Determinantal sets, singularities and application to optimal control in medical imagery

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.

[1]  Jean-Michel Coron,et al.  Optimal Geometric Control Applied to the Protein Misfolding Cyclic Amplification Process , 2015 .

[2]  Bican Xia,et al.  A complete algorithm for automated discovering of a class of inequality-type theorems , 2001, Science in China Series F Information Sciences.

[3]  Mohab Safey El Din,et al.  Exact algorithms for linear matrix inequalities , 2015, SIAM J. Optim..

[4]  L. S. Pontryagin,et al.  Mathematical Theory of Optimal Processes , 1962 .

[5]  Yun Zhang,et al.  Geometric Optimal Control of the Contrast Imaging Problem in Nuclear Magnetic Resonance , 2012, IEEE Transactions on Automatic Control.

[6]  Jean-Charles Faugère,et al.  On the complexity of the generalized MinRank problem , 2011, J. Symb. Comput..

[7]  Jean-Charles Faugère,et al.  FGb: A Library for Computing Gröbner Bases , 2010, ICMS.

[8]  James H. Davenport,et al.  The complexity of quantifier elimination and cylindrical algebraic decomposition , 2007, ISSAC '07.

[9]  Mohab Safey El Din,et al.  Real root finding for determinants of linear matrices , 2014, J. Symb. Comput..

[10]  Masahiro Shiota,et al.  Thorn’s first isotopy lemma: a semialgebraic version, with uniform bound , 1995 .

[11]  John Marriott,et al.  Algebraic geometric classification of the singular flow in the contrast imaging problem in nuclear magnetic resonance , 2013 .

[12]  Jean-Charles Faugère,et al.  Critical points and Gröbner bases: the unmixed case , 2012, ISSAC.

[13]  Scott McCallum,et al.  On projection in CAD-based quantifier elimination with equational constraint , 1999, ISSAC '99.

[14]  Tony Crilly,et al.  Bifurcations and Catastrophes , 2000 .

[15]  George E. Collins,et al.  Quantifier elimination for real closed fields by cylindrical algebraic decomposition , 1975 .

[16]  Mohab Safey El Din,et al.  Real Root Finding for Rank Defects in Linear Hankel Matrices , 2015, ISSAC.

[17]  Pierre-Jean Spaenlehauer,et al.  On the Complexity of Computing Critical Points with Gröbner Bases , 2013, SIAM J. Optim..

[18]  Mohab Safey El Din,et al.  Variant quantifier elimination , 2012, J. Symb. Comput..

[19]  Fabrice Rouillier,et al.  Solving parametric polynomial systems , 2004, J. Symb. Comput..

[20]  Jose Israel Rodriguez,et al.  Data-Discriminants of Likelihood Equations , 2015, ISSAC.