768 research outputs found
Querying Probabilistic Neighborhoods in Spatial Data Sets Efficiently
In this paper we define the notion
of a probabilistic neighborhood in spatial data: Let a set of points in
, a query point , a distance metric \dist,
and a monotonically decreasing function be
given. Then a point belongs to the probabilistic neighborhood of with respect to with probability f(\dist(p,q)). We envision
applications in facility location, sensor networks, and other scenarios where a
connection between two entities becomes less likely with increasing distance. A
straightforward query algorithm would determine a probabilistic neighborhood in
time by probing each point in .
To answer the query in sublinear time for the planar case, we augment a
quadtree suitably and design a corresponding query algorithm. Our theoretical
analysis shows that -- for certain distributions of planar -- our algorithm
answers a query in time with high probability
(whp). This matches up to a logarithmic factor the cost induced by
quadtree-based algorithms for deterministic queries and is asymptotically
faster than the straightforward approach whenever .
As practical proofs of concept we use two applications, one in the Euclidean
and one in the hyperbolic plane. In particular, our results yield the first
generator for random hyperbolic graphs with arbitrary temperatures in
subquadratic time. Moreover, our experimental data show the usefulness of our
algorithm even if the point distribution is unknown or not uniform: The running
time savings over the pairwise probing approach constitute at least one order
of magnitude already for a modest number of points and queries.Comment: The final publication is available at Springer via
http://dx.doi.org/10.1007/978-3-319-44543-4_3
Discrete denoising of heterogenous two-dimensional data
We consider discrete denoising of two-dimensional data with characteristics
that may be varying abruptly between regions.
Using a quadtree decomposition technique and space-filling curves, we extend
the recently developed S-DUDE (Shifting Discrete Universal DEnoiser), which was
tailored to one-dimensional data, to the two-dimensional case. Our scheme
competes with a genie that has access, in addition to the noisy data, also to
the underlying noiseless data, and can employ different two-dimensional
sliding window denoisers along distinct regions obtained by a quadtree
decomposition with leaves, in a way that minimizes the overall loss. We
show that, regardless of what the underlying noiseless data may be, the
two-dimensional S-DUDE performs essentially as well as this genie, provided
that the number of distinct regions satisfies , where is the total
size of the data. The resulting algorithm complexity is still linear in both
and , as in the one-dimensional case. Our experimental results show that
the two-dimensional S-DUDE can be effective when the characteristics of the
underlying clean image vary across different regions in the data.Comment: 16 pages, submitted to IEEE Transactions on Information Theor
Visual-hint Boundary to Segment Algorithm for Image Segmentation
Image segmentation has been a very active research topic in image analysis
area. Currently, most of the image segmentation algorithms are designed based
on the idea that images are partitioned into a set of regions preserving
homogeneous intra-regions and inhomogeneous inter-regions. However, human
visual intuition does not always follow this pattern. A new image segmentation
method named Visual-Hint Boundary to Segment (VHBS) is introduced, which is
more consistent with human perceptions. VHBS abides by two visual hint rules
based on human perceptions: (i) the global scale boundaries tend to be the real
boundaries of the objects; (ii) two adjacent regions with quite different
colors or textures tend to result in the real boundaries between them. It has
been demonstrated by experiments that, compared with traditional image
segmentation method, VHBS has better performance and also preserves higher
computational efficiency.Comment: 45 page
Implementation and application of adaptive mesh refinement for thermochemical mantle convection studies
Numerical modeling of mantle convection is challenging. Owing to the multiscale nature of mantle
dynamics, high resolution is often required in localized regions, with coarser resolution being sufficient
elsewhere. When investigating thermochemical mantle convection, high resolution is required to resolve
sharp and often discontinuous boundaries between distinct chemical components. In this paper, we present
a 2-D finite element code with adaptive mesh refinement techniques for simulating compressible thermochemical
mantle convection. By comparing model predictions with a range of analytical and previously
published benchmark solutions, we demonstrate the accuracy of our code. By refining and coarsening
the mesh according to certain criteria and dynamically adjusting the number of particles in each element,
our code can simulate such problems efficiently, dramatically reducing the computational requirements
(in terms of memory and CPU time) when compared to a fixed, uniform mesh simulation. The resolving
capabilities of the technique are further highlighted by examining plume‐induced entrainment in a thermochemical
mantle convection simulation
A limit process for partial match queries in random quadtrees and -d trees
We consider the problem of recovering items matching a partially specified
pattern in multidimensional trees (quadtrees and -d trees). We assume the
traditional model where the data consist of independent and uniform points in
the unit square. For this model, in a structure on points, it is known that
the number of nodes to visit in order to report the items matching
a random query , independent and uniformly distributed on ,
satisfies , where and
are explicit constants. We develop an approach based on the analysis of
the cost of any fixed query , and give precise estimates
for the variance and limit distribution of the cost . Our results
permit us to describe a limit process for the costs as varies in
; one of the consequences is that ; this settles a question of
Devroye [Pers. Comm., 2000].Comment: Published in at http://dx.doi.org/10.1214/12-AAP912 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org). arXiv admin note: text
overlap with arXiv:1107.223
Down the Rabbit Hole: Robust Proximity Search and Density Estimation in Sublinear Space
For a set of points in , and parameters and \eps, we present
a data structure that answers (1+\eps,k)-\ANN queries in logarithmic time.
Surprisingly, the space used by the data-structure is \Otilde (n /k); that
is, the space used is sublinear in the input size if is sufficiently large.
Our approach provides a novel way to summarize geometric data, such that
meaningful proximity queries on the data can be carried out using this sketch.
Using this, we provide a sublinear space data-structure that can estimate the
density of a point set under various measures, including:
\begin{inparaenum}[(i)]
\item sum of distances of closest points to the query point, and
\item sum of squared distances of closest points to the query point.
\end{inparaenum}
Our approach generalizes to other distance based estimation of densities of
similar flavor. We also study the problem of approximating some of these
quantities when using sampling. In particular, we show that a sample of size
\Otilde (n /k) is sufficient, in some restricted cases, to estimate the above
quantities. Remarkably, the sample size has only linear dependency on the
dimension
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