Computational Aspects of Tomographic Imaging : Models to Inversions

Course #: EE 659
Proposers: Naren Naik (EE)
Units: 3-0-0-0-[4]
Prerequisites: Consent of instructor

Description
Tomographic imaging is the science of "reconstructing" ("seeing" inside) a body/surface, without cutting it open. This is typically done by obtaining data reflecting the interaction of the object of interest with suitable stimulation (radiations such as X-ray, optical, microwave, or currents and voltages), and then mathematically/computationally "inverting" that data to quantitatively characterize the object's property of interest. The computational aspect of this extremely applicable science is a unique mix of mathematical modelling and signal processing.

Among the most visible examples of this science are the CT and MRI scanners in hospitals. The need to quantitatively image objects of interest from indirect data arises in several applications of sub/behind-surface investigations in biomedical engineering, geophysical prospecting, humanitarian de-mining, archaeology, battlefield-surveillance and nondestructive evaluation, to name only a few.

The course gives a consolidated view of the tomographic reconstruction process which includes typical mathematical models used, deterministic and stochastic inversion methodologies, and concepts of approximate tomography for limited data settings. Various modalities of subsurface and biomedical imaging will be addressed in the course to give a feel for their applications.

Content (# of hours)
  1. Motivations and overview of tomography, limited data settings, approximate tomography, multimodal tomography.
  2. Linear tomography: Straight path tomography, Born and Rytov approximations in diffraction tomography, algebraic reconstruction techniques.
  3. Nonlinear tomography: Problem formulations, generic solution scheme, Frechet derivative calculations , method of adjoints.
  4. Regularized linear and nonlinear least squares solutions.
  5. Approximate tomography: Overview of shape based tomography, linear sampling.
  6. Introduction to stochastic reconstruction schemes, maximum likelihood and Bayesian methods, posterior sampling.
  7. Applications: Diffuse optical tomography, electrical impedance tomography, refraction tomography, electromagnetic wave tomography, elastography, multimodal tomography.
References:
  1. Jari P. Kaipio, Erkki Somersalo, Statistical and Computational Inverse Problems, Springer-Verlag, 2001.
  2. Curtis Vogel, Computational Inverse Problems, SIAM 2002.
  3. A. C. Kak and Malcolm Slaney, Principles of Computerized Tomographic Imaging, SIAM 2001 (also IEEE Press, 1988).
  4. Relevant literature.