hausdorff

Warning

As of PDAL v2.6.0, the hausdorff command is marked as DEPRECATED. It will be removed from the default install in PDAL v2.7 and removed completely in PDAL v2.8.

The following Python code can be used with the PDAL Python bindings to compute the Hausdorff distance.

import pdal
import numpy as np

def hausdorff_distance(arr1, arr2):
    max_min_distance_1_to_2 = 0
    max_min_distance_2_to_1 = 0

    points1 = np.column_stack((arr1['X'], arr1['Y'], arr1['Z']))
    points2 = np.column_stack((arr2['X'], arr2['Y'], arr2['Z']))

    # Compute distance from each point in arr1 to arr2
    for p1 in points1:
        distances = np.sqrt(np.sum((points2 - p1)**2, axis=1))
        min_distance = np.min(distances)
        max_min_distance_1_to_2 = max(max_min_distance_1_to_2, min_distance)

    # Compute distance from each point in arr2 to arr1
    for p2 in points2:
        distances = np.sqrt(np.sum((points1 - p2)**2, axis=1))
        min_distance = np.min(distances)
        max_min_distance_2_to_1 = max(max_min_distance_2_to_1, min_distance)

    return max(max_min_distance_1_to_2, max_min_distance_2_to_1)

pipeline1 = pdal.Reader("/path/to/input1.laz").pipeline()
pipeline1.execute()
arr1 = pipeline1.arrays[0]

pipeline2 = pdal.Reader("/path/to/input2.laz").pipeline()
pipeline2.execute()
arr2 = pipeline2.arrays[0]

# Compute Hausdorff distance
result = hausdorff_distance(arr1, arr2)
print("Hausdorff Distance:", result)

SciPy can be used to simplify this function even further, as shown below.

import pdal
import numpy as np
from scipy.spatial.distance import directed_hausdorff

def hausdorff_distance(arr1, arr2):
    points1 = np.column_stack((arr1['X'], arr1['Y'], arr2['Z']))
    points2 = np.column_stack((arr2['X'], arr2['Y'], arr2['Z']))

    # Compute directed Hausdorff distances
    d1 = directed_hausdorff(points1, points2)[0]
    d2 = directed_hausdorff(points2, points1)[0]

    return max(d1, d2)

pipeline1 = pdal.Reader("/path/to/input1.laz").pipeline()
pipeline1.execute()
arr1 = pipeline1.array[0]

pipeline2 = pdal.Reader("/path/to/input2.laz").pipeline()
pipeline2.execute()
arr2 = pipeline2.array[0]

# Compute Hausdorff distance
result = hausdorff_distance(arr1, arr2)
print("Hausdorff Distance:", result)

The hausdorff command is used to compute the Hausdorff distance between two point clouds. In this context, the Hausdorff distance is the greatest of all Euclidean distances from a point in one point cloud to the closest point in the other point cloud.

More formally, for two non-empty subsets \(X\) and \(Y\), the Hausdorff distance \(d_H(X,Y)\) is

\[ d_H(X,Y) = \operatorname*{max} \big\{ \operatorname*{sup}_{x \in X} \operatorname*{inf}_{y \in Y} d(x,y), \operatorname*{sup}_{y \in Y} \operatorname*{inf}_{x \in X} d(x,y)\big\} \]

where \(\operatorname*{sup}\) and \(\operatorname*{inf}\) are the supremum and infimum respectively.

$ pdal hausdorff <source> <candidate>

--source arg     Source filename
--candidate arg  Candidate filename

The algorithm makes no distinction between source and candidate files (i.e., they can be transposed with no affect on the computed distance).

The command returns 0 along with a JSON-formatted message summarizing the PDAL version, source and candidate filenames, and the Hausdorff distance. Identical point clouds will return a Hausdorff distance of 0.

$ pdal hausdorff source.las candidate.las
{
  "filenames":
  [
    "\/path\/to\/source.las",
    "\/path\/to\/candidate.las"
  ],
  "hausdorff": 1.303648726,
  "pdal_version": "1.3.0 (git-version: 191301)"
}

Note

The hausdorff is computed for XYZ coordinates only and as such says nothing about differences in other dimensions or metadata.