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Inverse Problems & Imaging

2013 , Volume 7 , Issue 2

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On the optimal control of the free boundary problems for the second order parabolic equations. I. Well-posedness and convergence of the method of lines
Ugur G. Abdulla
2013, 7(2): 307-340 doi: 10.3934/ipi.2013.7.307 +[Abstract](781) +[PDF](522.0KB)
Abstract:
We develop a new 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 consists 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. We prove well-posedness in Sobolev spaces framework and convergence of discrete optimal control problems to the original problem both with respect to cost functional and control.
Stable determination of surface impedance on a rough obstacle by far field data
Giovanni Alessandrini, Eva Sincich and Sergio Vessella
2013, 7(2): 341-351 doi: 10.3934/ipi.2013.7.341 +[Abstract](577) +[PDF](369.5KB)
Abstract:
We treat the stability of determining the boundary impedance of an obstacle by scattering data, with a single incident field. A previous result by Sincich (SIAM J. Math. Anal. 38, (2006), 434-451) showed a log stability when the boundary of the obstacle is assumed to be $C^{1,1}$-smooth. We prove that, when the obstacle boundary is merely Lipschitz, a log-log type stability still holds.
Inverse diffusion from knowledge of power densities
Guillaume Bal, Eric Bonnetier, François Monard and Faouzi Triki
2013, 7(2): 353-375 doi: 10.3934/ipi.2013.7.353 +[Abstract](473) +[PDF](335.4KB)
Abstract:
This paper concerns the reconstruction of a diffusion coefficient in an elliptic equation from knowledge of several power densities. The power density is the product of the diffusion coefficient with the square of the modulus of the gradient of the elliptic solution. The derivation of such internal functionals comes from perturbing the medium of interest by acoustic (plane) waves, which results in small changes in the diffusion coefficient. After appropriate asymptotic expansions and (Fourier) transformation, this allow us to construct the power density of the equation point-wise inside the domain. Such a setting finds applications in ultrasound modulated electrical impedance tomography and ultrasound modulated optical tomography.
    We show that the diffusion coefficient can be uniquely and stably reconstructed from knowledge of a sufficient large number of power densities. Explicit expressions for the reconstruction of the diffusion coefficient are also provided. Such results hold for a large class of boundary conditions for the elliptic equation in the two-dimensional setting. In three dimensions, the results are proved for a more restrictive class of boundary conditions constructed by means of complex geometrical optics solutions.
Near-field imaging of the surface displacement on an infinite ground plane
Gang Bao and Junshan Lin
2013, 7(2): 377-396 doi: 10.3934/ipi.2013.7.377 +[Abstract](471) +[PDF](532.5KB)
Abstract:
This paper is concerned with the inverse diffraction problem for an unbounded obstacle which is a ground plane with some local disturbance. The data is collected in the near-field regime with a distance above the surface displacement that is smaller than the wavelength. In this regime, the evanescent modes carried by the scattered wave are significant, which makes it different from the far-field measurement. We formulate explicitly the connection between the evanescent wave modes and the high frequency components of the surface displacement, and present a new numerical scheme to reconstruct the surface displacement from the boundary measurements. By extracting the information carried by the evanescent modes effectively, it is shown that the resolution of the reconstructed image is significantly improved in the near field. Numerical examples show that images with a resolution of $\lambda/10$ are obtained.
Gaussian Markov random field priors for inverse problems
Johnathan M. Bardsley
2013, 7(2): 397-416 doi: 10.3934/ipi.2013.7.397 +[Abstract](527) +[PDF](531.7KB)
Abstract:
In this paper, our focus is on the connections between the methods of (quadratic) regularization for inverse problems and Gaussian Markov random field (GMRF) priors for problems in spatial statistics. We begin with the most standard GMRFs defined on a uniform computational grid, which correspond to the oft-used discrete negative-Laplacian regularization matrix. Next, we present a class of GMRFs that allow for the formation of edges in reconstructed images, and then draw concrete connections between these GMRFs and numerical discretizations of more general diffusion operators. The benefit of the GMRF interpretation of quadratic regularization is that a GMRF is built-up from concrete statistical assumptions about the values of the unknown at each pixel given the values of its neighbors. Thus the regularization term corresponds to a concrete spatial statistical model for the unknown, encapsulated in the prior. Throughout our discussion, strong ties between specific GMRFs, numerical discretizations of diffusion operators, and corresponding regularization matrices, are established. We then show how such GMRF priors can be used for edge-preserving reconstruction of images, in both image deblurring and medical imaging test cases. Moreover, we demonstrate the effectiveness of GMRF priors for data arising from both Gaussian and Poisson noise models.
