Related papers: Problems and related results in Algebraic Vision a…
Grassmann tensors arise from classical problems of scene reconstruction in computer vision. Trifocal Grassmann tensors, related to three projections from a projective space of dimension k onto view-spaces of varying dimensions are studied…
Grassmann tensors arise from classical problems of scene reconstruction in computer vision. In particular, bifocal Grassmann tensors, related to a pair of projections from a projective space onto view-spaces of varying dimensions,…
In Algebraic Vision, the projective reconstruction of the position of each camera and scene point from the knowledge of many enough corresponding points in the views is called the structure from motion problem. It is known that the…
Multi-view Geometry is reviewed from an Algebraic Geometry perspective and multi-focal tensors are constructed as equivariant projections of the Grassmannian. A connection to the principal minor assignment problem is made by considering…
Linear projections from P^k to P^h appear in computer vision as models of images of dynamic or segmented scenes. Given multiple projections of the same scene, the identification of many enough correspondences between the images allows, in…
The Grassmann manifold of linear subspaces is important for the mathematical modelling of a multitude of applications, ranging from problems in machine learning, computer vision and image processing to low-rank matrix optimization problems,…
We study algebraic varieties associated with the camera resectioning problem. We characterize these resectioning varieties' multigraded vanishing ideals using Gr\"obner basis techniques. As an application, we derive and re-interpret…
In this survey article, we present interactions between algebraic geometry and computer vision, which have recently come under the header of algebraic vision. The subject has given new insights in multiple view geometry and its application…
Estimating positions of world points from features observed in images is a key problem in 3D reconstruction, image mosaicking,simultaneous localization and mapping and structure from motion. We consider a special instance in which there is…
Multiview geometry is the study of two-dimensional images of three-dimensional scenes, a foundational subject in computer vision. We determine a universal Groebner basis for the multiview ideal of n generic cameras. As the cameras move, the…
We introduce a new formalism and a number of new results in the context of geometric computational vision. The classical scope of the research in geometric computer vision is essentially limited to static configurations of points and lines…
The purpose of this article is to introduce projective geometry over composition algebras : the equivalent of projective spaces and Grassmannians over them are defined. It will follow from this definition that the projective spaces are in…
The ultimate goal of many image-based modeling systems is to render photo-realistic novel views of a scene without visible artifacts. Existing evaluation metrics and benchmarks focus mainly on the geometric accuracy of the reconstructed…
A proper scene representation is central to the pursuit of spatial intelligence where agents can robustly reconstruct and efficiently understand 3D scenes. A scene representation is either metric, such as landmark maps in 3D reconstruction,…
Critical loci for projective reconstruction from three views in four dimensional projective space are defined by an ideal generated by maximal minors of suitable $4 \times 3$ matrices, $N,$ of linear forms. Such loci are classified in this…
We study the critical points over an algebraic variety of an optimization problem defined by a quadratic objective that is degenerate. This scenario arises in machine learning when the dataset size is small with respect to the model, and is…
We study the following problem in computer vision from the perspective of algebraic geometry: Using $m$ pinhole cameras we take $m$ pictures of a line in $\mathbb P^3$. This produces $m$ lines in $\mathbb P^2$ and the question is which…
This paper addresses the problem of recovering projective camera matrices from collections of fundamental matrices in multiview settings. We make two main contributions. First, given ${n \choose 2}$ fundamental matrices computed for $n$…
Grassmann and flag varieties lead many lives in pure and applied mathematics. Here we focus on the algebraic complexity of solving various problems in linear algebra and statistics as optimization problems over these varieties. The measure…
An introductory overview of vector spaces, algebras, and linear geometries over an arbitrary commutative field is given. Quotient spaces are emphasized and used in constructing the exterior and the symmetric algebras of a vector space.…