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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 numpy as np
from scipy import sparse, ndimage as ndi
from .._shared.utils import warn
# executive summary for next code block: try to import umfpack from
# scipy, but make sure not to raise a fuss if it fails since it's only
# needed to speed up a few cases.
# See discussions at:
from scipy.sparse.linalg.dsolve import umfpack
old_del = umfpack.UmfpackContext.__del__
def new_del(self):
except AttributeError:
umfpack.UmfpackContext.__del__ = new_del
UmfpackContext = umfpack.UmfpackContext()
UmfpackContext = None
from pyamg import ruge_stuben_solver
amg_loaded = True
except ImportError:
amg_loaded = False
from ..util import img_as_float
from ..filters import rank_order
from scipy.sparse.linalg import cg
import scipy
from distutils.version import LooseVersion as Version
import functools
if Version(scipy.__version__) >= Version('1.1'):
cg = functools.partial(cg, atol=0)
# -----------Laplacian--------------------
def _make_graph_edges_3d(n_x, n_y, n_z):
"""Returns a list of edges for a 3D image.
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
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, spacing, beta=130, eps=1.e-6,
# Weight calculation is main difference in multispectral version
# Original gradient**2 replaced with sum of gradients ** 2
gradients = 0
for channel in range(0, data.shape[-1]):
gradients += _compute_gradients_3d(data[..., channel],
spacing) ** 2
# All channels considered together in this standard deviation
beta /= 10 * data.std()
if multichannel:
# New final term in beta to give == results in trivial case where
# multiple identical spectra are passed.
beta /= np.sqrt(data.shape[-1])
gradients *= beta
weights = np.exp(- gradients)
weights += eps
return weights
def _compute_gradients_3d(data, spacing):
gr_deep = np.abs(data[:, :, :-1] - data[:, :, 1:]).ravel() / spacing[2]
gr_right = np.abs(data[:, :-1] - data[:, 1:]).ravel() / spacing[1]
gr_down = np.abs(data[:-1] - data[1:]).ravel() / spacing[0]
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, copy=False):
X = X.astype(labels.dtype)
if copy:
labels = np.copy(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(),
mask1 = np.hstack((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.astype(np.int64)]
return edges, weights
def _build_laplacian(data, spacing, mask=None, beta=50,
l_x, l_y, l_z = tuple(data.shape[i] for i in range(3))
edges = _make_graph_edges_3d(l_x, l_y, l_z)
weights = _compute_weights_3d(data, spacing, 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
def _check_isolated_seeds(labels):
Prune isolated seed pixels to prevent labeling errors, and
return coordinates and label values of isolated seeds, so
that it is possible to put labels back in random walker output.
fill = ndi.binary_propagation(labels == 0, mask=(labels >= 0))
isolated = np.logical_and(labels > 0, np.logical_not(fill))
inds = np.nonzero(isolated)
values = labels[inds]
labels[inds] = -1
return inds, values
def _unchanged_labels(labels, return_full_prob=False):
Return the input array of labels, unless ``return_full_prob`` is True,
in which case a mask of pixels is returned for each unique label, with the
last dimension corresponding to unique labels.
if return_full_prob:
# Find and iterate over valid labels
unique_labels = np.unique(labels)
unique_labels = unique_labels[unique_labels > 0]
out_labels = np.empty(labels.shape + (len(unique_labels),),
for n, i in enumerate(unique_labels):
out_labels[..., n] = (labels == i)
out_labels = labels
return out_labels
# ----------- Random walker algorithm --------------------------------
def random_walker(data, labels, beta=130, mode='bf', tol=1.e-3, copy=True,
multichannel=False, return_full_prob=False, spacing=None):
"""Random walker algorithm for segmentation from markers.
