What is Cone-Beam CT and How Does it Work?

A duality in integral geometry A duality for symmetric spaces The fourier transform on a symmetric space The Radon transform on $X$ and on $X_o$. Range questions Differential equations on symmetric spaces Eigenspace representations Solutions to exercises Bibliography Symbols frequently used Index.

Geometric Analysis on Symmetric Spaces

This paper treats the linearized inverse scattering problem for the case of variable background velocity and for an arbitrary configuration of sources and receivers. The linearized inverse scattering problem is formulated in terms of an integral equation in a form which covers wave propagation in fluids with constant and variable densities and in elastic solids. This integral equation is connected with the causal generalized Radon transform (GRT), and an asymptotic expansion of the solution of the integral equation is obtained using an inversion procedure for the GRT. The first term of this asymptotic expansion is interpreted as a migration algorithm. As a result, this paper contains a rigorous derivation of migration as a technique for imaging discontinuities of parameters describing a medium. Also, a partial reconstruction operator is explicitly derived for a limited aperture. When specialized to a constant background velocity and specific source–receiver geometries our results are directly related to some known migration algorithms.

Imaging of discontinuities in the inverse scattering problem by inversion of a causal generalized Radon transform

The theory of pseudo differential operators, discussed in § 1, is well suited for investigating various problems connected with elliptic differential equations. However, this theory fails to be adequate for studying equations of hyperbolic type, and one is then forced to examine a wider class of operators, the so-called Fourier integral operators (Egorov [1975], Hormander [1968, 1971, 1983, 1985], Kumano-go [1982], Shubin [1978], Taylor [1981], Treves [1980]).

Fourier Integral Operators

Suppose D is a bounded, connected, open set in Rn and f is a smooth function on Rn with support in $\overD$. We study the recovery of f from the mean values of f over spheres centered on a part or the whole boundary of D. For strictly convex $\overline{D}$, we prove uniqueness when the centers are restricted to an open subset of the boundary. We provide an inversion algorithm (with proof) when the mean values are known for all spheres centered on the boundary of D, with radii in the interval [0, diam(D)/2]. We also give an inversion formula when D is a ball in Rn, $n \geq 3$ and odd, and the mean values are known for all spheres centered on the boundary.

http://www.math.udel.edu/~rakesh/RESEARCH/meanvalue.pdf

Determining a function from its mean values over a family of spheres

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.

Local sobolev estimates of a function by means of its radon transform

Computed tomography and rockets

Generalized Radon transforms and applications in tomography

We present a paradigm for characterization of artifacts in limited data tomography problems. In particular, we use this paradigm to characterize artifacts that are generated in reconstructions from limited angle data with generalized Radon transforms and general filtered backprojection type operators. In order to find when visible singularities are imaged, we calculate the symbol of our reconstruction operator as a pseudodifferential operator.

A paradigm for the characterization of artifacts in tomography

Electron micrography (EM) is an important method for determining the three-dimensional (3D) structure of macromolecular complexes and biological specimens. But there are several limitations such as poor signal-to-noise, limitation on range of tilt angles and sub-region subject to electron exposure, unintentional movements of the specimen, with EM systems that make the reconstruction procedure a severely ill-posed problem. A different choice of reconstruction method may lead to different results and create different additional artifacts in reconstructed images. In this paper, we combined the artifacts reduction strategy and the iterative reconstruction algorithm using a Mumford-Shah model. With the combined method, one can not only regularize the ill-posedness and provide segmentation simultaneously but also smooth additional artifacts due to the limited data. We applied the method to both simulated data from the Shepp-Logan phantom and cryo-specimen tomography. The results demonstrate the performance of the method in reducing the noise and artifacts while preserving and enhancing the edges in the reconstructed image.

Eric Todd Quinto

Papers

Local sobolev estimates of a function by means of its radon transform

Computed tomography and rockets

Generalized Radon transforms and applications in tomography

A paradigm for the characterization of artifacts in tomography

Simultaneous reconstruction and segmentation with the Mumford-Shah functional for electron tomography