Study of noise effects in electrical impedance tomography with resistor networks
Liliana Borcea, Fernando Guevara Vasquez and Alexander V. Mamonov
2013, 7(2): 417-443 doi: 10.3934/ipi.2013.7.417 +[Abstract](512) +[PDF](3241.4KB)
Abstract:
We present a study of the numerical solution of the two dimensional electrical impedance tomography problem, with noisy measurements of the Dirichlet to Neumann map. The inversion uses parametrizations of the conductivity on optimal grids. The grids are optimal in the sense that finite volume discretizations on them give spectrally accurate approximations of the Dirichlet to Neumann map. The approximations are Dirichlet to Neumann maps of special resistor networks, that are uniquely recoverable from the measurements. Inversion on optimal grids has been proposed and analyzed recently, but the study of noise effects on the inversion has not been carried out. In this paper we present a numerical study of both the linearized and the nonlinear inverse problem. We take three different parametrizations of the unknown conductivity, with the same number of degrees of freedom. We obtain that the parametrization induced by the inversion on optimal grids is the most efficient of the three, because it gives the smallest standard deviation of the maximum a posteriori estimates of the conductivity, uniformly in the domain. For the nonlinear problem we compute the mean and variance of the maximum a posteriori estimates of the conductivity, on optimal grids. For small noise, we obtain that the estimates are unbiased and their variance is very close to the optimal one, given by the Cramér-Rao bound. For larger noise we use regularization and quantify the trade-off between reducing the variance and introducing bias in the solution. Both the full and partial measurement setups are considered.
Far field model for time reversal and application to selective focusing on small dielectric inhomogeneities
Corinna Burkard, Aurelia Minut and Karim Ramdani
2013, 7(2): 445-470 doi: 10.3934/ipi.2013.7.445 +[Abstract](474) +[PDF](1975.0KB)
Abstract:
Based on the time-harmonic far field model for small dielectric inclusions in $3$D, we study the so-called DORT method (DORT is the French acronym for ``Diagonalization of the Time Reversal Operator''). The main observation is to relate the eigenfunctions of the time-reversal operator to the location of small scattering inclusions. For non penetrable sound-soft acoustic scatterers, this observation has been rigorously proved for $2$ and $3$ dimensions by Hazard and Ramdani in [21] for small scatterers. In this work, we consider the case of small dielectric inclusions with far field measurements. The main difference with the acoustic case is related to the magnetic permeability and the related polarization tensors. We show that in the regime $kd\rightarrow \infty$ ($k$ denotes here the wavenumber and $d$ the minimal distance between the scatterers), each inhomogeneity gives rise to -at most- 4 distinct eigenvalues (one due to the electric contrast and three to the magnetic one) while each corresponding eigenfunction generates an incident wave focusing selectively on one of the scatterers. The method has connections to the MUSIC algorithm known in Signal Processing and the Factorization Method of Kirsch.
Source identification from line integral measurements and simple atmospheric models
Brittan Farmer, Cassandra Hall and Selim Esedoḡlu
2013, 7(2): 471-490 doi: 10.3934/ipi.2013.7.471 +[Abstract](386) +[PDF](1499.2KB)
Abstract:
We consider the problem of estimating the sparse initial condition of a solution to the advection-diffusion equation based on line integrals of the solution at a later time. We propose models for locating a single and multiple point sources. We also propose algorithms for the efficient implementation of these models. In practice, the models are relevant also for reconstructing the solution of the PDE at the observation time from a very sparse Radon transform; in this case, our models improve on more standard Radon inversion techniques by utilizing the specialized information about how the observed function was generated.
A note on analyticity properties of far field patterns
Roland Griesmaier, Nuutti Hyvönen and Otto Seiskari
2013, 7(2): 491-498 doi: 10.3934/ipi.2013.7.491 +[Abstract](518) +[PDF](336.6KB)
Abstract:
In scattering theory the far field pattern describes the directional dependence of a time-harmonic wave scattered by an obstacle or inhomogeneous medium, when observed sufficiently far away from these objects. Considering plane wave excitations, the far field pattern can be written as a function of two variables, namely the direction of propagation of the incident plane wave and the observation direction, and it is well-known to be separately real analytic with respect to each of them. We show that the far field pattern is in fact a jointly real analytic function of these two variables.
A geometry guided image denoising scheme
Weihong Guo and Jing Qin
2013, 7(2): 499-521 doi: 10.3934/ipi.2013.7.499 +[Abstract](666) +[PDF](1172.1KB)
Abstract:
During image denoising, it is often difficult to balance between the removal of noise and the preservation of contrast and fine features, especially when the noise is excessive. We propose to efficiently balance the two using segmentation and more general geometry extraction transforms. Explained in the nonlocal-means (NL-means) framework, we introduce a mutual position function to ensure the averaging is only taken over pixels in the same segmentation phase, and provide selection schemes for convolution kernel and weight function to further improve the performance. To address unreliable segmentation due to more excessive noise, we use a feature extraction transform that is more general than segmentation and less sensitive to noise. Unlike most denoising approaches that only work for one type of noise and/or involve heuristic parameter tuning, the proposed method comes with an automatic parameter selection scheme, and can be easily adapted for various types of noise, ranging from Gaussian, Poisson, Rician to ultrasound noise. Comparison with the original NL-means as well as ROF, BM3D, and K-SVD on various simulated data, MRI and SEM images, indicates potentials of the proposed method.
