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Inverse Problems & Imaging

August 2015 , Volume 9 , Issue 3

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Identifying defects in an unknown background using differential measurements
Lorenzo Audibert, Alexandre Girard and Houssem Haddar
2015, 9(3): 625-643 doi: 10.3934/ipi.2015.9.625 +[Abstract](701) +[PDF](722.8KB)
Abstract:
We present a new qualitative imaging method capable of selecting defects in complex and unknown background from differential measurements of farfield operators: i.e. far measurements of scattered waves in the cases with and without defects. Indeed, the main difficulty is that the background physical properties are unknown. Our approach is based on a new exact characterization of a scatterer domain in terms of the far field operator range and the link with solutions to so-called interior transmission problems. We present the theoretical foundations of the method and some validating numerical experiments in a two dimensional setting.
Determining a distributed conductance parameter for a neuronal cable model defined on a tree graph
Sergei Avdonin and Jonathan Bell
2015, 9(3): 645-659 doi: 10.3934/ipi.2015.9.645 +[Abstract](738) +[PDF](460.2KB)
Abstract:
In this paper we solve the inverse problem of recovering a single spatially distributed conductance parameter in a cable equation model (one-dimensional diffusion) defined on a metric tree graph that represents a dendritic tree of a neuron. Dendrites of nerve cells have membranes with spatially distributed densities of ionic channels and hence non-uniform conductances. We employ the boundary control method that gives a unique reconstruction and an algorithmic approach.
Periodic spline-based frames for image restoration
Amir Averbuch, Pekka Neittaanmäki and Valery Zheludev
2015, 9(3): 661-707 doi: 10.3934/ipi.2015.9.661 +[Abstract](655) +[PDF](6445.8KB)
Abstract:
We present a design scheme that generates tight and semi-tight frames in discrete-time periodic signals space originated from four-channel perfect reconstruction periodic filter banks. Filter banks are derived from interpolating and quasi-interpolating polynomial and discrete splines. Each filter bank comprises one linear phase low-pass filter (in most cases interpolating) and one high-pass filter, whose magnitude's response mirrors that of a low-pass filter. These filter banks comprise two band-pass filters. We introduce local discrete vanishing moments (LDVM). When the frame is tight, analysis framelets coincide with their synthesis counterparts. However, for semi-tight frames, we swap LDVM between synthesis and analysis framelets. The design scheme is generic and it enables us to design framelets with any number of LDVM. The computational complexity of the the framelet transforms, which consists of calculating the forward and the inverse FFTs, does not depend on the number of LDVM and does depend on the size of the the impulse response fi lters. The designed frames are used for image restoration tasks, which were degraded by blurring, random noise and missing pixels. The images were restored by the application of the Split Bregman Iterations method. The frames performances are evaluated. A potential application of this methodology is the design of a snapshot hyperspectral imager that is based on a regular digital camera. All these imaging applications are described.
Stability and uniqueness for a two-dimensional inverse boundary value problem for less regular potentials
Eemeli Blåsten, Oleg Yu. Imanuvilov and Masahiro Yamamoto
2015, 9(3): 709-723 doi: 10.3934/ipi.2015.9.709 +[Abstract](702) +[PDF](387.4KB)
Abstract:
We consider inverse boundary value problems for the Schrödinger equations in two dimensions. Within less regular classes of potentials, we establish a conditional stability estimate of logarithmic order. Moreover we prove the uniqueness within $L^p$-class of potentials with $p>2$.
The perturbation of transmission eigenvalues for inhomogeneous media in the presence of small penetrable inclusions
Fioralba Cakoni, Shari Moskow and Scott Rome
2015, 9(3): 725-748 doi: 10.3934/ipi.2015.9.725 +[Abstract](502) +[PDF](416.5KB)
Abstract:
This paper concerns the transmission eigenvalue problem for an inhomogeneous media of compact support containing small penetrable homogeneous inclusions. Assuming that the inhomogeneous background media is known and smooth, we investigate how these small volume inclusions affect the real transmission eigenvalues. Note that for practical applications the real transmission eigenvalues are important since they can be measured from the scattering data. In particular, in addition to proving the convergence rate for the eigenvalues corresponding to the perturbed media as inclusions' volume goes to zero, we also provide the explicit first correction term in the asymptotic expansion for simple eigenvalues. The correction terms involves the eigenvalues and eigenvectors of the unperturbed known background as well as information about the location, size and refractive index of small inhomogeneities. Thus, our asymptotic formula has the potential to be used to recover information about small inclusions from a knowledge of real transmission eigenvalues.
