The Experts below are selected from a list of 2028 Experts worldwide ranked by ideXlab platform
Eckhard Hitzer - One of the best experts on this subject based on the ideXlab platform.
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Exponential Factorization of Multivectors in Cl(p,q), P+q < 3
viXra, 2020Co-Authors: Eckhard HitzerAbstract:In this paper we consider general multivector elements of Clifford algebras Cl(p,q), p+q
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exponential factorization of Multivectors in cl p q p q 3
viXra, 2020Co-Authors: Eckhard HitzerAbstract:In this paper we consider general multivector elements of Clifford algebras Cl(p,q), p+q <3, and study multivector factorization into products of exponentials and idempotents, where the exponents are blades of grades zero (scalar) to n (pseudoscalar).
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Exponential Factorization and Polar Decomposition of Multivectors in $Cl(p,q)$, $p+q \leq 3$
viXra, 2019Co-Authors: Eckhard Hitzer, Stephen J. SangwineAbstract:In this paper we consider general multivector elements of Clifford algebras $Cl(p,q)$, $n=p+q \leq 3$, and study multivector equivalents of polar decompositions and factorization into products of exponentials, where the exponents are frequently blades of grades zero (scalar) to $n$ (pseudoscalar).
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exponential factorization and polar decomposition of Multivectors in cl p q p q leq 3
viXra, 2019Co-Authors: Eckhard Hitzer, Stephen J. SangwineAbstract:In this paper we consider general multivector elements of Clifford algebras $Cl(p,q)$, $n=p+q \leq 3$, and study multivector equivalents of polar decompositions and factorization into products of exponentials, where the exponents are frequently blades of grades zero (scalar) to $n$ (pseudoscalar).
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Construction of Multivector Inverse for Clif Ford Algebras Over 2m+1-Dimensional Vector Spaces from Multivector Inverse for Clifford Algebras Over 2m-Dimensional Vector Spaces
Advances in Applied Clifford Algebras, 2019Co-Authors: Eckhard Hitzer, Stephen J. SangwineAbstract:Assuming known algebraic expressions for multivector inverses in any Clifford algebra over an even dimensional vector space $$\mathbb {R}^{p',q'}$$ , $$n'=p'+q'=2m$$ , we derive a closed algebraic expression for the multivector inverse over vector spaces one dimension higher, namely over $$\mathbb {R}^{p,q}$$ , $$n=p+q=p'+q'+1=2m+1$$ . Explicit examples are provided for dimensions $$n'=2,4,6$$ , and the resulting inverses for $$n=n'+1=3,5,7$$ . The general result for $$n=7$$ appears to be the first ever reported closed algebraic expression for a multivector inverse in Clifford algebras Cl(p, q), $$n=p+q=7$$ , only involving a single addition of multivector products in forming the determinant.
Marian Mrozek - One of the best experts on this subject based on the ideXlab platform.
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persistence of the conley index in combinatorial dynamical systems
arXiv: Algebraic Topology, 2020Co-Authors: Marian Mrozek, Ryan SlechtaAbstract:A combinatorial framework for dynamical systems provides an avenue for connecting classical dynamics with data-oriented, algorithmic methods. Combinatorial vector fields introduced by Forman and their recent generalization to multivector fields have provided a starting point for building such a connection. In this work, we strengthen this relationship by placing the Conley index in the persistent homology setting. Conley indices are homological features associated with so-called isolated invariant sets, so a change in the Conley index is a response to perturbation in an underlying multivector field. We show how one can use zigzag persistence to summarize changes to the Conley index, and we develop techniques to capture such changes in the presence of noise. We conclude by developing an algorithm to track features in a changing multivector field.
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Conley-Morse-Forman theory for generalized combinatorial multivector fields on finite topological spaces.
arXiv: Dynamical Systems, 2019Co-Authors: Michal Lipinski, Marian Mrozek, Jacek Kubica, Thomas WannerAbstract:We generalize and extend the Conley-Morse-Forman theory for combinatorial multivector fields introduced in \cite{Mr2017}. The generalization consists in dropping the restrictive assumption in \cite{Mr2017} that every multivector has a unique maximal element. The extension is from the setting of Lefschetz complexes to the more general situation of finite topological spaces. We define isolated invariant sets, isolating neighbourhoods, Conley index and Morse decompositions. We also establish the additivity property of the Conley index and the Morse inequalities.
