222 research outputs found
Cone fields and topological sampling in manifolds with bounded curvature
Often noisy point clouds are given as an approximation of a particular
compact set of interest. A finite point cloud is a compact set. This paper
proves a reconstruction theorem which gives a sufficient condition, as a bound
on the Hausdorff distance between two compact sets, for when certain offsets of
these two sets are homotopic in terms of the absence of {\mu}-critical points
in an annular region. Since an offset of a set deformation retracts to the set
itself provided that there are no critical points of the distance function
nearby, we can use this theorem to show when the offset of a point cloud is
homotopy equivalent to the set it is sampled from. The ambient space can be any
Riemannian manifold but we focus on ambient manifolds which have nowhere
negative curvature. In the process, we prove stability theorems for
{\mu}-critical points when the ambient space is a manifold.Comment: 20 pages, 3 figure
Geometric Inference on Kernel Density Estimates
We show that geometric inference of a point cloud can be calculated by
examining its kernel density estimate with a Gaussian kernel. This allows one
to consider kernel density estimates, which are robust to spatial noise,
subsampling, and approximate computation in comparison to raw point sets. This
is achieved by examining the sublevel sets of the kernel distance, which
isomorphically map to superlevel sets of the kernel density estimate. We prove
new properties about the kernel distance, demonstrating stability results and
allowing it to inherit reconstruction results from recent advances in
distance-based topological reconstruction. Moreover, we provide an algorithm to
estimate its topology using weighted Vietoris-Rips complexes.Comment: To appear in SoCG 2015. 36 pages, 5 figure
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
Skeletal representations of orthogonal shapes
In this paper we present two skeletal representations applied to orthogonal shapes of R^n : the cube axis and a family of skeletal representations provided by the scale cube axis. Orthogonal shapes are a subset of polytopes, where the hyperplanes of the bounding facets are restricted to be axis aligned. Both skeletal representations rely on the Lâ metric and are proven to be homotopically equivalent to its shape. The resulting skeleton is composed of n â 1 dimensional facets. We also provide an efficient and robust algorithm to compute the scale cube axis in the plane and compare the resulting skeleton with other skeletal representations.Postprint (published version
Multi-Dimensional Medial Geometry: Formulation, Computation, and Applications
Medial axis is a classical shape descriptor. It is a piece of geometry that lies in the middle of the original shape. Compared to the original shape representation, the medial axis is always one dimension lower and it carries many intrinsic shape properties explicitly. Therefore, it is widely used in a large amount of applications in various fields. However, medial axis is unstable to the boundary noise, often referred to as its instability. A small amount of change on the object boundary can cause a dramatic change in the medial axis. To tackle this problem, a significance measure is often associated with the medial axis, so that medial points with small significance are removed and only the stable part remains. In addition to this problem, many applications prefer even lower dimensional medial forms, e.g., shape centers of 2D shapes, and medial curves of 3D shapes. Unfortunately, good significance measures and good definitions of lower dimensional medial forms are still lacking. In this dissertation, we extended Blum\u27s grassfire burning to the medial axis in both 2D and 3D to define a significance measure as a distance function on the medial axis. We show that this distance function is well behaved and it has nice properties. In 2D, we also define a shape center based on this distance function. We then devise an iterative algorithm to compute the distance function and the shape center. We demonstrate usefulness of this distance function and shape center in various applications. Finally we point out the direction for future research based on this dissertation
Improved Locally Adaptive Sampling Criterion for Topology Preserving Reconstruction of Multiple Regions
Volume based digitization processes often deal with non-manifold shapes. Even though many reconstruction algorithms have been proposed for non-manifold surfaces, they usually donât preserve topological properties. Only recently, methods were presented whichâgiven a finite set of surface sample pointsâresult in a mesh representation of the original boundary preserving all or certain neighbourhood relations, even if the sampling is sparse and highly noise corrupted. We show that the required sampling conditions of the algorithm called ârefinement reductionâ limit the guaranteed correctness of the outcome to a small class of shapes. We define new locally adaptive sampling conditions that depend on our new pruned medial axis and finally prove without any restriction on shapes that under these new conditions, the result of ârefinement reductionâ corresponds to a superset of a topologically equivalent mesh
Estimating the Reach of a Manifold
Various problems in manifold estimation make use of a quantity called the
reach, denoted by , which is a measure of the regularity of the
manifold. This paper is the first investigation into the problem of how to
estimate the reach. First, we study the geometry of the reach through an
approximation perspective. We derive new geometric results on the reach for
submanifolds without boundary. An estimator of is
proposed in a framework where tangent spaces are known, and bounds assessing
its efficiency are derived. In the case of i.i.d. random point cloud
, is showed to achieve uniform
expected loss bounds over a -like model. Finally, we obtain
upper and lower bounds on the minimax rate for estimating the reach
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