Betti numbers, updated
I promised to update the article about Betti numbers in the wiki. So here it is.
Betti numbers count the number of topological features in the image.
For example, in the first image on the right the number of objects is 6, so the 0th Betti number B0 is 6. The number of holes is 2, so the 1st Betti number B1 is 2 as well.
More generally, the 0th, 1st and 2nd dimensional topological features are:
- objects or connected components – dimension 0,
- holes or tunnels – dimension 1, and
- voids or cavities – dimension 2.
These numbers in each dimension are captured by the Betti numbers, B0, B1, and B2. Examples are in the table below.
| B0 (parts) | B1 (holes) | B2 (voids) | |
| Letter O | 1 | 1 | 0 |
| Two letters O | 2 | 2 | 0 |
| Letter B | 1 | 2 | 0 |
| Donut | 1 | 1 | 0 |
| Tire | 1 | 2 | 1 |
| Ball | 1 | 0 | 1 |
The tire (torus) has two tunnels represented by these two “cycles”.
This cycle is trivial and not to be counted.In practice, i.e., computer vision and image analysis, you can’t count topological features. Instead you count “cycles” that capture them. Counting 0-cycles and 1-cycles in 2D is fairly simple.
Let’s now consider 1-cycles, i.e., circular curves, in 3D. In the torus, there are two kinds of cycles: the latitudes (one is red) and the longitudes (one is blue). The latitudes capture the tunnel inside the tire while the longitudes capture the tunnel through the tire.
Clearly there are many cycles that capture the same topological feature. So, how do you avoid overcounting them? The answer is: if two cycles are homologous to each other, they are counted as one.
One way to explain homology is this. Two cycles are homologous if they together form the boundary of a surface.
For example, any latitude is homologous to the red cycle because they are the two ends of a tube cut from the tire. Any longitude is homologous to the blue cycle because they are the two edges of a strip cut from the tire. This is why B1 = 2.
In the sphere the blue cycle is homologous to a point, a trivial cycle, that’s why B1 = 0. Another way to see that it is trial is to imagine how it can contract to a point like a rubber band. The same thing happens with the donut.
Betti numbers are combined together (as the alternating sum) to produce the well-known Euler number (aka Euler characteristics).
BTW, Bn = rank Hn.
