# Point Cloud Metrics

Point Cloud Utils has functions to compute a number of commonly used metrics between point clouds.

## Chamfer Distance

The Chamfer distance between two point clouds \(P_1 = \{x_i \in \mathbb{R}^3\}_{i=1}^n\) and \(P_2 = \{x_j \in \mathbb{R}^3\}_{j=1}^m\) is defined as the average distance between pairs of nearest neighbors between \(P_1\) and \(P_2\) *i.e.*
$$
\text{chamfer}(P_1, P_2) = \frac{1}{2n} \sum_{i=1}^n |x_i - \text{NN}(x_i, P_2)| + \frac{1}{2m} \sum_{j=1}^n |x_j - \text{NN}(x_j, P_1)|
$$
and \(\text{NN}(x, P) = \text{argmin}_{x' \in P} \|x - x'\|\) is the nearest neighbor function.

The following code computes the Chamfer distance between two point clouds:

```
import point_cloud_utils as pcu
# p1 is an (n, 3)-shaped numpy array containing one point per row
p1 = pcu.load_mesh_v("point_cloud_1.ply")
# p2 is an (m, 3)-shaped numpy array containing one point per row
p2 = pcu.load_mesh_v("point_cloud_2.ply")
# Compute the chamfer distance between p1 and p2
cd = pcu.chamfer_distance(p1, p2)
```

## Hausdorff distance

The Hausdorff distance between two point clouds \(P_1 = \{x_i \in \mathbb{R}^3\}_{i=1}^n\) and \(P_2 = \{x_j \in \mathbb{R}^3\}_{j=1}^m\) is defined as the maxmimum distance between any pair of nearest neighbors between \(P_1\) and \(P_2\) *i.e.*
$$
\text{hausdorff}(P_1, P_2) = \frac{1}{2} \max_{x \in P_1} |x - \text{NN}(x, P_2)| + \frac{1}{2} \max_{x' \in P_2} |x' - \text{NN}(x', P_1)|
$$
and \(\text{NN}(x, P) = \text{argmin}_{x' \in P} \|x - x'\|\) is the nearest neighbor function.

The following code computes the Hausdorff distance between two point clouds:

```
import point_cloud_utils as pcu
# p1 is an (n, 3)-shaped numpy array containing one point per row
p1 = pcu.load_mesh_v("point_cloud_1.ply")
# p2 is an (m, 3)-shaped numpy array containing one point per row
p2 = pcu.load_mesh_v("point_cloud_2.ply")
# Compute the chamfer distance between p1 and p2
hd = pcu.hausdorff_distance(p1, p2)
```

### One sided Hausdorff distance

In some applications, one only needs the *one-sided Hausdorff distance* between \(P_1\) and \(P_2\), *i.e.*

The following code computes the one-sided Hausdorff distance between two point clouds:

```
import point_cloud_utils as pcu
# p1 is an (n, 3)-shaped numpy array containing one point per row
p1 = pcu.load_mesh_v("point_cloud_1.ply")
# p2 is an (m, 3)-shaped numpy array containing one point per row
p2 = pcu.load_mesh_v("point_cloud_2.ply")
# Compute the chamfer distance between p1 and p2
hd_p1_to_p2 = pcu.one_sided_hausdorff_distance(p1, p2)
```

Note

To get the \(P_2 \rightarrow P_1\) Hausdorff distance, just swap the arguments to `pcu.one_sided_hausdorff_distance`

## Earth-Mover's (Sinkhorn) distance

The Earth Mover's distance between two point clouds \(P = \{p_i \in \mathbb{R}^3\}_{i=1}^n\) and \(Q = \{q_j \in \mathbb{R}^3\}_{j=1}^m\) is computed as the average distance between pairs of points according to an optimal correspondence \(\pi \in \Pi(P, Q)\), where \(\Pi(P, Q)\) is the set of \(n \times m\) matrices where the rows and columns sum to one. The assignment \(\pi\) is thus a matrix where \(\Pi_{i,j}\) is a number between \(0\) and \(1\) denoting how much point \(p_i\) and \(q_j\) correspond. We can write the EMD formally as: $$ \text{EMD}(P, Q) = \min_{\pi \in \Pi(P, Q)} \sum_{i = 1}^n \sum_{j = 1}^m \pi_{i,j} |p_i - q_j| % \langle \pi, D \rangle \qquad D_{ij} = |p_i - q_j| $$

Point Cloud Utils implements the sinkhorn algorithm for computing the (approximate) Earth Mover's Distance. To compute the EMD, run:

```
import point_cloud_utils as pcu
# p1 is an (n, 3)-shaped numpy array containing one point per row
p1 = pcu.load_mesh_v("point_cloud_1.ply")
# p2 is an (m, 3)-shaped numpy array containing one point per row
p2 = pcu.load_mesh_v("point_cloud_2.ply")
# Compute the chamfer distance between p1 and p2
emd, pi = pcu.earth_movers_distance(p1, p2)
```