def linear_operator_from_mesh(mesh_coord,
mesh_triangles,
mask=None,
offset=0,
weights=None):
"""Generates the linear operator for the total variation Nesterov function
from a mesh.
Parameters
----------
mesh_coord : Numpy array [n, 3] of float.
mesh_triangles : Numpy array, n_triangles-by-3. The (integer) indices of
the three nodes forming the triangle.
mask : Numpy array (shape (n,)) of integers/boolean. Non-null values
correspond to columns of X. Groups may be defined using different
values in the mask. TV will be applied within groups of the same
value in the mask.
offset : Non-negative integer. The index of the first column, variable,
where TV applies. This is different from penalty_start which
define where the penalty applies. The offset defines where TV
applies within the penalised variables.
Example: X := [Intercept, Age, Weight, Image]. Intercept is
not penalized, TV does not apply on Age and Weight but only on
Image. Thus: penalty_start = 1, offset = 2 (skip Age and
Weight).
weights : Numpy array. The weight put on the gradient of every point.
Default is weight 1 for each point, or equivalently, no weight. The
weights is a numpy array of the same shape as mask.
Returns
-------
out1 : List or sparse matrices. Linear operator for the total variation
Nesterov function computed over a mesh.
out2 : Integer. The number of compacts.
Examples
--------
>>> import numpy as np
>>> import parsimony.functions.nesterov.l1tv as tv_helper
>>> mesh_coord = np.array([[0, 0], [1, 0], [0, 1], [1, 1], [0, 2], [1, 2]])
>>> mesh_triangles = np.array([[0 ,1, 3], [0, 2 ,3], [2, 3, 5], [2, 4, 5]])
>>> A = tv_helper.linear_operator_from_mesh(mesh_coord,mesh_triangles)
"""
Atv = tv.linear_operator_from_mesh(mesh_coord=mesh_coord,
mesh_triangles=mesh_triangles,
mask=mask,
offset=offset,
weights=weights)
num_variables = mask.sum() if not mask is None else mesh_coord.shape[0]
Al1 = l1.linear_operator_from_variables(num_variables,
penalty_start=offset)
A = LinearOperator(Al1[0], *Atv)
return A
def linear_operator_from_mask(mask, num_variables, penalty_start=0):
"""Generates the linear operator for the total variation Nesterov function
from a mask for a 3D image.
Parameters
----------
mask : Numpy array. The mask. The mask does not involve any intercept
variables.
num_variables : Positive integer. The total number of variables, including
the intercept variable(s).
penalty_start : Non-negative integer. The number of variables to exempt
from penalisation. Equivalently, the first index to be penalised.
Default is 0, all variables are included.
"""
Atv = tv.linear_operator_from_mask(mask)
Al1 = l1.A_from_variables(num_variables, penalty_start=penalty_start)
A = LinearOperator(Al1[0], *Atv)
return A
def linear_operator_from_shape(shape, num_variables, penalty_start=0):
"""Generates the linear operator for the total variation Nesterov function
from the shape of a 3D image.
Parameters
----------
shape : List or tuple with 1, 2 or 3 elements. The shape of the 1D, 2D or
3D image. shape has the form (Z, Y, X), where Z is the number of
"layers", Y is the number of rows and X is the number of columns.
The shape does not involve any intercept variables.
num_variables : Positive integer. The total number of variables, including
the intercept variable(s).
penalty_start : Non-negative integer. The number of variables to exempt
from penalisation. Equivalently, the first index to be penalised.
Default is 0, all variables are included.
"""
Atv = tv.linear_operator_from_shape(shape)
Al1 = l1.linear_operator_from_variables(num_variables, penalty_start=penalty_start)
A = LinearOperator(Al1[0], *Atv)
return A
def linear_operator_from_mesh(mesh_coord, mesh_triangles, mask=None, offset=0,
weights=None):
"""Generates the linear operator for the total variation Nesterov function
from a mesh.