A three-dimensional inverse gravimetry problem for ice with snow caps
Victor Isakov, Shingyu Leung and Jianliang Qian
2013, 7(2): 523-544 doi: 10.3934/ipi.2013.7.523 +[Abstract](501) +[PDF](1093.4KB)
Abstract:
We propose a model for the gravitational field of a floating iceberg $D$ with snow on its top. The inverse problem of interest in geophysics is to find $D$ and snow thickness $g$ on its known (visible) top from remote measurements of derivatives of the gravitational potential. By modifying the Novikov's orthogonality method we prove uniqueness of recovering $D$ and $g$ for the inverse problem. We design and test two algorithms for finding $D$ and $g$. One is based on a standard regularized minimization of a misfit functional. The second one applies the level set method to our problem. Numerical examples validate the theory and demonstrate effectiveness of the proposed algorithms.
Imaging acoustic obstacles by singular and hypersingular point sources
Jingzhi Li, Hongyu Liu, Hongpeng Sun and Jun Zou
2013, 7(2): 545-563 doi: 10.3934/ipi.2013.7.545 +[Abstract](414) +[PDF](1016.9KB)
Abstract:
We investigate a qualitative method for imaging acoustic obstacles in two and three dimensions by boundary measurements corresponding to hypersingular point sources. Rigorous mathematical justification of the imaging method is established, and numerical experiments are presented to illustrate the effectiveness of the proposed imaging scheme.
Total variation and wavelet regularization of orientation distribution functions in diffusion MRI
Yuyuan Ouyang, Yunmei Chen and Ying Wu
2013, 7(2): 565-583 doi: 10.3934/ipi.2013.7.565 +[Abstract](600) +[PDF](955.2KB)
Abstract:
We introduce a variational model and a numerical method for simultaneous ODF smoothing and reconstruction. The model uses the sparsity of MR images in finite difference domain and wavelet domain as the spatial regularization means in ODF's reconstruction. The model also incorporates angular regularization using Laplace-Beltrami operator on the unit sphere. A primal-dual scheme is applied to solve the model efficiently. The experimental results indicate that with spatial and angular regularization in the process of reconstruction, we can get better directional structures of reconstructed ODFs.
Local singularity reconstruction from integrals over curves in $\mathbb{R}^3$
Eric Todd Quinto and Hans Rullgård
2013, 7(2): 585-609 doi: 10.3934/ipi.2013.7.585 +[Abstract](453) +[PDF](735.9KB)
Abstract:
We define a general curvilinear Radon transform in $\mathbb{R}^3$, and we develop its microlocal properties. We prove that singularities can be added (or masked) in any backprojection reconstruction method for this transform. We use the microlocal properties of the transform to develop a local backprojection reconstruction algorithm that decreases the effect of the added singularities and reconstructs the shape of the object. This work was motivated by new models in electron microscope tomography in which the electrons travel over curves such as helices or spirals, and we provide reconstructions for a specific transform motivated by this electron microscope tomography problem.
Constructing continuous stationary covariances as limits of the second-order stochastic difference equations
Lassi Roininen, Petteri Piiroinen and Markku Lehtinen
2013, 7(2): 611-647 doi: 10.3934/ipi.2013.7.611 +[Abstract](498) +[PDF](556.9KB)
Abstract:
In Bayesian statistical inverse problems the a priori probability distributions are often given as stochastic difference equations. We derive a certain class of stochastic partial difference equations by starting from second-order stochastic partial differential equations in one and two dimensions. We discuss discretisation schemes on uniform lattices of these stationary continuous-time stochastic processes and convergence of the discrete-time processes to the continuous-time processes. A special emphasis is given to an analytical calculation of the covariance kernels of the processes. We find a representation for the covariance kernels in a simple parametric form with controllable parameters: correlation length and variance. In the discrete-time processes the discretisation step is also given as a parameter. Therefore, the discrete-time covariances can be considered as discretisation-invariant. In the two-dimensional cases we find rotation-invariant and anisotropic representations of the difference equations and the corresponding continuous-time covariance kernels.
Perfect pulse-compression coding via ARMA algorithms and unimodular transfer functions
Lassi Roininen and Markku S. Lehtinen
2013, 7(2): 649-661 doi: 10.3934/ipi.2013.7.649 +[Abstract](693) +[PDF](429.8KB)
Abstract:
We propose a method to construct perfect pulse-compression codes with autoregressive moving average algorithms. We first show the relation between the study of coding and decoding techniques in radar engineering and the study of unimodular polynomials with constrained coefficients. Then we extend the study to unimodular Fourier series and unimodular rational functions. We use the Fourier series and rational functions as transfer functions in the autoregressive moving average algorithms. We show that by a suitable choice of the coefficients, the autoregressive moving average algorithms are realisable, stable and causal. We show examples of some almost perfect codes, i.e. numerically truncated perfect codes. We end by proposing perfect code design principles for practical radar engineering purposes.

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