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:
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