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Random walker algo for segmentation + example

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emmanuelle
2012-01-15 17:22:59 +01:00
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doc/examples/plot_random_walker_segmentation.py
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"""
==========================
Random walker segmentation
==========================
The random walker algorithm (*Random walks for image segmentation*, Leo Grady,
IEEE Trans Pattern Anal Mach Intell. 2006 Nov;28(11):1768-83) determines the
segmentation of an image from a set of markers labeling several phases (2 or
more). An anisotropic diffusion equation is solved with tracers initiated
at the markers' position. The local diffusivity coefficient is greater if
neighboring pixels have similar values, so that diffusion is difficult across
high gradients. The label of each unknown pixel is attributed to the label
of the known marker that has the highest probability to be reached first
during this diffusion process.
In this example, two phases are clearly visible, but the data are too
noisy to perform the segmentation from the histogram only. We determine
markers of the two phases from the extreme tails of the histogram of gray
values, and use the random walker for the segmentation.
"""
print __doc__
import numpy as np
from scipy import ndimage
from skimage.segmentation import random_walker
import matplotlib.pyplot as plt
def microstructure(l=256):
"""
Synthetic binary data: binary microstructure with blobs.
Parameters
----------
l: int, optional
linear size of the returned image
"""
n = 5
x, y = np.ogrid[0:l, 0:l]
mask_outer = (x - l / 2) ** 2 + (y - l / 2) ** 2 < (l / 2) ** 2
mask = np.zeros((l, l))
generator = np.random.RandomState(1)
points = l * generator.rand(2, n ** 2)
mask[(points[0]).astype(np.int), (points[1]).astype(np.int)] = 1
mask = ndimage.gaussian_filter(mask, sigma=l / (4. * n))
return (mask > mask.mean()).astype(np.float)
# Generate noisy synthetic data
data = microstructure(l=128)
data += 0.35 * np.random.randn(*data.shape)
markers = np.zeros(data.shape, dtype=np.uint)
markers[data < -0.3] = 1
markers[data > 1.3] = 2
# Run random walker algorithm
labels = random_walker(data, markers, beta=10, mode='bf')
# Plot results
plt.figure(figsize=(9, 3.5))
plt.subplot(131)
plt.imshow(data, cmap='gray', interpolation='nearest')
plt.axis('off')
plt.title('Noisy data')
plt.subplot(132)
plt.imshow(markers, cmap='hot', interpolation='nearest')
plt.axis('off')
plt.title('Markers')
plt.subplot(133)
plt.imshow(labels, cmap='gray', interpolation='nearest')
plt.axis('off')
plt.title('Segmentation')
plt.subplots_adjust(hspace=0.01, wspace=0.01, top=1, bottom=0, left=0,
right=1)
plt.show()
skimage/segmentation/random_walker.py
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"""
Random walker segmentation algorithm
from *Random walks for image segmentation*, Leo Grady, IEEE Trans
Pattern Anal Mach Intell. 2006 Nov;28(11):1768-83.
Dependencies:
* numpy >= 1.4, scipy
* optional: pyamg
Installing pyamg and using the 'cg_mg' mode of random_walker improves
significantly the performance.
"""
# Author: Emmanuelle Gouillart <emmanuelle.gouillart@normalesup.org>
# Copyright (c) 2009-2011, Emmanuelle Gouillart
# License: BSD
import warnings
import numpy as np
from scipy import sparse, ndimage
try:
from scipy.sparse.linalg.dsolve import umfpack
u = umfpack.UmfpackContext()
except:
warnings.warn("""Scipy was built without UMFPACK. Consider rebuilding
Scipy with UMFPACK, this will greatly speed up the random walker
functions. You may also install pyamg and run the random walker function
in cg_mg mode (see the docstrings)
""")
try:
from pyamg import ruge_stuben_solver
amg_loaded = True
except ImportError:
amg_loaded = False
from scipy.sparse.linalg import cg
#-----------Laplacian--------------------
def _make_edges_3d(n_x, n_y, n_z):
"""
Returns a list of edges for a 3D image.
