Discrete Space, Voxelization and Distance Fields

1. Discrete Space, Voxelization and Distance Fields Jian Huang, CS 594, Spring 2002

2. Papers • Huang et al, ‘Accurate Voxelization of Polygonal Meshes’, IEEE Symposium on Volume Visualization, 1998 • Huang et al, ‘CDFR’, IEEE Conference on Visualization, 2001

3. Discrete Space • A 3D discrete space Z3is a set of integer grid points in a 3D • Euclidean space denoted by S. A 3D grid point is a zero dimensional object defined by its Cartesian coordinate (x,y,z). • The Voronoi neighborhood of grid point p is the set of all points in the Euclidean space that are closer to p than to any other grid point. • The Voronoi neighborhood of a 3D grid point is a unit cube around it, known also as a voxel.

4. Discrete Space • The aggregate of all voxels is a tessellation of 3D Euclidean space. • A voxel’s value is mapped into the set {0,1}: • voxels assigned the value “1'' are called “black'' or “non-empty'' voxels • those assigned the value “0'' are called “white'' or “empty'' voxels

5. N-Adjacency • In 3D discrete space • Two voxels are 26-adjacent if they share a vertex or an edge or a face • 26 such adjacent voxels for any voxel • Two voxels are 18-adjacent if they share an edge or a face • 18 such adjacent voxels for any voxel • Two voxels are 6-adjacent if they share a face • 6 such adjacent voxels for any voxel • In 2D discrete space, similarly, 4-adjacency and 8-adjacency.

6. N-Neighborhood • The set of 2D pixels that are N-adjacent to the dark pixel where N Î{4, 8} • The set of 3D voxels that are N-adjacent to the voxel at the center where N Î{6, 18, 26}

7. N-Path • An N-path is a sequence of black voxels such that consecutive pairs are N-adjacent • Two black voxels are said to be N-connected in S if there exists a connecting N-path consisting only of black voxels • A (closed) N-curve is an N-path P that either contains a single voxel or each voxel in P has exactly two N-adjacent voxels also in P • An open N-curve is an N-curve with two exceptions called endpoints, each of which has only one N-adjacent voxel in P

8. Separability • In continuous space, it is impossible to pass from the region enclosed by a curve to the region outside the curve without crossing the curve itself. • In discrete space, however, the opposite is possible. • To avoid this discrepancy, define opposite types of connectivity for white and black sets. • Opposite types in 2D space are 4 and 8 • In 3D space, 6 is “opposite” to 26 and 18

9. Separability • Let A, B and C be three disjoint sets of voxels. A is said to N-separate B and C if any N-path P between a voxel in B and a voxel in C meets A • Separability is a topological property 4-separating and 8-separating curves

10. Minimality • A voxel belonging to an N-separating surface is called an N-simple voxel if deleting it will not affect the surface separability. • A surface is N-minimal if it does not contain any N-simple voxels Examples of a 4-minimal curve (left), 8-simple point (center), and a 4-simple point (right).

11. Voxelization • To convert continuous surface representations (e.g. polygon mesh, parametric surfaces) into voxel representations • Need to preserve separability and minimality

12. Pixelizing a Line • For 4-separable or 8-separable, assuming the normal vector is normalized, need to include all pixels with distance to the line between:

13. Voxelizing a Plane • For 6-separable or 26-separable, assuming the normal vector is normalized, all voxels with distance to the line between: abs(Ax + By + Cz + D) <= t

14. Voxelizing a Polygon Mesh • Edges and vertices needs special handling for separability and minimality • Let t denote the desired connectivity distance, either t6or t26. • Rc = L/2 for 6-separability, for 26-separability:

15. Distance Field • Discrete distance field • Each element in a distance field specifies its minimum distance to a surface geometry • Positive and negative distances are used to distinguish outside and inside of the shape • negative values on the outside • positive values on the inside.

16. Getting a Distance Field (1) • First, voxelize the 3D mesh to a binary surface volume [Kaufman, Cohen, Huang…] • Second, run a distance transform on the surface volume to obtain a solid distance volume • Euclidean Distance • Chamfer Distance • Face, edge, vertex sharing • Manhattan Distance • Face sharing Chamfer distance

17. Getting a Distance Field (2) • Brute force: For every voxel in the volume, compute the minimal distance to the geometric surface • Euclidean distance • Doable with triangle meshes, but hard problem in general • Time consuming

18. Hierarchical Distance Field • Distance fields can be stored hierarchically in Quadtree or Octree structures • Aka: adaptively sampled distance field (ADF) • Use a smaller voxel size in areas of higher details

19. Disadvantages of Conventional Distance Fields • Need to choose an initial volume resolution (the high limit of error tolerance) • When the user picks a tighter tolerance, have to do everything from scratch again • The conventional distance volume is aliased • Real data sets are not smooth, thus not band-limited

