1. Surface Reconstruction Some figures by Turk, Curless, Amenta, et al.

2. Two Related Problems Given a point cloud, construct a surfaceGiven a point cloud, construct a surface Given several aligned scans (range images), construct a surfaceGiven several aligned scans (range images), construct a surface

3. Surface Reconstruction from Point Clouds Most techniques figure out how to connect up “nearby” pointsMost techniques figure out how to connect up “nearby” points Need sufficiently dense sampling, little noiseNeed sufficiently dense sampling, little noise Delaunay triangulation: connect nearest pointsDelaunay triangulation: connect nearest points – Officially, a triangle is in the Delaunay triangulation iff its circumcircle does not contain any points

4. The “Crust” Algorithm Amenta et al., 1998Amenta et al., 1998 Medial axis: set of points equidistant from 2 original pointsMedial axis: set of points equidistant from 2 original points In 2D:In 2D:

5. Medial Axes in 3D May contain surfaces as well as edges and verticesMay contain surfaces as well as edges and vertices

6. Voronoi Diagrams Partitioning of plane according to closest point (in a discrete point set)Partitioning of plane according to closest point (in a discrete point set) A subset of Voronoi vertices is an approximation to medial axisA subset of Voronoi vertices is an approximation to medial axis

7. The “Crust” Algorithm Compute Voronoi diagramCompute Voronoi diagram Compute Delaunay triangulation of original points + Voronoi vertices

8. Voronoi Cells in 3D Some Voronoi vertices lie neither near the surface nor near the medial axisSome Voronoi vertices lie neither near the surface nor near the medial axis Keep the “poles”Keep the “poles”

9. Crust Results 36K vertices36K vertices 23 minutes (1998)23 minutes (1998)

10. Crust Problems Problems with sharp cornersProblems with sharp corners – Medial axis touches surface – Theoretically need infinitely high sampling – In practice, heuristics to choose poles Topological problemsTopological problems

11. The Ball Pivoting Algorithm Bernardini et al., 1999Bernardini et al., 1999 Roll ball around surface, connect what it hitsRoll ball around surface, connect what it hits

12. Alpha Shapes

13. Problems With Reconstruction from Point Clouds

14. Surface Reconstruction from Range Images Often an easier problem than reconstruction from arbitrary point cloudsOften an easier problem than reconstruction from arbitrary point clouds – Implicit information about adjacency, connectivity – Roughly uniform spacing

15. Surface Reconstruction From Range Images First, construct surface from each range imageFirst, construct surface from each range image Then, merge resulting surfacesThen, merge resulting surfaces – Obtain average surface in overlapping regions – Control point density

16. Range Image Tesselation Given a range image, connect up the neighborsGiven a range image, connect up the neighbors

17. Range Image Tesselation Caveat #1: can’t be too aggressiveCaveat #1: can’t be too aggressive – Introduce distance threshold for tesselation

18. Caveat #2: Which way to triangulate?Caveat #2: Which way to triangulate? Possible heuristics:Possible heuristics: – Shorter diagonal – Dihedral angle closer to 180  – Maximize smallest angle in both triangles – Always the same way (best triangle strips) Range Image Tesselation

19. Scan Merging Using Zippering Turk & Levoy, 1994Turk & Levoy, 1994 Erode geometry in overlapping areasErode geometry in overlapping areas Stitch scans together along seamStitch scans together along seam Re-introduce all dataRe-introduce all data – Weighted average

20. Zippering

21. Point Weighting Higher weights to points facing the cameraHigher weights to points facing the camera – Favor higher sampling rates

22. Point Weighting Lower weights (tapering to 0) near boundariesLower weights (tapering to 0) near boundaries – Smooth blends between views

23. Point Weighting

24. Consensus Geometry

25. Zippering Example