Graph Based Representations in different application Domains Luc Brun and Myriam Mokhtari L.E.R.I. Reims University (France)

Content of the talk n 2D Segmentation n Shape Representation n 3D Segmentation n Links

2D Segmentation n Encoding the 2D partition

2D Segmentation n Encoding the 2D partition Simple graph [Piroddi 01]

2D Segmentation n Encoding the 2D partition Simple graph

2D Segmentation n Encoding the 2D partition Simple graph

2D Segmentation n Encoding the 2D partition Simple graph

2D Segmentation n Encoding the 2D partition Simple graph Dual graph

2D Segmentation n Encoding the 2D partition Simple graph Dual graph

2D Segmentation n Simple Graph n Efficient encoding n Dual Graph n Encodes topological features: multiple boundaries, surrounding relationships. May be encoded by usual graph libraries. n 2D Combinatorial maps n Encodes topological features. Allows an implicit encoding of the dual graph. May be extended to higher dimensions

Shape Representation n Encoding the shape of an object: Pattern Recognition, Matching –Skeleton –Medial Axis

Shape Representation n Medial graph Illustration from [Salotti 01]

Shape Representation n Crest graph/River Graph –Crest Graph n Vertex: Hill altitude n edge: lowest altitude of the paths between two hills –River Graph n Dual vertex: Basin altitude n dual edge: higher altitude of the paths between two basins.

Shape Representation n Crest Graph

Shape Representation n Crest Graph Original ImageSobel operatorCrest Graph Illustration from [Glantz 01]

Shape Representation n Shock Graph Illustration from [Siddiqi98]

n Medial graph n Exact reconstruction n Crest graph n Not restricted to distance map, support a parallel implementation, does not encode narrowing dead ends. n Shock Graph n Exact reconstruction, Highly structurated graph. Shape Representation

3D Segmentation n Encoding a 3D scene CUBE CUBE, HOLE 3HBAR, HOLE BOLT HEAD [Bauckhage 01]

3D Segmentation     -4 n Encoding 3D partition : 3D combi. maps n G=(D, ,  )  G=(D, , ,  ) Example from [Braquelaire98]

3D Segmentation n Two approaches: – Splitting scheme: Braquelaire, Desbarats, Domenger, Wüthrich n Tracking of surface boundaries –Bottom-up scheme: Bertrand, Damiand, Fiorio n Successive simplifications of the 3D map encoding the partition

Links n 3D dual graph - 3D combinatorial maps n Medial graph, Crest Graph, Shock Graph: which choice for which application ? n 3D Irregular Pyramids based on 3D combinatorial maps n 3D Split & Merge algorithms based on a combination of [Braquelaire] and [Bertrand]