Algebraic geometric classification of the singular flow in the contrast imaging problem in nuclear magnetic resonance

Bonnard, Bernard; Chyba, Monique; Jacquemard, Alain; Marriott, John

doi:10.3934/mcrf.2013.3.397

Cited by 6 publications

(17 citation statements)

References 17 publications

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“…The contrast imaging problem was analyzed in a series of recent articles using geometric optimal control [8,9] or numerical indirect methods [11], and leads to experimental work described in [12,26], but restricted mainly to the single-input case. In such studies, the contrast problem is essentially reduced to the analysis of the socalled singular trajectories of the system modeling the problem.…”

Section: A Terminal Manifold To Reachmentioning

confidence: 99%

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Geometric and Numerical Methods in the Contrast Imaging Problem in Nuclear Magnetic Resonance

et al. 2014

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In this article, the contrast imaging problem 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 techniques of geometric control. This leads to a numerical investigation based on so-called indirect methods using the HamPath software. The results are then compared with a direct method implemented within the Bocop toolbox. Finally lmi techniques are used to estimate a global optimum.

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Section: A Terminal Manifold To Reachmentioning

confidence: 99%

“…The geometric control methods are efficient to stratify the set of extremals, and they are complemented by the geometric analysis of the extremal curves. This is explained next based on preliminary works [7,16] for the first point and [9] for the second point.…”

Section: Classification Of the Extremal Curvesmentioning

confidence: 99%

Geometric and Numerical Methods in the Contrast Imaging Problem in Nuclear Magnetic Resonance

et al. 2014

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“…Using the symmetry of revolution [2] which allows to eliminate one state variable for each matter, we obtain the system…”

Section: Modeling the Dynamicsmentioning

confidence: 99%

“…We may restrict to the open subset of V r,u where Y (2) is invertible, then eliminating Y (1) yields that…”

Section: Incidence Varietiesmentioning

confidence: 99%

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Determinantal Sets, Singularities and Application to Optimal Control in Medical Imagery

Bonnard

Faugère

Jacquemard

et al. 2016

Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation

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Control theory has recently been involved in the field of nuclear magnetic resonance imagery. The goal is to control the magnetic field optimally in order to improve the contrast between two biological matters on the pictures. Geometric optimal control leads us here to analyze meromorphic vector fields depending upon physical parameters, and having their singularities defined by a determinantal variety. The involved matrix has polynomial entries with respect to both the state variables and the parameters. Taking into account the physical constraints of the problem, one needs to classify, with respect to the parameters, the number of real singularities lying in some prescribed semialgebraic set. We develop a dedicated algorithm for real root classification of the singularities of the rank defects of a polynomial matrix, cut with a given semi-algebraic set. The algorithm works under some genericity assumptions which are easy to check. These assumptions are not so restrictive and are satisfied in the aforementioned application. As more general strategies for real root classification do, our algorithm needs to compute the critical loci of some maps, intersections with the boundary of the semi-algebraic domain, etc. In order to compute these objects, the determinantal structure is exploited through a stratification by the rank of the polynomial matrix. This speeds up the computations by a factor 100. Furthermore, our implementation is able to solve the application in medical imagery, which was out of reach of more general algorithms for real root classification. For instance, computational results show that the contrast problem where one of the matters is water is partitioned into three distinct classes.

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Classification of Hyperbolic Singularities in Interval 3-Dimensional Linear Differential Systems

Mizukoshi¹,

Jacquemard²,

Lodwick³

2020

Information Processing and Management of Uncertainty in Knowledge-Based Systems

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Algebraic geometric classification of the singular flow in the contrast imaging problem in nuclear magnetic resonance

Cited by 6 publications

References 17 publications

Geometric and Numerical Methods in the Contrast Imaging Problem in Nuclear Magnetic Resonance

Geometric and Numerical Methods in the Contrast Imaging Problem in Nuclear Magnetic Resonance

Determinantal Sets, Singularities and Application to Optimal Control in Medical Imagery

Classification of Hyperbolic Singularities in Interval 3-Dimensional Linear Differential Systems

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