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blob.py
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blob.py
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from __future__ import division
import numpy as np
from scipy.ndimage import gaussian_filter, gaussian_laplace
import math
from math import sqrt, log
from scipy import spatial
from ..util import img_as_float
from .peak import peak_local_max
from ._hessian_det_appx import _hessian_matrix_det
from ..transform import integral_image
from .._shared.utils import assert_nD
# This basic blob detection algorithm is based on:
# http://www.cs.utah.edu/~jfishbau/advimproc/project1/ (04.04.2013)
# Theory behind: http://en.wikipedia.org/wiki/Blob_detection (04.04.2013)
def _compute_disk_overlap(d, r1, r2):
"""
Compute surface overlap between two disks of radii ``r1`` and ``r2``,
with centers separated by a distance ``d``.
Parameters
----------
d : float
Distance between centers.
r1 : float
Radius of the first disk.
r2 : float
Radius of the second disk.
Returns
-------
vol: float
Volume of the overlap between the two disks.
"""
ratio1 = (d ** 2 + r1 ** 2 - r2 ** 2) / (2 * d * r1)
ratio1 = np.clip(ratio1, -1, 1)
acos1 = math.acos(ratio1)
ratio2 = (d ** 2 + r2 ** 2 - r1 ** 2) / (2 * d * r2)
ratio2 = np.clip(ratio2, -1, 1)
acos2 = math.acos(ratio2)
a = -d + r2 + r1
b = d - r2 + r1
c = d + r2 - r1
d = d + r2 + r1
area = (r1 ** 2 * acos1 + r2 ** 2 * acos2 -
0.5 * sqrt(abs(a * b * c * d)))
return area / (math.pi * (min(r1, r2) ** 2))
def _compute_sphere_overlap(d, r1, r2):
"""
Compute volume overlap between two spheres of radii ``r1`` and ``r2``,
with centers separated by a distance ``d``.
Parameters
----------
d : float
Distance between centers.
r1 : float
Radius of the first sphere.
r2 : float
Radius of the second sphere.
Returns
-------
vol: float
Volume of the overlap between the two spheres.
Notes
-----
See for example http://mathworld.wolfram.com/Sphere-SphereIntersection.html
for more details.
"""
vol = (math.pi / (12 * d) * (r1 + r2 - d)**2 *
(d**2 + 2 * d * (r1 + r2) - 3 * (r1**2 + r2**2) + 6 * r1 * r2))
return vol / (4./3 * math.pi * min(r1, r2) ** 3)
def _blob_overlap(blob1, blob2):
"""Finds the overlapping area fraction between two blobs.
Returns a float representing fraction of overlapped area.
Parameters
----------
blob1 : sequence of arrays
A sequence of ``(row, col, sigma)`` or ``(pln, row, col, sigma)``,
where ``row, col`` (or ``(pln, row, col)``) are coordinates
of blob and ``sigma`` is the standard deviation of the Gaussian kernel
which detected the blob.
blob2 : sequence of arrays
A sequence of ``(row, col, sigma)`` or ``(pln, row, col, sigma)``,
where ``row, col`` (or ``(pln, row, col)``) are coordinates
of blob and ``sigma`` is the standard deviation of the Gaussian kernel
which detected the blob.
Returns
-------
f : float
Fraction of overlapped area (or volume in 3D).
"""
n_dim = len(blob1) - 1
root_ndim = sqrt(n_dim)
# extent of the blob is given by sqrt(2)*scale
r1 = blob1[-1] * root_ndim
r2 = blob2[-1] * root_ndim
d = sqrt(np.sum((blob1[:-1] - blob2[:-1])**2))
if d > r1 + r2:
return 0
# one blob is inside the other, the smaller blob must die
if d <= abs(r1 - r2):
return 1
if n_dim == 2:
return _compute_disk_overlap(d, r1, r2)
else: # http://mathworld.wolfram.com/Sphere-SphereIntersection.html
return _compute_sphere_overlap(d, r1, r2)
def _prune_blobs(blobs_array, overlap):
"""Eliminated blobs with area overlap.
Parameters
----------
blobs_array : ndarray
A 2d array with each row representing 3 (or 4) values,
``(row, col, sigma)`` or ``(pln, row, col, sigma)`` in 3D,
where ``(row, col)`` (``(pln, row, col)``) are coordinates of the blob
and ``sigma`` is the standard deviation of the Gaussian kernel which
detected the blob.
