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Kevin M. Sahr Department of Computer Science
Southern Oregon University
Central Place Indexing: Optimal Location Representation for
Digital Earth
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The Situation
• Geospatial computing has achieved many impressive results
• But it now faces unprecedented challenges
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Advent of “Digital Earth”• exemplifies the challenges facing geospatial
computing, combining in one platform:
✦ “mother of all (geospatial) databases”
✦ simulation, interactive 3D visualization, & analysis of vast quantities of diverse distributed global geospatial “big data”
✦ integrates real-time location update and manipulation
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The Key: Location Representation
• to implement this vision in totality a revolution in our fundamental approaches to geospatial computing is required
• at the core of all geospatial applications are data structures that represent location
✦ even minor efficiency improvements in location representation can lead to substantial performance increases
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Fundamental Location Representation Types• digital earth systems must provide data
structures for representing:
✦ raster/pixels for ✤ imagery ✤ discrete simulation ✤ “gridded” data analysis
✦ vector/point locations
✦ spatial databases/spatial indexes
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Traditional Raster Location
Representation • raster of square pixels
• addressed using 2-tuple of integers
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Traditional Image Processing Model
• traditional raster representation supports image processing based on a conceptual model of:
✦ input from square raster of sensors
✦ stored internally as matrix of pixels
✦ displayed one-to-one on a square raster of display pixels
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Digital Earth Reality• image processing in digital earth
systems breaks this mold
✦ processed satellite image pixels rarely correspond to individual sensors
✦ must support whole-earth image sets
✤ spherical topology
✦ internal pixels mapped to virtual 3D surface for display
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A Superior Alternative• numerous researchers have proposed using
hexagon-shaped pixels, arranged in a hexagonal raster
• the human eye uses a hexagonal arrangement of photoreceptors
• compared to square rasters, hexagon rasters
✦ are 13% more efficient at sampling
✦ 25% to 50% more efficient for common image processing algorithms
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Discrete Simulation• hexagonal grids also have numerous
advantages over square grids for discrete simulation
✦ superior angular resolution
✦ discrete distance metric better approximates cartesian distance
✦ display uniform unambiguous adjacency
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Traditional Vector Location Representation
• 3- or 2-tuples of floating point values
• attempt to mimic the real number coordinates used in pre-computer scientific analysis and 2D mapping
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But…
• vectors of real numbers
✦ are continuous and infinite
• tuples of floating point values
✦ are discrete and finite
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Problems• the simplest operations (e.g., equality
test) can result in profound semantic errors
• bounding the rounding error on individual operations can be difficult
✦ on complex systems can be impossible
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The Reality• floating point values are no more “exact” than
integer values
• given n bits, we can distinguish 2n unique values
✦ all other points must be quantized to these
• all computer representations of location — both raster and vector — are necessarily discrete
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A Superior Alternative• the human brain represents location using a
hexagonal arrangement of neurons
• quantization to the points of a hexagonal lattice is optimal using multiple formulations
✦ least average quantization error
✦ covering problem
✦ packing problem
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Traditional Spatial DBs• traditional raster and vector representations
are inefficient for many common spatial operations
• spatial DBs add a linear spatial index
✦ correspond to buckets containing locations, providing
✤ more efficient spatial queries
✤ coarse filter for specific algorithms
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Traditional Spatial DBs• underlying vector/raster representation
retained for
✦ final stage of many algorithms
✦ arbitrary spatial operations
• form of spatial DBs based on traditional vector/raster representational forms
✦ structured: square quad tree
✦ semi-structured: rectangular buckets (e.g. R-tree)
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Spatial Queries• traditional primary spatial query type:
window/axes-aligned rectangle
• but primary query type in modern geospatial systems is proximity
✦ recall that hexagonal discrete distance metric better approximates cartesian distance
✤ hexagon buckets provide more efficient proximity coarse filter
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The Task• design a hierarchical integer index for
hexagon lattices that can be used for:
✦ multi-precision vector location
✦ multi-resolution raster location
✦ structured spatial index
• must be explicitly spherical
• Digital Earth Primary Spatial Index:
✦ One Index to Rule Them All
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Hexagon Coordinate Systems
• single resolution hexagon grids have three natural axes spaced at 120° angles�
i�axis
j�axis
k�axis
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Hexagonal Multi-Res• regular multi-precision/resolution hexagon
lattices can be created with an infinite number of apertures
✦ aperture: ratio of cell areas between resolutions
• research has focused on the Central Place (Christaller, 1966) apertures 3, 4, and 7
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Central Place Apertures
aperture 3 aperture 4 aperture 7
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Prefix Codes• hierarchical prefix codes have many
advantages for hierarchical spatial indexes
✦ each digit in index corresponds to a single precision in the representation
✤ provides locality preserving total ordering
✤ implicitly encodes precision
✤ provides efficient generalization via truncation
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Aperture 7 Case• note that each hexagon is naturally associated
with 7 hexagons at the next finer resolution
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GBT• Generalized Balanced Ternary (GBT)
(Gibson & Lucas, 1982) is a hierarchical prefix code system for aperture 7 grids
• each indexing child adds the appropriate digit to their parent’s index
• single digits correspond to each possible hexagonal direction
0
i
j
k
4
2 6
3
1 5
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Apertures 3 and 4
• note that in apertures 3 and 4 each cell also naturally has 7 finer precision potential indexing children
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Central Place Indexing• we can apply the GBT arrangement to the
aperture 3 and 4 cases
• we call the result Central Place Indexing (CPI)
✦ provides uniform indexing for all 3 apertures
✦ allows for indexing mixed-aperture resolution sequences
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Pixel/Bucket Indexing• cells in aperture 3 and 4 resolutions can
have multiple parents cells and therefore multiple valid CPI indexes
✦ aperture 3 example:
A BB2A4
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Pixel/Bucket Indexing• if the cells represent pixels or DB buckets,
then a single unique index must be chosen for each cell
✦ a consistent choice of child assignment must be made
✦ example aperture 3 solutions:
i
b
i
b+1
i
b
i
b+1
i
b
i
b+1
G
123
G
5
G G
12
GG
56
G
1
G
3
G
56
i
b
=i
b+1
i
b
=i
b+1
G
1
G
3
G
5
G GGG
3
G
56
i
b
=i
b+1
i
b
=i
b+1
GGG
345
G G
1
G
34
GG
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Vector Indexing
• in apertures 3 and 4 point quantization can be performed at each successive resolution
B2A4A B
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Vector Indexing• thus aperture 3 and 4 grids effectively
address cell sub-regions
✦ provides true multi-precision point quantization
✦ truncation and rounding are equivalent
✦ indexes can be progressively transmitted
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CPI Algorithms• we have implemented planar CPI algorithms
for
✦ forward & inverse quantization
✦ addition/translation
✦ subtraction
✦ metric distance
• implemented using efficient per-digit table lookups
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The Sphere• we can apply any CPI system
to a spherical icosahedron to index a hexagonal Discrete Global Grid System (DGGS)
• note that cells centered on the icosahedral vertices are pentagons
✦ we can apply CPI indexing to them by deleting one of the non-centroid indexing sub-sequences
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Conclusions• multi-resolution hexagonal DGGSs provide
the best known basis for raster, vector, and spatial DB location representation for digital earth systems
• CPI provides a unified efficient hierarchical indexing for all types of location representation on hexagonal DGGSs
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2
www.discreteglobalgrids.org
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