Random walker algorithm is implemented for gray-level or multichannel
data : array_like
Image to be segmented in phases. Gray-level `data` can be two- or
three-dimensional; multichannel data can be three- or four-
dimensional (multichannel=True) with the highest dimension denoting
channels. Data spacing is assumed isotropic unless the `spacing`
keyword argument is used.
labels : array of ints, of same shape as `data` without channels dimension
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). If labels are not
consecutive integers, the labels array will be transformed so that
labels are consecutive. In the multichannel case, `labels` should have
the same shape as a single channel of `data`, i.e. without the final
dimension denoting channels.
beta : float, optional
Penalization coefficient for the random walker motion
(the greater `beta`, the more difficult the diffusion).
mode : string, available options {'cg_mg', 'cg', 'bf'}
Mode for solving the linear system in the random walker algorithm.
If no preference given, automatically attempt to use the fastest
option available ('cg_mg' from pyamg >> 'cg' with UMFPACK > 'bf').
- 'bf' (brute force): an LU factorization of the Laplacian is
computed. This is fast for small images (<1024x1024), but very slow
and memory-intensive for large images (e.g., 3-D volumes).
- '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 ( is
installed. For images of size > 512x512, this is the recommended
(fastest) mode.
tol : float, optional
tolerance to achieve when solving the linear system, in
cg' and 'cg_mg' modes.
copy : bool, optional
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.
multichannel : bool, optional
If True, input data is parsed as multichannel data (see 'data' above
for proper input format in this case).
return_full_prob : bool, optional
If True, the probability that a pixel belongs to each of the labels
will be returned, instead of only the most likely label.
spacing : iterable of floats, optional
Spacing between voxels in each spatial dimension. If `None`, then
the spacing between pixels/voxels in each dimension is assumed 1.
output : ndarray
* If `return_full_prob` is False, array of ints of same shape as
`data`, in which each pixel has been labeled according to the marker
that reached the pixel first by anisotropic diffusion.
* If `return_full_prob` is True, array of floats of shape
`(nlabels, data.shape)`. `output[label_nb, i, j]` is the probability
that label `label_nb` reaches the pixel `(i, j)` first.
See also
skimage.morphology.watershed: watershed segmentation
A segmentation algorithm based on mathematical morphology
and "flooding" of regions from markers.
Multichannel inputs are scaled with all channel data combined. Ensure all
channels are separately normalized prior to running this algorithm.
The `spacing` argument is specifically for anisotropic datasets, where
data points are spaced differently in one or more spatial dimensions.
Anisotropic data is commonly encountered in medical imaging.
The algorithm was first proposed in [1]_.
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
L = M B.T
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.
.. [1] Leo Grady, Random walks for image segmentation, IEEE Trans Pattern
Anal Mach Intell. 2006 Nov;28(11):1768-83.
>>> np.random.seed(0)
>>> a = np.zeros((10, 10)) + 0.2 * np.random.rand(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]], dtype=int32)
# Parse input data
if mode is None:
if amg_loaded:
mode = 'cg_mg'
elif UmfpackContext is not None:
mode = 'cg'
mode = 'bf'
elif mode not in ('cg_mg', 'cg', 'bf'):
raise ValueError("{mode} is not a valid mode. Valid modes are 'cg_mg',"
" 'cg' and 'bf'".format(mode=mode))
if UmfpackContext is None and mode == 'cg':
warn('"cg" mode will be used, but it may be slower than '
'"bf" because SciPy was built without UMFPACK. Consider'
' rebuilding SciPy with UMFPACK; this will greatly '
'accelerate the conjugate gradient ("cg") solver. '
'You may also install pyamg and run the random_walker '
'function in "cg_mg" mode (see docstring).')
if (labels != 0).all():
warn('Random walker only segments unlabeled areas, where '
'labels == 0. No zero valued areas in labels were '
'found. Returning provided labels.')
