alpha_shapes

A Python package for reconstructing the shape of a 2D point cloud on the plane.

Table of Contents

• Introduction
• Usage
• Features
  • Optimization
  • Normalization
• Inspiration

Introduction

Given a finite set of points (or point cloud) in the Euclidean plane, alpha shapes are members of a family of closed polygons on the 2D plane associated with the shape of this point cloud. Each alpha shape is associated with a single non negative parameter α.

Intuitively an alpha shape can be conceptualized as follows. Imagine carving out the plane using a cookie scoop of radius 1/α, without removing any of the points in the point cloud. The shape that remains is the shape of the point cloud. If we replace the arc-like edges, due to the circular rim of the scoop, with straight segments, we are left with the alpha shape of parameter α.

Usage

Imports:

import matplotlib.pyplot as plt
from descartes import PolygonPatch
from alpha_shapes.alpha_shapes import Alpha_Shaper

Define a set of points:

points = [(0.,     0.),    (0.,    1.),    (1.,     1.1),
          (1.,     0.),    (0.25,  0.15),  (0.65,   0.45),
          (0.75,   0.75),  (0.5,   0.5),   (0.5,    0.25),
          (0.5,    0.75),  (0.25,  0.5),   (0.75,   0.25),
          (0.,     2.),    (0.,    2.1),   (1.,     2.1),
          (0.5,    2.5),   (-0.5,  1.5),   (-0.25,  1.5),
          (-0.25,  1.25),  (0,     1.25),  (1.5,    1.5),
          (1.25,   1.5),   (1.25,  1.25),  (1,      1.25),
          (0.5,    2.25),  (1.,    2.),    (0.25,   2.15),
          (0.65,   2.45),  (0.75,  2.75),  (0.5,    2.25),
          (0.5,    2.75),  (0.25,  2.5),   (0.75,   2.25)]

Create the alpha shaper:

shaper = Alpha_Shaper(points)

For the alpha shape to be calculated, the user must choose a value for the alpha parameter. Here, let us set alpha to 3.0:

alpha = 3.0
alpha_shape = shaper.get_shape(alpha=alpha)

Visualize the result:

fig, (ax0, ax1) = plt.subplots(1, 2)
ax0.scatter(*zip(*points))
ax0.set_title('data')
ax1.scatter(*zip(*points))
ax1.add_patch(PolygonPatch(alpha_shape, alpha=0.2, color='r'))
ax1.set_title(f"$\\alpha={alpha:.2}$")

for ax in (ax0, ax1):
    ax.set_aspect('equal')
plt.show()

The resulting shape is only a rough outline of the figure formed by the point set. However, increasing the value of alpha to 4.5 yields much better results.

alpha = 4.5
alpha_shape = shaper.get_shape(alpha=alpha)

Features

Optimization

A satisfactory calculation of the alpha shape requires a successful guess of the alpha parameter. While trial and error might work well in some cases, users can let the Alpha_Shaper choose a value for them. That is what the optimize method is about.

>>> alpha_opt, alpha_shape = shaper.optimize()
>>> alpha_opt
5.331459512629295

The optimize method runs efficiently for relatively large point clouds. Here we calculate the optimal alpha shape of an ensemble of 1000 random points uniformly distributed on the unit square.

>>> from time import time
>>> points = np.random.random((1000, 2))
>>> alpha_shaper = Alpha_Shaper(points)
>>> ts = time()
>>> alpha_opt, alpha_shape = alpha_shaper.optimize()
>>> te = time()
>>> print(f'optimization took: {te-ts:.2} sec')
optimization took: 0.41 sec

Normalization

Before calculating the alpha shape, Alpha_Shaper normalizes by default the input points so that they are distributed on the unit square. When there is a scale separation along the x and y direction, deactivating this feature may yield surprising results.

import numpy as np
# Scale the points along the x-dimension
x_scale = 1e-3
points = np.array(points)
points[:, 0] *= x_scale

#  Create the alpha shape without accounting for the x and y scale separation
unnormalized_shaper = Alpha_Shaper(points, normalize=False)
_, alpha_shape_unscaled = unnormalized_shaper.optimize()


# If the characteristic scale along each axis varies significantly,
# it may make sense to turn on the `normalize` option.
shaper = Alpha_Shaper(points, normalize=True)
_, alpha_shape_scaled = shaper.optimize()

# Scale the points along the x-dimension
x_scale = 1e-3
points = np.array(points)
points[:, 0] *= x_scale

#  Create the alpha shape without accounting for the x and y scale separation
unnormalized_shaper = Alpha_Shaper(points, normalize=False)
_, alpha_shape_unscaled = unnormalized_shaper.optimize()


# If the characteristic scale along each axis varies significantly,
# it may make sense to turn on the `normalize` option.
shaper = Alpha_Shaper(points, normalize=True)
_, alpha_shape_scaled = shaper.optimize()

Inspiration

This library is inspired by the alphashape library.