For example, hausdroff distance

Given the two points in the Euclidean space, the Hausdorff distance is used to measure the distance between the two points.

Where ,,. The bidirectional hausdroff distance is called the unidirectional hausdroff distance from Point A to Point Set B. Correspondingly, it is called the one-way hausdroff distance from point B to Point Set.

The following example illustrates the hausdroff distance.

The path A, B, C, and D are given. Specifically, path a is (16-17-18-19-20 ), path B is (1-2-3-4-9-10 ). If the hausdroff distance is required, the one-way hausdroff distance and must be obtained first.

For example, in point 16 in a, in all vertices in path B, the distance closest to Point 16 is point 1, and the distance is 3. That is. The same reason diagram can be found ,,,. Among them, the maximum value is 3.

Likewise, you can ,.

So.

Similarly, we can find the one-way hausdroff distance between the four paths, as shown in the following table:

 

The two-way hausdoff distance is the one-way hausdroff distance and the larger of the two. Obviously, it measures the maximum degree of mismatch between the two sets.

When both A and B are closed sets, the hausdroff distance satisfies the three theorem of measurement:

(1) When and only when a = B ,.

(2)

(3)

If the Convex Set A and B meet the conditions and are respectively the vertex set of the and B boundary, then the hausdroff distance of A and B is equal to the hausdroff distance.

 

The hausdroff distance is vulnerable to sudden noise.

When the image is contaminated by noise or blocked, the original haudroff distance may cause incorrect matching. Therefore, in 1933, hutenlocher proposed the concept of some hausdroff distances.

To put it simply, the distance between Set B containing q points and the hausdroff of set a is to select K (and) points in B, then obtain the minimum distance from the K points to the set and sort them. Then, the K value after sorting is the one-way hausdroff distance from the part from the B to the set. The formula is as follows:

Correspondingly, some bidirectional hausdroff distances are defined:

For example, hausdroff distance