This array must not have a dimension of size 0.
overlap : float
A value between 0 and 1. If the fraction of area overlapping for 2
blobs is greater than `overlap` the smaller blob is eliminated.
Returns
-------
A : ndarray
`array` with overlapping blobs removed.
"""
sigma = blobs_array[:, -1].max()
distance = 2 * sigma * sqrt(blobs_array.shape[1] - 1)
tree = spatial.cKDTree(blobs_array[:, :-1])
pairs = np.array(list(tree.query_pairs(distance)))
if len(pairs) == 0:
return blobs_array
else:
for (i, j) in pairs:
blob1, blob2 = blobs_array[i], blobs_array[j]
if _blob_overlap(blob1, blob2) > overlap:
if blob1[-1] > blob2[-1]:
blob2[-1] = 0
else:
blob1[-1] = 0
return np.array([b for b in blobs_array if b[-1] > 0])
def blob_dog(image, min_sigma=1, max_sigma=50, sigma_ratio=1.6, threshold=2.0,
overlap=.5,):
"""Finds blobs in the given grayscale image.
Blobs are found using the Difference of Gaussian (DoG) method [1]_.
For each blob found, the method returns its coordinates and the standard
deviation of the Gaussian kernel that detected the blob.
Parameters
----------
image : 2D or 3D ndarray
Input grayscale image, blobs are assumed to be light on dark
background (white on black).
min_sigma : float, optional
The minimum standard deviation for Gaussian Kernel. Keep this low to
detect smaller blobs.
max_sigma : float, optional
The maximum standard deviation for Gaussian Kernel. Keep this high to
detect larger blobs.
sigma_ratio : float, optional
The ratio between the standard deviation of Gaussian Kernels used for
computing the Difference of Gaussians
threshold : float, optional.
The absolute lower bound for scale space maxima. Local maxima smaller
than thresh are ignored. Reduce this to detect blobs with less
intensities.
overlap : float, optional
A value between 0 and 1. If the area of two blobs overlaps by a
fraction greater than `threshold`, the smaller blob is eliminated.
Returns
-------
A : (n, image.ndim + 1) ndarray
A 2d array with each row representing 3 values for a 2D image,
and 4 values for a 3D image: ``(r, c, sigma)`` or ``(p, r, c, sigma)``
where ``(r, c)`` or ``(p, r, c)`` are coordinates of the blob and
``sigma`` is the standard deviation of the Gaussian kernel which
detected the blob.
References
----------
.. [1] http://en.wikipedia.org/wiki/Blob_detection#The_difference_of_Gaussians_approach
Examples
--------
>>> from skimage import data, feature
>>> feature.blob_dog(data.coins(), threshold=.5, max_sigma=40)
array([[ 267. , 359. , 16.777216],
[ 267. , 115. , 10.48576 ],
[ 263. , 302. , 16.777216],
[ 263. , 245. , 16.777216],
[ 261. , 173. , 16.777216],
[ 260. , 46. , 16.777216],
[ 198. , 155. , 10.48576 ],
[ 196. , 43. , 10.48576 ],
[ 195. , 102. , 16.777216],
[ 194. , 277. , 16.777216],
[ 193. , 213. , 16.777216],
[ 185. , 347. , 16.777216],
[ 128. , 154. , 10.48576 ],
[ 127. , 102. , 10.48576 ],
[ 125. , 208. , 10.48576 ],
[ 125. , 45. , 16.777216],
[ 124. , 337. , 10.48576 ],
[ 120. , 272. , 16.777216],
[ 58. , 100. , 10.48576 ],
[ 54. , 276. , 10.48576 ],
[ 54. , 42. , 16.777216],
[ 52. , 216. , 16.777216],
[ 52. , 155. , 16.777216],
[ 45. , 336. , 16.777216]])
Notes
-----
The radius of each blob is approximately :math:`\sqrt{2}\sigma` for
a 2-D image and :math:`\sqrt{3}\sigma` for a 3-D image.
