nonlinear perturbations
Ru-Yu Lai and Laurel Ohm
2022, 16(2): 305-323 doi: 10.3934/ipi.2021051 +[Abstract](1012) +[HTML](339) +[PDF](434.83KB)
Abstract:

We study the inverse problem for the fractional Laplace equation with multiple nonlinear lower order terms. We show that the direct problem is well-posed and the inverse problem is uniquely solvable. More specifically, the unknown nonlinearities can be uniquely determined from exterior measurements under suitable settings.

Fourier method for reconstructing elastic body force from the coupled-wave field
Xianchao Wang, Jiaqi Zhu, Minghui Song and Wei Wu
2022, 16(2): 325-340 doi: 10.3934/ipi.2021052 +[Abstract](1081) +[HTML](397) +[PDF](1957.76KB)
Abstract:

This paper is concerned with the inverse source problem of the time-harmonic elastic waves. A novel non-iterative reconstruction scheme is proposed for determining the elastic body force by using the multi-frequency Fourier expansion. The key ingredient of the approach is to choose appropriate admissible frequencies and establish an relationship between the Fourier coefficients and the coupled-wave field of compressional wave and shear wave. Both theoretical justifications and numerical examples are presented to verify the validity and robustness of the proposed method.

PCA reduced Gaussian mixture models with applications in superresolution
Johannes Hertrich, Dang-Phuong-Lan Nguyen, Jean-Francois Aujol, Dominique Bernard, Yannick Berthoumieu, Abdellatif Saadaldin and Gabriele Steidl
2022, 16(2): 341-366 doi: 10.3934/ipi.2021053 +[Abstract](1315) +[HTML](287) +[PDF](1582.44KB)
Abstract:

Despite the rapid development of computational hardware, the treatment of large and high dimensional data sets is still a challenging problem. The contribution of this paper to the topic is twofold. First, we propose a Gaussian mixture model in conjunction with a reduction of the dimensionality of the data in each component of the model by principal component analysis, which we call PCA-GMM. To learn the (low dimensional) parameters of the mixture model we propose an EM algorithm whose M-step requires the solution of constrained optimization problems. Fortunately, these constrained problems do not depend on the usually large number of samples and can be solved efficiently by an (inertial) proximal alternating linearized minimization algorithm. Second, we apply our PCA-GMM for the superresolution of 2D and 3D material images based on the approach of Sandeep and Jacob. Numerical results confirm the moderate influence of the dimensionality reduction on the overall superresolution result.

A fuzzy edge detector driven telegraph total variation model for image despeckling
Sudeb Majee, Subit K. Jain, Rajendra K. Ray and Ananta K. Majee
2022, 16(2): 367-396 doi: 10.3934/ipi.2021054 +[Abstract](1014) +[HTML](382) +[PDF](6261.66KB)
Abstract:

Speckle noise suppression is a challenging and crucial pre-processing stage for higher-level image analysis. In this work, a new attempt has been made using telegraph total variation equation and fuzzy set theory for image despeckling. The intuitionistic fuzzy divergence function has been used to distinguish between edges and noise. To the best of the authors' knowledge, most of the studies on the multiplicative speckle noise removal process focus only on diffusion-based filters, and little attention has been paid to the study of fuzzy set theory. The proposed approach enjoys the benefits of both telegraph total variation equation and fuzzy edge detector, which is robust to noise and preserves image structural details. Moreover, we establish the existence and uniqueness of weak solutions of a regularized version of the present system using the Schauder fixed point theorem. With the proposed technique, despeckling is carried out on natural, real synthetic aperture radar, and real ultrasound images. The experimental results computed by the suggested method are reported, which are found better in terms of noise elimination and detail/edge preservation, concerning the existing approaches.

An inverse source problem for the stochastic wave equation
Xiaoli Feng, Meixia Zhao, Peijun Li and Xu Wang
2022, 16(2): 397-415 doi: 10.3934/ipi.2021055 +[Abstract](1069) +[HTML](284) +[PDF](494.87KB)
Abstract:

This paper is concerned with an inverse source problem for the stochastic wave equation driven by a fractional Brownian motion. Given the random source, the direct problem is to study the solution of the stochastic wave equation. The inverse problem is to determine the statistical properties of the source from the expectation and covariance of the final-time data. For the direct problem, it is shown to be well-posed with a unique mild solution. For the inverse problem, the uniqueness is proved for a certain class of functions and the instability is characterized. Numerical experiments are presented to illustrate the reconstructions by using a truncation-based regularization method.

