We study the relation between the bigraded Castelnuovo-Mumford regularity of a bihomogeneous ideal I in the coordinate ring of the product of two projective spaces and the bidegrees of a Gröbner basis of I with respect to the degree reverse lexicographical monomial order in generic coordinates. For the single-graded case, Bayer and Stillman unraveled all aspects of this relationship forty years ago and these results led to complexity estimates for computations with Gröbner bases. We build on this work to introduce a bounding region of the bidegrees of minimal generators of bihomogeneous Gröbner bases for I. We also use this region to certify the presence of some minimal generators close to its boundary. Finally, we show that, up to a certain shift, this region is related to the bigraded Castelnuovo-Mumford regularity of I.

Accurate gait events detection is imperative for reliable assessment of normal and pathological gaits. However, this detection becomes challenging in the absence of force plates. Hence, this research introduces two geometric models integrated into an automatic algorithm (O-GEST) for overground gait events detection using kinematic data.O-GEST employs B-Spline-based geometric models to represent the horizontal trajectory of foot landmarks. It leverages gait-dependent thresholds, along with optimal coefficients to detect events and compute spatiotemporal parameters on healthy and pathological gaits. To validate the proposed algorithm, timing differences in the detected events using the force plates and O-GEST were calculated and also compared between different methods on the gait data of 390 subjects. This dataset includes 200 healthy subjects, 100 subjects with unilateral hip osteoarthritis, 50 stroke survivors, 26 individuals diagnosed with Parkinson’s disease, and 14 children with cerebral palsy.The validation results show that O-GEST detects gait events with an overall accuracy of 13.5 ms for foot-strike and 12.6 ms for foot-off. It also demonstrates significantly more accurate results than the common deep learning-based and kinematic-based methods.O-GEST offers several advantages, including its applicability for events detection across various pathologies, capability to handle noisy trajectories, and usability in the absence of certain foot landmarks. Development of such algorithms could lead to enhanced accuracy and reliability of gait analysis in force-plate-less environments, especially in markerless gait analysis setups.

We study bihomogeneous systems defining, non-zero dimensional, biprojective varieties for which the projection onto the first group of variables results in a finite set of points. To compute (with) the 0-dimensional projection and the corresponding quotient ring, we introduce linear maps that greatly extend the classical multiplication maps for zero-dimensional systems, but are not those associated to the elimination ideal; we also call them multiplication maps. We construct them using linear algebra on the restriction of the ideal to a carefully chosen bidegree or, if available, from an arbitrary Grobner basis. The multiplication maps allow us to compute the elimination ideal of the projection, by generalizing FGLM algorithm to bihomogenous, non-zero dimensional, varieties. We also study their properties, like their minimal polynomials and the multiplicities of their eigenvalues, and show that we can use the eigenvalues to compute numerical approximations of the zero-dimensional projection. Finally, we establish a single exponential complexity bound for computing multiplication maps and Grobner bases, that we express in terms of the bidegrees of the generators of the corresponding bihomogeneous ideal.

We provide effective methods to construct and manipulate trilinear birational maps & ccedil; : (P1)3--\dashrightarrow P3 by establishing a novel connection between birationality and tensor rank. These yield four families of nonlinear birational transformations between 3D spaces that can be operated with enough flexibility for applications in computer-aided geometric design. More precisely, we describe the geometric constraints on the defining control points of the map that are necessary for birationality and present constructions for such configurations. For adequately constrained control points, we prove that birationality is achieved if and only if a certain 2 \times 2 \times 2 tensor has rank one. As a corollary, we prove that the locus of weights that ensure birationality is P1 \times P1 \times P1. Additionally, we provide formulas for the inverse \phi-1 as well as the explicit defining equations of the irreducible components of the base loci. Finally, we introduce a notion of ``distance to birationality"" for trilinear rational maps and explain how to continuously deform birational maps.

