Lasse Deleuran 1/37 Homotopic Polygonal Line Simplification Lasse Deleuran PhD student

Lasse Deleuran 2/37 Content  Motivation  Our Results  Restricted simplification  Unrestricted simplification  Simplifying massive data

Lasse Deleuran 3/37 Motivation – Contour Lines

Lasse Deleuran 4/37 Motivation – Contour Lines

Lasse Deleuran 5/37 Motivation – Contour Lines

Lasse Deleuran 6/37 Motivation – Contour Lines

Lasse Deleuran 7/37 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

Lasse Deleuran 8/37 Definitions  Polygonal line: P=p 1, p 2, …,p n |P| = n-1  Homotopic poly. lines:

Lasse Deleuran 9/37 Homotopic X-Shortest Paths  n polygonal paths of combined size m  Endpoints are the obstacles  Compute x-shortest path while maintaining homotopy

Lasse Deleuran 10/37 Shortest Paths – Previous Work  Efrat et al. ’06: expected time O(nlog ε+1 n+mlogn)  Bespamyathnikh ’03: O(nlog ε+1 n+mlogn)  2-part approach:1) Compute homotopic x-shortest paths 2) Compute homotopic shortest paths  Use their 2. part to achieve O(nlog ε+1 n+m)

Lasse Deleuran 11/37 Restricted - Problem Definition  Given 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:

Lasse Deleuran 12/37 Restricted - Previous Results Previous Results  Imai & Iri ‘88: Framework for the problem  Hausdorff: Chan & Chin ’92 O(n 2 )  Frechét Distance: Alt & Goday ’95 O(n 3 )  L 1 and Uniform metric: Agarwal & Varadarajan ’00 O(n 4/3+ 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(n 6 m 2 )

Lasse Deleuran 13/37  Previous Results  Problems  Our Algorithm  Experimental Results Simplifying a Massive Ammount of Polygons

Lasse Deleuran 14/37 Previous Results: Terrain vs Polygons Simplifying 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

Lasse Deleuran 15/37 Challenges when Simplifying  Too many details  Massive data  Maintain precision  Maintain topology  Prevent intersections

Lasse Deleuran 16/37 Challenges – Too Many Details

Lasse Deleuran 17/37 Challenges - Massive Data  Denmark: 26B LIDAR points 12.4B grid cells

Lasse Deleuran 18/37 Challenges - Massive Data: I/O Model  RAM/internal memory size M  Unbounded disk/external memory  Transfer in blocks of size B  CPU only works on internal memory M B CPU Disk RAM

Lasse Deleuran 19/37 Massive Data - Practical Assumptions  Any polygon fits in memory (smaller than M)  Segments intersecting any vertical line is < M

Lasse Deleuran 20/37 Challenges - Maintain Precision  Simplification algorithms typically only consider movement in the plane (x and y)

Lasse Deleuran 21/37 Challenges - Maintain Topology  Topology: Parent / child relationships  Maintain topology through homotopy

Lasse Deleuran 22/37 Problems – Prevent Intersections  Intersections with other polygons / self intersections

Lasse Deleuran 23/37 Algorithm Overview  1: Collect polygons (I/O-efficient)  2: I/O efficient polygon visiting (I/O-efficient)  3: Simplify polygons (internal)

Lasse Deleuran 24/37 1/3 - Collect Polygons

Lasse Deleuran 25/37  Polygons are neighbors if no other poly. divides them  A polygon must be considered together w. neighbors 2/3 - I/O Efficient Polygon Visiting

Lasse Deleuran 26/37 3/3 Simplify Polygons  Basic Algorithm: Douglas Peucker

Lasse Deleuran 27/37 Problems with Douglass Peucker  Running time O(n 2 ), but O(nlogn) in practice  No z-constraint / not constrained by other polygons  Introduces self intersections, no homotopy

Lasse Deleuran 28/37 Simplifying – Adding Boundaries  Construct Trapezoidal decomposition O(nlogn)  Continue DP until inside of decomposition O(n 2 logn)  Add contraint for z

Lasse Deleuran 29/37 Simplifying – Removing Intersections  Sweep to find intersections O(nlogn)  Continue DP on intersecting segments  Repeat => O(n 2 logn)

Lasse Deleuran 30/37 Simplifying – Maintain Homotopy  Trapezoidal sequence: ABCDCFCDEDE  Contract XYX -> X  Canonical Sequence: ABCDE

Lasse Deleuran 31/37 Simplifying – Maintain Homotopy  Check segment for homotopy: O(n)  => O(n 2 logn)

Lasse Deleuran 32/37 Practical Optimizations

Lasse Deleuran 33/37 Optimization 1 - Conditioning the Terrain  Fill up all holes with depth of less than 0.5m  Do so for small hills too.

Lasse Deleuran 34/37 Optimization 2 - Exploit Bounding Boxes

Lasse Deleuran 35/37 Optimization 3 - Minimal Decompositions  Too much time will be spent constructing decompositions  Only use edges that intersect bounding box

Lasse Deleuran 36/37 Setup  Code in C++ using TPIE and TerraSTREAM  Machine:  8-core Intel Xenon 3.2GHz  12GB of RAM  disk speed: 400MB/s

Lasse Deleuran 37/37 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  self intersections  8.2% of the points remained after simplifying Thank You