Small defects reconstruction in waveguides from multifrequency one-side scattering data
Éric Bonnetier, Angèle Niclas, Laurent Seppecher and Grégory Vial
2022, 16(2): 417-450 doi: 10.3934/ipi.2021056 +[Abstract](821) +[HTML](285) +[PDF](861.81KB)
Abstract:

Localization and reconstruction of small defects in acoustic or electromagnetic waveguides is of crucial interest in nondestructive evaluation of structures. The aim of this work is to present a new multi-frequency inversion method to reconstruct small defects in a 2D waveguide. Given one-side multi-frequency wave field measurements of propagating modes, we use a Born approximation to provide a \begin{document}$ \text{L}^2 $\end{document}-stable reconstruction of three types of defects: a local perturbation inside the waveguide, a bending of the waveguide, and a localized defect in the geometry of the waveguide. This method is based on a mode-by-mode spacial Fourier inversion from the available partial data in the Fourier domain. Indeed, in the available data, some high and low spatial frequency information on the defect are missing. We overcome this issue using both a compact support hypothesis and a minimal smoothness hypothesis on the defects. We also provide a suitable numerical method for efficient reconstruction of such defects and we discuss its applications and limits.

Weighted area constraints-based breast lesion segmentation in ultrasound image analysis
Qianting Ma, Tieyong Zeng, Dexing Kong and Jianwei Zhang
2022, 16(2): 451-466 doi: 10.3934/ipi.2021057 +[Abstract](1106) +[HTML](231) +[PDF](3906.9KB)
Abstract:

Breast ultrasound segmentation is a challenging task in practice due to speckle noise, low contrast and blurry boundaries. Although numerous methods have been developed to solve this problem, most of them can not produce a satisfying result due to uncertainty of the segmented region without specialized domain knowledge. In this paper, we propose a novel breast ultrasound image segmentation method that incorporates weighted area constraints using level set representations. Specifically, we first use speckle reducing anisotropic diffusion filter to suppress speckle noise, and apply the Grabcut on them to provide an initial segmentation result. In order to refine the resulting image mask, we propose a weighted area constraints-based level set formulation (WACLSF) to extract a more accurate tumor boundary. The major contribution of this paper is the introduction of a simple nonlinear constraint for the regularization of probability scores from a classifier, which can speed up the motion of zero level set to move to a desired boundary. Comparisons with other state-of-the-art methods, such as FCN-AlexNet and U-Net, show the advantages of our proposed WACLSF-based strategy in terms of visual view and accuracy.

Counterexamples to inverse problems for the wave equation
Tony Liimatainen and Lauri Oksanen
2022, 16(2): 467-479 doi: 10.3934/ipi.2021058 +[Abstract](968) +[HTML](210) +[PDF](399.14KB)
Abstract:

We construct counterexamples to inverse problems for the wave operator on domains in \begin{document}$ \mathbb{R}^{n+1} $\end{document}, \begin{document}$ n \ge 2 $\end{document}, and on Lorentzian manifolds. We show that non-isometric Lorentzian metrics can lead to same partial data measurements, which are formulated in terms certain restrictions of the Dirichlet-to-Neumann map. The Lorentzian metrics giving counterexamples are time-dependent, but they are smooth and non-degenerate. On \begin{document}$ \mathbb{R}^{n+1} $\end{document} the metrics are conformal to the Minkowski metric.

Addendum to: "Variational source conditions for inverse Robin and flux problems by partial measurements"
De-Han Chen, Daijun Jiang, Irwin Yousept and Jun Zou
2022, 16(2): 481-481 doi: 10.3934/ipi.2022003 +[Abstract](5290) +[HTML](67) +[PDF](139.72KB)
Abstract:

2021 Impact Factor: 1.483
5 Year Impact Factor: 1.462
2021 CiteScore: 2.6

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