# American Institute of Mathematical Sciences

ISSN:
1930-8337

eISSN:
1930-8345

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## Inverse Problems & Imaging

November 2007 , Volume 1 , Issue 4

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2007, 1(4): 593-608 doi: 10.3934/ipi.2007.1.593 +[Abstract](1072) +[PDF](552.6KB)
Abstract:
In this paper, a theoretical framework for the conditional diffusion of digital images is presented. Different approaches have been proposed to solve this problem by extrapolating the idea of the anisotropic diffusion for a grey level images to vector-valued images. Then, the diffusion of each channel is conditioned to a direction which normally takes into account information from all channels. In our approach, the diffusion model assumes the a priori knowledge of the diffusion direction during all the process.
The consistency of the model is shown by proving the existence and uniqueness of solution for the proposed equation from the viscosity solutions theory. Also a numerical scheme adapted to this equation based on the neighborhood filter is proposed. Finally, we discuss several applications and we compare the corresponding numerical schemes for the proposed model.
2007, 1(4): 609-622 doi: 10.3934/ipi.2007.1.609 +[Abstract](1300) +[PDF](167.4KB)
Abstract:
We consider the inverse problem of time-harmonic acoustic wave scattering where the shape of an obstacle is reconstructed from a given incident field and the modulus of the far field pattern of the scattered field. Our approach is based on a pair of nonlinear and ill-posed integral equations to be solved for the shape of the unknown boundary. This approach is an extension of the method suggested by Kress and Rundell [5] for an inverse boundary value problem for the Laplace equation. Since the modulus of far field pattern is invariant under translations [4] we can reconstruct the shape of the obstacle but not the location.
The numerical implementation of the method is described and it is illustrated by numerical examples that the method yields satisfactory reconstructions both for sound-soft and sound-hard obstacles, also in the case when the modulus is given in a limited aperture.
2007, 1(4): 623-642 doi: 10.3934/ipi.2007.1.623 +[Abstract](1030) +[PDF](267.4KB)
Abstract:
We consider uniqueness of two-dimensional parallel beam tomography with unknown view angles. We show that infinitely many projections at unknown view angles of a sufficiently asymmetric object determine the object uniquely. An explicit expression for the required asymmetry is given in terms of the object's geometric moments. We also show that under certain assumptions finitely many projections guarantee uniqueness for the unknown view angles. Compared to previous results about uniqueness of view angles, our result reduces the minimum number of required projections to approximately half and is applicable to a larger set of objects. Our analysis is based on algebraic geometric properties of a certain system of homogeneous polynomials determined by the Helgason-Ludwig consistency conditions.
2007, 1(4): 643-660 doi: 10.3934/ipi.2007.1.643 +[Abstract](1603) +[PDF](428.2KB)
Abstract:
In this paper we consider the inverse scattering problem of determining the shape of one or more objects embedded in an inhomogeneous background from Cauchy data measured on the boundary of a domain containing the objects in its interior. Following [1], we use the reciprocity gap functional method. In an inhomogeneous background medium the use of a Herglotz wave function in the reciprocity gap functional is no longer permissable. Instead we propose to use a finite element representation. We provide analysis to support the method, and also describe implementation issues. Numerical examples are given showing the performance of the method.
2007, 1(4): 661-672 doi: 10.3934/ipi.2007.1.661 +[Abstract](1150) +[PDF](255.3KB)
Abstract:
We study the restoration of a sparse signal or an image with a sparse gradient from a relatively small number of linear measurements which are additionally corrupted by a small amount of white Gaussian noise and outliers. We minimize $\l_1-\l_1$ and $\l_1-TV$ regularization functionals using various algorithms and present numerical results for different measurement matrices as well as different sparsity levels of the initial signal/image and of the outlier vector.
2007, 1(4): 673-690 doi: 10.3934/ipi.2007.1.673 +[Abstract](1201) +[PDF](461.6KB)
Abstract:
Detecting optical flow means to find the apparent displacement field in a sequence of images. As starting point for many optical flow methods serves the so called optical flow constraint (OFC), that is the assumption that the gray value of a moving point does not change over time. Variational methods are amongst the most popular tools to compute the optical flow field. They compute the flow field as minimizer of an energy functional that consists of a data term to comply with the OFC and a smoothness term to obtain uniqueness of this underdetermined problem. In this article we replace the smoothness term by projecting the solution to a finite dimensional, affine subspace in the spatial variables which leads to a smoothing and gives a unique solution as well. We explain the mathematical details for the quadratic and nonquadratic minimization framework, and show how alternative model assumptions such as constancy of the brightness gradient can be incorporated. As basis functions we consider tensor products of B-splines. Under certain smoothness assumptions for the global minimizer in Sobolev scales, we prove optimal convergence rates in terms of the energy functional. Experiments are presented that demonstrate the feasibility of our approach.
2007, 1(4): 691-712 doi: 10.3934/ipi.2007.1.691 +[Abstract](1115) +[PDF](927.0KB)
Abstract:
We present a hybrid method to numerically solve the inverse acoustic sound-soft obstacle scattering problem in $\R^3$, given the far-field pattern for one incident direction. This method combines ideas of both iterative and decomposition methods, inheriting advantages of each of them, such as getting good reconstructions and not needing a forward solver at each step. A related Newton method is presented to show convergence of the method and numerical results show its feasibility.
2007, 1(4): 713-772 doi: 10.3934/ipi.2007.1.713 +[Abstract](1252) +[PDF](702.8KB)
Abstract:
This is a comprehensive review of the uses of potential theory in studying the spectral theory of orthogonal polynomials. Much of the article focuses on the Stahl--Totik theory of regular measures, especially the case of OPRL and OPUC. Links are made to the study of ergodic Schrödinger operators where one of our new results implies that, in complete generality, the spectral measure is supported on a set of zero Hausdorff dimension (indeed, of capacity zero) in the region of strictly positive Lyapunov exponent. There are many examples and some new conjectures and indications of new research directions. Included are appendices on potential theory and on Fekete--Szegő theory.

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