477 research outputs found
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
Pattern Recognition
Pattern recognition is a very wide research field. It involves factors as diverse as sensors, feature extraction, pattern classification, decision fusion, applications and others. The signals processed are commonly one, two or three dimensional, the processing is done in real- time or takes hours and days, some systems look for one narrow object class, others search huge databases for entries with at least a small amount of similarity. No single person can claim expertise across the whole field, which develops rapidly, updates its paradigms and comprehends several philosophical approaches. This book reflects this diversity by presenting a selection of recent developments within the area of pattern recognition and related fields. It covers theoretical advances in classification and feature extraction as well as application-oriented works. Authors of these 25 works present and advocate recent achievements of their research related to the field of pattern recognition
Heat Kernel Voting with Geometric Invariants
Here we provide a method for comparing geometric objects. Two objects of interest are embedded into an infinite dimensional Hilbert space using their Laplacian eigenvalues and eigenfunctions, truncated to a finite dimensional Euclidean space, where correspondences between the objects are searched for and voted on. To simplify correspondence finding, we propose using several geometric invariants to reduce the necessary computations. This method improves on voting methods by identifying isometric regions including shapes of genus greater than 0 and dimension greater than 3, as well as almost retaining isometry
Canonical tensor model through data analysis -- Dimensions, topologies, and geometries --
The canonical tensor model (CTM) is a tensor model in Hamilton formalism and
is studied as a model for gravity in both classical and quantum frameworks. Its
dynamical variables are a canonical conjugate pair of real symmetric
three-index tensors, and a question in this model was how to extract spacetime
pictures from the tensors. We give such an extraction procedure by using two
techniques widely known in data analysis. One is the tensor-rank (or CP, etc.)
decomposition, which is a certain generalization of the singular value
decomposition of a matrix and decomposes a tensor into a number of vectors. By
regarding the vectors as points forming a space, topological properties can be
extracted by using the other data analysis technique called persistent
homology, and geometries by virtual diffusion processes over points. Thus, time
evolutions of the tensors in the CTM can be interpreted as topological and
geometric evolutions of spaces. We have performed some initial investigations
of the classical equation of motion of the CTM in terms of these techniques for
a homogeneous fuzzy circle and homogeneous two- and three-dimensional fuzzy
spheres as spaces, and have obtained agreement with the general relativistic
system obtained previously in a formal continuum limit of the CTM. It is also
demonstrated by some concrete examples that the procedure is general for any
dimensions and topologies, showing the generality of the CTM.Comment: 44 pages, 16 figures, minor correction
Trends in Mathematical Imaging and Surface Processing
Motivated both by industrial applications and the challenge of new problems, one observes an increasing interest in the field of image and surface processing over the last years. It has become clear that even though the applications areas differ significantly the methodological overlap is enormous. Even if contributions to the field come from almost any discipline in mathematics, a major role is played by partial differential equations and in particular by geometric and variational modeling and by their numerical counterparts. The aim of the workshop was to gather a group of leading experts coming from mathematics, engineering and computer graphics to cover the main developments
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Shape theory and mathematical design of a general geometric kernel through regular stratified objects
This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.This dissertation focuses on the mathematical design of a unified shape kernel for geometric computing, with possible applications to computer aided design (CAM) and manufacturing (CAM), solid geometric modelling, free-form modelling of curves and surfaces, feature-based modelling, finite element meshing, computer animation, etc.
The generality of such a unified shape kernel grounds on a shape theory for objects in some Euclidean space. Shape does not mean herein only geometry as usual in geometric modelling, but has been extended to other contexts, e. g. topology, homotopy, convexity theory, etc. This shape theory has enabled to make a shape analysis of the current geometric kernels. Significant deficiencies have been then identified in how these geometric kernels represent shapes from different applications.
This thesis concludes that it is possible to construct a general shape kernel capable of representing and manipulating general specifications of shape for objects even in higher-dimensional Euclidean spaces, regardless whether such objects are implicitly or parametrically defined, they have ‘incomplete boundaries’ or not, they are structured with more or less detail or subcomplexes, which design sequence has been followed in a modelling session, etc. For this end, the basic constituents of such a general geometric kernel, say a combinatorial data structure and respective Euler operators for n-dimensional regular stratified objects, have been introduced and discussed
The art of building a smooth cosmic distance ladder in a perturbed universe
How does a smooth cosmic distance ladder emerge from observations made from a
single location in a lumpy Universe? Distances to Type Ia supernovae in the
Hubble flow are anchored on local distance measurements to sources that are
very nearby. We described how this configuration could be built in a perturbed
universe where lumpiness is described as small perturbations on top of a flat
Friedmann-Lema{\i}tre Robertson-Walker spacetime. We show that there is a
non-negligible modification (about 11\%) to the background
Friedmann-Lema{\i}tre Robertson-Walker area distance due to the presence of
inhomogeneities in the immediate neighbourhood of an observer. We find that the
modification is sourced by the electric part of the Weyl tensor indicating a
tidal deformation of the local spacetime of the observer. We show in detail how
it could impact the calibration of the Type Ia supernova absolute magnitude in
the Hubble flow. We show that it could potentially resolve the Type Ia
supernova absolute magnitude and Hubble tensions simultaneously without the
need for early or late dark energy.Comment: Version accepted for publication by JCA
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