We provide effective methods to construct and manipulate trilinear birational maps $\phi:(\mathbb{P}^1)^3\dashrightarrow \mathbb{P}^3$ by establishing a novel connection between birationality and tensor rank. These yield four families of nonlinear birational transformations between 3D spaces that can be operated with enough flexibility for applications in computer-aided geometric design. More precisely, we describe the geometric constraints on the defining control points of the map that are necessary for birationality, and present constructions for such configurations. For adequately constrained control points, we prove that birationality is achieved if and only if a certain $2\times 2\times 2$ tensor has rank one. As a corollary, we prove that the locus of weights that ensure birationality is $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$. Additionally, we provide formulas for the inverse $\phi^{-1}$ as well as the explicit defining equations of the irreducible components of the base loci. Finally, we introduce a notion of "distance to birationality" for trilinear rational maps, and explain how to continuously deform birational maps.

Despite the advancements in developing markerless gait analysis systems, they still demonstrate lower accuracy compared to gold-standard systems. Hence, in this research, a novel approach is presented to improve the lower limb kinematics accuracy in markerless gait analysis. This approach refines the 3D lower-limb skeletons obtained by AI-based pose estimation algorithms in a subject-specific geometric manner, preserves skeleton links' length, benefits from gait phases information that adds biomechanical awareness to the algorithm, and utilizes an embedded trajectory smoothing. Validation of the proposed method shows that it reduces 12.6-43.5 % of root mean square error (RMSE) and significantly improves kinematic curves' similarity to the gold-standard ones. Results also prove the feasibility of more accurate lower limb kinematics calculation using a single (2.02-7.57 degrees RMSE) or dual RGB-D camera (1.66-7.25 degrees RMSE). Development of such algorithms could result in requirement of fewer cameras that deliver comparable or even superior measurement accuracy compared to multi-camera approaches.

In this paper, we investigate the structure of the saturation of ideals generated by sparse homogeneous polynomials over a projective toric variety X with respect to the irrelevant ideal of X. As our main results, we establish a duality property and make it explicit by introducing toric Sylvester forms, under a certain positivity assumption on X. In particular, we prove that toric Sylvester forms yield bases of some graded components of Isat/I, where I denotes an ideal generated by n+1 generic forms, n is the dimension of X and Isat is the saturation of I with respect to the irrelevant ideal of the Cox ring of X. Then, to illustrate the relevance of toric Sylvester forms we provide three consequences in elimination theory over smooth toric varieties: (1) we introduce a new family of elimination matrices that can be used to solve sparse polynomial systems by means of linear algebra methods, including overdetermined polynomial systems; (2) by incorporating toric Sylvester forms to the classical Koszul complex associated to a polynomial system, we obtain new expressions of the sparse resultant as a determinant of a complex; (3) we explore the computation of the toric residue of the product of two forms.

A tri-linear rational map in dimension three is a rational map ϕ : ( P C 1 ) 3 ⇢ P C 3 \phi : (\mathbb {P}_\mathbb {C}^1)^3 \dashrightarrow \mathbb {P}_\mathbb {C}^3 defined by four tri-linear polynomials without a common factor. If ϕ \phi admits an inverse rational map ϕ − 1 \phi ^{-1} , it is a tri-linear birational map. In this paper, we address computational and geometric aspects about these transformations. We give a characterization of birationality based on the first syzygies of the entries. More generally, we describe all the possible minimal graded free resolutions of the ideal generated by these entries. With respect to geometry, we show that the set B i r ( 1 , 1 , 1 ) \mathfrak {Bir}_{(1,1,1)} of tri-linear birational maps, up to composition with an automorphism of P C 3 \mathbb {P}_\mathbb {C}^3 , is a locally closed algebraic subset of the Grassmannian of 4 4 -dimensional subspaces in the vector space of tri-linear polynomials, and has eight irreducible components. Additionally, the group action on B i r ( 1 , 1 , 1 ) \mathfrak {Bir}_{(1,1,1)} given by composition with automorphisms of ( P C 1 ) 3 (\mathbb {P}_\mathbb {C}^1)^3 defines 19 orbits, and each of these orbits determines an isomorphism class of the base loci of these transformations.