"""
image = img_as_float(image)
# k such that min_sigma*(sigma_ratio**k) > max_sigma
k = int(log(float(max_sigma) / min_sigma, sigma_ratio)) + 1
# a geometric progression of standard deviations for gaussian kernels
sigma_list = np.array([min_sigma * (sigma_ratio ** i)
for i in range(k + 1)])
gaussian_images = [gaussian_filter(image, s) for s in sigma_list]
# computing difference between two successive Gaussian blurred images
# multiplying with standard deviation provides scale invariance
dog_images = [(gaussian_images[i] - gaussian_images[i + 1])
* sigma_list[i] for i in range(k)]
image_cube = np.stack(dog_images, axis=-1)
# local_maxima = get_local_maxima(image_cube, threshold)
local_maxima = peak_local_max(image_cube, threshold_abs=threshold,
footprint=np.ones((3,) * (image.ndim + 1)),
threshold_rel=0.0,
exclude_border=False)
# Catch no peaks
if local_maxima.size == 0:
return np.empty((0, 3))
# Convert local_maxima to float64
lm = local_maxima.astype(np.float64)
# Convert the last index to its corresponding scale value
lm[:, -1] = sigma_list[local_maxima[:, -1]]
return _prune_blobs(lm, overlap)
def blob_log(image, min_sigma=1, max_sigma=50, num_sigma=10, threshold=.2,
overlap=.5, log_scale=False):
"""Finds blobs in the given grayscale image.
Blobs are found using the Laplacian of Gaussian (LoG) method [1]_.
For each blob found, the method returns its coordinates and the standard
deviation of the Gaussian kernel that detected the blob.
Parameters
----------
image : 2D or 3D ndarray
Input grayscale image, blobs are assumed to be light on dark
background (white on black).
min_sigma : float, optional
The minimum standard deviation for Gaussian Kernel. Keep this low to
detect smaller blobs.
max_sigma : float, optional
The maximum standard deviation for Gaussian Kernel. Keep this high to
detect larger blobs.
num_sigma : int, optional
The number of intermediate values of standard deviations to consider
between `min_sigma` and `max_sigma`.
threshold : float, optional.
The absolute lower bound for scale space maxima. Local maxima smaller
than thresh are ignored. Reduce this to detect blobs with less
intensities.
overlap : float, optional
A value between 0 and 1. If the area of two blobs overlaps by a
fraction greater than `threshold`, the smaller blob is eliminated.
log_scale : bool, optional
If set intermediate values of standard deviations are interpolated
using a logarithmic scale to the base `10`. If not, linear
interpolation is used.
Returns
-------
A : (n, image.ndim + 1) ndarray
A 2d array with each row representing 3 values for a 2D image,
and 4 values for a 3D image: ``(r, c, sigma)`` or ``(p, r, c, sigma)``
where ``(r, c)`` or ``(p, r, c)`` are coordinates of the blob and
``sigma`` is the standard deviation of the Gaussian kernel which
detected the blob.
References
----------
.. [1] http://en.wikipedia.org/wiki/Blob_detection#The_Laplacian_of_Gaussian
Examples
--------
>>> from skimage import data, feature, exposure
>>> img = data.coins()
>>> img = exposure.equalize_hist(img) # improves detection
>>> feature.blob_log(img, threshold = .3)
array([[ 266. , 115. , 11.88888889],
[ 263. , 302. , 17.33333333],
[ 263. , 244. , 17.33333333],
[ 260. , 174. , 17.33333333],
[ 198. , 155. , 11.88888889],
[ 198. , 103. , 11.88888889],
[ 197. , 44. , 11.88888889],
[ 194. , 276. , 17.33333333],
[ 194. , 213. , 17.33333333],
[ 185. , 344. , 17.33333333],
[ 128. , 154. , 11.88888889],
[ 127. , 102. , 11.88888889],
[ 126. , 208. , 11.88888889],
[ 126. , 46. , 11.88888889],
[ 124. , 336. , 11.88888889],
[ 121. , 272. , 17.33333333],
[ 113. , 323. , 1. ]])
Notes
-----
The radius of each blob is approximately :math:`\sqrt{2}\sigma` for
a 2-D image and :math:`\sqrt{3}\sigma` for a 3-D image.
"""
image = img_as_float(image)
if log_scale:
start, stop = log(min_sigma, 10), log(max_sigma, 10)
sigma_list = np.logspace(start, stop, num_sigma)
else:
sigma_list = np.linspace(min_sigma, max_sigma, num_sigma)
# computing gaussian laplace
# s**2 provides scale invariance
gl_images = [-gaussian_laplace(image, s) * s ** 2 for s in sigma_list]
image_cube = np.stack(gl_images, axis=-1)
local_maxima = peak_local_max(image_cube, threshold_abs=threshold,
footprint=np.ones((3,) * (image.ndim + 1)),
threshold_rel=0.0,
exclude_border=False)
# Catch no peaks
if local_maxima.size == 0:
return np.empty((0, 3))
# Convert local_maxima to float64
lm = local_maxima.astype(np.float64)
# Convert the last index to its corresponding scale value
lm[:, -1] = sigma_list[local_maxima[:, -1]]
return _prune_blobs(lm, overlap)
def blob_doh(image, min_sigma=1, max_sigma=30, num_sigma=10, threshold=0.01,
overlap=.5, log_scale=False):
"""Finds blobs in the given grayscale image.
