Euler Calculus and Topological Data Management
author:
Robert Ghrist,
Department of Mathematics, University of Illinois at Urbana-Champaign
Description
This talk covers the basic of an integral calculus based on Euler characteristic, and its utility in data problems, particularly in aggregation of redundant data and inverse problems over networks. This calculus is a blend of integral-geometric and sheaf theoretic techniques, and leads to surprisingly practical algorithms and computations. Qualitative versions of integral transforms for signal processing will be stressed.
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| Slides | |
| 0:00 | euler calculus & data |
| 0:49 | motivation |
| 1:09 | tools |
| 2:06 | euler calculus (1) |
| 2:47 | euler calculus (2) |
| 6:06 | sheaves |
| 7:47 | ∫h dχ |
| 9:57 | integration (1) |
| 14:17 | integration (2) |
| 16:56 | problem (1) |
| 18:09 | problem (2) |
| 19:11 | counting |
| 23:12 | computation |
| 23:46 | example (1) |
| 24:46 | example (2) |
| 25:10 | some applications in minimal sensing |
| 25:20 | waves |
| 26:58 | wheels (1) |
| 28:03 | wheels (2) |
| 29:19 | numerical integration (1) |
| 29:32 | numerical integration (2) |
| 30:43 | ad hoc networks |
| 34:23 | get real… |
| 34:34 | real-valued integrands (1) |
| 38:09 | real-valued integrands (2) |
| 40:10 | real-valued integrands (3) |
| 40:33 | incomplete data (1) |
| 41:57 | incomplete data (2) |
| 42:49 | expected values |
| 44:20 | integral transforms |
| 44:31 | inversion |
| 46:14 | fourier transform |
| 47:20 | radon transform |
| 47:41 | bessel transform |
| 52:20 | open questions |
| 53:46 | topological network topology |
| 53:57 | closing credits… |
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