20. Volume Anti-aliasing • Non binary pre-filtered volume [Sramek & Kaufmann] • Need higher order smoothing filters for reconstruction • No idea how much detail is gone in geometric sense • Not exactly sure about how geometric details are defined • Corners • Holes

21. Impasse • Sampling rate is limited • Distance fields of complex geometric models are not band-limited • Impasse: would desire to keep all the geometric details in a volumetric distance field • Geometric details at <0.1% of an object’s dimension

22. Observation • In spatial domain, if all that we want to capture are the distances to a set of finite polygons • Place an anchor point somewhere, and record the distances from the anchor to each of the finite polygons

23. Need New Distance Field Representation • Generalize volume representation from a discretization of a continuous domain entity to a spatial data structure • Try to build a spatial data structure • Every voxel to have all the information necessary to capture the exact local distance field within the span of that voxel • To answer a query of ‘what’s the thickness of an interior point, pnt,’ we only have to deal with the corresponding local voxel

24. CDFR • The spatial data structure is named: CDFR • A Complete Distance Field Representation • In the CDFR, deal with signed Euclidean distances from 3D points to finite triangles only • Each spatial point has a base triangle, which is used to determine the sign of the distance value

25. Base Triangle • Need to decide which triangle is the base triangle of a point, pnt • If pnt is closest to a triangle which pnt orthogonally projects into, then this triangle is the base triangle • Otherwise, if pnt is closest to 2 triangles sharing an edge, then compute pntproj on this edge, connect pnt and pntproj to form a vector V • Otherwise, pnt is closest to several triangles sharing a common vertex, connect pnt and this vertex to form the vector V

26. CDFR • In a CDFR, the center of each voxel serves as an anchor point that captures information for the local distance field in its span • First idea: each voxel stores the id of the base triangle of its center and the corresponding signed distance • Not enough • Don’t have the distance information for other locations in the span of each voxel

27. How much information do we need on each voxel? • Theorem (please refer to paper) • No triangle can be the base triangle to any location in the span of a voxel, v, if its distance to the center of v is larger than thickness(v) + sqrt(3) x voxel_size

28. Constructing a CDFR • First, pick an initial volume resolution • Only affect the performance, not accuracy • Second, voxelize the geometry into a surface volume • On each surface voxel, store those triangles that intersect that voxel • We store on each voxel a list of tuples: {triangle_id, signed distance} //CDD tuples • Third, an iterative contour-by-contour distance transform to obtain the final solid CDFR

29. Distance Transform • Loop: • Each voxel not on the surface volume inherits the CDD list from its 26-neighbors • For all the new triangles that it sees, compute its distance to each new triangle • Update the curr_min_dist • Discard all CDD tuples that have a distance larger than curr_min_dist + sqrt(3)*voxelsize • This loop ends until no new updates take place in the CDFR

30. Answering a Query • What is the minimal distance from an arbitrary 3D point to the surface geometry • Find out which voxel the point resides in, grab all triangles on that voxel • Compute the distance values from that point to all those triangles • The distance value with the minimal absolute value is what we want

31. Extracting a Distance Contour • A ‘Dividing Cube’ algorithm • Extract a point-based distance contour of thickness, t, with an error tolerance, E. • Traverse the CDFR, grab all voxels with min_distance in the range: • Subdivide these voxels to size: • Compute the thickness values of all the sub-voxels and extract all sub-voxels whose thickness are within [t - E/2, t + E/2]

32. On Convex Test Models • 32x32x32 CDFR, 5123 conventional res

33. On Concave Test Models • 32x32x32 CDFR, 5123 conventional res

34. On Practical Parts • 128x128x128 CDFR, 10243 conventional res

35. On Practical Parts • 128x128x128 CDFR,10243 conventional res

36. Storage Size • Only store CDFR for surface and interior voxels. For exterior voxels, just store a tag denoting ‘empty’. • CDD list for each voxel takes: (5 + 4*triangle_cnt) bytes

37. CDFR Storage Size and Construction Time • Connector, 242 triangles, (6.9, 2.0, 2.9) inches • Brevi, 1812 triangles, (38.1, 34.9, 96.0) inches

38. Testing Platform • SGI Octane with 300MHz R12000 processor, 512 MB memory

39. Contour Extraction Time • Depends on which thickness you pick • For instance: brevi at 4 inches • Higher CDFR resolution, trade storage for shorter extraction time.

40. The Engine Cylinder Head • Can build a 250-res 37MBytes CDFR in 30 min • Can extract 0.137 mm accuracy at 8.5 mm thickness within 11 min (470K points, 1988x3500x1218) 2 frames/sec rendering of the point-based model and sorted triangle mesh in semi-transparent mode.