Artificial boundary conditions and domain truncation in electrical impedance tomography. Part I: Theory and preliminary results
Daniela Calvetti, Paul J. Hadwin, Janne M. J. Huttunen, David Isaacson, Jari P. Kaipio, Debra McGivney, Erkki Somersalo and Joseph Volzer
2015, 9(3): 749-766 doi: 10.3934/ipi.2015.9.749 +[Abstract](739) +[PDF](1346.8KB)
Abstract:
Artificial boundary conditions have long been an active research topic in numerical approximation of scattering waves: The truncation of the computational domain and the assignment of the conditions along the fictitious boundary must be done so that no spurious reflections occur. In inverse boundary value problems, a similar problem appears when the estimation of the unknowns is restricted to a domain that represents the whole domain of the solutions of a partial differential equation with unknown coefficient. This problem is significantly more challenging than general scattering problems, because the coefficients representing the unknown material parameter of interest are not known in the truncated portion and assigning suitable condition on the fictitious boundary is part of the problem also. The problem is addressed by defining a Dirichlet-to-Neumann map, or Steklov-Poincaré map, on the boundary of the domain truncation. In this paper we describe the procedure, provide a theoretical justification and illustrate with computed examples the limitations of imposing fixed boundary condition. Extensions of the proposed approach will be presented in a sequel article.
Artificial boundary conditions and domain truncation in electrical impedance tomography. Part II: Stochastic extension of the boundary map
Daniela Calvetti, Paul J. Hadwin, Janne M. J. Huttunen, Jari P. Kaipio and Erkki Somersalo
2015, 9(3): 767-789 doi: 10.3934/ipi.2015.9.767 +[Abstract](769) +[PDF](5596.0KB)
Abstract:
In [3], the authors discussed the electrical impedance tomography (EIT) problem, in which the computational domain with an unknown conductivity distribution comprises only a portion of the whole conducting body, and a boundary condition along the artificial boundary needs to be set so as to minimally disturbs the estimate in the domain of interest. It was shown that a partial Dirichlet-to-Neumann operator, or Steklov-Poincaré map, provides theoretically a perfect boundary condition. However, since the boundary condition depends on the conductivity in the truncated portion of the conductive body, it is itself an unknown that needs to be estimated along with the conductivity of interest. In this article, we develop further the computational methodology, replacing the unknown integral kernel with a low dimensional approximation. The viability of the approach is demonstrated with finite element simulations as well as with real phantom data.
PDE-constrained optimal control approach for the approximation of an inverse Cauchy problem
Lili Chang, Wei Gong, Guiquan Sun and Ningning Yan
2015, 9(3): 791-814 doi: 10.3934/ipi.2015.9.791 +[Abstract](707) +[PDF](1237.4KB)
Abstract:
This paper concerns the approximation of a Cauchy problem for the elliptic equation. The inverse problem is transformed into a PDE-constrained optimal control problem and these two problems are equivalent under some assumptions. Different from the existing literature which is also based on the optimal control theory, we consider the state equation in the sense of very weak solution defined by the transposition technique. In this way, it does not need to impose any regularity requirement on the given data. Moreover, this method can yield theoretical analysis simply and numerical computation conveniently. To deal with the ill-posedness of the control problem, Tikhonov regularization term is introduced. The regularized problem is well-posed and its solution converges to the non-regularized counterpart as the regularization parameter approaches zero. We establish the finite element approximation to the regularized control problem and the convergence of the discrete problem is also investigated. Then we discuss the first order optimality condition of the control problem further and obtain an efficient numerical scheme for the Cauchy problem via the adjoint state equation. The paper is ended with numerical experiments.
Nomonotone spectral gradient method for sparse recovery
Wanyou Cheng, Zixin Chen and Donghui Li
2015, 9(3): 815-833 doi: 10.3934/ipi.2015.9.815 +[Abstract](792) +[PDF](533.7KB)
Abstract:
In the paper, we present an algorithm framework for the more general problem of minimizing the sum $f(x)+\psi(x)$, where $f$ is smooth and $\psi$ is convex, but possible nonsmooth. At each step, the search direction of the algorithm is obtained by solving an optimization problem involving a quadratic term with diagonal Hessian and Barzilai-Borwein steplength plus $ \psi(x)$. The nonmonotone strategy is combined with -Borwein steplength to accelerate the convergence process. The method with the nomonotone line search techniques is showed to be globally convergent. In particular, if $f$ is convex, we show that the method shares a sublinear global rate of convergence. Moreover, if $f$ is strongly convex, we prove that the method converges R-linearly. Numerical experiments with compressive sense problems show that our approach is competitive with several known methods for some standard $l_2-l_1$ problems.
