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Inverse Problems & Imaging

2009 , Volume 3 , Issue 4

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Stability estimates in the inverse transmission scattering problem
Michele Di Cristo
2009, 3(4): 551-565 doi: 10.3934/ipi.2009.3.551 +[Abstract](385) +[PDF](202.3KB)
Abstract:
We consider the inverse transmission scattering problem with piecewise constant refractive index. Under mild a priori assumptions on the obstacle we establish logarithmic stability estimates.
On infinite-dimensional hierarchical probability models in statistical inverse problems
Tapio Helin
2009, 3(4): 567-597 doi: 10.3934/ipi.2009.3.567 +[Abstract](316) +[PDF](373.9KB)
Abstract:
In this article, the solution of a statistical inverse problem $M = AU+$ε by the Bayesian approach is studied where $U$ is a function on the unit circle $\T$, i.e., a periodic signal. The mapping $A$ is a smoothing linear operator and ε a Gaussian noise. The connection to the solution of a finite-dimensional computational model $M_{kn} = A_k U_n + $εk is discussed. Furthermore, a novel hierarchical prior model for obtaining edge-preserving conditional mean estimates is introduced. The convergence of the method with respect to finer discretization is studied and the posterior distribution is shown to converge weakly. Finally, theoretical findings are illustrated by a numerical example with simulated data.
Regularized D-bar method for the inverse conductivity problem
Kim Knudsen, Matti Lassas, Jennifer L. Mueller and Samuli Siltanen
2009, 3(4): 599-624 doi: 10.3934/ipi.2009.3.599 +[Abstract](623) +[PDF](451.7KB)
Abstract:
A strategy for regularizing the inversion procedure for the two-dimensional D-bar reconstruction algorithm based on the global uniqueness proof of Nachman [Ann. Math. 143 (1996)] for the ill-posed inverse conductivity problem is presented. The strategy utilizes truncation of the boundary integral equation and the scattering transform. It is shown that this leads to a bound on the error in the scattering transform and a stable reconstruction of the conductivity; an explicit rate of convergence in appropriate Banach spaces is derived as well. Numerical results are also included, demonstrating the convergence of the reconstructed conductivity to the true conductivity as the noise level tends to zero. The results provide a link between two traditions of inverse problems research: theory of regularization and inversion methods based on complex geometrical optics. Also, the procedure is a novel regularized imaging method for electrical impedance tomography.
Variational denoising of diffusion weighted MRI
Tim McGraw, Baba Vemuri, Evren Özarslan, Yunmei Chen and Thomas Mareci
2009, 3(4): 625-648 doi: 10.3934/ipi.2009.3.625 +[Abstract](482) +[PDF](1595.2KB)
Abstract:
In this paper, we present a novel variational formulation for restoring high angular resolution diffusion imaging (HARDI) data. The restoration formulation involves smoothing signal measurements over the spherical domain and across the 3D image lattice. The regularization across the lattice is achieved using a total variation (TV) norm based scheme, while the finite element method (FEM) was employed to smooth the data on the sphere at each lattice point using first and second order smoothness constraints. Examples are presented to show the performance of the HARDI data restoration scheme and its effect on fiber direction computation on synthetic data, as well as on real data sets collected from excised rat brain and spinal cord.
A family of inversion formulas in thermoacoustic tomography
Linh V. Nguyen
2009, 3(4): 649-675 doi: 10.3934/ipi.2009.3.649 +[Abstract](345) +[PDF](284.6KB)
Abstract:
We present a family of closed form inversion formulas in thermoacoustic tomography in the case of a constant sound speed. The formulas are presented in both time-domain and frequency-domain versions. As special cases, they imply most of the previously known filtered backprojection type formulas.
Semismooth Newton method for minimization of the LLT model
Zhi-Feng Pang and Yu-Fei Yang
2009, 3(4): 677-691 doi: 10.3934/ipi.2009.3.677 +[Abstract](429) +[PDF](224.8KB)
Abstract:
In this paper, we discuss the nonsmooth second-order regularization, suggested by Lysaker, Lundervold and Tai, and its application in image denoising. A function space $BV^2$ is given and the well-posedness of the LLT model is proved in this function space. By means of the Fisher-Burmeister NCP function, we reformulate the dual formula of the LLT model in discrete setting as a system of semismooth equations. Then we propose a semismooth Newton method for the LLT model to build up a Q-superlinearly convergent numerical scheme. The computational experiments are supplied to demonstrate the efficiency of the proposed method.
Multiscale image representation using novel integro-differential equations
Eitan Tadmor and Prashant Athavale
2009, 3(4): 693-710 doi: 10.3934/ipi.2009.3.693 +[Abstract](510) +[PDF](321.1KB)
Abstract:
Motivated by the hierarchical multiscale image representation of Tadmor et. al., [25], we propose a novel integro-differential equation (IDE) for a multiscale image representation. To this end, one integrates in inverse scale space a succession of refined, recursive 'slices' of the image, which are balanced by a typical curvature term at the finer scale. Although the original motivation came from a variational approach, the resulting IDE can be extended using standard techniques from PDE-based image processing. We use filtering, edge preserving and tangential smoothing to yield a family of modified IDE models with applications to image denoising and image deblurring problems. The IDE models depend on a user scaling function which is shown to dictate the BV properties of the residual error. Numerical experiments demonstrate application of the IDE approach to denoising and deblurring.
Model reduction and pollution source identification from remote sensing data
A Voutilainen and Jari P. Kaipio
2009, 3(4): 711-730 doi: 10.3934/ipi.2009.3.711 +[Abstract](404) +[PDF](979.4KB)
Abstract:
We consider a source identification problem related to determination of contaminant source parameters in lake environments using remote sensing measurements. We carry out a numerical example case study in which we employ the statistical inversion approach for the determination of the source parameters. In the simulation study a pipeline breaks on the bottom of a lake and only low-resolution remote sensing measurements are available. We also describe how model uncertainties and especially errors that are related to model reduction are taken into account in the overall statistical model of the system. The results indicate that the estimates may be heavily misleading if the statistics of the model errors are not taken into account.

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