Non-Commutative Affine Geometry and Symbol Calculus: Fourier Transform Magnetic Resonance Imaging and Wavelets

Abstract

Abstract. Magnetic Resonance Imaging (MRI) is one of the most significant advances in medical imaging in this century. The multiplanar capability and high-resolution imaging are unmatched by any other current clinical imaging technique. The physical and mathematical principles on which this non-invasive imaging technology is based are as complex as the computer-controlled synergy of the MRI scanner organization. The MRI scanner is a quantum electrodynamical (QED) device, the components of which implement and detect the wavelet interference and resonance phenomena which underly the symplectic MRI filter-bank processing. Notice the fact that QED has approved the semi-classical approach to quantum holography. In this paper, a semi-classical QED treatment of the basic MRI system organization is presented, with deep roots in the Keppler phase triangulation procedure of physical astronomy and in the symplectically invariant symbol calculus of pseudodifferential operators. It is based on non-commutative geometry of affine transvections and non-commutative Fourier analysis. This allows for modeling the interference patterns of phase coherent wavelets by distributional harmonic analysis on the Heisenberg nilpotent Lie group G of quantum mechanics. Geometric quantization yields the tomographic slices by the planar coadjoint orbit stratification O v,v ≠ 0, of the unitary dual Ĝ of the Heisenberg group G. The resonance of affine wavelets is treated by harmonic analysis on the affine solvable Lie group GA(ℝ), which is intrinsically operating on non-trivial blocks of G. It allows the acquisition of the coordinates of high-resolution tomographic scans within the tomographic slices by the Lauterbur spatial encoding technique of quantum holography.