New Geometric Methods for Computer Vision: An Application to Structure and Motion Estimation

@article{Lasenby2004NewGM,
  title={New Geometric Methods for Computer Vision: An Application to Structure and Motion Estimation},
  author={Joan Lasenby and William J. Fitzgerald and Anthony N. Lasenby and C. J. L. Doran},
  journal={International Journal of Computer Vision},
  year={2004},
  volume={26},
  pages={191-213}
}
We discuss a coordinate-free approach to the geometry of computer vision problems. The technique we use to analyse the three-dimensional transformations involved will be that of geometric algebra: a framework based on the algebras of Clifford and Grassmann. This is not a system designed specifically for the task in hand, but rather a framework for all mathematical physics. Central to the power of this approach is the way in which the formalism deals with rotations; for example, if we have two… 
Design of Algorithms of Robot Vision Using Conformal Geometric Algebra
TLDR
The authors use this compact system following the Occam's razor philosophy that mathematical ‘entities should not be multiplied unnecessarily’ to handle simulated and real tasks for perception-action systems, treated in a single and efficient way.
Geometric Algebra: A Powerful Tool for Solving Geometric Problems in Visual Computing
TLDR
This tutorial aims at introducing the fundamental concepts of GA as a powerful mathematical tool to describe and solve geometric problems in visual computing.
A 1d Up Approach to Conformal Geometric Algebra: Applications in Line Fitting and Quantum Mechanics
We discuss an alternative approach to the conformal geometric algebra (CGA) in which just a single extra dimension is necessary, as compared to the two normally used. This is made possible by working
A Geometric Approach for the Theory and Applications of 3D Projective Invariants
TLDR
This work shows that geometric algebra is a very elegant language for expressing projective invariants using n views and Illustrations of the application of such projectic invariants in visual guided grasping, camera self-localization and reconstruction of shape and motion complement the experimental part.
Conformal Geometric Algebra for Robotic Vision
TLDR
The authors believe that the framework of conformal geometric algebra can be, in general, of great advantage for applications using stereo vision, range data, laser, omnidirectional and odometry based systems.
A covariant approach to geometry using geometric algebra
TLDR
Using the mathematical framework of conformal geometric algebra – a 5-dimensional representation of 3-dimensional space – is shown to provide an elegant covariant approach to geometry, thus enabling us to deal simply with the projective and non-Euclidean cases.
Visually Guided Robotics Using Conformal Geometric Computing
1. Abstract Classical Geometry, as conceived by Euclid, was a plataform from which Mathematics started to build its actual form. However, since the XIX century, it was a language that was not
Geometric algebra of points, lines, planes and spheres for computer vision and robotics
TLDR
The authors show that CGA deals with the intuition and insight of the geometry and it helps to reduce considerably the computational burden of the problems.
2 Visually Guided Robotics Using Conformal Geometric Computing
Classical Geometry, as conceived by Euclid, was a plataform from which Mathematics started to build its actual form. However, since the XIX century, it was a language that was not evolving as the
Structure from Motion 13.1 Introduction
  • Computer Science
TLDR
In this chapter and the next, tools and techniques for obtaining information about the geometry of 3D scenes from 2D images are described, which describes a practical interactive system for recovering geometric models from photographs of architectural scenes.
...
...

References

SHOWING 1-10 OF 53 REFERENCES
The Dou8ble Algebra: An Effective Tool for Computing Invariants in Computer Vision
  • S. Carlsson
  • Mathematics
    Applications of Invariance in Computer Vision
  • 1993
TLDR
This paper shows how to compute linear invariants of general configurations points and lines observed in two images and polyhedral configurations observed in one image without reconstructing individual points and Lines.
On the geometry and algebra of the point and line correspondences between N images
TLDR
The formalism of the Grassmann-Cayley algebra is proposed to use as the simplest way to make both geometric and algebraic statements in a very synthetic and effective way (i.e. allowing actual computation if needed).
Closed-form solution of absolute orientation using orthonormal matrices
Finding the relationship between two coordinate systems by using pairs of measurements of the coordinates of a number of points in both systems is a classic photogrammetric task. The solution has
Estimation of motion from a pair of range images: A review
Closed-form solution of absolute orientation using unit quaternions
TLDR
A closed-form solution to the least-squares problem for three or more paints is presented, simplified by use of unit quaternions to represent rotation.
A computer algorithm for reconstructing a scene from two projections
A simple algorithm for computing the three-dimensional structure of a scene from a correlated pair of perspective projections is described here, when the spatial relationship between the two
Imaginary numbers are not real—The geometric algebra of spacetime
This paper contains a tutorial introduction to the ideas of geometric algebra, concentrating on its physical applications. We show how the definition of a “geometric product” of vectors in 2-and
Motion and Structure From Two Perspective Views: Algorithms, Error Analysis, and Error Estimation
TLDR
The presented approach to error estimation applies to a wide variety of problems that involve least-squares optimization or pseudoinverse and shows, among other things, that the errors are very sensitive to the translation direction and the range of field view.
Motion and structure from feature correspondences: a review
TLDR
Some of the mathematical techniques borrowed from algebraic geometry, projective geometry, and homotopy theory that are required to solve three-dimensional (3D) motion and structure of rigid objects when their corresponding features are known at different times or are viewed by different cameras are mentioned.
...
...