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Homotopic Polygonal Line Simplification

Lasse DeleuranPhD student

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ContentMotivationOur ResultsRestricted simplificationUnrestricted simplificationSimplifying massive data

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Motivation – Contour Lines

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Motivation – Contour Lines

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Motivation – Contour Lines

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Motivation – Contour Lines

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Our Results “Improving Homotopic Shortest Paths Using

Homotopic X-Shortest Paths”, M. Abam and L. Deleuran, TBS

“Computing Homotopic Line Simplification in a Plane”, M. Abam, S. Daneshpajouh, L. Deleuran, S. Ehsani and M. Ghodsi, EuroCG 2011, Submitted to CGTA 2012

“Simplifying Massive Contour Maps”, L. Arge, L. Deleuran, T. Mølhave, M. Revsbæk, and J. Truelsen, ESA, 2012

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Definitions Polygonal line: P=p1,p2, … ,pn |P| = n-1

Homotopic poly. lines:

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Homotopic X-Shortest Paths n polygonal paths of combined size m

Endpoints are the obstacles

Compute x-shortest path while maintaining homotopy

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Shortest Paths – Previous Work Efrat et al. ’06: expected time O(nlogε+1n+mlogn) Bespamyathnikh ’03: O(nlogε+1n+mlogn)

2-part approach: 1) Compute homotopic x-shortest paths2) Compute homotopic shortest paths

Use their 2. part to achieve O(nlogε+1n+m)

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Restricted - Problem DefinitionGiven a path of size n, compute the paths with fewest points while

Only using original points Maintaining some error constraint Maintaining homotopy to m obstacle points

Strong vs weak homotopy:

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Restricted - Previous ResultsPrevious ResultsImai & Iri ‘88: Framework for the problemHausdorff: Chan & Chin ’92 O(n2)Frechét Distance: Alt & Goday ’95 O(n3)L1 and Uniform metric: Agarwal & Varadarajan ’00 O(n4/3+ ε )

Our Results Compute strongly homotopic ”links”

X-monotone path in O(mlog(nm) + nlogn log(nm) + k) Any path in O(n(m + n)log(nm))

Compute homotopic shortest path in O(n6m2)

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Previous ResultsProblemsOur AlgorithmExperimental Results

Simplifying a Massive Ammount of Polygons

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Previous Results: Terrain vs PolygonsSimplifying terrain

Agarwal, et. al. ’98 (I/O efficient contour generation) Agarwal, et. al. ’08 (I/O efficient map generation) Carr, et. al. ’10 (DEM Simplification) Garland & Heckbert ’97 (Surface Simplification) Agarwal, et. al. ’06 (I/O efficient conditioning)

Simplifying polygons See surveys by Mitchell ‘97, ‘98

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Challenges when SimplifyingToo many detailsMassive dataMaintain precisionMaintain topologyPrevent intersections

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Challenges – Too Many Details

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Challenges - Massive Data Denmark: 26B LIDAR points

12.4B grid cells

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Challenges - Massive Data: I/O ModelRAM/internal memory size MUnbounded disk/external memoryTransfer in blocks of size BCPU only works on internal memory

M BCPU Disk

RAM

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Massive Data - Practical AssumptionsAny polygon fits in memory (smaller than M)Segments intersecting any vertical line is < M

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Challenges - Maintain Precision Simplification algorithms typically only consider

movement in the plane (x and y)

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Challenges - Maintain Topology Topology: Parent / child relationships

Maintain topology through homotopy

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Problems – Prevent Intersections Intersections with other polygons / self intersections

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Algorithm Overview 1: Collect polygons (I/O-efficient) 2: I/O efficient polygon visiting (I/O-efficient) 3: Simplify polygons (internal)

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1/3 - Collect Polygons

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Polygons are neighbors if no other poly. divides them A polygon must be considered together w. neighbors

2/3 - I/O Efficient Polygon Visiting

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3/3 Simplify Polygons Basic Algorithm: Douglas Peucker

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Problems with Douglass Peucker Running time O(n2), but O(nlogn) in practice No z-constraint / not constrained by other polygons Introduces self intersections, no homotopy

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Simplifying – Adding Boundaries Construct Trapezoidal decomposition O(nlogn) Continue DP until inside of decomposition O(n2logn) Add contraint for z

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Simplifying – Removing Intersections Sweep to find intersections O(nlogn) Continue DP on intersecting segments Repeat => O(n2logn)

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Simplifying – Maintain Homotopy Trapezoidal sequence: ABCDCFCDEDE Contract XYX -> X Canonical Sequence: ABCDE

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Simplifying – Maintain Homotopy Check segment for homotopy: O(n)

=> O(n2logn)

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Practical Optimizations

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Optimization 1 - Conditioning the Terrain

Fill up all holes with depth of less than 0.5m Do so for small hills too.

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Optimization 2 - Exploit Bounding Boxes

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Optimization 3 - Minimal Decompositions Too much time will be spent constructing

decompositions Only use edges that intersect bounding box

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Setup Code in C++ using TPIE and TerraSTREAM Machine:

8-core Intel Xenon CPU @ 3.2GHz 12GB of RAM disk speed: 400MB/s

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Results Results for Denmark dataset (12B points):

49 hours to simplify 4B segments on 7M contours Z-diff: 0.5m (Border contours: 0.2m) DP-error: 5m 600.000 self intersections

8.2% of the points remained after simplifying

Thank You