Minimizing the echo time in diffusion imaging using spiral readouts and a head gradient system

2020 ◽  
Vol 84 (6) ◽  
pp. 3117-3127 ◽  
Author(s):  
Bertram Jakob Wilm ◽  
Franciszek Hennel ◽  
Manuela Barbara Roesler ◽  
Markus Weiger ◽  
Klaas Paul Pruessmann
Keyword(s):  
Diffusion Imaging ◽  
Gradient System ◽  
Head Gradient ◽  
Echo Time

scholarly journals Clinical application of ultrashort echo-time MRI for lung pathologies in children

2021 ◽  
Author(s):  
J. Geiger ◽  
K.G. Zeimpekis ◽  
A. Jung ◽  
A. Moeller ◽  
C.J. Kellenberger
Keyword(s):  
Clinical Application ◽  
Ultrashort Echo Time ◽  
Echo Time

scholarly journals Relating a Rate-Independent System and a Gradient System for the Case of One-Homogeneous Potentials

2021 ◽  
Author(s):  
Alexander Mielke
Keyword(s):  
Gradient Flow ◽  
Careful Analysis ◽  
Flow Equation ◽  
Uniqueness Result ◽  
Gradient System ◽  
Independent System ◽  
Exact Relation ◽  
Flow Equations ◽  
Total Variation Flow ◽  
Energetic Solutions

AbstractWe consider a non-negative and one-homogeneous energy functional $${{\mathcal {J}}}$$ J on a Hilbert space. The paper provides an exact relation between the solutions of the associated gradient-flow equations and the energetic solutions generated via the rate-independent system given in terms of the time-dependent functional $${{\mathcal {E}}}(t,u)= t {{\mathcal {J}}}(u)$$ E ( t , u ) = t J ( u ) and the norm as a dissipation distance. The relation between the two flows is given via a solution-dependent reparametrization of time that can be guessed from the homogeneities of energy and dissipations in the two equations. We provide several examples including the total-variation flow and show that equivalence of the two systems through a solution dependent reparametrization of the time. Making the relation mathematically rigorous includes a careful analysis of the jumps in energetic solutions which correspond to constant-speed intervals for the solutions of the gradient-flow equation. As a major result we obtain a non-trivial existence and uniqueness result for the energetic rate-independent system.

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