Lower Dimensional Topological Information: Theory and Applications

Explore lower dimensional topological features in data analysis through this Applied Algebraic Topology Network lecture. Delve into the theory and applications of topological data analysis, focusing on detecting lower dimensional zero density regions within density function supports. Learn about a novel detection method using shrinking radii covering balls, and understand the sufficient conditions for successful detection. Examine how lower dimensional topological information, particularly object boundaries, can enhance image segmentation. Compare and combine topological and statistical shape analysis methods for improved segmentation results. Gain insights into the interaction between topological information and statistical modeling, and discover the potential of lower dimensional features in uncovering dataset structures.

Intro
Theory: Lower dimensional topological features
Topological Data Analysis and the dimensionality
Minkowski dimension
Support of a density function
Order of smoothness
Detection method: scaled covering scheme
Line segment example revisited
Regularity conditions
The idea of the proof
Generalizations
Two image segmentation models - 11
Combining Geometric and Topological Information in Image Segmentation (Luo & Strait, 2019)
Application: Topological boundary and image segmentation