Shock Scaffold

Definitions

In 3D, there are generically FIVE principal types of shocks [Giblin & Kimia: ICCV'01]:

A_1^2 : SHEETS

A_1^3 : Axial CURVES

A_3 : "Surface" CURVES
(e.g., ridges)

A_1^4 : "Internal" NODES
(e.g., Voronoi nodes)

A_1 A_3 : "Surface" NODES
(e.g., end of ridges)

Flow on surfaces : Radial expansion


2 skewed edges give rise to a saddle MA sheet.	Flow is initiated at a single point.

Flow on curves : Bi-directional expansion

Regular (1), Initial (2), Final (4) flow points. Degenerate (3) not shown.

Flow on nodes : Inward / Outward flows

Labeling based on inward (curve) flows.

Final classification of 18 possible shock points
based on contact with spheres, A^n_k,
and flow types.

Hypergraph representation of shape :
- Nodes : Isolated points (A_1 A_3 and A_1^4)
- Links : Curves (A_3 and A_1^3)
- Hyperlinks : Sheets (A_1^2)

The Shock Scaffold denotes the three-tier hierarchical representation :
- Retain all details.
- Ignore the geometry of sheets, but retain their existence and connectivity.
- Ignore the geometry of sheets and curves, but retain their existence and connectivity.

Additional structure in the Scaffold: Initial points of flow on sheets and curves

Conjecture 1 : The Scaffold represents the Voronoi diagram

Conjecture 2 : The Scaffold represents the CW-complex (or Molecular graph)

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