A Multi-Parameter Persistence Framework for Mathematical Morphology

Explore a 55-minute lecture on applying persistent homology to mathematical morphology for image processing. Delve into the integration of topological data analysis with classic morphological operations, forming multiparameter filtrations. Learn how this framework extracts topological and geometric information from images, enabling automated optimization of image structure analysis and rendering. Discover an unsupervised denoising approach for binary, grayscale, and color images, comparable to state-of-the-art supervised deep learning methods. Cover topics including opening and closing operations, shifting functions, persistence diagrams, multiparameter filtrations, sublevel sets, and computational complexity. Gain insights into the potential of this innovative framework for enhancing image processing techniques through the lens of applied algebraic topology.

Introduction
Background knowledge notation
What is mathematical morphology
Opening and closing operations
New filtration
Shifting function
Persistence diagram
Variants
Multiparameter filtration
Alternating closing
Alternating opening
Assumption
Sublevel set
Basic idea
Comparison
Summary
Questions
Algebraic structure
Randomization
Natural biofiltration
Opening and closing
Computational complexity
Short answer