## Taiwanese Journal of Mathematics

### QUANTUM HOLOGRAPHY AND MAGNETIC RESONANCE TOMOGRAPHY: AN ENSEMBLE QUANTUM COMPUTING APPROACH

Walter Schempp

#### Abstract

Coherent wavelets form a unified basis of the multichannel perfect reconstruction analysis-synthesis filter bank of high resolution radar imaging and clinical magnetic resonance imaging (MRI). The filter bank construction is performed by the Kepplerian temporospatial phase detection strategy which allows for the stroboscopic and synchronous cross sectional quadrature filtering of phase histories in local frequency encoding multichannels with respect to the rotating coordinate frame of reference. The Kepplerian strategy and the associated filter bank construction take place in symplectic affine planes which are immersed as coadjoint orbits of the Heisenberg two-step nilpotent Lie group $G$ into the foliated three-dimensional real projective space $\mbox{ P}\bigl(\mbox{ R}\times\mbox{Lie}(G)^{\star}\bigr)$. Due to the factorization of transvections into affine dilations of opposite ratio, the Heisenberg group $G$ under its natural sub-Riemannian metric acts on the line bundle realizing the projective space $\mbox{ P}\bigl(\mbox{ R}\times \mbox{ Lie}(G)^{\star}\bigr)$. Its elliptic non-Euclidean geometry without absolute quadric, associated to the unitary dual $\hat{G}$, governs the design of the coils inside the bore of the MRI scanner system. It determines the distributional reproducing kernel of the tracial read-out process of quantum holograms excited and coexisting in the MRI scanner system. Thus the pathway of this paper leads from Keppler's approach to projective geometry to the Heisenberg approach to the sub-Riemannian geometry of quantum physics, and finally to the enormously appealing topic of ensemble quantum computing.

#### Article information

Source
Taiwanese J. Math., Volume 2, Number 3 (1998), 257-286.

Dates
First available in Project Euclid: 18 July 2017

https://projecteuclid.org/euclid.twjm/1500406964

Digital Object Identifier
doi:10.11650/twjm/1500406964

Mathematical Reviews number (MathSciNet)
MR1641151

Zentralblatt MATH identifier
0911.92019

#### Citation

Schempp, Walter. QUANTUM HOLOGRAPHY AND MAGNETIC RESONANCE TOMOGRAPHY: AN ENSEMBLE QUANTUM COMPUTING APPROACH. Taiwanese J. Math. 2 (1998), no. 3, 257--286. doi:10.11650/twjm/1500406964. https://projecteuclid.org/euclid.twjm/1500406964