Congruences and concurrent lines in multi-view geometry

Abstract

We present a new framework for multi-view geometry in computer vision. A camera is a mapping between P 3 and a line congruence. This model, which ignores image planes and measurements, is a natural abstraction of traditional pinhole cameras. It includes two-slit cameras, pushbroom cameras, catadioptric cameras, and many more. We study the concurrent lines variety, which consists of n-tuples of lines in P 3 that intersect at a point. Combining its equations with those of various congruences, we derive constraints for corresponding images in multiple views. We also study photographic cameras which use image measurements and are modeled as rational maps from P 3 to P 2 or P 1 × P 1 .

Selectivity for mid‐level properties of faces and places in the fusiform face area and parahippocampal place area

“…These results show that the selectivity to faces and places in the FFA and PPA is to some extent determined by the image properties that are preserved by local-scrambling, such as the spatial envelope of the image. This fits with previous studies that have demonstrated selectivity in higherlevel regions of the ventral stream to spatial properties of the image (Bracci & Op de Beeck, 2016;Cichy et al, 2013;Golomb & Kanwisher, 2012;Levy, Hasson, Avidan, Hendler, & Malach, 2001;Ponce, Sturmfels, & Trager, 2017;Watson et al, 2016). More generally, these results are consistent with previous studies that have shown that patterns of response in high-level visual regions are sensitive to image properties (Rice et al, 2014;Watson et al, 2016;Xu, Yue, Lescroart, Biederman, & Kim, 2009;Yue, Tjan, & Biederman, 2006).…”

The Multidegree of the Multi-Image Variety

“…Roughly speaking, a multi-view variety parametrizes all the possible images that can be taken by a fixed collection of cameras. See [1,11,14] for more details on multi-view varieties. See [13] for a survey on various camera models.…”

“…The work [11] presents a new point of view to study the multi-view varieties. A photographic camera maps a point in the scene to a point in the image.…”

“…A photographic camera maps a point in the scene to a point in the image. [11] defines a geometric camera, which maps a point in the scene not to a point, but to a viewing ray. More precisely, a photographic camera is a map P 3 P 2 or P 3 P 1 × P 1 , whereas a geometric camera is a map P 3 Gr(1, P 3 ) such that the image of each point is a line containing the point.…”

“…Let C i be the closure of the image of φ i inside the i-th Gr(1, P 3 ). Then under some assumptions, [11,Theorem 5.1] shows that…”