The Chern-Mather class of the multiview variety. (English) Zbl 1422.14016
Summary: The multiview varietyassociated to a collection of \(N\) cameras records which sequences of image points in \(\mathbb P^{2N}\) can be obtained by taking pictures of a given world point \(x\in\mathbb P^3\) with the cameras. In order to reconstruct a scene from its picture under the different cameras, it is important to be able to find the critical points of the function which measures the distance between a general point \(u\in\mathbb P^{2N}\) and the multiview variety. In this paper we calculate a specific degree 3 polynomial that computes the number of critical points as a function of \(N\). In order to do this, we construct a resolution of the multiview variety and use it to compute its Chern-Mather class.
MSC:
14C17 | Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry |
14E05 | Rational and birational maps |
14N10 | Enumerative problems (combinatorial problems) in algebraic geometry |
51N35 | Questions of classical algebraic geometry |
57R20 | Characteristic classes and numbers in differential topology |
14Q15 | Computational aspects of higher-dimensional varieties |
65D18 | Numerical aspects of computer graphics, image analysis, and computational geometry |
Keywords:
Chern-Mather class; Chern-Schwartz-MacPherson class; computer vision; Euclidean distance degree; higher discriminantsReferences:
[1] | Agarwal, S.; Furukawa, Y.; Snavely, N.; Simon, I.; Curless, B.; Seitz, S. M.; Szeliski, R., Building Rome in a day, Communications of the ACM, 54, 105-112, (2011) |
[2] | Aluffi, P., Projective duality and a Chern-Mather involution, Trans. Amer. Math. Soc., 370, 3, 1803-1822, (2018) · Zbl 1422.14052 |
[3] | Brasselet, J.-P.; Lê Dũng Tráng, .; Seade, J., Euler obstruction and indices of vector fields, Topology. An International Journal of Mathematics, 39, 6, 1193-1208, (2000) · Zbl 0983.32030 |
[4] | Draisma, J.; Horobeţ, E.; Ottaviani, G.; Sturmfels, B.; Thomas, R. R., The Euclidean distance degree of an algebraic variety, Found. Comput. Math., 16, 1, 99-149, (2016) · Zbl 1370.51020 |
[5] | Fulton, W., Intersection Theory, (1984), Springer-Verlag, Berlin · Zbl 0541.14005 |
[6] | Keel, S., Intersection theory of moduli space of stable n-pointed curves of genus zero, Trans. Amer. Math. Soc., 330, 2, 545-574, (1992) · Zbl 0768.14002 |
[7] | Laufer, H., Taut two-dimensional singularities, Math. Ann., 205, 2, 131-164, (1973) · Zbl 0281.32010 |
[8] | MacPherson, R., Chern classes for singular algebraic varieties, Ann. Math., 100, 2, 423-432, (1974) · Zbl 0311.14001 |
[9] | Migliorini, L.; Shende, V., Higher discriminants and the topology of algebraic maps, Algebr. Geom., 5, 1, 114-130, (2017) · Zbl 1406.14005 |
[10] | Stewenius, H.; Schaffalitzky, F.; Nister, D., How hard is 3-view triangulation really? In Tenth IEEE International Conference on Computer Vision (ICCV’05), (2005) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.