0000000000464718
AUTHOR
Olivier Cots
Geometric and numerical methods for the contrast and saturation problems in Magnetic Resonance Imaging
Talk; International audience; In this talk, we present the time minimal control problem about the saturation of a pair of spins of the same species but with inhomogeneities on the applied RF-magnetic field, in relation with the contrast problem in MRI. We make a complete analysis based on geometric control to classify the optimal syntheses in the single spin case, to pave the road to analyze the case of two spins. This points out the phenomenon of bridge, which consists in linking two singular arcs by a bang arc to bypass some singularities of the singular extremal flow. In the case of two spins, the question about global optimality is more intricate. The Bocop software is used to determine…
Energy minimization in two-level dissipative quantum control: The integrable case
International audience
A combination of algebraic, geometric and numerical methods in the contrast problem by saturation in magnetic resonance imaging
In this article, the contrast imaging problem by saturation in nuclear magnetic resonance is modeled as a Mayer problem in optimal control. The optimal solution can be found as an extremal solution of the Maximum Principle and analyzed with the recent advanced techniques of geometric optimal control. This leads to a numerical investigation based on shooting and continuation methods implemented in the HamPath software. The results are compared with a direct approach to the optimization problem and implemented within the Bocop toolbox. In complement lmi techniques are used to estimate a global optimum. It is completed with the analysis of the saturation problem of an ensemble of spin particle…
Skeleton-Based Multiview Reconstruction
International audience; The advantage of skeleton-based 3D reconstruction is to completely generate a single 3D object from well chosen views. Having numerous views is necessary for a reliable reconstruction but projections of skeletons lead to different topologies. We reconstruct 3D objects with curved medial axis (whose topology is a tree) from the perspective skeletons on an arbitrary number of calibrated acquisitions. The main contribution is to estimate the 3D skeleton, from multiple images: its topology is chosen as the closest to those of the perspective skeletons on the set of images, which means that the number of topology changes to map the 3D skeleton topology to topologies on im…
Comparison of Numerical Methods in the Contrast Imaging Problem in NMR
International audience; In this article, the contrast imaging problem in nuclear magnetic resonance is modeled as a Mayer problem in optimal control. A first synthesis of locally optimal solutions is given in the single-input case using geometric methods based on Pontryagin's maximum principle. We then compare these results using direct methods and a moment-based approach, and make a first step towards global optimality. Finally, some preliminary results are given in the bi-input case.
The energy minimization problem for two-level dissipative quantum systems
In this article, we study the energy minimization problem of dissipative two-level quantum systems whose dynamics is governed by the Kossakowski–Lindblad equations. In the first part, we classify the extremal curve solutions of the Pontryagin maximum principle. The optimality properties are analyzed using the concept of conjugate points and the Hamilton–Jacobi–Bellman equation. This analysis completed by numerical simulations based on adapted algorithms allows a computation of the optimal control law whose robustness with respect to the initial conditions and dissipative parameters is also detailed. In the final section, an application in nuclear magnetic resonance is presented.
Geometric and numerical methods in optimal control for the time minimal saturation in Magnetic Resonance
International audience
Algebraic-geometric techniques for the feedback classification and robustness of the optimal control of a pair of Bloch equations with application to Magnetic Resonance Imaging
The aim of this article is to classify the singular trajectories associated with the optimal control problems of a pair of controlled Bloch equations. The motivation is to analyze the robustness of the optimal solutions to the contrast and the time-minimal saturation problem, in magnetic resonance imaging, with respect to the parameters and B1-inhomogeneity. For this purpose, we use various computer algebra algorithms and methods to study solutions of polynomial systems of equations and inequalities which are used for classification issues: Gröbner basis, cylindrical algebraic decomposition of semi-algebraic sets, Thom's isotopy lemma.
Geometric and numerical techniques to compute conjugate and cut loci on Riemannian surfaces
International audience; We combine geometric and numerical techniques - the Hampath code - to compute conjugate and cut loci on Riemannian surfaces using three test bed examples: ellipsoids of revolution, general ellipsoids, and metrics with singularities on S2 associated to spin dynamics.