Parameters
===========
n_x: integer
The size of the grid in the x direction.
n_y: integer
The size of the grid in the y direction
n_z: integer
The size of the grid in the z direction
"""
vertices = np.arange(n_x * n_y * n_z).reshape((n_x, n_y, n_z))
edges_deep = np.vstack((vertices[:, :, :-1].ravel(),
vertices[:, :, 1:].ravel()))
edges_right = np.vstack((vertices[:, :-1].ravel(),
vertices[:, 1:].ravel()))
edges_down = np.vstack((vertices[:-1].ravel(), vertices[1:].ravel()))
edges = np.hstack((edges_deep, edges_right, edges_down))
return edges
def _compute_weights_3d(edges, data, beta=130, eps=1.e-6):
l_x, l_y, l_z = data.shape
gradients = _compute_gradients_3d(data) ** 2
beta /= 10 * data.std()
gradients *= beta
weights = np.exp(- gradients)
weights += eps
return weights
def _compute_gradients_3d(data):
l_x, l_y, l_z = data.shape
gr_deep = np.abs(data[:, :, :-1] - data[:, :, 1:]).ravel()
gr_right = np.abs(data[:, :-1] - data[:, 1:]).ravel()
gr_down = np.abs(data[:-1] - data[1:]).ravel()
return np.r_[gr_deep, gr_right, gr_down]
def _make_laplacian_sparse(edges, weights):
"""
Sparse implementation
"""
pixel_nb = edges.max() + 1
diag = np.arange(pixel_nb)
i_indices = np.hstack((edges[0], edges[1]))
j_indices = np.hstack((edges[1], edges[0]))
data = np.hstack((-weights, -weights))
lap = sparse.coo_matrix((data, (i_indices, j_indices)),
shape=(pixel_nb, pixel_nb))
connect = - np.ravel(lap.sum(axis=1))
lap = sparse.coo_matrix((np.hstack((data, connect)),
(np.hstack((i_indices, diag)), np.hstack((j_indices, diag)))),
shape=(pixel_nb, pixel_nb))
return lap.tocsr()
def _clean_labels_ar(X, labels):
labels = np.ravel(labels)
labels[labels == 0] = X
return labels
def _buildAB(lap_sparse, labels):
"""
Build the matrix A and rhs B of the linear system to solve
"""
l_x, l_y, l_z = labels.shape
labels = labels[labels >= 0]
indices = np.arange(labels.size)
unlabeled_indices = indices[labels == 0]
seeds_indices = indices[labels > 0]
# The following two lines take most of the time
B = lap_sparse[unlabeled_indices][:, seeds_indices]
lap_sparse = lap_sparse[unlabeled_indices][:, unlabeled_indices]
nlabels = labels.max()
rhs = []
for lab in range(1, nlabels + 1):
mask = (labels[seeds_indices] == lab)
fs = sparse.csr_matrix(mask)
fs = fs.transpose()
rhs.append(B * fs)
return lap_sparse, rhs
def _trim_edges_weights(edges, weights, mask):
mask0 = np.hstack((mask[:, :, :-1].ravel(), mask[:, :-1].ravel(),
mask[:-1].ravel()))
mask1 = np.hstack((mask[:, :, 1:].ravel(), mask[:, 1:].ravel(),
mask[1:].ravel()))
ind_mask = np.logical_and(mask0, mask1)
edges, weights = edges[:, ind_mask], weights[ind_mask]
maxval = edges.max()
order = np.searchsorted(np.unique(edges.ravel()), np.arange(maxval + 1))
edges = order[edges]
return edges, weights
def _build_laplacian(data, mask=None, beta=50):
l_x, l_y, l_z = data.shape
edges = _make_edges_3d(l_x, l_y, l_z)
weights = _compute_weights_3d(edges, data, beta=beta, eps=1.e-10)
if mask is not None:
edges, weights = _trim_edges_weights(edges, weights, mask)
lap = _make_laplacian_sparse(edges, weights)
del edges, weights
return lap
#----------- Random walker algorithm --------------------------------
def random_walker(data, labels, beta=130, mode='bf', tol=1.e-3, copy=True):
"""
Random walker algorithm for segmentation from markers.
Parameters
----------
data : array_like
Image to be segmented in phases. `data` can be two- or
three-dimensional.
labels : array of ints, of same shape as `data`
Array of seed markers labeled with different positive integers
for different phases. Zero-labeled pixels are unlabeled pixels.
Negative labels correspond to inactive pixels that are not taken
into account (they are removed from the graph).
beta : float
Penalization coefficient for the random walker motion
(the greater `beta`, the more difficult the diffusion).
mode : {'bf', 'cg_mg', 'cg'} (default: 'bf')
Mode for solving the linear system in the random walker
algorithm.
- 'bf' (brute force, default): an LU factorization of the
Laplacian is computed. This is fast for small images (<1024x1024),
but very slow (due to the memory cost) and memory-consuming for
big images (in 3-D for example).