Parameters
----------
mesh_coord : Numpy array [n, 3] of float.
mesh_triangles : Numpy array, n_triangles-by-3. The (integer) indices of
the three nodes forming the triangle.
mask : Numpy array (shape (n,)) of integers/boolean. Non-null values
correspond to columns of X. Groups may be defined using different
values in the mask. TV will be applied within groups of the same
value in the mask.
offset : Non-negative integer. The index of the first column, variable,
where TV applies. This is different from penalty_start which
define where the penalty applies. The offset defines where TV
applies within the penalised variables.
Example: X := [Intercept, Age, Weight, Image]. Intercept is
not penalized, TV does not apply on Age and Weight but only on
Image. Thus: penalty_start = 1, offset = 2 (skip Age and
Weight).
weights : Numpy array. The weight put on the gradient of every point.
Default is weight 1 for each point, or equivalently, no weight. The
weights is a numpy array of the same shape as mask.
Returns
-------
out1 : List or sparse matrices. Linear operator for the total variation
Nesterov function computed over a mesh.
out2 : Integer. The number of compacts.
Examples
--------
>>> import numpy as np
>>> import parsimony.functions.nesterov.tv as tv_helper
>>> mesh_coord = np.array([[0, 0], [1, 0], [0, 1], [1, 1], [0, 2], [1, 2]])
>>> mesh_triangles = np.array([[0 ,1, 3], [0, 2 ,3], [2, 3, 5], [2, 4, 5]])
>>> A = tv_helper.linear_operator_from_mesh(mesh_coord, mesh_triangles)
"""
if mask is None:
mask = np.ones(mesh_coord.shape[0], dtype=bool)
assert mask.shape[0] == mesh_coord.shape[0]
mask_bool = mask != 0
mask_idx = np.where(mask_bool)[0]
# Mapping from full array to masked array.
map_full2masked = np.zeros(mask.shape, dtype=int)
map_full2masked[:] = -1
map_full2masked[mask_bool] = np.arange(np.sum(mask_bool)) + offset
## 1) Associate edges to nodes
nodes_with_edges = [[] for i in xrange(mesh_coord.shape[0])]
def connect_edge_to_node(node_idx1, node_idx2, nodes_with_edges):
# Attach edge to first node.
if np.sum(mesh_coord[node_idx1] - mesh_coord[node_idx2]) >= 0:
edge = [node_idx1, node_idx2]
if not edge in nodes_with_edges[node_idx1]:
nodes_with_edges[node_idx1].append(edge)
else: # attach edge to second node
edge = [node_idx2, node_idx1]
if not edge in nodes_with_edges[node_idx2]:
nodes_with_edges[node_idx2].append(edge)
for i in xrange(mesh_triangles.shape[0]):
t = mesh_triangles[i, :]
connect_edge_to_node(t[0], t[1], nodes_with_edges)
connect_edge_to_node(t[0], t[2], nodes_with_edges)
connect_edge_to_node(t[1], t[2], nodes_with_edges)
max_connectivity = np.max(np.array([len(n) for n in nodes_with_edges]))
# 3. build sparse matrices
# 1..max_connectivity of i, j and value
A = [[[], [], []] for i in xrange(max_connectivity)]
n_compacts = 0
for node_idx in mask_idx:
#node_idx = 0
found = False
node = nodes_with_edges[node_idx]
for i, v in enumerate(node):
found = False
if weights is not None:
w = weights[i]
else:
w = 1.0
#print i, v
node1_idx, node2_idx = v
if mask_bool[node1_idx] and mask_bool[node2_idx]:
found = True
A[i][0] += [map_full2masked[node1_idx],
map_full2masked[node1_idx]]
A[i][1] += [map_full2masked[node1_idx],
map_full2masked[node2_idx]]
A[i][2] += [-w, w]
if found:
n_compacts += 1
p = mask.sum()
A = [sparse.csr_matrix((A[i][2], (A[i][0], A[i][1])),
shape=(p, p)) for i in xrange(len(A))]
A = LinearOperator(*A)
A.n_compacts = n_compacts
return A
def linear_operator_from_shape(shape, weights=None):
"""Generates the linear operator for the total variation Nesterov function
from the shape of a 1D, 2D or 3D image.
