Computed Tomography

Computed tomography (CT) is one of the most important diagnostic tools in modern medicine. While the term computed tomography was initially reserved for x-ray based CT scanners, it nowadays covers various non-invasive imaging technologies, where mathematics plays a major role for obtaining diagnostic images. Examples include x-ray CT, single photon emission computed tomography (SPECT), positron emission tomography (PET), magnetic resonance tomography (MRT), ultrasound tomography, electrical impedance tomography, optical imaging, as well as photo- and thermoacoustic tomography (see, for example, [HalPer14, Her09, Kuc14, Nat01] for some general introductions).

A unifying element of all tomographic applications is that only indirect information about the quantity of interest (usually modelled as a function or some high dimensional parameter vector) can be collected when scanning the patient. Due to the modeling imperfections, measurement errors and statistical uncertainties, the data are additionally corrupted by deterministic or random noise. Such types of applications are most conveniently studied in the framework of inverse problems, where the reconstruction problem is formulated as a system of linear or non-linear equations in high or infinite dimensional spaces. In our research we work on various theoretical and practical for the solution of such tomographic image reconstruction problems. Examples include general iterative reconstruction methods (see, for example, [HKLS07, HLR09]), or special, problem adapted techniques for photoacoustic tomography (see, for example [FHR07, Hal14, Kow14]).

References:

[FHR07] Inversion of spherical means and the wave equation in even dimensions. D. Finch, M. Haltmeier, Rakesh SIAM J. Appl. Math. 68(2): 392-412, 2007.

[Hal14] Universal inversion formulas for recovering a function from spherical means. M. Haltmeier. SIAM J. Math. Anal. 46(1): 214–232, 2014. [arxiv.org]

[HalPer14] M. Haltmeier, S. Pereverzyev Jr. Introduction to the mathematics of computed tomography, Internat. Math. Nachrichten 226:29-51, 2014. [pdf ]

[Her09] G. T. Herman. Fundamentals of computerized tomography, second edition. Springer, 2009.

[HKLS07] Kaczmarz methods for regularizing nonlinear ill-posed equations II: Applications M. Haltmeier, R. Kowar, A. Leitao, O. Scherzer. Inverse Probl. Imaging 1(3): 507-523, 2007.

[HLR09] M. Haltmeier, A. Leitao, E. Resmerita. On regularization methods of EM-Kaczmarz type. Inverse Problems 25(7): 075008, 2009.

[Kow14] On time reversal in photoacoustic tomography for tissue similar to water. R. Kowar SIAM J. Imaging Sciences 7(1): 509-527, 2014. [pdf ] [arxiv.org]

[Kuc14] P. Kuchment. The Radon transform and medical imaging. SIAM, 2014.

[Nat01] F. Natterer. The Mathematics of Computerized Tomography. SIAM, 2001.