Geometrical Homography
A Geometrical Homography is a Invertible Transformation on projective spaces that maps lines to lines.
- AKA: Homography, Projectivity, Projective Transformation, Projective Collineation.
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- Example(s)
- Counter-Example(s)
- See: Homography Estimation Task, Desargues's Theorem, Projective Geometry, Isomorphism, Projective Space, Vector Space, Bijection, Collineation, Projective Geometry Theorem, Perspectivity, Projection, Euclidean Geometry.
References
2018a
- (Wikipedia, 2018) ⇒ https://en.wikipedia.org/wiki/Homography Retrieved:2018-5-27.
- In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. It is a bijection that maps lines to lines, and thus a collineation. In general, some collineations are not homographies, but the fundamental theorem of projective geometry asserts that is not so in the case of real projective spaces of dimension at least two. Synonyms include projectivity, projective transformation, and projective collineation.
Historically, homographies (and projective spaces) have been introduced to study perspective and projections in Euclidean geometry, and the term homography, which, etymologically, roughly means "similar drawing" date from this time. At the end of the 19th century, formal definitions of projective spaces were introduced, which differed from extending Euclidean or affine spaces by adding points at infinity. The term "projective transformation" originated in these abstract constructions. These constructions divide into two classes that have been shown to be equivalent. A projective space may be constructed as the set of the lines of a vector space over a given field (the above definition is based on this version); this construction facilitates the definition of projective coordinates and allows using the tools of linear algebra for the study of homographies. The alternative approach consists in defining the projective space through a set of axioms, which do not involve explicitly any field (incidence geometry, see also synthetic geometry); in this context, collineations are easier to define than homographies, and homographies are defined as specific collineations, thus called "projective collineations".
For sake of simplicity, unless otherwise stated, the projective spaces considered in this article are supposed to be defined over a (commutative) field. Equivalently Pappus's hexagon theorem and Desargues's theorem are supposed to be true. A large part of the results remain true, or may be generalized to projective geometries for which these theorems do not hold.
- In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. It is a bijection that maps lines to lines, and thus a collineation. In general, some collineations are not homographies, but the fundamental theorem of projective geometry asserts that is not so in the case of real projective spaces of dimension at least two. Synonyms include projectivity, projective transformation, and projective collineation.
2018b
- (MathWorld, 2018) ⇒ Weisstein, Eric W. "Homography." From MathWorld -- A Wolfram Web Resource. http://mathworld.wolfram.com/Homography.html Retrieved:2018-5-27.
- QUOTE: A circle-preserving transformation composed of an even number of inversions.
2018b
- (Wikipedia, 2018) ⇒ https://en.wikipedia.org/wiki/Homography_(computer_vision) Retrieved:2018-5-27.
- In the field of computer vision, any two images of the same planar surface in space are related by a homography (assuming a pinhole camera model). This has many practical applications, such as image rectification, image registration, or computation of camera motion — rotation and translation — between two images. Once camera rotation and translation have been extracted from an estimated homography matrix, this information may be used for navigation, or to insert models of 3D objects into an image or video, so that they are rendered with the correct perspective and appear to have been part of the original scene (see Augmented reality).
2018c =
- (Nguyen et al., 2018) ⇒ Ty Nguyen, Steven W. Chen, Shreyas S. Shivakumar, Camillo J. Taylor, Vijay Kumar (2018). "Unsupervised Deep Homography: A Fast and Robust Homography Estimation Model". IEEE Robotics and Automation Letters.
- ABSTRACT: Homography estimation between multiple aerial images can provide relative pose estimation for collaborative autonomous exploration and monitoring. The usage on a robotic system requires a fast and robust homography estimation algorithm. In this study, we propose an unsupervised learning algorithm that trains a Deep Convolutional Neural Network to estimate planar homographies. We compare the proposed algorithm to traditional feature-based and direct methods, as well as a corresponding supervised learning algorithm. Our empirical results demonstrate that compared to traditional approaches, the unsupervised algorithm achieves faster inference speed, while maintaining comparable or better accuracy and robustness to illumination variation. In addition, on both a synthetic dataset and representative real-world aerial dataset, our unsupervised method has superior adaptability and performance compared to the supervised deep learning method.