Discrete Space, Voxelization and Distance Fields

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Discrete Space, Voxelization and Distance Fields

• Jian Huang, CS 594, Spring 2002

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Papers

• Huang et al, Accurate Voxelization of Polygonal
  Meshes, IEEE Symposium on Volume Visualization,
  1998
• Huang et al, CDFR, IEEE Conference on
  Visualization, 2001

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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.

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Discrete Space

• The aggregate of all voxels is a tessellation of
  3D Euclidean space.
• A voxels 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

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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.

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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

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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

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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

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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
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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).
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Voxelization

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

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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

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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) lt t

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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

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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.

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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
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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

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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

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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

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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

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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 lt0.1 of an objects
  dimension

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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

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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 whats the thickness of an
  interior point, pnt, we only have to deal with
  the corresponding local voxel

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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

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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

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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
• Dont have the distance information for other
  locations in the span of each voxel

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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

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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

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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

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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

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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

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On Convex Test Models

• 32x32x32 CDFR, 5123 conventional res

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On Concave Test Models

• 32x32x32 CDFR, 5123 conventional res

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On Practical Parts

• 128x128x128 CDFR, 10243 conventional res

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On Practical Parts

• 128x128x128 CDFR,10243 conventional res

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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 4triangle_cnt) bytes

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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
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Testing Platform

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

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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
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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.