2,734 research outputs found
Revisiting Clifford algebras and spinors III: conformal structures and twistors in the paravector model of spacetime
This paper is the third of a series of three, and it is the continuation of
math-ph/0412074 and math-ph/0412075. After reviewing the conformal spacetime
structure, conformal maps are described in Minkowski spacetime as the twisted
adjoint representation of the group Spin_+(2,4), acting on paravectors.
Twistors are then presented via the paravector model of Clifford algebras and
related to conformal maps in the Clifford algebra over the lorentzian R{4,1}$
spacetime. We construct twistors in Minkowski spacetime as algebraic spinors
associated with the Dirac-Clifford algebra Cl(1,3)(C) using one lower spacetime
dimension than standard Clifford algebra formulations, since for this purpose
the Clifford algebra over R{4,1} is also used to describe conformal maps,
instead of R{2,4}. Although some papers have already described twistors using
the algebra Cl(1,3)(C), isomorphic to Cl(4,1), the present formulation sheds
some new light on the use of the paravector model and generalizations.Comment: 17 page
M\"obius Invariants of Shapes and Images
Identifying when different images are of the same object despite changes
caused by imaging technologies, or processes such as growth, has many
applications in fields such as computer vision and biological image analysis.
One approach to this problem is to identify the group of possible
transformations of the object and to find invariants to the action of that
group, meaning that the object has the same values of the invariants despite
the action of the group. In this paper we study the invariants of planar shapes
and images under the M\"obius group , which arises
in the conformal camera model of vision and may also correspond to neurological
aspects of vision, such as grouping of lines and circles. We survey properties
of invariants that are important in applications, and the known M\"obius
invariants, and then develop an algorithm by which shapes can be recognised
that is M\"obius- and reparametrization-invariant, numerically stable, and
robust to noise. We demonstrate the efficacy of this new invariant approach on
sets of curves, and then develop a M\"obius-invariant signature of grey-scale
images
Spectral Generalized Multi-Dimensional Scaling
Multidimensional scaling (MDS) is a family of methods that embed a given set
of points into a simple, usually flat, domain. The points are assumed to be
sampled from some metric space, and the mapping attempts to preserve the
distances between each pair of points in the set. Distances in the target space
can be computed analytically in this setting. Generalized MDS is an extension
that allows mapping one metric space into another, that is, multidimensional
scaling into target spaces in which distances are evaluated numerically rather
than analytically. Here, we propose an efficient approach for computing such
mappings between surfaces based on their natural spectral decomposition, where
the surfaces are treated as sampled metric-spaces. The resulting spectral-GMDS
procedure enables efficient embedding by implicitly incorporating smoothness of
the mapping into the problem, thereby substantially reducing the complexity
involved in its solution while practically overcoming its non-convex nature.
The method is compared to existing techniques that compute dense correspondence
between shapes. Numerical experiments of the proposed method demonstrate its
efficiency and accuracy compared to state-of-the-art approaches
An Optimized Architecture for CGA Operations and Its Application to a Simulated Robotic Arm
Conformal geometric algebra (CGA) is a new geometric computation tool that is attracting growing attention in many research fields, such as computer graphics, robotics, and computer vision. Regarding the robotic applications, new approaches based on CGA have been proposed to efficiently solve problems as the inverse kinematics and grasping of a robotic arm. The hardware acceleration of CGA operations is required to meet real-time performance requirements in embedded robotic platforms. In this paper, we present a novel embedded coprocessor for accelerating CGA operations in robotic tasks. Two robotic algorithms, namely, inverse kinematics and grasping of a human-arm-like kinematics chain, are used to prove the effectiveness of the proposed approach. The coprocessor natively supports the entire set of CGA operations including both basic operations (products, sums/differences, and unary operations) and complex operations as rigid body motion operations (reflections, rotations, translations, and dilations). The coprocessor prototype is implemented on the Xilinx ML510 development platform as a complete system-on-chip (SoC), integrating both a PowerPC processing core and a CGA coprocessing core on the same Xilinx Virtex-5 FPGA chip. Experimental results show speedups of 78x and 246x for inverse kinematics and grasping algorithms, respectively, with respect to the execution on the PowerPC processor
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