Group of Homography in Real Projective Plane

Using the Mizar system [2], we formalized that homographies of the projective real plane (as defined in [5]), form a group. Then, we prove that, using the notations of Borsuk and Szmielew in [3] “Consider in space ℝℙ2 points P1, P2, P3, P4 of which three points are not collinear and points Q1,Q2,Q3,Q4 each three points of which are also not collinear. There exists one homography h of space ℝℙ2 such that h(Pi) = Qi for i = 1, 2, 3, 4. ”(Existence Statement 52 and Existence Statement 53) [3]. Or, using notations of Richter [11] “Let [a], [b], [c], [d] in ℝℙ2 be four points of which no three are collinear and let [a′],[b′],[c′],[d′] in ℝℙ2 be another four points of which no three are collinear, then there exists a 3 × 3 matrix M such that [Ma] = [a′], [Mb] = [b′], [Mc] = [c′], and [Md] = [d′] ”Makarios has formalized the same results in Isabelle/Isar (the collineations form a group, lemma statement52-existence and lemma statement 53-existence) and published it in Archive of Formal Proofs [10], [9].Rue de la Brasserie 5, 7100, La Louvière, BelgiumGrzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Grzegorz Bancerek, Czesław Bylinski, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pak, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261-279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1.Karol Borsuk and Wanda Szmielew. Foundations of Geometry. North Holland, 1960.Czesław Byliński. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661-668, 1990.Roland Coghetto. Homography in RP2. Formalized Mathematics, 24(4):239-251, 2016.Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991.Kanchun, Hiroshi Yamazaki, and Yatsuka Nakamura. Cross products and tripple vector products in 3-dimensional Euclidean space. Formalized Mathematics, 11(4):381-383, 2003.Wojciech Leończuk and Krzysztof Prazmowski. A construction of analytical projective space. Formalized Mathematics, 1(4):761-766, 1990.Timothy James McKenzie Makarios. The independence of Tarski’s Euclidean Axiom. Archive of Formal Proofs, October 2012.Timothy James McKenzie Makarios. A mechanical verification of the independence of Tarski’s Euclidean Axiom. Victoria University of Wellington, New Zealand, 2012. Master’s thesis.Jürgen Richter-Gebert. Perspectives on projective geometry: a guided tour through real and complex geometry. Springer Science & Business Media, 2011

Detecting shadows and low-lying objects in indoor and outdoor scenes using homographies

Many computer vision applications apply background suppression techniques for the detection and segmentation of moving objects in a scene. While these algorithms tend to work well in controlled conditions they often fail when applied to unconstrained real-world environments. This paper describes a system that detects and removes erroneously segmented foreground regions that are close to a ground plane. These regions include shadows, changing background objects and other low-lying objects such as leaves and rubbish. The system uses a set-up of two or more cameras and requires no 3D reconstruction or depth analysis of the regions. Therefore, a strong camera calibration of the set-up is not necessary. A geometric constraint called a homography is exploited to determine if foreground points are on or above the ground plane. The system takes advantage of the fact that regions in images off the homography plane will not correspond after a homography transformation. Experimental results using real world scenes from a pedestrian tracking application illustrate the effectiveness of the proposed approach

Automated Top View Registration of Broadcast Football Videos

In this paper, we propose a novel method to register football broadcast video frames on the static top view model of the playing surface. The proposed method is fully automatic in contrast to the current state of the art which requires manual initialization of point correspondences between the image and the static model. Automatic registration using existing approaches has been difficult due to the lack of sufficient point correspondences. We investigate an alternate approach exploiting the edge information from the line markings on the field. We formulate the registration problem as a nearest neighbour search over a synthetically generated dictionary of edge map and homography pairs. The synthetic dictionary generation allows us to exhaustively cover a wide variety of camera angles and positions and reduce this problem to a minimal per-frame edge map matching procedure. We show that the per-frame results can be improved in videos using an optimization framework for temporal camera stabilization. We demonstrate the efficacy of our approach by presenting extensive results on a dataset collected from matches of football World Cup 2014

Radially-Distorted Conjugate Translations

This paper introduces the first minimal solvers that jointly solve for affine-rectification and radial lens distortion from coplanar repeated patterns. Even with imagery from moderately distorted lenses, plane rectification using the pinhole camera model is inaccurate or invalid. The proposed solvers incorporate lens distortion into the camera model and extend accurate rectification to wide-angle imagery, which is now common from consumer cameras. The solvers are derived from constraints induced by the conjugate translations of an imaged scene plane, which are integrated with the division model for radial lens distortion. The hidden-variable trick with ideal saturation is used to reformulate the constraints so that the solvers generated by the Grobner-basis method are stable, small and fast. Rectification and lens distortion are recovered from either one conjugately translated affine-covariant feature or two independently translated similarity-covariant features. The proposed solvers are used in a \RANSAC-based estimator, which gives accurate rectifications after few iterations. The proposed solvers are evaluated against the state-of-the-art and demonstrate significantly better rectifications on noisy measurements. Qualitative results on diverse imagery demonstrate high-accuracy undistortions and rectifications. The source code is publicly available at https://github.com/prittjam/repeats