- 'cg' (conjugate gradient): the linear system is solved
iteratively using the Conjugate Gradient method from
scipy.sparse.linalg. This is less memory-consuming than the
brute force method for large images, but it is quite slow.
- 'cg_mg' (conjugate gradient with multigrid preconditioner):
a preconditioner is computed using a multigrid solver, then
the solution is computed with the Conjugate Gradient method.
This mode requires that the pyamg module
(http://code.google.com/p/pyamg/) is installed. For images of
size > 512x512, this is the recommended (fastest) mode.
tol : tolerance to achieve when solving the linear system, in
cg' and 'cg_mg' modes.
copy : bool
If copy is False, the `labels` array will be overwritten with
the result of the segmentation. Use copy=False if you want to
save on memory.
Returns
-------
output : ndarray of ints
Array in which each pixel has been labeled according to the marker
that reached the pixel first by anisotropic diffusion.
Notes
-----
The algorithm was first proposed in *Random walks for image
segmentation*, Leo Grady, IEEE Trans Pattern Anal Mach Intell.
2006 Nov;28(11):1768-83.
Examples
--------
>>> a = np.zeros((10, 10)) + 0.2*np.random.random((10, 10))
>>> a[5:8, 5:8] += 1
>>> b = np.zeros_like(a)
>>> b[3,3] = 1 #Marker for first phase
>>> b[6,6] = 2 #Marker for second phase
>>> random_walker(a, b)
array([[ 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.],
[ 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.],
[ 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.],
[ 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.],
[ 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.],
[ 1., 1., 1., 1., 1., 2., 2., 2., 1., 1.],
[ 1., 1., 1., 1., 1., 2., 2., 2., 1., 1.],
[ 1., 1., 1., 1., 1., 2., 2., 2., 1., 1.],
[ 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.],
[ 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.]])
"""
# We work with 3-D arrays
data = np.atleast_3d(data)
if copy:
labels = np.copy(labels)
labels = labels.astype(np.int)
# If the array has pruned zones, be sure that no isolated pixels
# exist between pruned zones (they could not be determined)
if np.any(labels < 0):
filled = ndimage.binary_propagation(labels > 0, mask=labels >= 0)
labels[np.logical_and(np.logical_not(filled), labels == 0)] = -1
del filled
labels = np.atleast_3d(labels)
if np.any(labels < 0):
lap_sparse = _build_laplacian(data, mask=labels >= 0, beta=beta)
else:
lap_sparse = _build_laplacian(data, beta=beta)
lap_sparse, B = _buildAB(lap_sparse, labels)
# We solve the linear system
# lap_sparse X = B
# where X[i, j] is the probability that a marker of label i arrives
# first at pixel j by anisotropic diffusion.
if mode == 'cg':
X = _solve_cg(lap_sparse, B, tol=tol)
if mode == 'cg_mg':
if not amg_loaded:
warnings.warn(
"""pyamg (http://code.google.com/p/pyamg/)) is needed to use
this mode, but is not installed. The 'cg' mode will be used
instead.""")
X = _solve_cg(lap_sparse, B, tol=tol)
else:
X = _solve_cg_mg(lap_sparse, B, tol=tol)
if mode == 'bf':
X = _solve_bf(lap_sparse, B)
X = _clean_labels_ar(X + 1, labels)
data = np.squeeze(data)
return X.reshape(data.shape)
def _solve_bf(lap_sparse, B):
"""
solves lap_sparse X_i = B_i for each phase i. An LU decomposition
of lap_sparse is computed first. For each pixel, the label i
corresponding to the maximal X_i is returned.
"""
lap_sparse = lap_sparse.tocsc()
solver = sparse.linalg.factorized(lap_sparse.astype(np.double))
X = np.array([solver(np.array((-B[i]).todense()).ravel())\
for i in range(len(B))])
X = np.argmax(X, axis=0)
return X
def _solve_cg(lap_sparse, B, tol):
"""
solves lap_sparse X_i = B_i for each phase i, using the conjugate
gradient method. For each pixel, the label i corresponding to the
maximal X_i is returned.
"""
lap_sparse = lap_sparse.tocsc()
X = []
for i in range(len(B)):
x0 = cg(lap_sparse, -B[i].todense(), tol=tol)[0]
X.append(x0)
X = np.array(X)
X = np.argmax(X, axis=0)
return X
def _solve_cg_mg(lap_sparse, B, tol):
"""
solves lap_sparse X_i = B_i for each phase i, using the conjugate
gradient method with a multigrid preconditioner (ruge-stuben from
pyamg). For each pixel, the label i corresponding to the maximal
X_i is returned.
