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from math import sqrt
import numpy as np
from scipy import ndimage as ndi
STREL_4 = np.array([[0, 1, 0],
[1, 1, 1],
[0, 1, 0]], dtype=np.uint8)
STREL_8 = np.ones((3, 3), dtype=np.uint8)
# Coefficients from
# Ohser J., Nagel W., Schladitz K. (2002) The Euler Number of Discretized Sets
# - On the Choice of Adjacency in Homogeneous Lattices.
# In: Mecke K., Stoyan D. (eds) Morphology of Condensed Matter. Lecture Notes
# in Physics, vol 600. Springer, Berlin, Heidelberg.
# The value of coefficients correspond to the contributions to the Euler number
# of specific voxel configurations, which are themselves encoded thanks to a
# LUT. Computing the Euler number from the addition of the contributions of
# local configurations is possible thanks to an integral geometry formula
# (see the paper by Ohser et al. for more details).
EULER_COEFS2D_4 = [0, 1, 0, 0, 0, 0, 0, -1, 0, 1, 0, 0, 0, 0, 0, 0]
EULER_COEFS2D_8 = [0, 0, 0, 0, 0, 0, -1, 0, 1, 0, 0, 0, 0, 0, -1, 0]
EULER_COEFS3D_26 = np.array([0, 1, 1, 0, 1, 0, -2, -1,
1, -2, 0, -1, 0, -1, -1, 0,
1, 0, -2, -1, -2, -1, -1, -2,
-6, -3, -3, -2, -3, -2, 0, -1,
1, -2, 0, -1, -6, -3, -3, -2,
-2, -1, -1, -2, -3, 0, -2, -1,
0, -1, -1, 0, -3, -2, 0, -1,
-3, 0, -2, -1, 0, 1, 1, 0,
1, -2, -6, -3, 0, -1, -3, -2,
-2, -1, -3, 0, -1, -2, -2, -1,
0, -1, -3, -2, -1, 0, 0, -1,
-3, 0, 0, 1, -2, -1, 1, 0,
-2, -1, -3, 0, -3, 0, 0, 1,
-1, 4, 0, 3, 0, 3, 1, 2,
-1, -2, -2, -1, -2, -1, 1,
0, 0, 3, 1, 2, 1, 2, 2, 1,
1, -6, -2, -3, -2, -3, -1, 0,
0, -3, -1, -2, -1, -2, -2, -1,
-2, -3, -1, 0, -1, 0, 4, 3,
-3, 0, 0, 1, 0, 1, 3, 2,
0, -3, -1, -2, -3, 0, 0, 1,
-1, 0, 0, -1, -2, 1, -1, 0,
-1, -2, -2, -1, 0, 1, 3, 2,
-2, 1, -1, 0, 1, 2, 2, 1,
0, -3, -3, 0, -1, -2, 0, 1,
-1, 0, -2, 1, 0, -1, -1, 0,
-1, -2, 0, 1, -2, -1, 3, 2,
-2, 1, 1, 2, -1, 0, 2, 1,
-1, 0, -2, 1, -2, 1, 1, 2,
-2, 3, -1, 2, -1, 2, 0, 1,
0, -1, -1, 0, -1, 0, 2, 1,
-1, 2, 0, 1, 0, 1, 1, 0, ])
def euler_number(image, connectivity=None):
"""Calculate the Euler characteristic in binary image.
For 2D objects, the Euler number is the number of objects minus the number
of holes. For 3D objects, the Euler number is obtained as the number of
objects plus the number of holes, minus the number of tunnels, or loops.
Parameters
----------
image: (N, M) ndarray or (N, M, D) ndarray.
2D or 3D images.
If image is not binary, all values strictly greater than zero
are considered as the object.
connectivity : int, optional
Maximum number of orthogonal hops to consider a pixel/voxel
as a neighbor.
Accepted values are ranging from 1 to input.ndim. If ``None``, a full
connectivity of ``input.ndim`` is used.
4 or 8 neighborhoods are defined for 2D images (connectivity 1 and 2,
respectively).
6 or 26 neighborhoods are defined for 3D images, (connectivity 1 and 3,
respectively). Connectivity 2 is not defined.
Returns
-------
euler_number : int
Euler characteristic of the set of all objects in the image.
