“The Road and the River Cross at the Bridge” “The Road and the River Cross at the Bridge” Problem:Problem:

Internal and Relative Topology in an MRDBInternal and Relative Topology in an MRDB

Barbara P. Buttenfieldbabs@colorado.edu

Eric Wolfebwolf@colorado.edu

Funding from NSF BCS 04-51509

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The Road, River & Bridge Problem The Road, River & Bridge Problem “Conflation Error” “Conflation Error”

Roads & Streams 25k Streams 25k, Roads 100k

Variable data resolutions in an MRDB don’t always match

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Outline of PresentationOutline of Presentation

� Quickly review the paper� Pyramid architecture for vectors -- MRVIN

� Implementation and status

� Demonstrate how MRVIN preserves topology� internal (vectors must not self-cross)

� relative (road & river must cross at the bridge)

� Metadata about feature geometry across scale� Implementation in MRVIN

� Why it’s useful

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MRVIN ArchitectureMRVIN Architecture

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MRVIN ArchitectureMRVIN Architecture

� Implementation and Status

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Requirements for MRDB vectorsRequirements for MRDB vectors

� Reliable measurements for modeling� Length, area, perimeter etc.

� Local coordinate density

� Visual semantics� Protect cognitive / perceptual expectations for mapping

(Preserving reliable measurements handles most of this)

� Preserve topology� Internal (vectors must not self-cross)

� Relative (road & river must cross at the bridge)

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Vector data is heterogeneous - non-stationary

Reflects local geomorphology and geology

Local Density of DetailLocal Density of Detail

Lake Eriesimple coast

Chesapeake Bayhigh frequency crenulations

Brownsville Texasspace-filling curve

Variations must be preserved in all representationsVariations must be preserved in all representations

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Building the DatabaseBuilding the Database

Deconstruct the line � vector pyramid

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Building the DatabaseBuilding the Database

� Topological conditions� Vectors must not self-cross

� Isolated vectors must remain separate

� Generate convex hulls

“If two convex hulls do not overlap, the contents of those hulls will not overlap.”

(Saalfeld, 1999)

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Preserve Internal TopologyPreserve Internal Topology

� On retrieving vectors:� Retrieve one row of pyramid

� Correct topology by retrieving extra detail locally but only as needed

� This works!

Single vectors do not self-cross

Compound vectors need extra step

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Internal Topology for Compound VectorsInternal Topology for Compound Vectors

� Compound vectors include� Stream tributaries, road networks

� Also called “multi-part vectors”

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Internal Topology for Compound VectorsInternal Topology for Compound Vectors

� Data stored in a grove of trees� Convex hull around entire network

� Convex Hulls around each treeat tributary confluence nodes

• Proceed as for individual vectors (topological checks and corrections)

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� Portion of stream tributary from Boulder County, with hulls. Pyramid rows are displayed for the first five rows of the grove.

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Preserve Relative TopologyPreserve Relative Topology

� Generate three data groves (and correct internal topology)

� Construct a forest of groves� Intersect groves as follows � …

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The road, the river and the bridgeThe road, the river and the bridge

� Overlay convex hulls� Generate new vectors where features intersect � Insert pseudo-nodes

� Final result same asconventional arc-node topology but nodes are consistent across data layers

� Nodes protect correct intersections atevery resolution, even when representations are intermixed

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Metadata about feature geometry across Metadata about feature geometry across scalescale

� Where it’s stored in MRVIN

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MetadataMetadata

Olen = length orig coord string

Alen = length of gen’lzd line

Dev = vector displacement

Original

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Tree ID 44 Level Strips Node

0 1 2 1 2 3 2 4 5 3 8 9 4 16 17 5 28 29 6 40 41 7 50 51 8 55 56 9 58 59 10 59 60

Tree 11 117 60

Length Rep min max mean std dev Length

4585.685 4585.685 4585.685 0 4585.68

3172.129 3681.271 3426.7 207.856 6853.40

867.057 2922.701 1750.413 714.921 7001.65

264.553 1962.464 917.255 587.781 7338.04

64.11 1125.441 463.021 326.921 7408.32

25.049 881.25 266.297 210.143 7456.32

75.46 762.838 187.09 138.006 7483.61

26.114 564.711 149.888 97.848 7494.42

60.533 472.314 136.298 76.871 7496.41

35.833 323.151 129.258 59.394 7496.96

69.986 169.507 127.07 57.859 7497.10

7497.10

Tree ID Deviation 44 Level Strips Node min max mean std dev

0 1 2 2530.851 2530.851 2530.851 0 1 2 3 242.764 373.422 308.093 65.329 2 4 5 83.257 386.349 204.367 124.03 3 8 9 18.866 164.825 58.88 45.611 4 16 17 3.583 84.931 27.433 22.016 5 28 29 0.069 49.655 17.706 12.213 6 40 41 0.001 20.545 11.276 5.478 7 50 51 1.798 15.807 7.025 4.674 8 55 56 2.142 7.653 5.044 2.259 9 58 59 3.748 3.748 3.748 0 10 59 60 0 0 0 0

Tree 11 117 60

Original

Level 2Level 5Level 6

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Metadata Metadata –– Why it’s UsefulWhy it’s Useful

Leibniz law: (x)(y)[ (x = y) ≡ (F)(Fx ≡ Fy) ]

If A and B are identical, then they have the same properties.

If A and B have different properties, then they cannot be identical.

(Stevenson, 1972; Sober, 2001)

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Metadata Metadata –– Why it’s UsefulWhy it’s Useful

� Monitor geometry and topology metrics across range of resolutions and datasets

� Exploit metrics to test Leibniz’ Law:Are two data resolutions essentially providing the same versions of a geographical feature?

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Metadata Metadata –– Why it’s UsefulWhy it’s Useful

� Avoid retrieving more data than is necessary

� Comparing metrics involves far less effort than comparing complete datasets, and could inform the choice of resolution in analysis and multi-scale mapping.

� Opens the door to a host of interesting questions about equivalence and data representation