Data Acquisition. Mesh Reconstruction. Sensors. Scan Conversion. Scanner types Laser Imaging (2D/3D) Probing Mixed

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1 Mesh Reconstruction Data Acquisition Show Scan Conversion Main phases: Sensing - capture raw data Point set Boundary contours Voxels Conversion to polygonal model Followed by Mesh improvement Smoothing Simplification Freeform surface reconstruction Scanner types Laser Imaging (2D/3D) Probing Mixed Sensors

2 Sensing Technologies - Probing Probing position probe on object record the location Sensing Technologies - Imaging Capture multiple 2D images Carve space from silhouettes Obtain visual hull Output: point cloud data Problematic Labour intensive Error prone 3D Imaging Energy-based sensors Ultrasound Magnetic Resonance Imaging (MRI) X-Ray Computed Tomography (CT) Alternative - slice object, take photographs of slices Outputs volumetric data (voxels) contour lines (use imaging techniques) Range Scanners Laser/Optical range scanner provides 2D array of depth data Some capture colour (texture) Multiple views for complete object scan: Rotate object Rotate sensor Output point set

3 Triangulating Point Clouds General Idea Connect neighboring points into triangles Curve from Points - Connect the Dots Given unordered set of points P connect them by linear segments Issues Connectivity manifold? connected? Efficiency Curve from Points - Connect the Dots Can be ambiguous Harder when topology not known Curve from Points - Connect the Dots Use Voronoi diagram Construct Delaunay triangulation Which edges to choose?

4 Medial Axis Medial axis of (d-1)-dimensional surface in R d - set of points with more than one closest point on the surface Alternative definition: locus of centers of maximal inscribed spheres Voronoi diagram of set of points on curve approximates Medial Axis if points sampled densely enough Medial Axis & VD 2D 3D Sampling Criterion Good sample - sampling density (at least) inversely proportional to distance from medial axis r-sample : distance from any point on surface to nearest sample point r x distance from point to medial axis In general, r (0,1] r=0.5 good enough Sampling Criterion (cont.) Inherently takes into account curvature of the surface proximity of other parts of the surface r = 0.5

5 Crust Algorithm (2D) Crust Algorithm (2D) Idea adopt Delaunay edges which are far from MA Idea adopt Delaunay edges which are far from MA Approximate MA by Voronoi vertices Crust Algorithm (2D) Crust Algorithm (2D) Idea adopt Delaunay edges which are far from MA Approximate MA by Voronoi vertices Delaunay edge e in crust circumcircle of e is empty of other sample points or Voronoi vertices of S Idea adopt Delaunay edges which are far from MA Approximate MA by Voronoi vertices Delaunay edge e in crust circumcircle of e is empty of other sample points or Voronoi vertices of S

6 Crust Algorithm (2D) Crust Algorithm Idea adopt Delaunay edges which are far from MA Approximate MA by Voronoi vertices Delaunay edge e in crust circumcircle of e is empty of other sample points or Voronoi vertices of S 1. Compute Voronoi diagram of S -let V be set of Voronoi vertices S - Samples Crust Algorithm Crust Algorithm 1. Compute Voronoi diagram of S -let V be set of Voronoi vertices 1. Compute Voronoi diagram of S -let V be set of Voronoi vertices 2. Compute Delaunay triangulation of S U V S - Samples V - Voronoi vertices S - Samples V - Voronoi vertices

7 Crust Algorithm 1. Compute Voronoi diagram of S -let V be set of Voronoi vertices 2. Compute Delaunay triangulation of S U V Theorem: The crust of an r-sample from a smooth curve F, for r 0.25, connects only adjacent samples of F 3. Return all Delaunay edges between points of S Smooth curve F demo S - Samples V - Voronoi vertices Theorem: The crust of an r-sample from a smooth curve F, for r 0.25 connects only adjacent samples of F Theorem: The crust of an r-sample from a smooth curve F, for r 0.25 connects only adjacent samples of F Smooth curve F r-sample (r 0.25) Smooth curve F r-sample (r 0.25) The Crust

8 Theorem: The crust of an r-sample from a smooth curve F, for r 0.25 connects only adjacent samples of F Smooth curve F r-sample (r > 0.25) The Crust

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10 3D Crust Algorithm Extend 2D approach Voronoi cells are poyhedra Voronoi vertex is equidistant from 4 sample points BUT in 3D not all Voronoi vertices are near medial axis (regardless of sampling density) 3D Crust Algorithm But some vertices of the Voronoi cell are near medial axis Intuitively cell is closed not just from the sides but also from both sides of the surface. 3D Crust Algorithm Solution: use only two farthest vertices of V s - one on each side of the surface Call vertices poles of s: p + (s), p - (s) 3D Algorithm (basic) Compute Voronoi diagram of S For each s S, identify the poles p + (s) and p - (s) p + (s) is the vertex of V s most distant from s. p - p - (s) is the vertex of V s most distant from s in the opposite direction. Let P be the set of all poles p + Compute Delaunay triangulation T of S U P Add to crust all triangles in T with vertices only in S

11 Reconstruction Example Crust of set of points and poles used in its reconstruction Time Complexity Time complexity is O(n2) where n = S Upper bound on 3D Delaunay triangulation Almost never arises in practice Other steps are linear (but expensive) demo Reconstruction Examples