Parameters
----------
shape : List or tuple with 1, 2 or 3 integers. The shape of the 1D, 2D or
3D image. shape has the form X, (X,), (Y, X) or (Z, Y, X), where Z
is the number of "layers", Y is the number of rows and X is the
number of columns. The shape does not involve any intercept
variables.
weights : Sequence, e.g. list or numpy (p-by-1) array. Weights put on the
groups. Default is weight 1 for each group, i.e. no weight.
"""
if not isinstance(shape, (list, tuple)):
shape = [shape]
while len(shape) < 3:
shape = tuple([1] + list(shape))
nz = shape[0]
ny = shape[1]
nx = shape[2]
p = nx * ny * nz
ind = np.arange(p).reshape((nz, ny, nx))
if weights is not None:
weights = np.array(weights)
weights = weights.ravel()
# w = sparse.spdiags(weights.ravel(), 0, p, p)
if nx > 1:
if weights is not None:
Ax = sparse.spdiags(weights, -1, p, p).T - \
sparse.spdiags(weights, 0, p, p)
Ax = Ax.tocsr()
else:
Ax = sparse.eye(p, p, 1, format='csr') - \
sparse.eye(p, p)
zind = ind[:, :, -1].ravel()
for i in zind:
Ax.data[Ax.indptr[i]: \
Ax.indptr[i + 1]] = 0
Ax.eliminate_zeros()
else:
Ax = sparse.csr_matrix((p, p), dtype=float)
if ny > 1:
if weights is not None:
Ay = sparse.spdiags(weights, -nx, p, p).T - \
sparse.spdiags(weights, 0, p, p)
Ay = Ay.tocsr()
else:
Ay = sparse.eye(p, p, nx, format='csr') - \
sparse.eye(p, p)
yind = ind[:, -1, :].ravel()
for i in yind:
Ay.data[Ay.indptr[i]: \
Ay.indptr[i + 1]] = 0
Ay.eliminate_zeros()
else:
Ay = sparse.csr_matrix((p, p), dtype=float)
if nz > 1:
if weights is not None:
Az = sparse.spdiags(weights, -(ny * nx), p, p).T - \
sparse.spdiags(weights, 0, p, p)
Az = Az.tocsr()
else:
Az = (sparse.eye(p, p, ny * nx, format='csr') - \
sparse.eye(p, p))
xind = ind[-1, :, :].ravel()
for i in xind:
Az.data[Az.indptr[i]: \
Az.indptr[i + 1]] = 0
Az.eliminate_zeros()
else:
Az = sparse.csr_matrix((p, p), dtype=float)
A = LinearOperator(Ax, Ay, Az)
A.n_compacts = (nz * ny * nx - 1)
return A
def linear_operator_from_subset_mask(mask, weights=None):
"""Generates the linear operator for the total variation Nesterov function
from a mask for a 3D image.
The binary mask marks a subset of the variables that are supposed to be
smoothed. The mask has the same size as the input and output image.
Parameters
----------
mask : Numpy array. The mask. The mask does not involve any intercept
variables.
weights : Numpy array. The weight put on the gradient of every point.
Default is weight 1 for each point, or equivalently, no weight. The
weights is a numpy array of the same shape as mask.
"""
while len(mask.shape) < 3:
mask = mask[np.newaxis, :]
if weights is not None:
while len(weights.shape) < 3:
weights = weights[np.newaxis, :]
nz, ny, nx = mask.shape
mask = mask.astype(bool)
zyx_mask = np.where(mask)
Ax_i = list()
Ax_j = list()
Ax_v = list()
Ay_i = list()
Ay_j = list()
Ay_v = list()
Az_i = list()
Az_j = list()
Az_v = list()