Inspired by the strengths of quadric error metrics initially designed for mesh decimation, we propose a concise mesh reconstruction approach for 3D point clouds. Our approach proceeds by clustering the input points enriched with quadric error metrics, where the generator of each cluster is the optimal 3D point for the sum of its quadric error metrics. This approach favors the placement of generators on sharp features, and tends to equidistribute the error among clusters. We reconstruct the output surface mesh from the adjacency between clusters and a constrained binary solver. We combine our clustering process with an adaptive refinement driven by the error. Compared to prior art, our method avoids dense reconstruction prior to simplification and produces immediately an optimized mesh.

A d-dimensional tensor A of format n x n x...x n defines naturally a rational map Psi from the projective space Pn-1 to itself and its eigenscheme is then the subscheme of Pn-1 of fixed points of Psi. The eigendiscriminant is an irreducible polynomial in the coefficients of A that vanishes for a given tensor if and only if its eigenscheme is singular. In this paper, we contribute two formulas for the computation of eigendiscriminants in the cases n = 3 and n = 4. In particular, by restriction to symmetric tensors, we obtain closed formulas for the eigendiscriminants of plane curves and surfaces in P-3 as the ratio of some determinants of resultant matrices.

This book intends to overview the wide topic of algebraic curves and surfaces and is addressed to graduate students.

The emergence of RGB-D cameras and the development of pose estimation algorithms offer opportunities in biomechanics. However, some challenges still remain when using them for gait analysis, including noise which leads to misidentification of gait events and inaccuracy. Therefore, we present a novel kinematic-geometric model for spatio-temporal gait analysis, based on ankles' trajectory in the frontal plane and distance-to-camera data (depth). Our approach consists of three main steps: identification of the gait pattern and modeling via parameterized curves, development of a fitting algorithm, and computation of locomotive indices. The proposed fitting algorithm applies on both ankles' depth data simultaneously, by minimizing through numerical optimization some geometric and biomechanical error functions. For validation, 15 subjects were asked to walk inside the walkway of the OptoGait, while the OptoGait and an RGB-D camera (Microsoft Azure Kinect) were both recording. Then, the spatio-temporal parameters of both feet were computed using the OptoGait and the proposed model. Validation results show that the proposed model yields good to excellent absolute statistical agreement (0.86 ≤ Rc ≤ 0.99). Our kinematic-geometric model offers several benefits: (1) It relies only on the ankles' depth trajectory both for gait events extraction and spatio-temporal parameters' calculation; (2) it is usable with any kind of RGB-D camera or even with 3D marker-based motion analysis systems in absence of toes' and heels' markers; and (3) it enables improving the results by denoising and smoothing the ankles' depth trajectory. Hence, the proposed kinematic-geometric model facilitates the development of portable markerless systems for accurate gait analysis.

In this paper, we investigate the structure of the saturation of ideals generated by square systems of sparse homogeneous polynomials over a toric variety X with respect to the irrelevant ideal of X . As our main results, we establish a duality property and make it explicit by introducing toric Sylvester forms, under a certain positivity assumption on X . In particular, we prove that toric Sylvester forms yield bases of some graded components of I sat /I , where I denotes an ideal generated by n + 1 generic forms, n is the dimension of X and I sat the saturation of I with respect to the irrelevant ideal of the Cox ring of X . Then, to illustrate the relevance of toric Sylvester forms we provide three consequences in elimination theory: (1) we introduce a new family of elimination matrices that can be used to solve sparse polynomial systems by means of linear algebra methods, including overdetermined polynomial systems; (2) by incorporating toric Sylvester forms to the classical Koszul complex associated to a polynomial system, we obtain new expressions of the sparse resultant as a determinant of a complex; (3) we prove a new formula for computing toric residues of the product of two forms.

