American Institute of Mathematical Sciences

Piecewise constant signals and images are an important kind of data. Typical examples include bar code signals, logos, cartoons, QR codes (Quick Response codes), and text images, which are widely used in both general commercial and automotive industry use. One previous work called a general selective averaging method (GSAM) was introduced to remove noise from them. It chooses homogeneous neighbors from the two closest pixels (one pixel at each side) to update the current pixel. One limitation is that it suffered from appearing sparse noisy pixels in the denoised result when the noise level is high. In this paper, we try to solve this problem by proposing a selective averaging method with multiple neighbors. To update the intensity value at each pixel, the proposed algorithm averages more homogeneous neighbors selected from a large domain, which is based on the property of the local geometry of signals and images. This greatly reduces sparse noisy pixels left in the final result by GSAM. Similarly, our method adopts the Neumann boundary condition at edges, and thus preserves edges well. In 1D case, some theoretical results are given to guarantee the convergence of our algorithm. In 2D case, except eliminating additive Gaussian noise, this algorithm can be used for restoring noisy images corrupted by speckle noise. Intensive experiments on both gray and color image denoising demonstrate that the proposed method is quite effective for piecewise constant image denoising and achieves superior performance visually and quantitatively.

Signals and images recovered from edge-sparsity based reconstruction methods may not truely be sparse in the edge domain, and often result in poor quality reconstruction. Iteratively reweighted methods provide some improvement in accuracy, but at the cost of extended runtime. This paper examines such methods when data are acquired as non-uniform Fourier samples, and then presents a new non-iterative weighted regularization method that first pre-processes the data to determine the precise locations of the non-zero values in the edge domain. Our new method is both accurate and efficient, and outperforms reweighted regularization methods in several numerical experiments.

An inverse obstacle problem for the wave governed by the wave equation in a two layered medium is considered under the framework of the time domain enclosure method. The wave is generated by an initial data supported on a closed ball in the upper half-space, and observed on the same ball over a finite time interval. The unknown obstacle is penetrable and embedded in the lower half-space. It is assumed that the propagation speed of the wave in the upper half-space is greater than that of the wave in the lower half-space, which is excluded in the previous study: Ikehata and Kawashita, Inverse Problems and Imaging 12 (2018), no.5, 1173-1198. In the present case, when the reflected waves from the obstacle enter the upper half-space, the total reflection phenomena occur, which give singularities to the integral representation of the fundamental solution for the reduced transmission problem in the background medium. This fact makes the problem more complicated. However, it is shown that these waves do not have any influence on the leading profile of the indicator function of the time domain enclosure method.

We derive bounds of solutions of the Cauchy problem for general elliptic partial differential equations of second order containing parameter (wave number) \begin{document}$ k $\end{document} which are getting nearly Lipschitz for large \begin{document}$ k $\end{document}. Proofs use energy estimates combined with splitting solutions into low and high frequencies parts, an associated hyperbolic equation and the Fourier-Bros-Iagolnitzer transform to replace the hyperbolic equation with an elliptic equation without parameter \begin{document}$ k $\end{document}. The results suggest a better resolution in prospecting by various (acoustic, electromagnetic, etc) stationary waves with higher wave numbers without any geometric assumptions on domains and observation sites.

The problem of recovering a diffusion coefficient \begin{document}$ a $\end{document} in a second-order elliptic partial differential equation from a corresponding solution \begin{document}$ u $\end{document} for a given right-hand side \begin{document}$ f $\end{document} is considered, with particular focus on the case where \begin{document}$ f $\end{document} is allowed to take both positive and negative values. Identifiability of \begin{document}$ a $\end{document} from \begin{document}$ u $\end{document} is shown under mild smoothness requirements on \begin{document}$ a $\end{document}, \begin{document}$ f $\end{document}, and on the spatial domain \begin{document}$ D $\end{document}, assuming that either the gradient of \begin{document}$ u $\end{document} is nonzero almost everywhere, or that \begin{document}$ f $\end{document} as a distribution does not vanish on any open subset of \begin{document}$ D $\end{document}. Further results of this type under essentially minimal regularity conditions are obtained for the case of \begin{document}$ D $\end{document} being an interval, including detailed information on the continuity properties of the mapping from \begin{document}$ u $\end{document} to \begin{document}$ a $\end{document}.

In this note we discuss the conditional stability issue for the finite dimensional Calderón problem for the fractional Schrödinger equation with a finite number of measurements. More precisely, we assume that the unknown potential \begin{document}$ q \in L^{\infty}(\Omega) $\end{document} in the equation \begin{document}$ ((- \Delta)^s+ q)u = 0 \mbox{ in } \Omega\subset \mathbb{R}^n $\end{document} satisfies the a priori assumption that it is contained in a finite dimensional subspace of \begin{document}$ L^{\infty}(\Omega) $\end{document}. Under this condition we prove Lipschitz stability estimates for the fractional Calderón problem by means of finitely many Cauchy data depending on \begin{document}$ q $\end{document}. We allow for the possibility of zero being a Dirichlet eigenvalue of the associated fractional Schrödinger equation. Our result relies on the strong Runge approximation property of the fractional Schrödinger equation.

We show that the knowledge of Dirichlet to Neumann map for rough \begin{document}$ A $\end{document} and \begin{document}$ q $\end{document} in \begin{document}$ (-\Delta)^m +A\cdot D +q $\end{document} for \begin{document}$ m \geq 2 $\end{document} for a bounded domain in \begin{document}$ \mathbb{R}^n $\end{document}, \begin{document}$ n \geq 3 $\end{document} determines \begin{document}$ A $\end{document} and \begin{document}$ q $\end{document} uniquely. This unique identifiability is proved via construction of complex geometrical optics solutions with sufficient decay of remainder terms, by using property of products of functions in Sobolev spaces.

A new numerical method to solve an inverse source problem for the Helmholtz equation in inhomogenous media is proposed. This method reduces the original inverse problem to a boundary value problem for a coupled system of elliptic PDEs, in which the unknown source function is not involved. The Dirichlet boundary condition is given on the entire boundary of the domain of interest and the Neumann boundary condition is given on a part of this boundary. To solve this problem, the quasi-reversibility method is applied. Uniqueness and existence of the minimizer are proven. A new Carleman estimate is established. Next, the convergence of those minimizers to the exact solution is proven using that Carleman estimate. Results of numerical tests are presented.

A boundary integral based method for the stable reconstruction of missing boundary data is presented for the biharmonic equation. The solution (displacement) is known throughout the boundary of an annular domain whilst the normal derivative and bending moment are specified only on the outer boundary curve. A recent iterative method is applied for the data completion solving mixed problems throughout the iterations. The solution to each mixed problem is represented as a biharmonic single-layer potential. Matching against the given boundary data, a system of boundary integrals is obtained to be solved for densities over the boundary. This system is discretised using the Nyström method. A direct approach is also given representing the solution of the ill-posed problem as a biharmonic single-layer potential and applying the similar techniques as for the mixed problems. Tikhonov regularization is employed for the solution of the corresponding discretised system. Numerical results are presented for several annular domains showing the efficiency of both data completion approaches.

We study the properties of a regularization method for inverse problems with joint Kullback-Leibler data term and regularization when the data and the operator are corrupted by some noise. We show the convergence of the method and we obtain convergence rates for the approximate solution of the inverse problem and for the operator when it is characterized by some kernel, under the assumption that some source conditions are satisfied. Numerical results showing the effect of the noise levels on the reconstructed solution are provided for Spectral Computerized Tomography.