num_compacts = 0
# p = np.sum(mask)
# Mapping from image coordinate to flat masked array.
def im2flat(sub, dims):
return sub[0] * dims[2] * dims[1] + \
sub[1] * dims[2] + \
sub[2]
# im2flat = np.zeros(mask.shape, dtype=int)
# im2flat[:] = -1
# im2flat[mask] = np.arange(p)
# im2flat[np.arange(p)] = np.arange(p)
for pt in xrange(len(zyx_mask[0])):
found = False
z, y, x = zyx_mask[0][pt], zyx_mask[1][pt], zyx_mask[2][pt]
i_pt = im2flat((z, y, x), mask.shape)
if weights is not None:
w = weights[z, y, x]
else:
w = 1.0
if z + 1 < nz and mask[z + 1, y, x]:
found = True
Az_i += [i_pt, i_pt]
Az_j += [i_pt, im2flat((z + 1, y, x), mask.shape)]
Az_v += [-w, w]
if y + 1 < ny and mask[z, y + 1, x]:
found = True
Ay_i += [i_pt, i_pt]
Ay_j += [i_pt, im2flat((z, y + 1, x), mask.shape)]
Ay_v += [-w, w]
if x + 1 < nx and mask[z, y, x + 1]:
found = True
Ax_i += [i_pt, i_pt]
Ax_j += [i_pt, im2flat((z, y, x + 1), mask.shape)]
Ax_v += [-w, w]
if found:
num_compacts += 1
p = np.prod(mask.shape)
Az = sparse.csr_matrix((Az_v, (Az_i, Az_j)), shape=(p, p))
Ay = sparse.csr_matrix((Ay_v, (Ay_i, Ay_j)), shape=(p, p))
Ax = sparse.csr_matrix((Ax_v, (Ax_i, Ax_j)), shape=(p, p))
A = LinearOperator(Ax, Ay, Az)
A.n_compacts = num_compacts
return A
def linear_operator_from_mask(mask, offset=0, weights=None):
"""Generates the linear operator for the total variation Nesterov function
from a mask for a 3D image.
Parameters
----------
mask : Numpy array of integers. The mask has the same shape as the original
data. Non-null values correspond to columns of X. Groups may be
defined using different values in the mask. TV will be applied
within groups of the same value in the mask.
offset: Non-negative integer. The index of the first column, variable,
where TV applies. This is different from penalty_start which
define where the penalty applies. The offset defines where TV
applies within the penalised variables.
Example: X := [Intercept, Age, Weight, Image]. Intercept is
not penalized, TV does not apply on Age and Weight but only on
Image. Thus: penalty_start = 1, offset = 2 (skip Age and
Weight).
weights : Numpy array. The weight put on the gradient of every point.
Default is weight 1 for each point, or equivalently, no weight. The
weights is a numpy array of the same shape as mask.
"""
while len(mask.shape) < 3:
mask = mask[..., np.newaxis]
if weights is not None:
while len(weights.shape) < 3:
weights = weights[..., np.newaxis]
nx, ny, nz = mask.shape
mask_bool = mask != 0
xyz_mask = np.where(mask_bool)
Ax_i = list()
Ax_j = list()
Ax_v = list()
Ay_i = list()
Ay_j = list()
Ay_v = list()
Az_i = list()
Az_j = list()
Az_v = list()
n_compacts = 0
p = np.sum(mask_bool) + offset
# Mapping from image coordinate to flat masked array.
im2flat = np.zeros(mask.shape, dtype=int)
im2flat[:] = -1
im2flat[mask_bool] = np.arange(np.sum(mask_bool)) + offset
for pt in xrange(len(xyz_mask[0])):
found = False
x, y, z = xyz_mask[0][pt], xyz_mask[1][pt], xyz_mask[2][pt]
i_pt = im2flat[x, y, z]
val = mask[x, y, z]
if weights is not None:
w = weights[x, y, z]
else:
w = 1.0
if x + 1 < nx and (mask[x + 1, y, z] == val):
found = True
Ax_i += [i_pt, i_pt]
Ax_j += [i_pt, im2flat[x + 1, y, z]]
Ax_v += [-w, w]
if y + 1 < ny and (mask[x, y + 1, z] == val):
found = True
Ay_i += [i_pt, i_pt]
Ay_j += [i_pt, im2flat[x, y + 1, z]]
Ay_v += [-w, w]
if z + 1 < nz and (mask[x, y, z + 1] == val):
found = True
Az_i += [i_pt, i_pt]
Az_j += [i_pt, im2flat[x, y, z + 1]]
Az_v += [-w, w]
if found:
n_compacts += 1
Ax = sparse.csr_matrix((Ax_v, (Ax_i, Ax_j)), shape=(p, p))
Ay = sparse.csr_matrix((Ay_v, (Ay_i, Ay_j)), shape=(p, p))
Az = sparse.csr_matrix((Az_v, (Az_i, Az_j)), shape=(p, p))
A = LinearOperator(Ax, Ay, Az)
A.n_compacts = n_compacts
return A
def linear_operator_from_mesh(mesh_coord,
mesh_triangles,
mask=None,
offset=0,
weights=None):
"""Generates the linear operator for the total variation Nesterov function
from a mesh.
