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The Role of the Essential Manifold in Data Mining – An Introductory Approach
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Abstract(s)
Interpolating data and the application of data mining tech niques in nonlinear manifolds plays a significant role in different areas of
knowledge, ranging from computer vision and robotics, to industrial and
medical requests, and these growing number of applications have sparked
the research interest of the scientific community to these topics. The Gen eralized Essential manifold, briefly, Essential manifold, consisting of the
product of the Grassmann manifold of all k-dimensional subspaces of
Rn and the Lie group of rotations in Rn, for instance, plays an impor tant role in the problem of recovering the structure and motion from a
sequence of images, also known as stereo matching, which is a crucial
problem in image processing and computer vision. A well-known recur sive procedure to generate interpolating polynomial curves in Euclidean
spaces is the classical De Casteljau algorithm, which is a simple and pow erful tool widely used in the field of Computer Aided Geometric Design,
particularly because it is essentially geometrically based. This algorithm
has been generalized to geodesically complete Riemannian manifolds.
Thus, having this in mind, in this work we present all the ingredients for
a detailed implementation of the generalized De Casteljau algorithm to
generate geometric cubic polynomials in the Essential manifold preparing
the ground to solve different real interpolation problems in this manifold
Description
Keywords
Cubic polynomials Essential manifold De Casteljau algorithm Geodesics Data mining Interpolating data