Robinson Professor of Mathematics
Department of Mathematics
503 Boston Avenue
Medford, MA 02155
Email @tufts.edu: todd.quinto
Both applied and pure mathematics: tomography and integral geometry
My research is in the applied area of tomography and its pure
cousin, integral geometry. Tomography is an inverse problem, and the
goal of tomography is to map the interior structure of objects using
indirect data such as from X-rays. Integral geometry is the
mathematics of averaging over curves and surfaces, and it is the
pure math behind many problems in tomography. Integral geometry
combines geometric intuition, harmonic analysis, and microlocal
analysis (the analysis of singularities and what Fourier integral
operators do to them). I have proven support theorems and properties
of transforms integrating over hyperplanes, circles and spheres in
Euclidean space and manifolds.
Because of the mentorship of Tufts physics professor and tomography
pioneer, Allan Cormack (Tufts' only Nobel Laureate) I developed
X-ray tomography algorithms for the nondestructive evaluation of
large objects such as rocket bodies, and this motivated my research
in limited data tomography
In limited data tomography problems, some tomographic data are
missing. I developed a paradigm to describe which features of the
object will be visible from limited tomographic data and which will
be invisible (or difficult to reconstruct). I proved the paradigm
using microlocal analysis. Often artifacts are added to tomographic
reconstructions from limited data, and colleagues and I recently
used microlocal analysis to prove the cause of these added artifacts
and to predict where they will occur.
Collaborators and I have developed local algorithms for electron
microscopy, emission tomography, Radar, Sonar, and ultrasound. In
each case we use microlocal analysis to determine the strengths and
weaknesses of the problem and to refine and improve the algorithms.
I have been pleased to have undergraduate and graduate students help
me with each of the projects described above.
More information is
available on my personal website.