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Conley–Morse–Forman Theory for Combinatorial Multivector Fields on Lefschetz Complexes
Foundations of Computational Mathematics, 2017Co-Authors: Marian MrozekAbstract:We introduce combinatorial multivector fields, associate with them multivalued dynamics and study their topological features. Our combinatorial multivector fields generalize combinatorial vector fields of Forman. We define isolated invariant sets, Conley index, attractors, repellers and Morse decompositions. We provide a topological characterization of attractors and repellers and prove Morse inequalities. The generalization aims at algorithmic analysis of dynamical systems through combinatorialization of flows given by differential equations and through sampling dynamics in physical and numerical experiments. We provide a prototype algorithm for such applications.
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Conley-Morse-Forman theory for combinatorial multivector fields
arXiv: Dynamical Systems, 2015Co-Authors: Marian MrozekAbstract:We introduce combinatorial multivector fields, associate with them multivalued dynamics and study their topological features. Our combinatorial multivector fields generalize combinatorial vector fields of Forman. We define isolated invariant sets, Conley index, attractors, repellers and Morse decompositions. We provide a topological characterization of attractors and repellers and prove Morse inequalities. The generalization aims at algorithmic analysis of dynamical systems through combinatorialization of flows given by differential equations and through sampling dynamics in physical and numerical experiments. We provide a prototype algorithm for such applications.
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Conley-Morse-Forman theory for combinatorial multivector fields on Lefschetz complexes
arXiv: Dynamical Systems, 2015Co-Authors: Marian MrozekAbstract:Working in the algebraic setting of free chain complexes with a distinguished basis (Lefschetz complexes) we introduce combinatorial multivector fields, associate with them multivalued dynamics and study their topological features with respect to the $T_0$ topology of Lefschetz complexes. Our combinatorial multivector fields generalize combinatorial vector fields of Forman. We define isolated invariant sets, Conley index, attractors, repellers and Morse decompositions. We provide a topological characterization of attractors and repellers and prove Morse inequalities. The generalization aims at algorithmic analysis of dynamical systems through combinatorization of flows given by differential equations and through sampling dynamics in physical and numerical experiments. We provide a prototype algorithm for such applications.
Gerald Sommer - One of the best experts on this subject based on the ideXlab platform.
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Advances in Applied Clifford Algebras Parameter Estimation from Uncertain Data in Geometric Algebra
2010Co-Authors: Christian Gebken, Christian Perwass, Gerald SommerAbstract:Abstract. We show how standard parameter estimation methods can be applied to Geometric Algebra in order to fit geometric entities and operators to uncertain data. These methods are applied to three selected problems. One of which is the perspective pose estimation problem. We show experiments with synthetic data and compare the results of our algorithm with standard approaches. In general, our aim is to find Multivectors that satisfy a particular constraint, which depends on a set of uncertain measurements. The specific problem and the type of multivector, representing a geometric entity or a geometric operator, determine the constraint. We consider the case of point measurements in Euclidian 3D-space, where the respective uncertainties are given by covariance matrices. We want to find a best fitting circle or line together with their uncertainty. This problem can be expressed in a linear manner, when it is embedded in the corresponding conformal space. In this space, it is also possible to evaluate screw motions and their uncertainty, in very much the same way. The parameter estimation method we use is a least-squares adjustment method, which is based on the so-called Gauss-Helmert model, also known as mixed model with constraints. For this linear model, we benefit from the implicit linearization when expressing our constraints in conformal space. The multivector representation of the entities we are interested in also allows their uncertainty to be expressed by covariance matrices. As a by-product, this method provides such covariance matrices
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Chapter 38 The Structure Multivector
2002Co-Authors: Michael Felsberg, Gerald SommerAbstract:The structure multivector is an operator for analysing the local structure of an image. It combines ideas from the structure tensor, steerable filters, and quadrature filters where the advantages of all three approaches are brought into a single method by means of geometric algebra. The proposed operator is efficient to implement and linear up to a final steering operation. In this paper we derive the structure multivector from the Laplace equation, which also introduces a new viewpoint on scale space. A phase approach for intrinsically 2D structures is derived and applications are presented which make use of the 2D part of the new operator.
