Central Place Indexing: Optimal Location Representation for Digital Earth. Kevin M. Sahr Department of Computer Science Southern Oregon University
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1 Central Place Indexing: Optimal Location Representation for Digital Earth Kevin M. Sahr Department of Computer Science Southern Oregon University 1 Kevin Sahr - October 6, 2014
2 The Situation Geospatial computing has achieved many impressive results But it now faces unprecedented challenges 2 Kevin Sahr - October 6, 2014
3 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 3 Kevin Sahr - October 6, 2014
4 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 4 Kevin Sahr - October 6, 2014
5 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 5 Kevin Sahr - October 6, 2014
6 Traditional Raster Location Representation raster of square pixels addressed using 2-tuple of integers 6 Kevin Sahr - October 6, 2014
7 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 7 Kevin Sahr - October 6, 2014
8 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 8 Kevin Sahr - October 6, 2014
9 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 9 Kevin Sahr - October 6, 2014
10 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 10 Kevin Sahr - October 6, 2014
11 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 11 Kevin Sahr - October 6, 2014
12 But vectors of real numbers are continuous and infinite tuples of floating point values are discrete and finite 12 Kevin Sahr - October 6, 2014
13 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 13 Kevin Sahr - October 6, 2014
14 The Reality floating point values are no more exact than integer values given n bits, we can distinguish 2 n unique values all other points must be quantized to these all computer representations of location both raster and vector are necessarily discrete 14 Kevin Sahr - October 6, 2014
15 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 15 Kevin Sahr - October 6, 2014
16 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 16 Kevin Sahr - October 6, 2014
17 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) 17 Kevin Sahr - October 6, 2014
18 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 18 Kevin Sahr - October 6, 2014
19 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 19 Kevin Sahr - October 6, 2014
20 Hexagon Coordinate Systems single resolution hexagon grids have three natural axes spaced at 120 angles j axis i axis k axis 20 Kevin Sahr - October 6, 2014
21 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 21 Kevin Sahr - October 6, 2014
22 Central Place Apertures aperture 3 aperture 4 aperture 7 22 Kevin Sahr - October 6, 2014
23 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 23 Kevin Sahr - October 6, 2014
24 Aperture 7 Case note that each hexagon is naturally associated with 7 hexagons at the next finer resolution 24 Kevin Sahr - October 6, 2014
25 GBT Generalized Balanced Ternary (GBT) (Gibson & Lucas, 1982) is a hierarchical prefix code system for aperture 7 grids j k 0 4 i 1 5 each indexing child adds the appropriate digit to their parent s index single digits correspond to each possible hexagonal direction 25 Kevin Sahr - October 6, 2014
26 Apertures 3 and 4 note that in apertures 3 and 4 each cell also naturally has 7 finer precision potential indexing children 26 Kevin Sahr - October 6, 2014
27 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 27 Kevin Sahr - October 6, 2014
28 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 A4 B2 B 28 Kevin Sahr - October 6, 2014
29 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+1 i b+1 i b+1 i b i b i b 29 Kevin Sahr - October 6, 2014
30 Vector Indexing in apertures 3 and 4 point quantization can be performed at each successive resolution A A4 B2 B 30 Kevin Sahr - October 6, 2014
31 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 31 Kevin Sahr - October 6, 2014
32 CPI Algorithms we have implemented planar CPI algorithms for forward & inverse quantization addition/translation subtraction metric distance implemented using efficient per-digit table lookups 32 Kevin Sahr - October 6, 2014
33 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 33 Kevin Sahr - October 6, 2014
34 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 34 Kevin Sahr - October 6, 2014
35 Kevin Sahr - October 6, 2014
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