52,650 research outputs found
Unified Heat Kernel Regression for Diffusion, Kernel Smoothing and Wavelets on Manifolds and Its Application to Mandible Growth Modeling in CT Images
We present a novel kernel regression framework for smoothing scalar surface
data using the Laplace-Beltrami eigenfunctions. Starting with the heat kernel
constructed from the eigenfunctions, we formulate a new bivariate kernel
regression framework as a weighted eigenfunction expansion with the heat kernel
as the weights. The new kernel regression is mathematically equivalent to
isotropic heat diffusion, kernel smoothing and recently popular diffusion
wavelets. Unlike many previous partial differential equation based approaches
involving diffusion, our approach represents the solution of diffusion
analytically, reducing numerical inaccuracy and slow convergence. The numerical
implementation is validated on a unit sphere using spherical harmonics. As an
illustration, we have applied the method in characterizing the localized growth
pattern of mandible surfaces obtained in CT images from subjects between ages 0
and 20 years by regressing the length of displacement vectors with respect to
the template surface.Comment: Accepted in Medical Image Analysi
An Emergent Space for Distributed Data with Hidden Internal Order through Manifold Learning
Manifold-learning techniques are routinely used in mining complex
spatiotemporal data to extract useful, parsimonious data
representations/parametrizations; these are, in turn, useful in nonlinear model
identification tasks. We focus here on the case of time series data that can
ultimately be modelled as a spatially distributed system (e.g. a partial
differential equation, PDE), but where we do not know the space in which this
PDE should be formulated. Hence, even the spatial coordinates for the
distributed system themselves need to be identified - to emerge from - the data
mining process. We will first validate this emergent space reconstruction for
time series sampled without space labels in known PDEs; this brings up the
issue of observability of physical space from temporal observation data, and
the transition from spatially resolved to lumped (order-parameter-based)
representations by tuning the scale of the data mining kernels. We will then
present actual emergent space discovery illustrations. Our illustrative
examples include chimera states (states of coexisting coherent and incoherent
dynamics), and chaotic as well as quasiperiodic spatiotemporal dynamics,
arising in partial differential equations and/or in heterogeneous networks. We
also discuss how data-driven spatial coordinates can be extracted in ways
invariant to the nature of the measuring instrument. Such gauge-invariant data
mining can go beyond the fusion of heterogeneous observations of the same
system, to the possible matching of apparently different systems
Recent advances in directional statistics
Mainstream statistical methodology is generally applicable to data observed
in Euclidean space. There are, however, numerous contexts of considerable
scientific interest in which the natural supports for the data under
consideration are Riemannian manifolds like the unit circle, torus, sphere and
their extensions. Typically, such data can be represented using one or more
directions, and directional statistics is the branch of statistics that deals
with their analysis. In this paper we provide a review of the many recent
developments in the field since the publication of Mardia and Jupp (1999),
still the most comprehensive text on directional statistics. Many of those
developments have been stimulated by interesting applications in fields as
diverse as astronomy, medicine, genetics, neurology, aeronautics, acoustics,
image analysis, text mining, environmetrics, and machine learning. We begin by
considering developments for the exploratory analysis of directional data
before progressing to distributional models, general approaches to inference,
hypothesis testing, regression, nonparametric curve estimation, methods for
dimension reduction, classification and clustering, and the modelling of time
series, spatial and spatio-temporal data. An overview of currently available
software for analysing directional data is also provided, and potential future
developments discussed.Comment: 61 page
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