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The Structure Multivector
Applications of Geometric Algebra in Computer Science and Engineering, 2002Co-Authors: Michael Felsberg, Gerald SommerAbstract:The structure multivector is an operator for analysing the local structure of an image. It combines ideas from the structure tensor, steerable filters, and quadrature filters where the advantages of all three approaches are brought into a single method by means of geometric algebra. The proposed operator is efficient to implement and linear up to a final steering operation. In this paper we derive the structure multivector from the Laplace equation, which also introduces a new viewpoint on scale space. A phase approach for intrinsically 2D structures is derived and applications are presented which make use of the 2D part of the new operator.
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Theoretical Foundations of Computer Vision - Structure Multivector for Local Analysis of Images
Multi-Image Analysis, 2001Co-Authors: Michael Felsberg, Gerald SommerAbstract:The structure multivector is a new approach for analyzing the local properties of a two-dimensional signal (e.g. image). It combines the classical concepts of the structure tensor and the analytic signal in a new way. This has been made possible using a representation in the algebra of quaternions. The resulting method is linear and of low complexity. The filter-response includes local phase, local amplitude and local orientation of intrinsically one-dimensional neighborhoods in the signal. As for the structure tensor, the structure multivector field can be used to apply special filters to it for detecting features in images.
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Structure multivector for local analysis of images
Lecture Notes in Computer Science, 2001Co-Authors: Michael Felsberg, Gerald SommerAbstract:The structure multivector is a new approach for analyzing the local properties of a two-dimensional signal (e.g. image). It combines the classical concepts of the structure tensor and the analytic signal in a new way. This has been made possible using a representation in the algebra of quaternions. The resulting method is linear and of low complexity. The filter-response includes local phase, local amplitude and local orientation of intrinsically one-dimensional neighborhoods in the signal. As for the structure tensor, the structure multivector field can be used to apply special filters to it for detecting features in images.
Bahri Mawardi - One of the best experts on this subject based on the ideXlab platform.
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Clifford Fourier Transform on Multivector Fields and Uncertainty Principles for Dimensions N = 2 (Mod 4) and N = 3 (Mod 4)
viXra, 2013Co-Authors: Eckhard Hitzer, Bahri MawardiAbstract:First, the basic concepts of the multivector functions, vector differential and vector derivative in geometric algebra are introduced. Second, we dene a generalized real Fourier transform on Clifford multivector-valued functions ( f : R^n -> Cl(n,0), n = 2,3 (mod 4) ). Third, we show a set of important properties of the Clifford Fourier transform on Cl(n,0), n = 2,3 (mod 4) such as dierentiation properties, and the Plancherel theorem, independent of special commutation properties. Fourth, we develop and utilize commutation properties for giving explicit formulas for f x^m; f Nabla^m and for the Clifford convolution. Finally, we apply Clifford Fourier transform properties for proving an uncertainty principle for Cl(n,0), n = 2,3 (mod 4) multivector functions. Keywords: Vector derivative, multivector-valued function, Clifford (geometric) algebra, Clifford Fourier transform, uncertainty principle.
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Clifford Fourier Transform on Multivector Fields and Uncertainty Principles for Dimensions n = 2 (mod 4) and n = 3 (mod 4)
Advances in Applied Clifford Algebras, 2008Co-Authors: Eckhard Hitzer, Bahri MawardiAbstract:First, the basic concepts of the multivector functions, vector differential and vector derivative in geometric algebra are introduced. Second, we define a generalized real Fourier transform on Clifford multivector-valued functions (\(f : {{\mathbb{R}}}^n \rightarrow Cl_{n,0}, n = 2, 3\) (mod 4)). Third, we show a set of important properties of the Clifford Fourier transform on Cln,0, n = 2, 3 (mod 4) such as differentiation properties, and the Plancherel theorem, independent of special commutation properties. Fourth, we develop and utilize commutation properties for giving explicit formulas for fxm, f ∇m and for the Clifford convolution. Finally, we apply Clifford Fourier transform properties for proving an uncertainty principle for Cln,0, n = 2, 3 (mod 4) multivector functions.
