# American Institute of Mathematical Sciences

ISSN:
1930-8337

eISSN:
1930-8345

All Issues

## Inverse Problems & Imaging

November 2016 , Volume 10 , Issue 4

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2016, 10(4): 869-898 doi: 10.3934/ipi.2016025 +[Abstract](2808) +[PDF](560.1KB)
Abstract:
We consider a variational formulation of the inverse Stefan problem, where information on the heat flux on the fixed boundary is missing and must be found along with the temperature and free boundary. We employ optimal control framework, where boundary heat flux and free boundary are components of the control vector, and optimality criteria consist of the minimization of the sum of $L_2$-norm declinations from the available measurement of the temperature flux on the fixed boundary and available information on the phase transition temperature on the free boundary. This approach allows one to tackle situations when the phase transition temperature is not known explicitly, and is available through measurement with possible error. It also allows for the development of iterative numerical methods of least computational cost due to the fact that for every given control vector, the parabolic PDE is solved in a fixed region instead of full free boundary problem. In Inverse Problems and Imaging, 7, 2(2013), 307-340 we proved well-posedness in Sobolev spaces framework and convergence of time-discretized optimal control problems. In this paper we perform full discretization and prove convergence of the discrete optimal control problems to the original problem both with respect to cost functional and control.
2016, 10(4): 899-914 doi: 10.3934/ipi.2016026 +[Abstract](1709) +[PDF](566.4KB)
Abstract:
Localized basis set functions related to spherical wave functions are constructed for scalar and vector 3D tomography. The functions are truncated in a spherical domain of radius $r_0$, so that these vanish outside the domain. The analytical form of the respective scalar and vector ray transforms are obtained. So, the inversion algorithm can be reduced to solving the linear system of equations because of orthogonality and completeness of the basis functions.
2016, 10(4): 915-941 doi: 10.3934/ipi.2016027 +[Abstract](1425) +[PDF](1802.7KB)
Abstract:
We study an inverse scattering problem for Maxwell's equations in terminating waveguides, where localized reflectors are to be imaged using a remote array of sensors. The array probes the waveguide with waves and measures the scattered returns. The mathematical formulation of the inverse scattering problem is based on the electromagnetic Lippmann-Schwinger integral equation and an explicit calculation of the Green tensor. The image formation is carried with reverse time migration and with $\ell_1$ optimization.
2016, 10(4): 943-975 doi: 10.3934/ipi.2016028 +[Abstract](2249) +[PDF](1120.8KB)
Abstract:
We present a systematic construction of FEM-based dimension-independent (discretization-invariant) Markov chain Monte Carlo (MCMC) approaches to explore PDE-constrained Bayesian inverse problems in infinite dimensional parameter spaces. In particular, we consider two frameworks to achieve this goal: Metropolize-then-discretize and discretize-then-Metropolize. The former refers to the method of discretizing function-space MCMC methods. The latter, on the other hand, first discretizes the Bayesian inverse problem and then proposes MCMC methods for the resulting discretized posterior probability density. In general, these two frameworks do not commute, that is, the resulting finite dimensional MCMC algorithms are not identical. The discretization step of the former may not be trivial since it involves both numerical analysis and probability theory, while the latter, perhaps easier'', may not be discretization-invariant using traditional approaches. This paper constructively develops finite element (FEM) discretization schemes for both frameworks and shows that both commutativity and discretization-invariance are attained. In particular, it shows how to construct discretize-then-Metropolize approaches for both Metropolis-adjusted Langevin algorithm and the hybrid Monte Carlo method that commute with their Metropolize-then-discretize counterparts. The key that enables this achievement is a proper FEM discretization of the prior, the likelihood, and the Bayes' formula, together with a correct definition of quantities such as the gradient and the covariance matrix in discretized finite dimensional parameter spaces. The implication is that practitioners can take advantage of the developments in this paper to straightforwardly construct discretization-invariant discretize-then-Metropolize MCMC for large-scale inverse problems. Numerical results for one- and two-dimensional elliptic inverse problems with up to $17899$ parameters are presented to support the proposed developments.
2016, 10(4): 977-1006 doi: 10.3934/ipi.2016029 +[Abstract](1751) +[PDF](655.0KB)
Abstract:
We are interested in a time harmonic acoustic problem in a waveguide containing flies. The flies are modelled by small sound soft obstacles. We explain how they should arrange to become invisible to an observer sending waves from $-\infty$ and measuring the resulting scattered field at the same position. We assume that the flies can control their position and/or their size. Both monomodal and multimodal regimes are considered. On the other hand, we show that any sound soft obstacle (non necessarily small) embedded in the waveguide always produces some non exponentially decaying scattered field at $+\infty$ for wavenumbers smaller than a constant that we explicit. As a consequence, for such wavenumbers, the flies cannot be made completely invisible to an observer equipped with a measurement device located at $+\infty$.
2016, 10(4): 1007-1036 doi: 10.3934/ipi.2016030 +[Abstract](2617) +[PDF](2812.9KB)
Abstract:
We provide a rigorous Bayesian formulation of the EIT problem in an infinite dimensional setting, leading to well-posedness in the Hellinger metric with respect to the data. We focus particularly on the reconstruction of binary fields where the interface between different media is the primary unknown. We consider three different prior models -- log-Gaussian, star-shaped and level set. Numerical simulations based on the implementation of MCMC are performed, illustrating the advantages and disadvantages of each type of prior in the reconstruction, in the case where the true conductivity is a binary field, and exhibiting the properties of the resulting posterior distribution.
2016, 10(4): 1037-1055 doi: 10.3934/ipi.2016031 +[Abstract](2207) +[PDF](1460.7KB)
Abstract:
In this paper, we study the problem of image colorization based on the propagation from given color pixels to the other grey-level pixels in grayscale images. We propose to use a coupled total variation model with curvature information of luminance channel to control the colorization process. There are two distinct advantages of the proposed model: (i) the involved optimization problem is convex and it is not sensitive to initial guess of colorization procedure; (ii) the proposed model makes use of curvature information to control the color diffusion process which is more effectively than that by using the gradient information. The existence of the minimizer of the proposed model can be shown, and the numerical solver based on convex programming techniques can be developed to solve the resulting model very efficiently. Experimental results are reported to demonstrate that the performance of the proposed model is better than those of the other color propagation models, especially when we deal with large regions of grayscale images for colorization.
2016, 10(4): 1057-1085 doi: 10.3934/ipi.2016032 +[Abstract](2178) +[PDF](2010.9KB)
Abstract:
In this paper, a reconstruction method for the spatially distributed dielectric constant of a medium from the back scattering wave field in the frequency domain is considered. Our approach is to propose a globally convergent algorithm, which does not require any knowledge of a small neighborhood of the solution of the inverse problem in advance. The Quasi-Reversibility Method (QRM) is used in the algorithm. The convergence of the QRM is proved via a Carleman estimate. The method is tested on both computationally simulated and experimental data.
2016, 10(4): 1087-1110 doi: 10.3934/ipi.2016033 +[Abstract](2600) +[PDF](1211.1KB)
Abstract:
We contribute to the mathematical modeling and analysis of magnetic particle imaging which is a promising new in-vivo imaging modality. Concerning modeling, we develop a structured decomposition of the imaging process and extract its core part which we reveal to be common to all previous contributions in this context. The central contribution of this paper is the development of reconstruction formulae for MPI in 2D and 3D. Until now, in the multivariate setup, only time consuming measurement-based approaches are available, whereas reconstruction formulae are only available in 1D. The 2D and the 3D (describing the real world) reconstruction formulae which we derive here are significantly different from the 1D situation -- in particular there is no Dirac property in dimensions greater than one in the high resolution limit. As a further result of our analysis, we conclude that the reconstruction problem in MPI is severely ill-posed. Finally, we obtain a model-based reconstruction algorithm.
2016, 10(4): 1111-1139 doi: 10.3934/ipi.2016034 +[Abstract](1708) +[PDF](528.4KB)
Abstract:
We examine the location of the eigenvalues of the generator $G$ of a semi-group $V(t) = e^{tG},\: t \geq 0,$ related to the wave equation in an unbounded domain $\Omega \subset \mathbb{R}^d$ with dissipative boundary condition $\partial_{\nu}u - \gamma(x) \partial_t u = 0$ on $\Gamma = \partial \Omega.$ We study two cases: $(A): \: 0 < \gamma(x) < 1,\: \forall x \in \Gamma$ and $(B):\: 1 < \gamma(x), \: \forall x \in \Gamma.$ We prove that for every $0 < \epsilon \ll 1,$ the eigenvalues of $G$ in the case $(A)$ lie in the region $\Lambda_{\epsilon} = \{ z \in \mathbb{C}:\: |Re z | \leq C_{\epsilon} (|Im z|^{\frac{1}{2} + \epsilon} + 1), \: Re z < 0\},$ while in the case $(B)$ for every $0 < \epsilon \ll 1$ and every $N \in \mathbb{N}$ the eigenvalues lie in $\Lambda_{\epsilon} \cup {\mathcal R}_N,$ where ${\mathcal R}_N = \{z \in \mathbb{C}:\: |Im z| \leq C_N (|Re z| + 1)^{-N},\: Re z < 0\}.$
2016, 10(4): 1141-1147 doi: 10.3934/ipi.2016035 +[Abstract](1815) +[PDF](335.5KB)
Abstract:
We show that given two hyperbolic Dirichlet to Neumann maps associated to two Riemannian metrics of a Riemannian manifold with boundary which coincide near the boundary are close then the lens data of the two metrics is the same. As a consequence, we prove uniqueness of recovery a conformal factor (sound speed) locally under some conditions on the latter.
2016, 10(4): 1149-1180 doi: 10.3934/ipi.2016036 +[Abstract](2392) +[PDF](1909.5KB)
Abstract:
We consider a geometric approach to graph partitioning based on the graph Beltrami energy, a discrete version of a functional that appears in classical minimal surface problems. More specifically, the optimality criterion is given by the sum of the minimal Beltrami energies of the partition components. Since the Beltrami energy interpolates between the Total Variation and Dirichlet energies, various results for optimal partitions for these two energies can be recovered. We adapt primal-dual convex optimization methods to solve for the minimal Beltrami energy for each component of a given partition. A rearrangement algorithm is proposed to find the graph partition to minimize a relaxed version of the objective. The method is applied to several clustering problems on graphs constructed from manifold discretizations, synthetic data, the MNIST handwritten digit dataset, and image segmentation. The model has a semisupervised extension and provides a natural representative for the clusters as well.

2018  Impact Factor: 1.469