26,212 research outputs found
A Bayesian Approach to Manifold Topology Reconstruction
In this paper, we investigate the problem of statistical reconstruction of piecewise linear manifold topology. Given a noisy, probably undersampled point cloud from a one- or two-manifold, the algorithm reconstructs an approximated most likely mesh in a Bayesian sense from which the sample might have been taken. We incorporate statistical priors on the object geometry to improve the reconstruction quality if additional knowledge about the class of original shapes is available. The priors can be formulated analytically or learned from example geometry with known manifold tessellation. The statistical objective function is approximated by a linear programming / integer programming problem, for which a globally optimal solution is found. We apply the algorithm to a set of 2D and 3D reconstruction examples, demon-strating that a statistics-based manifold reconstruction is feasible, and still yields plausible results in situations where sampling conditions are violated
Towards Persistence-Based Reconstruction in Euclidean Spaces
Manifold reconstruction has been extensively studied for the last decade or
so, especially in two and three dimensions. Recently, significant improvements
were made in higher dimensions, leading to new methods to reconstruct large
classes of compact subsets of Euclidean space . However, the complexities
of these methods scale up exponentially with d, which makes them impractical in
medium or high dimensions, even for handling low-dimensional submanifolds. In
this paper, we introduce a novel approach that stands in-between classical
reconstruction and topological estimation, and whose complexity scales up with
the intrinsic dimension of the data. Specifically, when the data points are
sufficiently densely sampled from a smooth -submanifold of , our
method retrieves the homology of the submanifold in time at most ,
where is the size of the input and is a constant depending solely on
. It can also provably well handle a wide range of compact subsets of
, though with worse complexities. Along the way to proving the
correctness of our algorithm, we obtain new results on \v{C}ech, Rips, and
witness complex filtrations in Euclidean spaces
Geometry Processing of Conventionally Produced Mouse Brain Slice Images
Brain mapping research in most neuroanatomical laboratories relies on
conventional processing techniques, which often introduce histological
artifacts such as tissue tears and tissue loss. In this paper we present
techniques and algorithms for automatic registration and 3D reconstruction of
conventionally produced mouse brain slices in a standardized atlas space. This
is achieved first by constructing a virtual 3D mouse brain model from annotated
slices of Allen Reference Atlas (ARA). Virtual re-slicing of the reconstructed
model generates ARA-based slice images corresponding to the microscopic images
of histological brain sections. These image pairs are aligned using a geometric
approach through contour images. Histological artifacts in the microscopic
images are detected and removed using Constrained Delaunay Triangulation before
performing global alignment. Finally, non-linear registration is performed by
solving Laplace's equation with Dirichlet boundary conditions. Our methods
provide significant improvements over previously reported registration
techniques for the tested slices in 3D space, especially on slices with
significant histological artifacts. Further, as an application we count the
number of neurons in various anatomical regions using a dataset of 51
microscopic slices from a single mouse brain. This work represents a
significant contribution to this subfield of neuroscience as it provides tools
to neuroanatomist for analyzing and processing histological data.Comment: 14 pages, 11 figure
Genus statistics using the Delaunay tessellation field estimation method: (I) tests with the Millennium Simulation and the SDSS DR7
We study the topology of cosmic large-scale structure through the genus
statistics, using galaxy catalogues generated from the Millennium Simulation
and observational data from the latest Sloan Digital Sky Survey Data Release
(SDSS DR7). We introduce a new method for constructing galaxy density fields
and for measuring the genus statistics of its isodensity surfaces. It is based
on a Delaunay tessellation field estimation (DTFE) technique that allows the
definition of a piece-wise continuous density field and the exact computation
of the topology of its polygonal isodensity contours, without introducing any
free numerical parameter. Besides this new approach, we also employ the
traditional approaches of smoothing the galaxy distribution with a Gaussian of
fixed width, or by adaptively smoothing with a kernel that encloses a constant
number of neighboring galaxies. Our results show that the Delaunay-based method
extracts the largest amount of topological information. Unlike the traditional
approach for genus statistics, it is able to discriminate between the different
theoretical galaxy catalogues analyzed here, both in real space and in redshift
space, even though they are based on the same underlying simulation model. In
particular, the DTFE approach detects with high confidence a discrepancy of one
of the semi-analytic models studied here compared with the SDSS data, while the
other models are found to be consistent.Comment: 14 pages, 9 figures, accepted by Ap
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