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clifford fourier transformation and uncertainty principle for the clifford geometric algebra cl3 0
Advances in Applied Clifford Algebras, 2006Co-Authors: Bahri Mawardi, Eckhard HitzerAbstract:First, the basic concept of the vector derivative in geometric algebra is introduced. Second, beginning with the Fourier transform on a scalar function we generalize to a real Fourier transform on Clifford multivector-valued functions \( (f:\user2{\mathbb{R}}^3 \to Cl_{3,0} ). \) Third, we show a set of important properties of the Clifford Fourier transform on Cl3,0 such as differentiation properties, and the Plancherel theorem. Finally, we apply the Clifford Fourier transform properties for proving an uncertainty principle for Cl3,0 multivector functions.
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Clifford Fourier Transformation and Uncertainty Principle for the Clifford Geometric Algebra Cl_3,0
Advances in Applied Clifford Algebras, 2006Co-Authors: Bahri Mawardi, Eckhard M. S. HitzerAbstract:First, the basic concept of the vector derivative in geometric algebra is introduced. Second, beginning with the Fourier transform on a scalar function we generalize to a real Fourier transform on Clifford multivector-valued functions $$ (f:\user2{\mathbb{R}}^3 \to Cl_{3,0} ). $$ Third, we show a set of important properties of the Clifford Fourier transform on Cl _3,0 such as differentiation properties, and the Plancherel theorem. Finally, we apply the Clifford Fourier transform properties for proving an uncertainty principle for Cl _3,0 multivector functions.
Patrick J. Byrne - One of the best experts on this subject based on the ideXlab platform.
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the multivector gracilis free functional muscle flap for facial reanimation
JAMA Facial Plastic Surgery, 2018Co-Authors: Kofi O Boahene, James A Owusu, Lisa Ishii, Masaru Ishii, Shaun C Desai, Patrick J. ByrneAbstract:ImportanceA multivector functional muscle flap that closely simulates the biomechanical effects of facial muscle groups is essential for complete smile restoration after facial paralysis. Objective...
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the multivector gracilis free functional muscle flap for facial reanimation
JAMA Facial Plastic Surgery, 2018Co-Authors: Kofi O Boahene, James A Owusu, Lisa Ishii, Masaru Ishii, Shaun C Desai, Patrick J. ByrneAbstract:Importance A multivector functional muscle flap that closely simulates the biomechanical effects of facial muscle groups is essential for complete smile restoration after facial paralysis. Objective To determine the feasibility of a multivector gracilis muscle flap design for reanimation after facial paralysis and to analyze the effect on the smile display zone. Design, Setting, and Participants Prospective analysis of patients who underwent a double paddle multivector gracilis flap for complete facial paralysis between June 2015 and December 2016 was carried out in a tertiary hospital. Interventions The gracilis muscle was harvested as a double paddle flap and inserted along 2 vectors for facial reanimation. Main Outcomes and Measures The primary outcome measures were: (1) dental display (the number of maxillary teeth displayed on paralyzed vs normal sides), (2) exposed maxillary gingival scaffold width, (3) interlabial gap at midline and canine, (4) facial asymmetry index (FAI), and (5) dynamic periorbital wrinkling. Results There were 10 women and 2 men between ages 20 and 64 years (mean [SD], 46 [15] years). Five flaps were reinnervated with facial and masseteric nerves, 5 with masseteric nerve only, and 2 with crossfacial nerve only. There was functional muscle recovery in all cases. On average there was additional 3.1 maxillary teeth exposed posttreatment when smiling (5.5 vs 8.6; CI, 7.9 to 16.6;P Conclusions and Relevance The gracilis flap can be reliably designed as a functional double paddle muscle flap for a multivector facial reanimation. The multivector gracilis flap design is effective in improving all components of the smile display zone and has the potential for producing periorbital-wrinkling characteristic of a Duchenne smile. Level of Evidence 4.
