Related Work

1
Related Work

• Bergman, Rogowitz, and Treinish(1995) - enhancing
  colormapped visualiztion.
• Rosenfeld and Kak(1982) - Histogram equalization
• Gershon(1992) - Generalized Animation
• Jones and Chen(1994), Lorensen and Cline(1987),
  Wilhelms and Gelder(1990) - isocontours in 2d and
  3d scalar data.
• Fowler and Little(1979) - Detecting
• ridges and valleys.
• McCormack, Gahegan, Roberts, Hogg, Hoyle(1993) -
  detecting drainage patterns in geographic
  terrain.
• Interrante, Fuchs, and Pizer(1995) - Surfaces.
• Itoh and Koyamada(1994) - Isocontour extraction.
• Helman and Hesselink(1991) - vector field
  topology.
• Bader(1990)- gradient fields in molecular systems.

2
Scalar Topology

• For scalar field S
• The local maxima of S
• The local minima of S
• The saddle points of S
• Selected critical curves joining each of the
  above
• Critical curves dxfy dyfx

3
Construction of Scalar Topology

• Detect stationary (critical) points in S.
• Classify stationary points.
• Integrate selected critical curves in gradient
  field.

4
Classification of Critical Points
5
Tracing Critical Curves

• Four critical curves are computed for each saddle
  point, two in the direction corresponding to the
  positive eigenvalue, and two in the direction
  corresponding to the negative eigenvalue.

6
Some Applications

• Data Correlation - Due in part to the invariance
  under translation and scaling, scalar topology is
  useful in visually determing linear correlation
  between multiple scalar variables.
• Image Co-registration - Scalar topology in
  adjacent planes provides a bacbone which may be
  used to aling the planes.
• Warping/Morphing - Editing of the scalar backbone
  may be used to apply a warping effect to an
  image, or to warp between the backbones of two
  similar images.
• Mesh Reduction - The scalar topology may server
  as a guide to aid in computation of reduced
  resolution meshes.
• Surface Triangulation - Adaptive triangulation of
  arbitrary mathematical surfaces by decomposition
  into montonic patches which may be subdivided to
  an arbitrary precision.
• Surface Reconstruction - Adaptive Triangulation
  of Space.