278 research outputs found
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
Linkless octree using multi-level perfect hashing
The standard C/C++ implementation of a spatial partitioning data structure, such as octree and quadtree, is often inefficient in terms of storage requirements particularly when the memory overhead for maintaining parent-to-child pointers is significant with respect to the amount of actual data in each tree node. In this work, we present a novel data structure that implements uniform spatial partitioning without storing explicit parent-to-child pointer links. Our linkless tree encodes the storage locations of subdivided nodes using perfect hashing while retaining important properties of uniform spatial partitioning trees, such as coarse-to-fine hierarchical representation, efficient storage usage, and efficient random accessibility. We demonstrate the performance of our linkless trees using image compression and path planning examples.postprin
Dynamic Smooth Compressed Quadtrees
We introduce dynamic smooth (a.k.a. balanced) compressed quadtrees with worst-case constant time updates in constant dimensions. We distinguish two versions of the problem. First, we show that quadtrees as a space-division data structure can be made smooth and dynamic subject to split and merge operations on the quadtree cells. Second, we show that quadtrees used to store a set of points in R^d can be made smooth and dynamic subject to insertions and deletions of points. The second version uses the first but must additionally deal with compression and alignment of quadtree components. In both cases our updates take 2^{O(d log d)} time, except for the point location part in the second version which has a lower bound of Omega(log n); but if a pointer (finger) to the correct quadtree cell is given, the rest of the updates take worst-case constant time. Our result implies that several classic and recent results (ranging from ray tracing to planar point location) in computational geometry which use quadtrees can deal with arbitrary point sets on a real RAM pointer machine
Quadtrees, transforms and image coding
Transforms and quadtrees are both methods of representing information in an
image in terms of the presence of information at differing length scales. This paper
presents a mathematical relationship between these two approaches to describing
images in the particular case when Walsh transforms are used. Furthermore, both
methods have been used for the compression of images for transmission. This paper
notes that under certain circumstances, quadtree compression produces identical
results to Walsh transform coding, but requires less computational effort to do so.
Remarks are also made about the differences between these approaches
Robust Proximity Search for Balls using Sublinear Space
Given a set of n disjoint balls b1, . . ., bn in IRd, we provide a data
structure, of near linear size, that can answer (1 \pm \epsilon)-approximate
kth-nearest neighbor queries in O(log n + 1/\epsilon^d) time, where k and
\epsilon are provided at query time. If k and \epsilon are provided in advance,
we provide a data structure to answer such queries, that requires (roughly)
O(n/k) space; that is, the data structure has sublinear space requirement if k
is sufficiently large
3D oil reservoir visualisation using octree compression techniques utilising logical grid co-ordinates
Octree compression techniques have been used for several years for compressing large three dimensional
data sets into homogeneous regions. This compression technique is ideally suited to datasets
which have similar values in clusters. Oil engineers represent reservoirs as a three dimensional grid
where hydrocarbons occur naturally in clusters. This research looks at the efficiency of storing these
grids using octree compression techniques where grid cells are broken into active and inactive regions.
Initial experiments yielded high compression ratios as only active leaf nodes and their ancestor, header
nodes are stored as a bitstream to file on disk. Savings in computational time and memory were possible
at decompression, as only active leaf nodes are sent to the graphics card eliminating the need of
reconstructing the original matrix. This results in a more compact vertex table, which can be loaded
into the graphics card quicker and generating shorter refresh delay times
Topographic map visualization from adaptively compressed textures
Raster-based topographic maps are commonly used in geoinformation systems to overlay geographic entities on top of digital terrain models. Using compressed texture formats for encoding topographic maps allows reducing latency times while visualizing large geographic datasets. Topographic maps encompass high-frequency content with large uniform regions, making current compressed texture formats inappropriate for encoding them. In this paper we present a method for locally-adaptive compression of topographic maps. Key elements include a Hilbert scan to maximize spatial coherence, efficient encoding of homogeneous image regions through arbitrarily-sized texel runs, a cumulative run-length encoding supporting fast random-access, and a compression algorithm supporting lossless and lossy compression. Our scheme can be easily implemented on current programmable graphics
hardware allowing real-time GPU decompression and rendering of bilinear-filtered topographic maps.Postprint (published version
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