Conformal Geometric Algebra in Stochastic Optimization Problems ...

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230 CHAPTER 8. APPLICATIONS IN OMNIDIRECTIONAL VISION Fig. 8.19: Estimated epipoles: setup 4x5 (see figure 8.20). The guess of the SVDmethod (lower red cross) and the refinement by the GH-method (upper green cross). Fig. 8.20: Estimated epipoles: magnification for the scenarios 4x5 (l), 3x5 (c) and 7x8 (r). The differences in pixel are 34, 7 and 37, respectively. Scenario 3x5 represents an average situation.
Chapter 9 Conclusion It was intended to demonstrate that combining conformal geometric algebra with the method of least squares adjustment is best suited for a wide range of applications, especially those from the field of computer vision. The main topic dealt with from this subject is pose estimation. But the individual contributions of the chapters shall be subsumed first. • A thorough and formal introduction to GA with vivid examples and several connections to the standard vector algebra is given. The outer product and its momentous consequence - the blade - is derived in every detail. Sophisticated algebra expressions are analyzed and broken down into intelligible representations. Operations as the reverse, the magnitude, the conjugate, the inverse, the projection or the rejection are explained. Vital concepts as duality and outermorphism are elucidated. The relevance of versors and null blades is discussed. • The conformal space and its underlying embedding is illustratively derived. Conformal geometric algebra, its rich subspace concept and the transformations from the conformal group, which can act on the geometric objects that live in the subspaces, are enlightened. The multivectors inhering with these properties are analyzed in detail with respect to their structure and their mutual relationships. An entirely algebraic factorization of motors (rigid body motions) into specific translational and rotational parts is proposed. • The principle of pose estimation is succinctly rephrased using CGA. Starting from a clear geometric concept, a new geometric view on the 3-point pose estimation is obtained. The solution so inspired amounts to finding a root of a well-behaved scalar valued function of an angle. An n-point problem is shown to be solvable by taking the algebraic group nature of motors into account; a technique called intrinsic mean builds a weighted average of several 3-point solutions exploiting the tight relationship to the Lie algebra of the motors. The resultant method is robust, sound and provides accurate estimates. 231
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INSTITUT FÜR INFORMATIK Conformal
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Conformal Geometric Algebra in Stoc
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Dedicated to my beloved wife, Steph
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ACKNOWLEDGMENT This thesis as is ca
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3 The Conformal Geometric Algebra 8
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8 Applications in Omnidirectional V
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2 CHAPTER 1. INTRODUCTION with a si
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6 CHAPTER 1. INTRODUCTION Spherical
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