Discrete Space, Voxelization and Distance Fields

Jian Huang, CS 594, Spring 2002

Papers

• Huang et al, ‘Accurate Voxelization of Polygonal Meshes’, IEEE Symposium on Volume Visualization, 1998

• Huang et al, ‘CDFR’, IEEE Conference on Visualization, 2001

Discrete Space• A 3D discrete space Z3 is 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.

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

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.

N-Neighborhood• The set of 2D pixels that are N-adjacent to the dark

pixel where N 4, 8

• The set of 3D voxels that are N-adjacent to the voxel at the center where N 6, 18, 26

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 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

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

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 between a voxel in B and a voxel in C meets A

• Separability is a topological property

4-separating and 8-separating curves

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).

Voxelization

• To convert continuous surface representations (e.g. polygon mesh, parametric surfaces) into voxel representations

• Need to preserve separability and minimality

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:

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

Voxelizing a Polygon Mesh

• Edges and vertices needs special handling for separability and minimality

• Let t denote the desired connectivity distance, either t6 or t26.

• Rc = L/2 for 6-separability, for 26-separability:

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.

• 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

Getting a Distance Field (1)

Chamfer distance

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

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

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

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

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

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

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

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

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

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

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

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

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

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

• 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]

Extracting a Distance Contour

On Convex Test Models

• 32x32x32 CDFR, 5123 conventional res

On Concave Test Models

• 32x32x32 CDFR, 5123 conventional res

On Practical Parts

• 128x128x128 CDFR, 10243 conventional res

On Practical Parts

• 128x128x128 CDFR,10243 conventional res

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

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

Voxel Cnt (K)

Avg tri/surf voxel

Avg tri/int voxel

CDFR Size

(KB)

Time

(sec)

128 con 319 1.95 2.91 970 8

256 con 2,322 1.43 2.53 7,548 82

128 brevi 373 3.01 4.66 3,459 52

256 brevi 2,659 1.96 3.69 25,260 448

Testing Platform

• SGI Octane with 300MHz R12000 processor, 512 MB memory

Contour Extraction Time

• Depends on which thickness you pick• For instance: brevi at 4 inches

• Higher CDFR resolution, trade storage for shorter extraction time.

Extraction time (sec)

512 f-res 768 f-res 1024 f-res

128 CDFR 23.66 74.65 174.79

256 CDFR 9.25 28.29 64.46

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.