Topological Filters - A Toolbox for Processing Dynamic Signals - Michael Robinson

Explore topological filters as a powerful toolbox for processing dynamic signals in this comprehensive lecture from the Workshop on Topology: Identifying Order in Complex Systems. Delve into the fundamentals of overlap constructs, sheaf theory, and finite topologies derived from partial orders. Learn how changing overlaps affects topology and discover the concept of consistency in sheaves. Examine practical applications, including separating sinusoids from noise and flight tracking, to understand the excellent performance of sheaf-based approaches. Investigate discrete-time LTI filters, comparing standard adaptive filters with topological filters like OPLPF. Gain insights into the application of these techniques in extracting musical structure from Afro-Cuban drumming. Presented by Michael Robinson from American University, this 53-minute talk offers a deep dive into advanced signal processing techniques using topological methods.

Intro
Overlap constructs topology
Changing overlaps changes the topology
Sheaves are about consistency
Finite topologies from partial orders
Topologizing a partial order
A sheaf on a poset is...
An assignment is...
A global section is...
Consistency radius is...
The space of global sections
Separating sinusoids from noise
Sheaves deliver excellent performance
More complex example: flight tracking
Sheaf model of the sensors . We can forma partial order of the sensors and
Known flight path
Minor RDF angle error
Major flight path error
Discrete-time LTI filters
Proof sketch: Input sheaf
Proof sketch: The internal state
OPLPF block diagram
How is this a topological filter?
Compare: standard adaptive filter
Filter performance comparison - OPLPF combines good noise removal with signal envelope stability
Context: Afro-Cuban drumming
Extracting musical structure
Some instruments are less clear