Lecture 3

Fundamental Algorithms (1 of 2)

(includes text and images from the 3rd edition of the VTK book)

Categorize algorithms according to:

• structure - effect transform has on topology
• type - type of dataset the transform operates on

structure

• Geometric transformations - alter the geometry but not the topology
  • eg rotate or scale polygonal dataset
• Topological transformation - alters topology but not geometry or attribute data
  • eg converting polygonal data to unstructured grid
  • very uncommon
• Attribute transformations - convert data attributes to another form or create new attributes from input data
  • structure of dataset remains unaffected
  • eg compute vector magnitude
• Combined transformations - change dataset structure and attribute data
  • eg computing contour lines or surfaces

type

• scalar - operate on scalar data - eg generating contour lines from temperature on map
• vector - operate on vector data - eg showing oriented arrows of airflow on the same map
• tensor - operate on tensor matrices - eg showing components of stress using oriented icons
• modelling - generate topology or geometry, normals, texture data - vtk includes lots of misc algorithms in this category

Scalar Algorithms

single data value associated with each point in dataset

Flow density coloured with different lookup tables

• Color Mapping - map scalar data to colours using a discrete colour lookup table
  • lookup table holds an array of colours
  • lookup table has min and max scalar range
  • scalar values serve as indices into lookup table
    • if scalar > max then scalar = max
    • if scalar < min then scalar = min
    • map scalar value into the lookup table
  • transfer function - continuous mapping from scalar data to colour
  • mapping may be linear or logarithmic, etc
  • careful choice of colours is VERY important - Interaction of Color by Josef Albers
    

  
  Contouring examples. UL: Marching squares used to generate contour lines, UR marching cubes surface of human bone, BL marching cubes surface of flow density, BR Marching cubes surface of iron-proteint
  

• Contouring - constructing boundaries between regions
  • 2D - contour lines of constant scalar value
    • eg weather maps annotated with lines of constant temperature (isotherms)
    • eg topological maps with lines of constant elevation
  • 3D - isosurfaces - surfaces of constant scalar value
    • eg skin or bone in a medical image showing a constant intensity
  • generating contours probably requires interpolation between the given points
  • generate a new set of points then connect them into a contour

  • Marching Squares (2D) / Marching Cubes (3D)
    • assumes a contour can pass through a cell in only a finite number of ways
    • can create a table enumerating all these possible topological states of cell
      • # of states depends on # of cell vertices and # of possible in/out relationships
      • vertex is 'inside' if scalar value > value of contour line
      • vertex is 'outside' if scalar value < value of contour line
      • e.g. if cell has 4 vertices and each can be in or out then 16 possibilities
        • can represent these by a 4 bit code
      • currently not interested in 'where' the contour passes through cell, just 'how'
    • treats each cell independently
    1. select cell
    2. calculate inside/outside for each vertex of the cell
    3. create 4-bit index from in/out values
    4. use index to look up topological state
    5. interpolate contour location for each edge

    
    dark vertex is above the contour value, white vertex is below

    

    • may create duplicate vertices at cell boundaries to be merged later on
    • for marching cubes 256 possibilities for a cell with 8 vertices can reduce this to 15 using symmetry
    • contouring ambiguity in both 2D and 3D cases
      • may generate holes, or fill holes that should exist
      • more serious in 3D case
      • can add 6 more complementary cases to the 15

Some contours from last time - the grand canyon - 2D contours and Iron Protein - 3D contours

Vector Algorithms

• 3D representation of direction and magnitude
• Hedgehog - drawing an oriented scaled line for each vector
• Oriented Glyphs - 2D or 3D geometric representations can be used instead of lines
• Warping - deforming geometry according to the vector field
• Displacement Plots - shows motion of an object in the direction perpendicular to its surface
  • motion is caused by applied vector field
  • vectors converted to scalars using surface normal (dot product) vector
• Time Animation (here there be math dragons)
• Streamlines - connect point positions over multiple timesteps
  • numerical approximation to a particle trace represented as a line
  • tracing trajectories in a vector field

Three different representations of venctor data: hedgehog, glyphs, streamlines

warping

displacement plot

167000 3D vectors

Coming Next Time

More Fundamental Algorithms