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Graph-Based Dimensionality Reduction
Published in Olivier Lézoray, Leo Grady, Image Processing and Analysis with Graphs, 2012
Let us imagine a manifold made of a curved sheet of paper, which is embedded in our physical three-dimensional world. The Swiss roll is an example of such manifold. Let us also consider the Euclidean distance between two relatively distant points of the manifold. Depending on the sheet curvature, the distance will vary, although the sheet keeps the same size (paper does not allow any stretching or shrinking). This shows that distance preservation makes little sense if the manifold to be embedded in a lower-dimensional space needs some unfolding or unrolling. This issue arises because Euclidean distances measure lengths along straight lines, whereas the manifold occupies a nonlinear subspace. The solution to this problem is obviously to compute distances as the ant crawls instead of as the crow flies. In other words, distances should be computed along the manifold or, more accurately, along manifold geodesic curves. In a smooth manifold, the geodesic curve between two points on the manifold is the smooth one-dimensional submanifold with the shortest length. The term geodesic distance refers to this length. In a manifold such as a sheet of paper, geodesic distances are invariant to curvature changes. Therefore, geodesic distances capture the internal structure of the manifold without influence from the way it is embedded.
The Suspensions of Maps to Flows
Published in Christos H. Skiadas, Charilaos Skiadas, Handbook of Applications of Chaos Theory, 2017
Let M be a smooth manifold, and g: M → V be a smooth invertible map. The quotient manifold Q = M × [0,1]/~ with M × 0 and M × 1 identified, is called a suspension manifold, and the flow ϕ(x, t) on Q with ϕ(x,1)=ϕ(g(x),0)∀x∈M is called a suspension flow.
Dual-Manipulator Testing Technique
Published in Chunguang Xu, Robotic Nondestructive Testing Technology, 2022
Lie group is applied to partial differential equations, ordinary differential equations (groups) and autonomous systems to some extent. Lie group is a smooth manifold G, as shown in Figure 5.16. It satisfies the group operation, that is, its multiplication operation and inversion operation are smooth mappings.
Minimal control effort and time Lie-group synchronisation design based on proportional-derivative control
Published in International Journal of Control, 2022
A smooth manifold that is also an algebraic group is termed a Lie group. An algebraic group is a structure () where m : is an associative binary operation (such as a multiplication), is an identity element with respect to m and i : is the function which represents the inversion with respect to the operation m, so that for each . A left translation on a Lie group is denoted by and is defined as .