Section: New Results

Algebraic algorithms for geometric computing

Implicit matrix representations of rational Bézier curves and surfaces

Participant : Laurent Busé.

In this work, we introduce and study a new implicit representation of rational Bézier curves and surfaces in the 3-dimensional space. Given such a curve or surface, this representation consists of a matrix whose entries depend on the space variables and whose rank drops exactly on this curve or surface. Our approach can be seen as an extension of the moving lines implicitization method introduced by Sederberg, from non-singular matrices to the more general context of singular matrices. First, we describe the construction of these new implicit matrix representations and their main geometric properties, in particular their ability to solve efficiently the inversion problem. Second, we show that these implicitization matrices adapt geometric problems, such as intersection problems, to the powerful tools of numerical linear algebra, in particular to one of the most important: the singular value decomposition. So, from the singular values of a given implicit matrix representation, we introduce a real evaluation function. We show that the variation of this function is qualitatively comparable to the Euclidean distance function. As an interesting consequence, we obtain a new determinantal formula for implicitizing a rational space curve or surface over the field of real numbers. Then, we show that implicit matrix representations can be used with numerical computations, in particular there is no need for symbolic computations to use them. We give some rigorous results explaining the numerical stability that we have observed in our experiments. We end the paper with a short illustration on ray tracing of parameterized surfaces.

This work has been accepted for presentation and publication at the SIAM conference on Geometric and Physical Modeling 2013 (Denver, USA, Nov. 11-14) [15] . It has been awarded the best paper price, 1st place.

Superfast solution of Toeplitz systems based on syzygy reduction

Participant : Bernard Mourrain.

In [22] , we present a new superfast algorithm for solving Toeplitz systems. This algorithm is based on a relation between the solution of such problems and syzygies of polynomials or moving lines. We show an explicit connection between the generators of a Toeplitz matrix and the generators of the corresponding module of syzygies. We show that this module is generated by two elements and the solution of a Toeplitz system $Tu=g$ can be reinterpreted as the remainder of a vector depending on $g$, by these two generators. We obtain these generators and this remainder with computational complexity $O\left(n{log}^{2}n\right)$ for a Toeplitz matrix of size $n\times n$.

This is a joint work with Houssam Khalil (Université Claude Bernard - Lyon I) and Michelle Schatzman (Institut Camille Jordan, Lyon).

Budan Tables of Real Univariate Polynomials

Participant : André Galligo.

The Budan table of $f$ collects the signs of the iterated derivative of $f$. We revisit the classical Budan-Fourier theorem for a univariate real polynomial $f$ and establish a new connexity property of its Budan table. In [18] , we use this property to characterize the virtual roots of $f$, (introduced by Gonzales-Vega, Lombardi, Mahe in 1998); they are continuous functions of the coefficients of $f$. We also consider a property (P) of a polynomial $f$, which is generically satisfied, it eases the topological-combinatorial description and study of the Budan tables. A natural extension of the information collected by the virtual roots provides alternative representations of (P)-polynomials; while an attached tree structure allows a strati fication of the space of (P)-polynomials.

A polynomial approach for extracting the extrema of a spherical function and its application in diffusion MRI

Participant : Bernard Mourrain.

Antipodally symmetric spherical functions play a pivotal role in diffusion MRI (Magnetic Resonance Imaging) in representing sub-voxel-resolution microstructural information of the underlying tissue. This information is described by the geometry of the spherical function. In [20] , we propose a method to automatically compute all the extrema of a spherical function. We then classify the extrema as maxima, minima and saddle-points to identify the maxima. We take advantage of the fact that a spherical function can be described equivalently in the spherical harmonic (SH) basis, in the symmetric tensor (ST) basis constrained to the sphere, and in the homogeneous polynomial (HP) basis constrained to the sphere. We extract the extrema of the spherical function by computing the stationary points of its constrained HP representation. Instead of using traditional optimization approaches, which are inherently local and require exhaustive search or re-initializations to locate multiple extrema, we use a novel polynomial system solver which analytically brackets all the extrema and refines them numerically, thus missing none and achieving high precision. To illustrate our approach we consider the Orientation Distribution Function (ODF). In diffusion MRI the ODF is a spherical function which represents a state-of-the-art reconstruction algorithm whose maxima are aligned with the dominant fiber bundles. It is, therefore, vital to correctly compute these maxima to detect the fiber bundle directions. To demonstrate the potential of the proposed polynomial approach we compute the extrema of the ODF to extract all its maxima. This polynomial approach is, however, not dependent on the ODF and the framework presented in this paper can be applied to any spherical function described in either the SH basis, ST basis or the HP basis.