Notes
-----
The Euler characteristic is an integer number that describes the
topology of the set of all objects in the input image. If object is
4-connected, then background is 8-connected, and conversely.
The computation of the Euler characteristic is based on an integral
geometry formula in discretized space. In practice, a neighborhood
configuration is constructed, and a LUT is applied for each
configuration. The coefficients used are the ones of Ohser et al.
It can be useful to compute the Euler characteristic for several
connectivities. A large relative difference between results
for different connectivities suggests that the image resolution
(with respect to the size of objects and holes) is too low.
References
----------
.. [1] S. Rivollier. Analyse d’image geometrique et morphometrique par
diagrammes de forme et voisinages adaptatifs generaux. PhD thesis,
2010. Ecole Nationale Superieure des Mines de Saint-Etienne.
https://tel.archives-ouvertes.fr/tel-00560838
.. [2] Ohser J., Nagel W., Schladitz K. (2002) The Euler Number of
Discretized Sets - On the Choice of Adjacency in Homogeneous
Lattices. In: Mecke K., Stoyan D. (eds) Morphology of Condensed
Matter. Lecture Notes in Physics, vol 600. Springer, Berlin,
Heidelberg.
Examples
--------
>>> import numpy as np
>>> SAMPLE = np.zeros((100,100,100));
>>> SAMPLE[40:60, 40:60, 40:60]=1
>>> euler_number(SAMPLE) # doctest: +ELLIPSIS
1...
>>> SAMPLE[45:55,45:55,45:55] = 0;
>>> euler_number(SAMPLE) # doctest: +ELLIPSIS
2...
>>> SAMPLE = np.array([[0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0],
... [0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0],
... [0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0],
... [0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0],
... [0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0],
... [0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0],
... [0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0],
... [1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0],
... [0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1],
... [0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1]])
>>> euler_number(SAMPLE) # doctest:
0
>>> euler_number(SAMPLE, connectivity=1) # doctest:
2
"""
# as image can be a label image, transform it to binary
image = (image > 0).astype(int)
image = np.pad(image, pad_width=1, mode='constant')
# check connectivity
if connectivity is None:
connectivity = image.ndim
# config variable is an adjacency configuration. A coefficient given by
# variable coefs is attributed to each configuration in order to get
# the Euler characteristic.
if image.ndim == 2:
config = np.array([[0, 0, 0], [0, 1, 4], [0, 2, 8]])
if connectivity == 1:
coefs = EULER_COEFS2D_4
else:
coefs = EULER_COEFS2D_8
bins = 16
else: # 3D images
if connectivity == 2:
raise NotImplementedError(
'For 3D images, Euler number is implemented '
'for connectivities 1 and 3 only')
config = np.array([[[0, 0, 0], [0, 0, 0], [0, 0, 0]],
[[0, 0, 0], [0, 1, 4], [0, 2, 8]],
[[0, 0, 0], [0, 16, 64], [0, 32, 128]]])
if connectivity == 1:
coefs = EULER_COEFS3D_26[::-1]
else:
coefs = EULER_COEFS3D_26
bins = 256
# XF has values in the 0-255 range in 3D, and in the 0-15 range in 2D,
# with one unique value for each binary configuration of the
# 27-voxel cube in 3D / 8-pixel square in 2D, up to symmetries
XF = ndi.convolve(image, config, mode='constant', cval=0)
h = np.bincount(XF.ravel(), minlength=bins)
if image.ndim == 2:
return coefs @ h
else:
return int(0.125 * coefs @ h)
def perimeter(image, neighborhood=4):
"""Calculate total perimeter of all objects in binary image.
Parameters
----------
image : (N, M) ndarray
2D binary image.
neighborhood : 4 or 8, optional
Neighborhood connectivity for border pixel determination. It is used to
compute the contour. A higher neighborhood widens the border on which
the perimeter is computed.
Returns
-------
perimeter : float
Total perimeter of all objects in binary image.
References
----------
.. [1] K. Benkrid, D. Crookes. Design and FPGA Implementation of
a Perimeter Estimator. The Queen's University of Belfast.
http://www.cs.qub.ac.uk/~d.crookes/webpubs/papers/perimeter.doc
Examples
--------
>>> from skimage import data, util
>>> from skimage.measure import label
>>> # coins image (binary)
>>> img_coins = data.coins() > 110
>>> # total perimeter of all objects in the image
>>> perimeter(img_coins, neighborhood=4) # doctest: +ELLIPSIS
7796.867...