Point-wise behavior of the Geman--McClure and the Hebert--Leahy image restoration models
Petteri Harjulehto, Peter Hästö and Juha Tiirola
2015, 9(3): 835-851 doi: 10.3934/ipi.2015.9.835 +[Abstract](825) +[PDF](410.5KB)
Abstract:
We present new continuous variants of the Geman--McClure model and the Hebert--Leahy model for image restoration, where the energy is given by the nonconvex function $x \mapsto x^2/(1+x^2)$ or $x \mapsto \log(1+x^2)$, respectively. In addition to studying these models' $\Gamma$-convergence, we consider their point-wise behaviour when the scale of convolution tends to zero. In both cases the limit is the Mumford-Shah functional.
The Cauchy problem for a nonlinear elliptic equation: Nash-game approach and application to image inpainting
Moez Kallel, Maher Moakher and Anis Theljani
2015, 9(3): 853-874 doi: 10.3934/ipi.2015.9.853 +[Abstract](636) +[PDF](685.7KB)
Abstract:
Image inpainting or disocclusion, which refers to the process of restoring a damaged image with missing information, has many applications in different fields. Different techniques can be applied to solve this problem. In particular, many variational models have appeared in the literature. These models give rise to partial differential equations for which Dirichlet boundary conditions are usually used. The basic idea of the algorithms that have been proposed in the literature is to fill-in damaged regions with available information from their surroundings. The aim of this work is to treat the case where this information is not available in a part of the boundary of the damaged region. We formulate the image inpainting problem as a nonlinear Cauchy problem. Then, we give a Nash-game formulation of this Cauchy problem and we present different numerical experiments using the finite-element method for solving the image inpainting problem.
A reweighted $l^2$ method for image restoration with Poisson and mixed Poisson-Gaussian noise
Jia Li, Zuowei Shen, Rujie Yin and Xiaoqun Zhang
2015, 9(3): 875-894 doi: 10.3934/ipi.2015.9.875 +[Abstract](795) +[PDF](2103.0KB)
Abstract:
We study weighted $l^2$ fidelity in variational models for Poisson noise related image restoration problems. Gaussian approximation to Poisson noise statistic is adopted to deduce weighted $l^2$ fidelity. Different from the traditional weighted $l^2$ approximation, we propose a reweighted $l^2$ fidelity with sparse regularization by wavelet frame. Based on the split Bregman algorithm introduced in [21], the proposed numerical scheme is composed of three easy subproblems that involve quadratic minimization, soft shrinkage and matrix vector multiplications. Unlike usual least square approximation of Poisson noise, we dynamically update the underlying noise variance from previous estimate. The solution of the proposed algorithm is shown to be the same as the one obtained by minimizing Kullback-Leibler divergence fidelity with the same regularization. This reweighted $l^2$ formulation can be easily extended to mixed Poisson-Gaussian noise case. Finally, the efficiency and quality of the proposed algorithm compared to other Poisson noise removal methods are demonstrated through denoising and deblurring examples. Moreover, mixed Poisson-Gaussian noise tests are performed on both simulated and real digital images for further illustration of the performance of the proposed method.
Oracle-type posterior contraction rates in Bayesian inverse problems
Kui Lin, Shuai Lu and Peter Mathé
2015, 9(3): 895-915 doi: 10.3934/ipi.2015.9.895 +[Abstract](678) +[PDF](436.1KB)
Abstract:
We discuss Bayesian inverse problems in Hilbert spaces. The focus is on a fast concentration of the posterior probability around the unknown true solution as expressed in the concept of posterior contraction rates. This concentration is dominated by a parameter which controls the variance of the prior distribution. Previous results determine posterior contraction rates based on known solution smoothness. Here we show that an oracle-type parameter choice is possible. This is done by relating the posterior contraction rate to the root mean squared estimation error. In addition we show that the tail probability, which usually is bounded by using the Chebyshev inequality, has exponential decay, at least for a priori parameter choices. These results implement the exponential concentration of Gaussian measures in Hilbert spaces.
Hyperspectral unmixing by the alternating direction method of multipliers
Russell E. Warren and Stanley J. Osher
2015, 9(3): 917-933 doi: 10.3934/ipi.2015.9.917 +[Abstract](700) +[PDF](1867.5KB)
Abstract:
We have developed a method for hyperspectral image data unmixing that requires neither pure pixels nor any prior knowledge about the data. Based on the well-established Alternating Direction Method of Multipliers, the problem is formulated as a biconvex constrained optimization with the constraints enforced by Bregman splitting. The resulting algorithm estimates the spectral and spatial structure in the image through a numerically stable iterative approach that removes the need for separate endmember and spatial abundance estimation steps. The method is illustrated on data collected by the SpecTIR imaging sensor.

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