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scikit-image/skimage/segmentation/random_walker_segmentation.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.
Installing pyamg and using the 'cg_mg' mode of random_walker improves
significantly the performance.
"""
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_graph_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
Returns
-------
edges : (2, N) ndarray
with the total number of edges N = n_x * n_y * (nz - 1) +
n_x * (n_y - 1) * nz +
(n_x - 1) * n_y * nz
Graph edges with each column describing a node-id pair.
"""
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(data, beta=130, eps=1.e-6):
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):
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.
A and B are two block of the laplacian of the image graph.
"""
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 in this function
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 _mask_edges_weights(edges, weights, mask):
"""
Remove edges of the graph connected to masked nodes, as well as
corresponding weights of the edges.
"""
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]
max_node_index = edges.max()
# Reassign edges labels to 0, 1, ... edges_number - 1
order = np.searchsorted(np.unique(edges.ravel()),
np.arange(max_node_index + 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_graph_edges_3d(l_x, l_y, l_z)
weights = _compute_weights_3d(data, beta=beta, eps=1.e-10)
if mask is not None:
edges, weights = _mask_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 : float
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.
See also
--------
skimage.morphology.watershed: watershed segmentation
A segmentation algorithm based on mathematical morphology
and "flooding" of regions from markers.
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.
The algorithm solves the diffusion equation at infinite times for
sources placed on markers of each phase in turn. A pixel is labeled with
the phase that has the greatest probability to diffuse first to the pixel.
The diffusion equation is solved by minimizing x.T L x for each phase,
where L is the Laplacian of the weighted graph of the image, and x is
the probability that a marker of the given phase arrives first at a pixel
by diffusion (x=1 on markers of the phase, x=0 on the other markers, and
the other coefficients are looked for). Each pixel is attributed the label
for which it has a maximal value of x. The Laplacian L of the image
is defined as:
- L_ii = d_i, the number of neighbors of pixel i (the degree of i)
- L_ij = -w_ij if i and j are adjacent pixels
The weight w_ij is a decreasing function of the norm of the local gradient.
This ensures that diffusion is easier between pixels of similar values.
When the Laplacian is decomposed into blocks of marked and unmarked pixels::
L = M B.T
B A
with first indices corresponding to marked pixels, and then to unmarked
pixels, minimizing x.T L x for one phase amount to solving::
A x = - B x_m
where x_m=1 on markers of the given phase, and 0 on other markers.
This linear system is solved in the algorithm using a direct method for
small images, and an iterative method for larger images.
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.intp)
# 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
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