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Applications of Geometric Algebra in Computer Vision by Christian B.U. Perwass
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About this book :-
Applications of Geometric Algebra in Computer Vision Christian Perwass.
There are four main areas discussed in this thesis: Geometric Algebra, Projective Geometry, Multiple View Tensors and 3D-Reconstruction. Our discussion of Geometric Algebra is similar to that of Hestenes and Sobczyk. Our construction differs mainly in how we define the inner and outer product, and how we work with the geometric product. The geometric algebra we construct here is finite dimensional, non-degenerate and universal.
Our discussion of projective geometry in terms of GA differs some what from Hestenes and Ziegler , in that we embed Euclidean space in projective space in a different way. Instead of using the projective split we employ reciprocal vectors to the same effect. Our approach is independent of the signature of an underlying orthonormal frame. We also use reciprocal frames to give concise descriptions of projections and intersections.
Book Detail :-
Title: Applications of Geometric Algebra in Computer Vision
Author(s): Christian B.U. Perwass
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Book Contents :-
Applications of Geometric Algebra in Computer Vision Christian Perwass cover the following topics.
Structure of the thesis, Geometric Algebra
A Construction of Geometric Algebra
Introduction, The Foundations, The Fundamental Axioms, The Commutator and Anti-Commutator Products, Vectors, Basis Blades, Inner and Outer Products, The Outer Product, The Inner Product, Further Development, The Grade-Projection Operator, The Reversion Operator, More on Vectors in Gn, Conclusions
Why Projective Geometry?, Fundamentals, Points, Lines and Planes, Points, Lines, Planes, Intersections, Intersection of Lines in P2, Intersection of Parallel Lines in P2, Intersections with Planes in P3, Intersection of Lines in P3, Reciprocal Vector Frames, Line Frames, Plane Frames, Determinants, Meet and Join, Cameras and Projections, Dual Representations of Lines and Points, Epipoles, Camera Matrices, Conclusions
The geometry of multiple view tensors
Introduction, The Fundamental Matrix, Derivation, Rank of F, Degrees of Freedom of F, Transferring Points with F, Epipoles of F, The Trifocal Tensor, Derivation, Transferring Lines, Transferring Points, Rank of T, Degrees of Freedom of T, Constraints on T, Relation between T and F, Second Order Constraints, Epipoles, The Quadfocal Tensor, Derivation, Transferring Lines, Rank of Q, Degrees of Freedom of Q, Constraints on Q, Relation between Q and T, Reconstruction and the Trifocal Tensor, Conclusions
Introduction, Image Plane Bases, Plane Collineation, Calculating the Collineation Tensor M, Rank of M, The Plane at Infinity, Vanishing Points and P8, Calculating Vanishing Points, ? 8 or M 8 from Vanishing Points, The Reconstruction Algorithm, The Geometry, The Minimisation Function, The Minimisation Routine, Experimental Results, Synthetic, Real Data, Conclusions
Summary of Important, Geometric Algebra, Projective Geometry, Multiple View Tensors, 3D-Reconstruction
The MVT Program
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