20、Betti Numbers and Homology Groups in Digital Image Analysis

Betti Numbers and Homology Groups in Digital Image Analysis

1. Introduction to Betti Numbers

In the realm of computational geometry and topology, Betti numbers play a crucial role in understanding the topological structure of shapes within digital images. Betti numbers provide a way to quantify the number of “holes” in various dimensions within a given shape. Specifically:

• ( b_0 ): The number of connected components.
• ( b_1 ): The number of one-dimensional loops or tunnels.
• ( b_2 ): The number of two-dimensional voids or cavities.

These numbers help in distinguishing and classifying shapes based on their topological properties. For instance, a donut and a coffee cup are topologically equivalent because they both