1,682 research outputs found
Riemannian mathematical morphology
This paper introduces mathematical morphology operators for real-valued images whose support space is a Riemannian manifold. The starting point consists in replacing the Euclidean distance in the canonic quadratic structuring function by the Riemannian distance used for the adjoint dilation/erosion. We then extend the canonic case to a most general framework of Riemannian operators based on the notion of admissible Riemannian structuring function. An alternative paradigm of morphological Riemannian operators involves an external structuring function which is parallel transported to each point on the manifold. Besides the definition of the various Riemannian dilation/erosion and Riemannian opening/closing, their main properties are studied. We show also how recent results on Lasry-Lions regularization can be used for non-smooth image filtering based on morphological Riemannian operators. Theoretical connections with previous works on adaptive morphology and manifold shape morphology are also considered. From a practical viewpoint, various useful image embedding into Riemannian manifolds are formalized, with some illustrative examples of morphological processing real-valued 3D surfaces
The Data Big Bang and the Expanding Digital Universe: High-Dimensional, Complex and Massive Data Sets in an Inflationary Epoch
Recent and forthcoming advances in instrumentation, and giant new surveys,
are creating astronomical data sets that are not amenable to the methods of
analysis familiar to astronomers. Traditional methods are often inadequate not
merely because of the size in bytes of the data sets, but also because of the
complexity of modern data sets. Mathematical limitations of familiar algorithms
and techniques in dealing with such data sets create a critical need for new
paradigms for the representation, analysis and scientific visualization (as
opposed to illustrative visualization) of heterogeneous, multiresolution data
across application domains. Some of the problems presented by the new data sets
have been addressed by other disciplines such as applied mathematics,
statistics and machine learning and have been utilized by other sciences such
as space-based geosciences. Unfortunately, valuable results pertaining to these
problems are mostly to be found only in publications outside of astronomy. Here
we offer brief overviews of a number of concepts, techniques and developments,
some "old" and some new. These are generally unknown to most of the
astronomical community, but are vital to the analysis and visualization of
complex datasets and images. In order for astronomers to take advantage of the
richness and complexity of the new era of data, and to be able to identify,
adopt, and apply new solutions, the astronomical community needs a certain
degree of awareness and understanding of the new concepts. One of the goals of
this paper is to help bridge the gap between applied mathematics, artificial
intelligence and computer science on the one side and astronomy on the other.Comment: 24 pages, 8 Figures, 1 Table. Accepted for publication: "Advances in
Astronomy, special issue "Robotic Astronomy
A Concept for Exploring Western Music Tonality in Physical Space
Musical theory about the structure and morphology of Western tonality is quite difficult to teach to young children, due to the relatively complex mathematical concepts behind tonality. Children usually grasp the concepts of musical harmony intuitively through listening to music examples. Placing the 12 notes of the well-tempered scale into a spatial arrangement, in which the proximity of these notes represents their mutual harmonic relationship, would allow to link physical motion through a spatial area with the exploration of music tonality. Music theorists have postulated the Circle of Fifth, the “Spiral Array”, and the “Tonnetz” as paradigms for spatial arrangements of music notes which allow mapping the distance between notes onto their “mutual consonance”. These approaches mostly have been of qualitative nature, leaving the actual numeric parameters of the spatial description undetermined. In this paper, these parameters have been determined, leading to a concrete numerical description of the planar Tonnetz. This allows the design of a physical space in which the music notes are distributed in space according to their musical consonance. Set up in an outdoor area, handheld devices (e.g. PDA) with integrated Global Positioning System can be used to play these notes at their actual physical location. This makes it possible for children to explore this musical space by moving through the real spatial area and experience the relationships of the notes through their proximity. Defining a range for each note as a circular area around each note location, consonant chords can be produced in those areas where those circles overlap. Using this concept, games can be developed in which the listeners have to perform certain tasks related to this musical space. This appears to be a promising approach for the music education of young children who can intuitively learn about music morphology without being explicitly taught about the complex theoretical mathematical background
Bayesian Estimation of White Matter Atlas from High Angular Resolution Diffusion Imaging
We present a Bayesian probabilistic model to estimate the brain white matter
atlas from high angular resolution diffusion imaging (HARDI) data. This model
incorporates a shape prior of the white matter anatomy and the likelihood of
individual observed HARDI datasets. We first assume that the atlas is generated
from a known hyperatlas through a flow of diffeomorphisms and its shape prior
can be constructed based on the framework of large deformation diffeomorphic
metric mapping (LDDMM). LDDMM characterizes a nonlinear diffeomorphic shape
space in a linear space of initial momentum uniquely determining diffeomorphic
geodesic flows from the hyperatlas. Therefore, the shape prior of the HARDI
atlas can be modeled using a centered Gaussian random field (GRF) model of the
initial momentum. In order to construct the likelihood of observed HARDI
datasets, it is necessary to study the diffeomorphic transformation of
individual observations relative to the atlas and the probabilistic
distribution of orientation distribution functions (ODFs). To this end, we
construct the likelihood related to the transformation using the same
construction as discussed for the shape prior of the atlas. The probabilistic
distribution of ODFs is then constructed based on the ODF Riemannian manifold.
We assume that the observed ODFs are generated by an exponential map of random
tangent vectors at the deformed atlas ODF. Hence, the likelihood of the ODFs
can be modeled using a GRF of their tangent vectors in the ODF Riemannian
manifold. We solve for the maximum a posteriori using the
Expectation-Maximization algorithm and derive the corresponding update
equations. Finally, we illustrate the HARDI atlas constructed based on a
Chinese aging cohort of 94 adults and compare it with that generated by
averaging the coefficients of spherical harmonics of the ODF across subjects
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