By Stephan Heuel (auth.)
Algebraic projective geometry, with its multilinear family and its embedding into Grassmann-Cayley algebra, has turn into the fundamental illustration of a number of view geometry, leading to deep insights into the algebraic constitution of geometric kinfolk, in addition to in effective and flexible algorithms for computing device imaginative and prescient and photo research.
This publication presents a coherent integration of algebraic projective geometry and spatial reasoning less than uncertainty with purposes in laptop imaginative and prescient. past systematically introducing the theoretical foundations from geometry and information and transparent principles for appearing geometric reasoning lower than uncertainty, the writer offers a suite of precise algorithms.
The ebook addresses researchers and complicated scholars drawn to algebraic projective geometry for snapshot research, in statistical illustration of gadgets and changes, or in standard instruments for trying out and estimating in the context of geometric multiple-view research.
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Additional resources for Uncertain Projective Geometry: Statistical Reasoning for Polyhedral Object Reconstruction
10) for hyperplanes. 2 Representation of Geometric Entities and Transformations 26 A line segment is a finite segment of an (infinite) line. It can be represented by a triple the infinite line a mid-point of the line segment and a length This is equivalent compared to defining two enclosing points on the line, cf. definition 1 on page 6. 12) with the angle-distance form: they are essentially the same up to a multiplication factor As the scalar refers to the distance of the line to the origin, we obtain Note that the angle is the angle of the normal of the line l.
24). 3 on page 117. Note that because we have subdivided all homogeneous vectors of geometric entities into a homogeneous and a Euclidean part, the conditioning of geometric entities is very simple and works consistently for homogeneous vectors for points, lines and planes within a coordinate frame. 5 Duality Principle The duality principle is an important feature of projective geometry as all propositions occur twice: taking the dual of a true proposition yields a new proposition, possibly with a different meaning.
The robot must be able to deal with uncertainty since it knows its position and the position of other objects only up to some error, so it is not certain about its position. Here, dealing with uncertainty means to (i) represent the errors of its position in an efficient manner, (ii) propagate these errors when moving the camera or changing the position and (iii) taking them into account when making decisions, such as “Can I go through this door? ”. Recently, more and more researchers acknowledge that precise logic alone is not capable of solving these and many other tasks in a robust way, cf.