return _unchanged_labels(labels, return_full_prob)
# This algorithm expects 4-D arrays of floats, where the first three
# dimensions are spatial and the final denotes channels. 2-D images have
# a singleton placeholder dimension added for the third spatial dimension,
# and single channel images likewise have a singleton added for channels.
# The following block ensures valid input and coerces it to the correct
# form.
if not multichannel:
if data.ndim not in (2, 3):
raise ValueError('For non-multichannel input, data must be of '
'dimension 2 or 3.')
dims = data.shape # To reshape final labeled result
data = np.atleast_3d(img_as_float(data))[..., np.newaxis]
if data.ndim not in (3, 4):
raise ValueError('For multichannel input, data must have 3 or 4 '
dims = data[..., 0].shape # To reshape final labeled result
data = img_as_float(data)
if data.ndim == 3: # 2D multispectral, needs singleton in 3rd axis
data = data[:, :, np.newaxis, :]
# Spacing kwarg checks
if spacing is None:
spacing = np.asarray((1.,) * 3)
elif len(spacing) == len(dims):
if len(spacing) == 2: # Need a dummy spacing for singleton 3rd dim
spacing = np.r_[spacing, 1.]
else: # Convert to array
spacing = np.asarray(spacing)
raise ValueError('Input argument `spacing` incorrect, should be an '
'iterable with one number per spatial dimension.')
if copy:
labels = np.copy(labels)
label_values = np.unique(labels)
# If some labeled pixels are isolated inside pruned zones, prune them
# as well and keep the labels for the final output
inds_isolated_seeds, isolated_values = _check_isolated_seeds(labels)
# Reorder label values to have consecutive integers (no gaps)
if np.any(np.diff(label_values) != 1):
mask = labels >= 0
labels[mask] = rank_order(labels[mask])[0].astype(labels.dtype)
labels = labels.astype(np.int32)
# 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 = ndi.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)
# No unlabeled pixel, so nothing to do
if (labels == 0).sum() == 0:
labels = np.squeeze(labels)
labels[inds_isolated_seeds] = isolated_values
warn('Random walker only segments unlabeled areas, where '
'labels == 0. No zero valued areas in labels were '
'found. Returning provided labels.')
return _unchanged_labels(labels, return_full_prob)
if np.any(labels < 0):
lap_sparse = _build_laplacian(data, spacing, mask=labels >= 0,
beta=beta, multichannel=multichannel)
lap_sparse = _build_laplacian(data, spacing, 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:
warn("""pyamg ( is needed to use
this mode, but is not installed. The 'cg' mode will be used
X = _solve_cg(lap_sparse, B, tol=tol,
X = _solve_cg_mg(lap_sparse, B, tol=tol,
if mode == 'bf':
X = _solve_bf(lap_sparse, B,
# Clean up results
if return_full_prob:
labels = labels.astype(np.float)
# Put back labels of isolated seeds
if len(isolated_values) > 0:
labels[inds_isolated_seeds] = isolated_values
X = np.array([_clean_labels_ar(Xline, labels, copy=True).reshape(dims)
for Xline in X])
for i in range(1, int(labels.max()) + 1):
mask_i = np.squeeze(labels == i)
X[:, mask_i] = 0
X[i - 1, mask_i] = 1
X = _clean_labels_ar(X + 1, labels).reshape(dims)
# Put back labels of isolated seeds
X[inds_isolated_seeds] = isolated_values
return X
def _solve_bf(lap_sparse, B, return_full_prob=False):
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]).toarray()).ravel())
for i in range(len(B))])
if not return_full_prob:
X = np.argmax(X, axis=0)
return X
def _solve_cg(lap_sparse, B, tol, return_full_prob=False):
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].toarray(), tol=tol)[0]
if not return_full_prob:
X = np.array(X)
X = np.argmax(X, axis=0)
return X
def _solve_cg_mg(lap_sparse, B, tol, return_full_prob=False):
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].toarray(), tol=tol, M=M, maxiter=30)[0]
if not return_full_prob:
X = np.array(X)
X = np.argmax(X, axis=0)
return X