The Rank of Trifocal Grassmann Tensors

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 in this work. A canonical form for the combined projection matrices is obtained. When the centers of projections satisfy a natural generality assumption, such canonical form gives a closed formula for the rank of the trifocal Grassmann tensors. The same approach is also applied to the case of two projections, confirming a previous result obtained with different methods in [6]. The rank of sequences of tensors converging to tensors associated with degenerate configurations of projection centers is also considered, giving concrete examples of a wide spectrum of phenomena that can happen.Comment: 18 page

The rank of trifocal grassmann tensors

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 in this work. A canonical form for the combined projection matrices is obtained. When the centers of projections satisfy a natural generality assumption, such canonical form gives a closed formula for the rank of trifocal Grassmann tensors. The same approach is also applied to the case of two projections, confirming a previous result obtained with different methods in [M. Bertolini, G. Besana, and C. Turrini, Ann. Mat. Pura Appl. (4), 196 (2016), pp. 539-553]. The rank of sequences of tensors converging to tensors associated with degenerate configurations of projection centers is also considered, giving concrete examples of a wide spectrum of phenomena that can happen

The varieties of bifocal Grassmann tensors

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, generalize the classical notion of fundamental matrices. In this paper, we study in full generality the variety of bifocal Grassmann tensors focusing on its birational geometry. To carry out this analysis, every object of multi-view geometry is described both from an algebraic and geometric point of view, e.g., the duality between the view spaces, and the space of rays is explicitly described via polarity. Next, we deal with the moduli of bifocal Grassmann tensors, thus showing that this variety is both birational to a suitable homogeneous space and endowed with a dominant rational map to a Grassmannian

Integrated Photogrammetric-Acoustic Technique for Qualitative Analysis of the Performance of Acoustic Screens in Sandy Soils

[EN] In this work, we present an integrated photogrammetric-acoustic technique that, together with the construction of a scaled wind tunnel, allows us to experimentally analyze the permeability behavior of a new type of acoustic screen based on a material called sonic crystal. Acoustic screens are devices used to reduce noise, mostly due to communication infrastructures, in its transmission phase from the source to the receiver. The main constructive difference between these new screens and the classic ones is that the first ones are formed by arrays of acoustic scatterers while the second ones are formed by continuous walls. This implies that, due to their geometry, screens based on sonic crystals are permeable to wind and water, unlike the classic ones. This fact may allow the use of these new screens in sandy soils, where sand would pass through the screen, avoiding the formation of sand dunes that are formed in classic screens and drastically reducing their acoustic performance. In this work, the movement of the sand and the resulting acoustic attenuation in these new screens are analyzed qualitatively, comparing the results with those obtained with the classic ones, and obtaining interesting results from the acoustic point of view.Bravo, JM.; Buchón Moragues, FF.; Redondo, J.; Ferri García, M.; Sánchez Pérez, JV. (2019). Integrated Photogrammetric-Acoustic Technique for Qualitative Analysis of the Performance of Acoustic Screens in Sandy Soils. Sensors. 19(22):1-17. https://doi.org/10.3390/s19224881S1171922Castiñeira-Ibañez, S., Rubio, C., & Sánchez-Pérez, J. V. (2015). Environmental noise control during its transmission phase to protect buildings. Design model for acoustic barriers based on arrays of isolated scatterers. Building and Environment, 93, 179-185. doi:10.1016/j.buildenv.2015.07.002Fredianelli, L., Del Pizzo, A., & Licitra, G. (2019). Recent Developments in Sonic Crystals as Barriers for Road Traffic Noise Mitigation. 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Problems and related results in Algebraic Vision and Multiview Geometry

The article presents a survey of results in algebraic vision and multiview geometry. The starting points is the study of projective algebraic varieties critical for scene reconstruction. Initially studied for reconstructing static scenes in three-dimensional spaces, these critical loci are later investigated for dynamic and segmented scenes in higher-dimensional projective spaces. The formal analysis of the ideals of critical loci employs Grassmann tensors, introduced as crucial tools for determining these ideals and aiding the reconstruction process away from critical loci. A long-term goal of the authors with other co-authors involves two main aspects: firstly studying properties of Grassmann tensors, as rank, multi-rank and core, along with the varieties that parameterize these tensors; concurrently conducting an analysis of families of critical loci in various scenarios

The joint image handbook

International audienceGiven multiple perspective photographs, point correspondences form the " joint image " , effectively a replica of three-dimensional space distributed across its two-dimensional projections. This set can be characterized by multilinear equations over image coordinates, such as epipolar and trifocal constraints. We revisit in this paper the geometric and algebraic properties of the joint image, and address fundamental questions such as how many and which multilinearities are necessary and/or sufficient to determine camera geometry and/or image correspondences. The new theoretical results in this paper answer these questions in a very general setting and, in turn, are intended to serve as a " handbook " reference about multilinearities for practitioners

Generalizations of the projective reconstruction theorem

We present generalizations of the classic theorem of projective reconstruction as a tool for the design and analysis of the projective reconstruction algorithms. Our main focus is algorithms such as bundle adjustment and factorization-based techniques, which try to solve the projective equations directly for the structure points and projection matrices, rather than the so called tensor-based approaches. First, we consider the classic case of 3D to 2D projections. Our new theorem shows that projective reconstruction is possible under a much weaker restriction than requiring, a priori, that all estimated projective depths are nonzero. By completely specifying possible forms of wrong configurations when some of the projective depths are allowed to be zero, the theory enables us to present a class of depth constraints under which any reconstruction of cameras and points projecting into given image points is projectively equivalent to the true camera-point configuration. This is very useful for the design and analysis of different factorization-based algorithms. Here, we analyse several constraints used in the literature using our theory, and also demonstrate how our theory can be used for the design of new constraints with desirable properties. The next part of the thesis is devoted to projective reconstruction in arbitrary dimensions, which is important due to its applications in the analysis of dynamical scenes. The current theory, due to Hartley and Schaffalitzky, is based on the Grassmann tensor, generalizing the notions of Fundamental matrix, trifocal tensor and quardifocal tensor used for 3D to 2D projections. We extend their work by giving a theory whose point of departure is the projective equations rather than the Grassmann tensor. First, we prove the uniqueness of the Grassmann tensor corresponding to each set of image points, a question that remained open in the work of Hartley and Schaffalitzky. Then, we show that projective equivalence follows from the set of projective equations, provided that the depths are all nonzero. Finally, we classify possible wrong solutions to the projective factorization problem, where not all the projective depths are restricted to be nonzero. We test our theory experimentally by running the factorization based algorithms for rigid structure and motion in the case of 3D to 2D projections. We further run simulations for projections from higher dimensions. In each case, we present examples demonstrating how the algorithm can converge to the degenerate solutions introduced in the earlier chapters. We also show how the use of proper constraints can result in a better performance in terms of finding a correct solution