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

Given a function f, the author specifies the singularities of f that are visible in a stable way from limited X-ray tomographic data. This determines which singularities of f can be stably recovere...

Singularities of the X-ray transform and limited data tomography in R 2 and R 3

Injectivity Sets for the Radon Transform over Circles and Complete Systems of Radial Functions

We consider the reconstruction problem for limited angle tomography using filtered backprojection (FBP) and lambda tomography. We use microlocal analysis to explain why the well-known streak artifacts are present at the end of the limited angular range. We explain how to mitigate the streaks and prove that our modified FBP and lambda operators are standard pseudodifferential operators, and so they do not add artifacts. We provide reconstructions to illustrate our mathematical results.

https://iopscience.iop.org/article/10.1088/0266-5611/29/12/125007/pdf

Characterization and reduction of artifacts in limited angle tomography

Support theorems for real-analytic Radon transforms

ABsmAcr. Guillemin proved that the generalized Radon transform R and its dual R' are Fourier integral operators and that R'R is an elliptic pseudodifferential operator. In this paper we investigate the dependence of the Radon transform on the defining measures. In the general case we calculate the symbol of R'R as a pseudodifferential operator in terms of the measures and give a necessary condition on the defining measures for R'R to be invertible by a differential operator. Then we examine the Radon transform on points and hyperplanes in RX with general measures and we calculate the symbol of R'R in terms of the defining measures. Finally, if R'R is a translation invariant operator on RI then we prove that R'R is invertible and that our condition is equivalent to (R'R)' being a differential operator.

https://www.ams.org/tran/1980-257-02/S0002-9947-1980-0552261-8/S0002-9947-1980-0552261-8.pdf

Eric Todd Quinto

Papers

Singularities of the X-ray transform and limited data tomography in R 2 and R 3

Injectivity Sets for the Radon Transform over Circles and Complete Systems of Radial Functions

Characterization and reduction of artifacts in limited angle tomography

Support theorems for real-analytic Radon transforms

The dependence of the generalized Radon transform on defining measures