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

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

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

Content Provider	ACM Digital Library
Author	Jacquemard, Alain Safey El Din, Mohab Faugère, Jean-Charles Verron, Thibaut Bonnard, Bernard
Abstract	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 semi-algebraic 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.
Starting Page	103
Ending Page	110
Page Count	8
File Format	PDF
ISBN	9781450343800
DOI	10.1145/2930889.2930916
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	2016-07-20
Publisher Place	New York
Access Restriction	Subscribed
Subject Keyword	Real geometry Applications Polynomial system solving
Content Type	Text
Resource Type	Article

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