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Special Seminar in IMI

Hold Date
2014-08-22 14:30〜2014-08-22 17:00
Seminar Room 6, Faculty of Mathematics building / IMI, Ito Campus
Object person
Hamish Carr (University of Leeds)

Special Seminar in IMI

Speaker: Hamish Carr (University of Leeds)
Event Date: August 22 (Fri), 14:30--15:30, 16:00--17:00
Room: Seminar room 6

(1) 14:30~15:30
Title: Multivariate Topology Simplification and Measure Persistence
(Joint work with Amit Chattopadhyay, David Duke & Zhao Geng)

Topological simplification of scalar and vector fields is well-established as an effective method for analysing and visualising complex data sets. For multi-field data, topological analysis requires simultaneous advances both mathematically and computationally. Mathematically, we show that the projection of the Jacobi Set of multi-variate data into the Reeb Space produces a Jacobi Structure that separates the Reeb Space into regions. We also show that the dual graph of these regions gives rise to a Reeb Skeleton that has properties similar to the scalar contour tree and Reeb Graph. We then show how to generalise scalar persistence using Measure Persistence in the range to give a scaling-invariant total ordering of features that can be used for simplification. Computationally, we show how to compute Jacobi Structure, Reeb Skeleton, and Measure Persistence in the Joint Contour Net, an approximation of the Reeb Space, and that these can be used for visualisation in a fashion similar to the contour tree and Reeb Graph.

(2) 16:00~17:00
Title: Tetrahedral Deflation: An Algorithm for R3->R2 Reeb Space Computation over Simplicial Complexes

The earliest efficient algorithm for computing the Reeb Graph for a scalar field on a general 2-manifold was based on quantising the range of the field, with subsequent algorithms generating correct Reeb Graphs either by reducing to the simpler case of contour tree computation, or more recently a streaming computation of the Reeb Graph by incremental addition of simplices from a complex.

In the case of the Reeb Space, initial work also quantised the range of the field to produce the Joint Contour Net. The question then arises whether more efficient computations are available through generalisations of other Reeb Graph algorithms, and in particular the streaming computation. This talk will show that this algorithm can be generalised through the use of the existing Projected Tetrahedra algorithm, collapsing each simplex to a range-space projection and incrementally building the correct Reeb Space. Some thoughts will also be presented on how to extend this to R4 -> R3, how to modify the computation for non-simplicial cell complexes, and how this can be exploited to analyse time-dependent data.

Organizer: Osamu Saeki (IMI)