Blobs are found using the Determinant of Hessian method [1]_. For each blob
found, the method returns its coordinates and the standard deviation
of the Gaussian Kernel used for the Hessian matrix whose determinant
detected the blob. Determinant of Hessians is approximated using [2]_.
Parameters
----------
image : 2D ndarray
Input grayscale image.Blobs can either be light on dark or vice versa.
min_sigma : float, optional
The minimum standard deviation for Gaussian Kernel used to compute
Hessian matrix. Keep this low to detect smaller blobs.
max_sigma : float, optional
The maximum standard deviation for Gaussian Kernel used to compute
Hessian matrix. Keep this high to detect larger blobs.
num_sigma : int, optional
The number of intermediate values of standard deviations to consider
between `min_sigma` and `max_sigma`.
threshold : float, optional.
The absolute lower bound for scale space maxima. Local maxima smaller
than thresh are ignored. Reduce this to detect less prominent blobs.
overlap : float, optional
A value between 0 and 1. If the area of two blobs overlaps by a
fraction greater than `threshold`, the smaller blob is eliminated.
log_scale : bool, optional
If set intermediate values of standard deviations are interpolated
using a logarithmic scale to the base `10`. If not, linear
interpolation is used.
Returns
-------
A : (n, 3) ndarray
A 2d array with each row representing 3 values, ``(y,x,sigma)``
where ``(y,x)`` are coordinates of the blob and ``sigma`` is the
standard deviation of the Gaussian kernel of the Hessian Matrix whose
determinant detected the blob.
References
----------
.. [1] http://en.wikipedia.org/wiki/Blob_detection#The_determinant_of_the_Hessian
.. [2] Herbert Bay, Andreas Ess, Tinne Tuytelaars, Luc Van Gool,
"SURF: Speeded Up Robust Features"
ftp://ftp.vision.ee.ethz.ch/publications/articles/eth_biwi_00517.pdf
Examples
--------
>>> from skimage import data, feature
>>> img = data.coins()
>>> feature.blob_doh(img)
array([[ 270. , 363. , 30. ],
[ 265. , 113. , 23.55555556],
[ 262. , 243. , 23.55555556],
[ 260. , 173. , 30. ],
[ 197. , 153. , 20.33333333],
[ 197. , 44. , 20.33333333],
[ 195. , 100. , 23.55555556],
[ 193. , 275. , 23.55555556],
[ 192. , 212. , 23.55555556],
[ 185. , 348. , 30. ],
[ 156. , 302. , 30. ],
[ 126. , 153. , 20.33333333],
[ 126. , 101. , 20.33333333],
[ 124. , 336. , 20.33333333],
[ 123. , 205. , 20.33333333],
[ 123. , 44. , 23.55555556],
[ 121. , 271. , 30. ]])
Notes
-----
The radius of each blob is approximately `sigma`.
Computation of Determinant of Hessians is independent of the standard
deviation. Therefore detecting larger blobs won't take more time. In
methods line :py:meth:`blob_dog` and :py:meth:`blob_log` the computation
of Gaussians for larger `sigma` takes more time. The downside is that
this method can't be used for detecting blobs of radius less than `3px`
due to the box filters used in the approximation of Hessian Determinant.
"""
assert_nD(image, 2)
image = img_as_float(image)
image = integral_image(image)
if log_scale:
start, stop = log(min_sigma, 10), log(max_sigma, 10)
sigma_list = np.logspace(start, stop, num_sigma)
else:
sigma_list = np.linspace(min_sigma, max_sigma, num_sigma)
hessian_images = [_hessian_matrix_det(image, s) for s in sigma_list]
image_cube = np.dstack(hessian_images)
local_maxima = peak_local_max(image_cube, threshold_abs=threshold,
footprint=np.ones((3,) * image_cube.ndim),
threshold_rel=0.0,
exclude_border=False)
# Catch no peaks
if local_maxima.size == 0:
return np.empty((0, 3))
# Convert local_maxima to float64
lm = local_maxima.astype(np.float64)
# Convert the last index to its corresponding scale value
lm[:, -1] = sigma_list[local_maxima[:, -1]]
return _prune_blobs(lm, overlap)