Volumetric and Blobby Objects

1
Volumetric and Blobby Objects

• Lecture 8 (Modelling)

2
Isometric Data

• Medical Scans (CT, MRI) produce image slices with
  a regular grid of sample values (for skin, bone,
  muscle, etc).
• The isosurface at a particular isovalue is very
  difficult to visualize from separate 2D scans.
• Solution convert to a 3D surface.
• Isosurface Extraction Methods
• Voxels an intermediate volumetric
  representation.
• Ray Casting direct rendering for ray tracing.
• Marching Cubes convert to a polygon mesh.

3
Voxels

• Spatial Occupancy Enumeration
• solid is decomposed into identical cells arranged
  in a rectangular grid.
• Elements are generally cubes, called voxels
  (volume elements).
• Scan to Voxel Conversion
• If image slices are stacked into pairs then
  voxels can be created with image sample values at
  the corners.
• A voxel is given the value 1 (solid) if all
  sample values are greater than the iso-value and
  0 (transparent) otherwise.
• Problems
• Aliasing and high storage costs.
• Still requires conversion to polygon-mesh for
  rendering.
• Solutions 3D anti-aliasing and octree data
  structure.

4
Voxels Octree Data Structure

• Octree
• A compact hierarchical encoding of a voxmap
  (voxel map).
• The 3D generalization of a quadtree.
• Leaves represent full or empty voxels of
  different volumes. The voxel dimensions depend on
  its depth in the tree.
• Can merge same value voxels into a single larger
  voxel.

5
Marching Cubes

• Purpose convert 2D isometric scans into a
  polygon mesh isosurface for a particular isovalue
  .
• Algorithm
• Pair up scans. Create cubes (cells) with sample
  values at the vertices (4 from the upper scan, 4
  from the lower scan).
• March over all cubes
• Topology Place triangles in a cube depending on
  the isovalues at the cell vertices.
• Geometry position the vertices of these
  triangles according to the isovalues.
• For efficiency exploit inter-cell coherence. Cell
  vertices (hence some topology and geometry) are
  shared between cells.
• Seminal Paper Lorensen, W. and Cline, H.
  Marching Cubes A High Resolution 3D Surface
  Construction Algorithm, SIGGRAPH 87, pp.
  163-169.

6
Marching Cubes Triangulating a Cell

• Binary coding A vertex with value
  is inside (1) or outside (0) as
  follows
• The binary codes for all vertices are
  placed in a bit string used as an index
  into the
  triangulation table.
• There are combinations.
• But
• Complementary cases inverting the
  cell bits does not change
  triangulation
  (reduce to 148 combinations).
• Rotational symmetry many patterns
  can be reached by rotating the cube
  (reduce to 14 unique combinations).

7
Marching Cubes Positioning Vertices

• Triangulated cells have triangle vertices placed
  along cell edges.
• The exact position and normal of these vertices
  must depend on the cell values.
• Solution Triangle vertices are positioned at
  along an edge using linear interpolation
• where is the value at the cell vertex inside
  the isosurface, and is the external vertex
  value.
• Normals can be similarly interpolated.
• More sophisticated interpolation is possible.

8
Blobby Objects

• Blobby (or Soft) objects are employed in the
  creation and animation of smooth shapes.
• Use implicit rather then
  parametric surfaces.
• A skeleton is defined by a set of key points,
  which radiate energy to the surrounding space.
• The object boundary is an isosurface at a
  particular energy level.
• Isosurface extraction (e.g. marching cubes) is
  used to find this boundary.
• The position and orientation of keys can be
  animated.
• Animation control is easier than for parametric
  surfaces.
• Other blobby parameters are amenable to
  animation.

9
Blobby Objects Parameters

• Blobby objects consist of a number of keys, each
  with
• A position of the key in 3D.
• An optional set of axes to orient the blob.
• An implicit field function , which
  determines the shape of the blob. depends
  on and . Superellipsoids are popular.
• Force at the key . Often set to
  (merges with other blobs) or (creates
  imprints on other blobs).
• A Decay function , which defines how
  intensity tails off from the key towards the blob
  boundary.

10
Blobby Objects Field Calculation

• Given a point , the field contribution from a
  key is calculated as follows
• Find .
• Calculate the distance where the field
  value turns to zero along the line by
  solving the field function .
• If then the keys contribution is
  zero. Otherwise, the field value at is
  
  (force times decay).
• A popular decay function is
  .
• Repeat and sum the field
  value for all keys.

11
Blobby Objects Examples