"""
X = []
ml = ruge_stuben_solver(lap_sparse)
M = ml.aspreconditioner(cycle='V')
for i in range(len(B)):
x0 = cg(lap_sparse, -B[i].todense(), tol=tol, M=M, maxiter=30)[0]
X.append(x0)
X = np.array(X)
X = np.argmax(X, axis=0)
return X
skimage/segmentation/tests/test_random_walker.py
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import numpy as np
from random_walker import random_walker
try:
import pyamg
amg_loaded = True
except ImportError:
amg_loaded = False
def make_2d_syntheticdata(lx, ly=None):
if ly is None:
ly = lx
data = np.zeros((lx, ly)) + 0.1 * np.random.randn(lx, ly)
small_l = int(lx / 5)
data[lx / 2 - small_l:lx / 2 + small_l,
ly / 2 - small_l:ly / 2 + small_l] = 1
data[lx / 2 - small_l + 1:lx / 2 + small_l - 1,
ly / 2 - small_l + 1:ly / 2 + small_l - 1] = \
0.1 * np.random.randn(2 * small_l - 2, 2 * small_l - 2)
data[lx / 2 - small_l, ly / 2 - small_l / 8:ly / 2 + small_l / 8] = 0
seeds = np.zeros_like(data)
seeds[lx / 5, ly / 5] = 1
seeds[lx / 2 + small_l / 4, ly / 2 - small_l / 4] = 2
return data, seeds
def make_3d_syntheticdata(lx, ly=None, lz=None):
if ly is None:
ly = lx
if lz is None:
lz = lx
data = np.zeros((lx, ly, lz)) + 0.1 * np.random.randn(lx, ly, lz)
small_l = int(lx / 5)
data[lx / 2 - small_l:lx / 2 + small_l,
ly / 2 - small_l:ly / 2 + small_l,
lz / 2 - small_l:lz / 2 + small_l] = 1
data[lx / 2 - small_l + 1:lx / 2 + small_l - 1,
ly / 2 - small_l + 1:ly / 2 + small_l - 1,
lz / 2 - small_l + 1:lz / 2 + small_l - 1] = 0
# make a hole
hole_size = np.max([1, small_l / 8])
data[lx / 2 - small_l,
ly / 2 - hole_size:ly / 2 + hole_size,\
lz / 2 - hole_size:lz / 2 + hole_size] = 0
seeds = np.zeros_like(data)
seeds[lx / 5, ly / 5, lz / 5] = 1
seeds[lx / 2 + small_l / 4, ly / 2 - small_l / 4, lz / 2 - small_l / 4] = 2
return data, seeds
def test_2d_bf():
lx = 70
ly = 100
data, labels = make_2d_syntheticdata(lx, ly)
labels_bf = random_walker(data, labels, beta=90, mode='bf')
assert (labels_bf[25:45, 40:60] == 2).all()
return data, labels_bf
def test_2d_cg():
lx = 70
ly = 100
data, labels = make_2d_syntheticdata(lx, ly)
labels_cg = random_walker(data, labels, beta=90, mode='cg')
assert (labels_cg[25:45, 40:60] == 2).all()
return data, labels_cg
def test_2d_cg_mg():
lx = 70
ly = 100
data, labels = make_2d_syntheticdata(lx, ly)
labels_cg_mg = random_walker(data, labels, beta=90, mode='cg_mg')
assert (labels_cg_mg[25:45, 40:60] == 2).all()
return data, labels_cg_mg
def test_2d_inactive():
lx = 70
ly = 100
data, labels = make_2d_syntheticdata(lx, ly)
labels[10:20, 10:20] = -1
labels[46:50, 33:38] = -2
labels = random_walker(data, labels, beta=90)
assert (labels.reshape((lx, ly))[25:45, 40:60] == 2).all()
return data, labels
def test_3d():
n = 30
lx, ly, lz = n, n, n
data, labels = make_3d_syntheticdata(lx, ly, lz)
labels = random_walker(data, labels, mode='cg')
assert (labels.reshape(data.shape)[13:17, 13:17, 13:17] == 2).all()
return data, labels
def test_3d_inactive():
n = 30
lx, ly, lz = n, n, n
data, labels = make_3d_syntheticdata(lx, ly, lz)
old_labels = np.copy(labels)
labels[5:25, 26:29, 26:29] = -1
after_labels = np.copy(labels)
labels = random_walker(data, labels, mode='cg')
assert (labels.reshape(data.shape)[13:17, 13:17, 13:17] == 2).all()
return data, labels, old_labels, after_labels