The central advantage of medical imaging technologies is that pathologies can be observed directly rather than inferred from symptoms and validated by invasive biopsies. Diagnostic imaging has been undergoing a dramatic evolution in the past few decades with the advent of faster, more accurate, and less invasive scanner technologies. This scientific progress in clinical and biomedical research is of fundamental importance for image-guided therapy delivery systems as well as image-guided surgical procedures. In the rapidly expanding area of neuroprosthetics, the scientific advances of neuroimaging are going to include the accurate pre-operative location of intracortical sensor implant sites [P. Jezzard, P. M. Matthews, and S. M. Smith (eds.), Functional MRI: An introduction to methods. Oxford Univ. Press, (2001); M. Lotze, U. Laubis-Herrmann, H. Topka, M. Erb, and W. Grodd, Reorganization in the primary motor cortex after spinal cord injury. A functional magnetic resonance (fMRI) study. Restor. Neurol. Neurosci. 14, 183–187 (1999); L. R. Hochberg, M. D. Serruya, G. M. Friehs, J. A. Mukand, M. Saleh, A. H. Caplan, A. Branner, D. Chen, R. D. Penn, and J. P. Donoghue, Neuronal ensemble control of prosthetic devices by a human with tetraplegia. Nature 442, 164–171 (2006)]. Therefore the translation of diagnostic imagery into a rigorous mathematical framwework has been an ongoing concern of the theoretical community. For the most part, the challenge has been to find an innovative way of placing appropriate and well-defined mathematical structures on various medical imaging modalities. Among these enterprises, the embedding of the spinor-orbit interaction plays an important role for magnetic resonance tomography and neurofunctional magnetic resonance imaging [W. Schempp, Harmonic analysis on the Heisenberg nilpotent Lie group, with applications to signal theory. (1986; Zbl 0632.43001); Magnetic resonance imaging. Mathematical foundations and applications. (1998; Zbl 0930.92015)].
In the past, scientific progress in the direction of an embedding into suitable mathematical structures has been impeded by the complexity of the various biomedical imaging procedures consisting of image planning, scanning, registration, and processing strategies [G. Lohmann, K. Müller, V. Bosch, H. Mentzel, S. Hessler, Lin Chen, S. Zysset, and D.Y. von Cramon, LIPSIA – A new software system for the evaluation of functional magnetic resonance images of the human brain. Computerized Medical Imaging and Graphics 25, 449–457 (2001)]. The present survey paper touches only those aspects of the mathematics of medical imaging which reflect the personal tastes of the authors. These include image smoothing, registration, and segmentation. Due to this limitation, the paper completely ignores the deep impact of non–local quantum field theory and mathematical cosmology onto the modality of clinical magnetic resonance imaging which has proven itself to be the premiere diagnostic imaging technology of the last two decades, exceeding X–ray computed tomography (CT) and ultrasonography (US) in its soft tissue contrast. In the context of multislice CT (MSCT) and US examinations, it is important to notice that even for the embryonic state there exist no approved indications that a routine MRI examination could be a dangerous clinical imaging modality.
The authors demonstrate how geometric partial differential equations and variational methods may be applied to address automated methods that generate patient–specific models of relevant anatomical morphology from medical images, and automated methods that align multiple data sets with each other. They emphasize that much of the software to improve the quality of medical imagery is based on novel methods utilizing geometric partial differential equations in conjunction with standard image processing techniques as well as computer graphics tools facilitating man/scanner interactions. However, the authors omit completely the symbolic calculus in the sense of pseudodifferential operators which is needed for a rigorous mathematical treatment of the image reconstruction procedure in symplectic spinor–response imaging [W.J. Schempp, Magnetic resonance imaging: Mathematical foundations and applications. (1998); P. Jezzard et al. (eds.), Functional MRI: An introduction to methods. (2001); W. Schempp, H. J. Caulfield and C. S. Vikram (eds.), New Directions in Holography and Speckles. The Fourier holographic encoding strategy of symplectic spinor visualization. (to appear)]. In this context it should be observed that an important feature of the interactive imaging procedures is that there is a highly trained clinical radiologist in the loop who serves as the ultimate judge of the utility of the image modality and who tunes the parameters of the procedure either on- or off–line. However, the recent dramatic increase in availability, diversity, and contrast resolution of diagnostic imaging protocols threatens to overwhelm the technicians and clinicians who have to determine the details of imaging data acquisition, of patient position, and application of an optional contrast agent, as well as to analyze the final results.