# American Institute of Mathematical Sciences

ISSN:
1930-8337

eISSN:
1930-8345

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## Inverse Problems & Imaging

November 2010 , Volume 4 , Issue 4

Special issue dedicated to Jan Boman
on the occasion of his 75th birthday

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2010, 4(4): i-iv doi: 10.3934/ipi.2010.4.4i +[Abstract](1209) +[PDF](440.1KB)
Abstract:
This volume contains the proceedings of the international conference
Integral Geometry and Tomography

held at Stockholm University, August 12-15, 2008. The meeting was dedicated to Jan Boman on the occasion of his 75-th birthday.
We are happy that so many of the participants have contributed to these proceedings with original research articles, some of which have been presented at the conference, others resulting from inspiring discussions during the meeting. A few contributions have also been written by colleagues who were invited but could not come to Stockholm.

2010, 4(4): 571-577 doi: 10.3934/ipi.2010.4.571 +[Abstract](1321) +[PDF](2219.4KB)
Abstract:
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2010, 4(4): 579-598 doi: 10.3934/ipi.2010.4.579 +[Abstract](1782) +[PDF](253.2KB)
Abstract:
The inverse problem for the Schrödinger operator on a star graph is investigated. It is proven that such Schrödinger operator, i.e. the graph, the real potential on it and the matching conditions at the central vertex, can be reconstructed from the Titchmarsh-Weyl matrix function associated with the graph boundary. The reconstruction is also unique if the spectral data include not the whole Titchmarsh-Weyl function but its principal block (the matrix reduced by one dimension). The same result holds true if instead of the Titchmarsh-Weyl function the dynamical response operator or just its principal block is known.
2010, 4(4): 599-618 doi: 10.3934/ipi.2010.4.599 +[Abstract](1501) +[PDF](222.3KB)
Abstract:
Considerations of the backscattering data for the Schrödinger operator $H_v= -\Delta+ v$ in $\RR^n$, where $n\ge 3$ is odd, give rise to an entire analytic mapping from $C_0^\infty ( \RRn)$ to $C^\infty (\RRn)$, the backscattering transformation. The aim of this paper is to give formulas for $B_2(v, w)$ where $B_2$ is the symmetric bilinear operator that corresponds to the quadratic part of the backscattering transformation and $v$ and $w$ are rotation invariant.
2010, 4(4): 619-630 doi: 10.3934/ipi.2010.4.619 +[Abstract](1704) +[PDF](199.1KB)
Abstract:
Using a vanishing theorem for microlocally real analytic distributions and a theorem on flatness of a distribution vanishing on infinitely many hyperplanes we give a new proof of an injectivity theorem of Bélisle, Massé, and Ransford for the ray transform on $\R^n$. By means of an example we show that this result is sharp. An extension is given where real analyticity is replaced by quasianalyticity.
2010, 4(4): 631-637 doi: 10.3934/ipi.2010.4.631 +[Abstract](1583) +[PDF](132.1KB)
Abstract:
We consider a weighted Radon transform in the plane, $R_m(\xi, \eta) = \int_{\R} f(x, \xi x + \eta) m(x,\xi,\eta) dx$, where $m(x,\xi,\eta)$ is a smooth, positive function. Using an extension of an argument of Strichartz we prove a local injectivity theorem for $R_m$ for essentially the same class of $m(x,\xi,\eta)$ that was considered by Gindikin in his article in this issue.
2010, 4(4): 639-647 doi: 10.3934/ipi.2010.4.639 +[Abstract](1469) +[PDF](119.5KB)
Abstract:
Special functions are functions that show up in several contexts. The most classical special functions are the monomials and the exponential functions. On the next level we find the hypergeometric functions, which appear in such varied contexts as partial differential equations, number theory, and group representations. The standard hypergeometric functions have power series which satisfy 2 term recursion relations. This leads to the usual expressions for the power series coefficients as quotionts of rational and factorial-like expressions. We have developed a "hierarchy" of special functions which satisfy higher order recursion relations. They generalize the classical Mathieu and Lamé functions. These classical functions satisfy 3 term recursion relations and our theory produces "Lamé - like" functions which satisfy recursions of any order.
2010, 4(4): 649-653 doi: 10.3934/ipi.2010.4.649 +[Abstract](2136) +[PDF](84.7KB)
Abstract:
We consider a class of weights on the plane for which the weighted Radon transform admits an inversion formula similar to the classical one. These transforms are naturally dual to the attenuated Radon.
2010, 4(4): 655-664 doi: 10.3934/ipi.2010.4.655 +[Abstract](1685) +[PDF](144.2KB)
Abstract:
