Visibility Preserving Terrain Simplification An Experimental Study Boaz Ben-Moshe (Ben-Gurion U.) Matthew Katz (Ben-Gurion U.) Joseph Mitchell (U. at Stony Brook) Yuval Nir (Ben-Gurion U.) 4/11/2019

Project Goals Define a visibility-based measure of quality of simplification. Develop a visibility preserving terrain simplification method - VPTS. Should preserve most of the visibility Should be efficient Experiment with VPTS, as well as with other TS methods, using the new quality measure. 4/11/2019

Motivation What is terrain simplification (TS), and Why is it needed (especially in the context of facility location). What are the common ways to measure quality of simplification. What types of facility location tasks or other tasks might use VPTS. 4/11/2019

Definitions Terrain T : A xy-monotone triangulation of a set P of points in 3-space. Simplification T’ of T : A xy-monotone triangulation of a subset P’ of P. Quality (error) measure : Determines how well T’ approximates T. 4/11/2019

Error measures Common error measures : The maximum vertical distance between T and T’ The volume of the region consisting of all points above T and below T’ or vice versa Our visibility-based quality measure : The expected similarity of the “views” from p and p’, respectively, where p in T and p’ in T’ are an arbitrary pair of corresponding points. 4/11/2019

Visibility-based measure Ideally, if two points in T see (do not see) each other, then the corresponding points in T’ should also see (not see) each other. Let X be a set of pairs of points. Let V be the set of pairs (a,b) in X for which T.los(a,b)=T’.los(a,b) . The quality of T’ (with respect to X) is |V| / |X| . 4/11/2019

Visibility-based measure Ideal measure : random set - all pairs. Transmitter-receiver measure : receivers and potential transmitters locations – all mixed pairs in given range. 4/11/2019

Visibility-Preserving TS - Overview Typically, the view from p is blocked by ridges Main stages: Compute the ridge network (a collection of chains of edges of T). Approximate the ridge network. The ridge network induces a subdivision of the terrain into patches. Simplify each patch (independently), using one of the standard TS methods. 4/11/2019

Defining the ridge network Three types of edges. Take all difluent edges. Two edges are connected if they share a vertex & no flow from one side of the 2-chain to the other. 4/11/2019

Defining the ridge network Special cases: flow-splitting triangles – take one of receiving edges a few more special cases (e.g., lakes). 4/11/2019

Defining the ridge network 4/11/2019

Approximating the ridge network Goal: Replace RN with an approx network of size k. Preliminary phase: divide RN into chains and assign to each chain a level of importance. Phase 1: collapse all chains - replace each chain c by a single-edge chain defined by the two endpoints of c. Phase 2: repeatedly drop a leaf edge of min importance from current network, until current size is k’. Phase 3: repeatedly refine the chain that “needs” it most, until the desired size k is achieved. 4/11/2019

Approximating the ridge network Original ridge network 4/11/2019

Approximating the ridge network Reducing the num of vertices from 31 to k=9 using k’=6: 4/11/2019

Approximating the ridge network Phase 1: collapsing all chains 4/11/2019

Approximating the ridge network Phase 2: Remove the least important leaf chain 4/11/2019

Approximating the ridge network 4/11/2019

Approximating the ridge network Phase 2: Remove the least important leaf chain 4/11/2019

Approximating the ridge network Phase 2: Remove the least important leaf chain 4/11/2019

Approximating the ridge network Phase 2: Remove the least important leaf chain 4/11/2019

Approximating the ridge network End of Phase 2 4/11/2019

Approximating the ridge network Phase 3: Refine the max dist chain 4/11/2019

Approximating the ridge network 4/11/2019

Approximating the ridge network Phase 3: Refine the max dist chain 4/11/2019

Approximating the ridge network 4/11/2019

Approximating the ridge network Phase 3: Refine the max dist chain 4/11/2019

Approximating the ridge network End of Phase 3: k =9 4/11/2019

Approximating the ridge network Original RN  Approximate RN 4/11/2019

Approximating the ridge network 4/11/2019

Approximating the ridge network 4/11/2019

Approximating the ridge network 4/11/2019

The main TS algorithm The (simplified) Ridge Network induces a subdivision of the terrain into regions: For each region (map[i]) in the subdivision If map[i] is “big” then recursively apply VPTS to map[i]. Else (map[i] is “small”) simplify map[i] using a “standard” simplification method (such as Garland’s “Terra”). 4/11/2019

Experimental Results Input: 20 terrains, each of size 15,000-20,000, representing different geographic regions. VPTS was implemented using CGAL. “Regular” TS methods: Terra [Heckbert-Garland], GcTin [Silva-Mitchell], Qslim [Garlad-Heckbert]. Each input terrain was ‘compressed’ to 1000, 500, 250,125 points. 4/11/2019

Tests Note: every test was repeated 4 times, for each of the 4*20*4 = 320 compressed terrains. Thus in total about 320*4*4 = 5120 quality of simplification evaluations were done. 4/11/2019

Results 4/11/2019

Triangulation representation 4/11/2019

Future work Compare VPTS with other TS methods. Implement a robust version of VPTS. Implement a version where regular visibility is replaced by “RF visibility”. 4/11/2019

Fin http://www.cs.bgu.ac.il/~benmoshe 4/11/2019

Conclusion: It is a good idea to compress terrains before applying facility location algorithms Original: 150,000 Object1: 10,000 – 99.5% Object2: 1,000 – 97.8 % 4/11/2019