Francois Monard
University of California-Santa Cruz, CA·Mathematics & Statistics
CT scanners, seismic imaging of Earth's interior and non-destructive testing of manufactured parts all reveal what is inside an object without opening it. They share one mathematical problem: rebuilding an unknown interior property from measurements taken along lines or curves through the object.
- Fields
Mathematics & Statistics
- Topics
integral geometry
inverse problems
X-ray transform
- Grants
- 1 known grant, through Jul 2028
What they work on
- Unify three complex-analytic methods for X-ray transforms, including transport twistor space, to derive exact reconstruction formulas in curved (Riemannian and non-Euclidean) geometries.
- Derive sharper characterizations of which measurement data are possible in partial-data X-ray transform problems, using complex geometry and microlocal analysis.
- Extend explicit tensor tomography solutions, which recover direction-dependent fields from curve measurements, to surfaces of varying curvature and geometries with conjugate points.
- Begin extending tensor tomography reconstructions to three-dimensional settings and test the theory's practical feasibility with proof-of-concept numerical implementations.
More about the research
The task becomes harder when the material is non-uniform, as in tissue, rock or lenses whose light-bending index varies, because the measurement paths then curve. Better mathematics could make these reconstructions more precise.
Why it matters
Stronger mathematical foundations for reconstruction from line and curve measurements could improve medical imaging such as X-ray CT. They could also improve seismic imaging used in natural-resource management and inspection of manufactured materials, especially when the imaged material is non-uniform and measurement paths bend.
Known grants
From NIH and NSF research grants. Labs often have other funding we can't see.
Closest labs
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Emory University, GA·Mathematics & Statistics
Recover the initial state of the Universe from the cosmic microwave background by combining an X-ray tomography approach with Einstein's equations, developing statistical inference and numerical reconstruction algorithms.
Jeremy Marzuola
University of North Carolina at Chapel Hill, NC·Mathematics & Statistics
Establish rigorous constructions of fundamental modes and resonances for model operators from condensed matter and topological physics, including light propagating in crystalline structures that contain localized defects.
Xuwen Zhu
Northeastern University, MA·Mathematics & Statistics
Study how gravitational instantons, four-dimensional spaces that solve Einstein's equations, degenerate and how their moduli spaces are structured, using geometric microlocal analysis and PDE techniques.
The school
University of California-Santa Cruz: 15 labs on PI Pages, $218M a year on research, 183 research doctorates a year.
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