In 1963, K.P.~Grotemeyer proved an interesting variant of the Gauss-Bonnet Theorem. Let $M$ be an oriented closed surface in the Euclidean space $\mathbb R^3$ with Euler characteristic $\chi(M)$, Gauss curvature $G$ and unit normal vector field $\vec n$. Grotemeyer's identity replaces the Gauss-Bonnet integrand $G$ by the normal moment $( \vec a \cdot \vec n )^2G$, where $a$ is a fixed unit vector: $\int_M(\vec a\cdot \vec n)^2 Gdv=\frac{2 \pi}{3}\chi(M)$. We generalize Grotemeyer's result to oriented closed even-dimensional hypersurfaces of dimension $n$ in an $(n+1)$-dimensional space form $N^{n+1}(k)$.
2010, 4(4): 665-673 doi: 10.3934/ipi.2010.4.665 +[Abstract](1877) +[PDF](383.2KB)
Abstract:
Several hybrid tomographic methods utilizing ultrasound modulation have been introduced lately. Success of these methods hinges on the feasibility of focusing ultrasound waves at an arbitrary point of interest. Such focusing, however, is difficult to achieve in practice. We thus propose a way to avoid the use of focused waves through what we call synthetic focusing, i.e. by reconstructing the would-be response to the focused modulation from the measurements corresponding to realistic unfocused waves. Examples of reconstructions from simulated data are provided. This non-technical paper describes only the general concept, while technical details will appear elsewhere.
2010, 4(4): 675-683 doi: 10.3934/ipi.2010.4.675 +[Abstract](1573) +[PDF](456.1KB)
Abstract:
As a consequence of very noisy tomographic data the reconstructed images are contaminated by severely amplified noise. Typically two remedies are considered. Firstly, the data are smoothed, this is called pre-whitening in the engineering literature. The disadvantage here is that the individually treated data sets could become inconsistent. Secondly, the image, reconstructed from the original data sets, is treated by methods of image smoothing. As example diffusion filters are mentioned. In this paper we present a method where the reconstruction of the smoothed image is performed in one step; i.e., we develop special reconstruction kernels, which directly compute the image smoothed by a diffusion filter. Examples from synthetic data are presented.
2010, 4(4): 685-691 doi: 10.3934/ipi.2010.4.685 +[Abstract](1903) +[PDF](370.6KB)
Abstract:
We study reflection imaging as an incomplete data problem in frequency domain. It turns out that this amounts to inverting the Fourier transform using only frequencies outside some set. By numerical simulations we show the effect of this incompleteness on concrete reconstruction problems. We try to complete the data by analytic continuation. An explicit formula is obtained by an inversion formula for the exponential Radon transform. We discuss the application to medical ultrasound tomography and to seismic imaging. We describe an alternative method based on the presence of reflectors.
2010, 4(4): 693-702 doi: 10.3934/ipi.2010.4.693 +[Abstract](2104) +[PDF](168.2KB)
Abstract:
We discuss properties of a generalized Minkowski-Funk transform defined for a family of hypersurfaces. We prove two-side estimates and show that the range conditions can be written in terms of the reciprocal Funk transform. Some applications to the spherical mean transform are considered.
2010, 4(4): 703-712 doi: 10.3934/ipi.2010.4.703 +[Abstract](1599) +[PDF](403.2KB)
Abstract:
In this paper we develop the numerical algorithm for solving the inverse problem for the wave equation by the Boundary Control method. The results of numerical experiments are presented.
2010, 4(4): 713-720 doi: 10.3934/ipi.2010.4.713 +[Abstract](1828) +[PDF](148.8KB)
Abstract:
Damek-Ricci spaces, also called harmonic $NA$ groups, make up a large class of harmonic Riemannian manifolds including all hyperbolic spaces. We prove here an inversion formula and a support theorem for the X-ray transform, i.e. integration along geodesics, on those spaces.
Using suitably chosen totally geodesic submanifolds we reduce the problems to similar questions on low-dimensional hyperbolic spaces.
2010, 4(4): 721-734 doi: 10.3934/ipi.2010.4.721 +[Abstract](1963) +[PDF](198.8KB)
Abstract:
In this article, we will define local and microlocal Sobolev seminorms and prove local and microlocal inverse continuity estimates for the Radon hyperplane transform in these seminorms. The relation between the Sobolev wavefront set of a function $f$ and of its Radon transform is well-known [18]. However, Sobolev wavefront is qualitative and therefore the relation in [18] is qualitative. Our results will make the relation between singularities of a function and those of its Radon transform quantitative. This could be important for practical applications, such as tomography, in which the data are smooth but can have large derivatives.

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