In this paper we study the equations of the elimination ideal associated with n+1 generic multihomogeneous polynomials defined over a product of projective spaces of dimension n. We first prove a duality property and then make this duality explicit by introducing multigraded Sylvester forms. These results provide a partial generalization of similar properties that are known in the setting of homogeneous polynomial systems defined over a single projective space. As an important consequence, we derive a new family of elimination matrices that can be used for solving zero-dimensional multiprojective polynomial systems by means of linear algebra methods.

Many global implicit surface reconstruction algorithms formulate the problem as a volumetric energy minimization, trading data fitting for geometric regularization. As a result, the output surfaces may be located arbitrarily far away from the input samples. This is amplified when considering i) strong regularization terms, ii) sparsely distributed samples or iii) missing data. This breaks the strong assumption commonly used by popular octree‐based and triangulation‐based approaches that the output surface should be located near the input samples. As these approaches refine during a pre‐process, their cells near the input samples, the implicit solver deals with a domain discretization not fully adapted to the final isosurface. We relax this assumption and propose a progressive coarse‐to‐fine approach that jointly refines the implicit function and its representation domain, through iterating solver, optimization and refinement steps applied to a 3D Delaunay triangulation. There are several advantages to this approach: the discretized domain is adapted near the isosurface and optimized to improve both the solver conditioning and the quality of the output surface mesh contoured via marching tetrahedra.

For a reduced hypersurface V(f)⊆Pn of degree d, the Castelnuovo-Mumford regularity of the Milnor algebra M(f) is well understood when V(f) is smooth, as well as when V(f) has isolated singularities. We study the regularity of M(f) when V(f) has a positive dimensional singular locus. In certain situations, we prove that the regularity is bounded by (d−2)(n+1), which is the degree of the Hessian polynomial of f. However, this is not always the case, and we prove that in Pn the regularity of the Milnor algebra can grow quadratically in d.

Parameterized algebraic curves and surfaces are widely used in geometric modeling and their manipulation is an important task in the processing of geometric models. In particular, the determination of the intersection loci between points, pieces of parameterized algebraic curves and pieces of algebraic surfaces is a key problem in this context. In this paper, we survey recent methods based on syzygies and blowup algebras for computing the image and the finite fibers of a curve or surface parameterization, more generally of a rational map. Conceptually, the main idea is to use elimination matrices, mainly built from syzygies, as representations of rational maps and to extract geometric informations from them. The construction and main properties of these matrices are first reviewed and then illustrated through several settings, each of them highlighting a particular feature of this approach that combines tools from commutative algebra, algebraic geometric and elimination theory.

For a reduced hypersurface V (f) ⊆ P of degree d, the Castelnuovo-Mumford regularity of the Milnor algebra M(f) is well understood when V (f) is smooth, as well as when V (f) has isolated singularities. We study the regularity of M(f) when V (f) has a positive dimensional singular locus. In certain situations, we prove that the regularity is bounded by (d− 2)(n+1), which is the degree of the Hessian polynomial of f . However, this is not always the case, and we prove that in P the regularity of the Milnor algebra can grow quadratically in d.

A tensor product surface $\mathscr{S}$ is an algebraic surface that is defined as the closure of the image of a rational map $\phi$ from $\mathbb{P}^1\times \mathbb{P}^1$ to $\mathbb{P}^3$. We provide new determinantal representations of $\mathscr{S}$ under the assumptions that $\phi$ is generically injective and its base points are finitely many and locally complete intersections. These determinantal representations are matrices that are built from the coefficients of linear relations (syzygies) and quadratic relations of the bihomogeneous polynomials defining $\phi$. Our approach relies on a formalization and generalization of the method of moving quadrics introduced and studied by David Cox and his co-authors.