>>> perimeter(img_coins, neighborhood=8) # doctest: +ELLIPSIS
8806.268...
"""
if image.ndim != 2:
raise NotImplementedError('`perimeter` supports 2D images only')
if neighborhood == 4:
strel = STREL_4
else:
strel = STREL_8
image = image.astype(np.uint8)
eroded_image = ndi.binary_erosion(image, strel, border_value=0)
border_image = image - eroded_image
perimeter_weights = np.zeros(50, dtype=np.float64)
perimeter_weights[[5, 7, 15, 17, 25, 27]] = 1
perimeter_weights[[21, 33]] = sqrt(2)
perimeter_weights[[13, 23]] = (1 + sqrt(2)) / 2
perimeter_image = ndi.convolve(border_image, np.array([[10, 2, 10],
[2, 1, 2],
[10, 2, 10]]),
mode='constant', cval=0)
# You can also write
# return perimeter_weights[perimeter_image].sum()
# but that was measured as taking much longer than bincount + np.dot (5x
# as much time)
perimeter_histogram = np.bincount(perimeter_image.ravel(), minlength=50)
total_perimeter = perimeter_histogram @ perimeter_weights
return total_perimeter
def perimeter_crofton(image, directions=4):
"""Calculate total Crofton perimeter of all objects in binary image.
Parameters
----------
image : (N, M) ndarray
2D image. If image is not binary, all values strictly greater than zero
are considered as the object.
directions : 2 or 4, optional
Number of directions used to approximate the Crofton perimeter. By
default, 4 is used: it should be more accurate than 2.
Computation time is the same in both cases.
Returns
-------
perimeter : float
Total perimeter of all objects in binary image.
Notes
-----
This measure is based on Crofton formula [1], which is a measure from
integral geometry. It is defined for general curve length evaluation via
a double integral along all directions. In a discrete
space, 2 or 4 directions give a quite good approximation, 4 being more
accurate than 2 for more complex shapes.
Similar to :func:`~.measure.perimeter`, this function returns an
approximation of the perimeter in continuous space.
References
----------
.. [1] https://en.wikipedia.org/wiki/Crofton_formula
.. [2] S. Rivollier. Analyse d’image geometrique et morphometrique par
diagrammes de forme et voisinages adaptatifs generaux. PhD thesis,
2010.
Ecole Nationale Superieure des Mines de Saint-Etienne.
https://tel.archives-ouvertes.fr/tel-00560838
Examples
--------
>>> from skimage import data, util
>>> from skimage.measure import label
>>> # coins image (binary)
>>> img_coins = data.coins() > 110
>>> # total perimeter of all objects in the image
>>> perimeter_crofton(img_coins, directions=2) # doctest: +ELLIPSIS
8144.578...
>>> perimeter_crofton(img_coins, directions=4) # doctest: +ELLIPSIS
7837.077...
"""
if image.ndim != 2:
raise NotImplementedError(
'`perimeter_crofton` supports 2D images only')
# as image could be a label image, transform it to binary image
image = (image > 0).astype(np.uint8)
image = np.pad(image, pad_width=1, mode='constant')
XF = ndi.convolve(image, np.array([[0, 0, 0], [0, 1, 4], [0, 2, 8]]),
mode='constant', cval=0)
h = np.bincount(XF.ravel(), minlength=16)
# definition of the LUT
if directions == 2:
coefs = [0, np.pi / 2, 0, 0, 0, np.pi / 2, 0, 0,
np.pi / 2, np.pi, 0, 0, np.pi / 2, np.pi, 0, 0]
else:
coefs = [0, np.pi / 4 * (1 + 1 / (np.sqrt(2))),
np.pi / (4 * np.sqrt(2)),
np.pi / (2 * np.sqrt(2)), 0,
np.pi / 4 * (1 + 1 / (np.sqrt(2))),
0, np.pi / (4 * np.sqrt(2)), np.pi / 4, np.pi / 2,
np.pi / (4 * np.sqrt(2)), np.pi / (4 * np.sqrt(2)),
np.pi / 4, np.pi / 2, 0, 0]
total_perimeter = coefs @ h
return total_perimeter