Eric Todd Quinto

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

The magnet of an electrodynamic clockwork is secured to a control lever connected by coupling means with a pendulum so that the magnet swings with the control lever and the pendulum to generate pulses in windings for driving the pendulum. The coupling means includes a fork embracing the pendulum and permitting the lower part of the same to assume a vertical position when the housing is displaced. Since the magnet is not carried by the pendulum, but by the control lever, the winding can be secured to the housing near the clockwork.

Local Tomographic Methods in Sonar

Tomographic methods are described that will reconstruct object boundaries in shallow water using sonar data. The basic ideas involve microlocal analysis, and they are valid under weak assumptions even if the data do not correspond exactly to our model.

Characterization and reduction of artifacts in limited angle tomography

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.

The invertibility of rotation invariant Radon transforms

Let Rμ denote the Radon transform on Rn that integrates a function over hyperplanes in given smooth positive measures μ depending on the hyperplane. We characterize the measures μ for which Rμ is rotation invariant. We prove rotation invariant transforms are all one-to-one and hence invertible on the domain of square integrable functions of compact support, L02(Rn). We prove the hole theorem: if f ϵ L02Rn and Rμf = 0 for hyperplanes not intersecting a ball centered at the origin, then f is zero outside of that ball. Using the theory of Fourier integral operators, we extend these results to the domain of distributions of compact support on Rn. Our results prove invertibility for a mathematical model of positron emission tomography and imply a hole theorem for the constantly attenuated Radon transform as well as invertibility for other Radon transforms.

Tomographic reconstructions from incomplete data-numerical inversion of the exterior Radon transform

X-rays passing through the beating heart or bone can create error in standard computer tomographic scans of the organs around these regions. Often doctors are interested in imaging only the organs surrounded in these regions, not the regions themselves. The authors algorithm reconstructs the organs around the heart or metal without using the X-rays that create the error. Reconstructions of mathematical phantoms using this algorithm are given, and a description of the completed algorithm is presented. A mathematical explanation is given for the difficulties inherent in any tomography problem with incomplete data. It predicts weaknesses inherent in all reconstruction algorithms that use incomplete data.

Mathematical Methods in Tomography

This is the seventh Oberwolfach conference on the mathematics of tomography, the first one taking place in 1980. Tomography is the most popular of a series of medical and scientific imaging techniques that have been developed since the mid seventies of the last century.

Singular value decompositions and inversion methods for the exterior Radon transform and a spherical transform

Abstract The classical Radon transform, R, maps an integrable function in Rn to its integrals over all n − 1 dimensional hyperplanes, and the exterior Radon transform is the transform R restricted to hyperplanes that do not intersect a given disc. A singular value decomposition for the exterior transform is given for spaces of square integrable functions on the exterior of the disc. This decomposition in orthogonal functions explicitly produces the null space and range of the exterior transform and gives a new method for inverting the transform modulo the null space. A modification of this method is given that will exactly invert functions of compact support. These results generalize theorems of R. M. Perry and the author. A singular value decomposition for the Radon transform that integrates over spheres in Rn containing the origin is also given. This follows from the singular value decomposition for R and yields the null space and a new inversion method for this transform.

Null spaces and ranges for the classical and spherical Radon transforms

Abstract Let R be the classical Radon transform that integrates a function over hyperplanes in R n and let SM be the transform that integrates a function over spheres containing the origin in R n . We prove continuity results for both transforms and explicitly give the null space of R for a class of square integrable functions on the exterior of a ball in R n as well as the null space of SM for square integrable functions on a ball. We show SM : L 2 ( R n ) → L 2 ( R n ) is one-one, and we characterize the range of SM on classes of smooth functions and square integrable functions by certain moment conditions. If g ( x ) is a Schwartz function on R n that is zero to infinite order at x = 0, we prove moment conditions sufficient for g to be in the range of SM ( C ∞ ( R n )). We apply our results on SM to existence and uniqueness theorems for solutions to a characteristic initial value problem for the Darboux partial differential equation.

LOCAL TOMOGRAPHY IN ELECTRON MICROSCOPY

We present a new local tomographic algorithm applicable to electron microscope tomography. Our algorithm applies to the standard data acquisition method, single-axis tilting, as well as to more arbitrary acquisition methods including double axis and conical tilt. Using microlocal analysis we put the reconstructions in a mathematical context, explaining which singularities are stably visible from the limited data given by the data collection protocol in the electron microscope. Finally, we provide reconstructions of real specimens from electron tomography data.

Artifacts in Incomplete Data Tomography with Applications to Photoacoustic Tomography and Sonar

We develop a paradigm using microlocal analysis that allows one to characterize the visible and added singularities in a broad range of incomplete data tomography problems. We give precise characterizations for photoacoustic and thermoacoustic tomography and sonar, and provide artifact reduction strategies. In particular, our theorems show that it is better to arrange sonar detectors so that the boundary of the set of detectors does not have corners and is smooth. To illustrate our results, we provide reconstructions from synthetic spherical mean data as well as from experimental photoacoustic data.