This is a joint work with Aurobrata Ghosh (Inria, EPI ATHENA), Elias Tsigaridas (Inria, EPI POLSYS), Rachid Deriche (Inria, EPI ATHENA).

The geometry of sound-source localization using non-coplanar microphone arrays

The paper [29] addresses the task of sound-source localization from time delay estimates using arbitrarily shaped non-coplanar microphone arrays. We fully exploit the direct path propagation model and our contribution is threefold: we provide a necessary and sufficient condition for a set of time delays to correspond to a sound source position, a proof of the uniqueness of this position, and a localization mapping to retrieve it. The time delay estimation task is casted into a non-linear multivariate optimization problem constrained by necessary and sufficient conditions on time delays. Two global optimization techniques to estimate time delays and localize the sound source are investigated. We report an extensive set of experiments and comparisons with state-of-the-art methods on simulated and real data in the presence of noise and reverberations.

This is a joint work with Xavier Alameda-Pineda (Inria, EPI PERCEPTION) and Radu Horaud (Inria, EPI PERCEPTION).

Rational Invariants of a Group Action

Participant : Evelyne Hubert.

The article [28] is based on introductory lectures delivered at the Journées Nationales de Calcul Formel that took place at the Centre International de Recherche en Mathématiques (2013) in Marseille. We introduce basic notions on algebraic group actions and their invariants. Based on geometric considerations, we present algebraic constructions for a generating set of rational invariants. In particular the use of sections and quasi-sections contribute to increased efficiency and reduced output size. The notion of sections is refined compared to the cross-sections used in [9] .

Rational Invariants of Finite Abelian groups

Participant : Evelyne Hubert.

In [36] we investigate the field of rational invariants of the linear action of a finite abelian group in the non modular case. By diagonalization, the group is accurately described by an integer matrix of exponents. We make use of linear algebra to compute a minimal generating set of invariants and the substitution to rewrite any invariant in terms of this generating set. We show that the generating set can be chosen to consist of polynomial invariants. As an application, we provide a symmetry reduction scheme for polynomial systems the solution set of which are invariant by the group action.

This is joint work with George Labahn, University of Waterloo, Ontario (Canada).

Exact relaxation for polynomial optimization on semi-algebraic sets

Participants : Marta Abril Bucero, Bernard Mourrain.

In [31] , we study the problem of computing by relaxation hierarchies the infimum of a real polynomial function $f$ on a closed basic semialgebraic set $S$ and the points where this infimum is reached, if they exist. We show that when the infimum is reached, a relaxation hierarchy constructed from the Karush-Kuhn-Tucker ideal is always exact and that the vanishing ideal of the KKT minimizer points is generated by the kernel of the associated moment matrix in that degree, even if this ideal is not zero-dimensional. We also show that this relaxation allows to detect when there is no KKT minimizer. We prove that the exactness of the relaxation depends only on the real points which satisfy these constraints. This exploits representations of positive polynomials as elements of the preordering modulo the KKT ideal, which only involves polynomials in the initial set of variables. The approach provides a uniform treatment of different optimization problems considered previously. Applications to global optimization, optimization on semialgebraic sets defined by regular sets of constraints, optimization on finite semialgebraic sets, real radical computation are given.