Title:Mathematics and Algorithms for 3D Imaging of Dynamic Processes
Scanning devices with the technology to provide 3D images of the interior of objects have been around for decades, initially driven by advances in medical imaging. X-ray Computed Tomography (CT) and Magnetic Resonance Imaging (MRI) are commonly used in hospitals, but also have a broad number of spin-offs in other scanning applications, including nondestructive testing, baggage inspection, and fundamental research of materials and biological specimens.
Mathematics, and in particular the field of Inverse Problems, plays a crucial role in these instruments. The scanning device captures a series of measurements, modelled by the forward operator (e.g. the Radon transform for CT). The key problem that must be solved to obtain the 3D image is the mathematical inversion of this forward operator. The field of inverse problems in imaging is a mature research area in mathematics.
In recent years, a growing interest has emerged in capturing the dynamics of the evolving object, creating a 4D (space + time) movie of the object dynamics as it undergoes the scan. This opens up a broad range of applications in science (in-situ observations of physical and chemical processes), medicine (creating movies of hart-beat, blood-flow, etc.), and industry (functional quality control of mechanical systems).
Most theory on inverse problems in 3D imaging has been developed under the assumption that the object is constant over time. To create a 3D movie of the object evolution, completely new mathematical techniques must be developed that combine the two worlds of Dynamical Systems theory (describing the object evolution) and Linear/Nonlinear Inverse Problems (which in itself builds on numerical analysis, combinatorial optimization, and has a rich algebraic structure).
In this talk, I will outline the challenges in creating computational methods for 3D imaging of dynamic processes. The basic ideas for solving these highly underdetermined inverse problems will be illustrated by some recent results on high-speed imaging of flows through porous media.