Curve Reconstruction

Transcription

1 Curve Reconstruction Ernst Althaus Tamal Dey Stefan Näher Edgar Ramos Ernst Althaus and Kurt Mehlhorn: Traveling Salesman-Based Curve Reconstruction in Polynomial Time, SIAM Journal on Comuting, 3, 27 66, 2002 MPI Informatik Kurt Mehlhorn

2 Inut: A finite samle S from a curve γ Curve Reconstruction Outut: G S E where xy E iff x and y are adjacent on γ. The Goal: Algorithms that come with guarantees: reconstuction succeeds if γ Γ (= a class of curves) and S satisfies a certain samling condition. efficiency Motivation: line drawings from raster images surface reconstruction MPI Informatik 2 Kurt Mehlhorn

3 State of the Art til 97 uniformly samled smooth closed curves 97 non-uniformly samled smooth closed curves, ABE, DK, Go 99 non-uniformly samled smooth oen and closed curves, DMR 99 TSP reconstructs uniformly samled non-smooth curves, Gi 99 TSP reconstructs non-uniformly samled non-smooth curves in olynomial time, AM smooth curve: tangent everywhere uniform samle: at least one samle from every curve segment of length ε. non-uniform samle: samling rate deends on local features of the curve. in the last two years: O n2 alg for non-smooth curves (Funke/Ramos), reconstruction of smooth surfaces (Amenta/Bern, Dey-et-al, Funke/Ramos) MPI Informatik 3 Kurt Mehlhorn

4 V CRUST (ABE). V = vertices of Voronoi diagram of S. 2. CRUST = DelaunayDiagram S S S void CRUST(list<oint> L, GRAPH<oint,int>& G) { ma<oint,bool> voronoi_oint(false); GRAPH<circle,oint> VD; VORONOI(L,VD); node v; oint ; forall_nodes(v,vd) { if (VD.outdeg(v) < 2) continue; L.aend( = VD[v].center()); voronoi_oint[] = true; } DELAUNAY_TRIANG(L,G); } forall_nodes(v,g) { if ( voronoi_oint[ G[v] ] ) G.del_node(v); }

5 Crust: An Examle generated with LEDA-demo geowin/voro demo. MPI Informatik 5 Kurt Mehlhorn

6 Weaknesses oen curves shar corners branching oints Oen Curves DMR (Comgeo 99): A variant of CRUST reconstructs non-uniformly samled oen and closed curves. (a) (b) (c) (d) (e) (f) DK ABE DMR MPI Informatik 6 Kurt Mehlhorn

7 S Shar Corners semi-regular curve: left and right tangents exist and turning angle less than π. YES NO Giesen (Comgeo99): TSP reconstructs uniformly-samled semi-regular curves, i.e., for every semi-regular curve γ there is an ε 0: if S contains at least one oint from every curve segment of length ε then TSP reconstructs γ. exact TSP is required, aroximate TSP will not do MPI Informatik 7 Kurt Mehlhorn

8 TSP does not work for turning angle equal to zero (x,x^2) (a,a^2) (0.,0 (2a,0) O = origin, x-axis, arabola x2 y let x be such that dist order on curve = 2a O 0 0 x x 2 dist a a2 0 O x 2a 0 x2 wrong order = 2a 0 a a2 0 0 x x2 wrong order gives shorter length than correct order for arbitrarily small a since a a2 lies on the wrong side of the angular bisector MPI Informatik 8 Kurt Mehlhorn

9 xe The Subtour LP x e variable for edge e uvce Euclidean length of uv e minimize subject to e c e x e uv ; u R R v x uv u x uv 2for all v S 2 for all R S with R /0 S 0 integral subtour LP, remove integrality constraint subtour LP can be solved in olynomial time otimal solution is (in general) fractional MPI Informatik 9 Kurt Mehlhorn

10 Fractional Otimal Solution /2 /2 /2 2 2 /2 /2 2 /2 left side: edges weightsright side: otimal solution to LP otimal tour has cost fractional solution has cost MPI Informatik 0 Kurt Mehlhorn

11 A Cutting Plane Algorithm Solving the LPexonentially many subtour elimination constraints solve LP without subtour elimination constraints check for violated subtour elimination constraint let xe be the solution of the LP minimum cut in S E x e /2 /2 /2 acut of value 2yields a violated constraint add and reeat /2 runs in olynomial time with Ellisoid method is ractically efficient with simlex method Solving the ILP When LP has fractional solution, branch on fractional variable MPI Informatik Kurt Mehlhorn

12 Exerimental observation: the rogram never branched Main Result Let γ be a semi-regular curve. If S is a sufficiently dense samle of γ then otimal solution of subtour LP is integral (and hence a tour) can be found in olynomial time Proof Idea exloit duality of subtour LP and Held-Kar bound show that Held-Kar bound yields a tour. MPI Informatik 2 Kurt Mehlhorn

13 How Dense is Sufficiently Dense?. for every corner (= discontinuity), let R be largest such that (a) legs of corner are basically straight (= turn by less than 0%) within B (b) curve is connected inside the disk R 2. must have at least one samle oint on each leg within B R 4 3. break curve into smooth ieces by removing the arts inside the disks B R 8 4. for every in one of the smooth arts, let R be largest such that (a) curve is basically straight (= turns by less than 60 degrees) within B (b) curve is connected inside the disk R 5. must have at least one samle oint within B R 4 MPI Informatik 3 Kurt Mehlhorn

14 T The Held-Kar Lower Bound T π : S traveling salesman ath with endoints a and b otential function c π x y xy π x π y, modified distance MST π (cost of) minimum sanning tree wrt c π Cπ T C T 2 x Sπ x π a π b MST π Cπ maxπ MST π 2 x S π x π a b T π C (Held-Kar lower bound) If MST π is a ath, then it is otimal. Held-Kar bound and Subtour LP have the same value. If there is a π such that MST π is a ath and MST π is unique, then Subtour LP has integral solution. MPI Informatik 4 Kurt Mehlhorn

15 MST and Shar Corners MST connects oints according to their Euclidean distances. works if Euclidean distances corresond to geodesic distances (at least locally) q q r r MST works for corners with interior angle π 2. cannot handle corners with interior angle π 2 but a simle otential function makes it work even for shar corners. MPI Informatik 5 Kurt Mehlhorn

16 A Potential Function for Shar Corners "# "#!! 0 y$mx y$ %mx otential of a node is equal to its x-coordinate as moves away from 0: Euclidean distance d 0 grows and otential of goes down otential goes down slower than distance grows when comes around the corner: Euclidean distance shrinks and otential of goes u otential grows faster than distance shrinks MPI Informatik 6 Kurt Mehlhorn

17 Exerimental Results (cometitive) running time TSP-algorithm is more robust 2 TSP DMR ABE Gould DK Delaunay 0 TSP-algorithm 8 Runtime/s 6 4 Dey-Kumar (because it makes global decisions) Periods 3000 oints MPI Informatik 7 Kurt Mehlhorn

18 Summary non-uniform samling suffices for TSP-based curve reconstruction. TSP-based reconstruction in olynomial time. theorem insired by exeriments. exeriments were easy to erform thanks to LEDA, CGAL, ABACUS, CPLEX, and SCIL. MPI Informatik 8 Kurt Mehlhorn