Parameters
----------
mesh_coord : Numpy array [n, 3] of float.
mesh_triangles : Numpy array, n_triangles-by-3. The (integer) indices of
the three nodes forming the triangle.
mask : Numpy array (shape (n,)) of integers/boolean. Non-null values
correspond to columns of X. Groups may be defined using different
values in the mask. TV will be applied within groups of the same
value in the mask.
offset : Non-negative integer. The index of the first column, variable,
where TV applies. This is different from penalty_start which
define where the penalty applies. The offset defines where TV
applies within the penalised variables.
Example: X := [Intercept, Age, Weight, Image]. Intercept is
not penalized, TV does not apply on Age and Weight but only on
Image. Thus: penalty_start = 1, offset = 2 (skip Age and
Weight).
weights : Numpy array. The weight put on the gradient of every point.
Default is weight 1 for each point, or equivalently, no weight. The
weights is a numpy array of the same shape as mask.
Returns
-------
out1 : List or sparse matrices. Linear operator for the total variation
Nesterov function computed over a mesh.
out2 : Integer. The number of compacts.
Examples
--------
>>> import numpy as np
>>> import parsimony.functions.nesterov.tv as tv_helper
>>> mesh_coord = np.array([[0, 0], [1, 0], [0, 1], [1, 1], [0, 2], [1, 2]])
>>> mesh_triangles = np.array([[0 ,1, 3], [0, 2 ,3], [2, 3, 5], [2, 4, 5]])
>>> A = tv_helper.linear_operator_from_mesh(mesh_coord, mesh_triangles)
"""
if mask is None:
mask = np.ones(mesh_coord.shape[0], dtype=bool)
assert mask.shape[0] == mesh_coord.shape[0]
mask_bool = mask != 0
mask_idx = np.where(mask_bool)[0]
# Mapping from full array to masked array.
map_full2masked = np.zeros(mask.shape, dtype=int)
map_full2masked[:] = -1
map_full2masked[mask_bool] = np.arange(np.sum(mask_bool)) + offset
## 1) Associate edges to nodes
nodes_with_edges = [[] for i in xrange(mesh_coord.shape[0])]
def connect_edge_to_node(node_idx1, node_idx2, nodes_with_edges):
# Attach edge to first node.
if np.sum(mesh_coord[node_idx1] - mesh_coord[node_idx2]) >= 0:
edge = [node_idx1, node_idx2]
if not edge in nodes_with_edges[node_idx1]:
nodes_with_edges[node_idx1].append(edge)
else: # attach edge to second node
edge = [node_idx2, node_idx1]
if not edge in nodes_with_edges[node_idx2]:
nodes_with_edges[node_idx2].append(edge)
for i in xrange(mesh_triangles.shape[0]):
t = mesh_triangles[i, :]
connect_edge_to_node(t[0], t[1], nodes_with_edges)
connect_edge_to_node(t[0], t[2], nodes_with_edges)
connect_edge_to_node(t[1], t[2], nodes_with_edges)
max_connectivity = np.max(np.array([len(n) for n in nodes_with_edges]))
# 3. build sparse matrices
# 1..max_connectivity of i, j and value
A = [[[], [], []] for i in xrange(max_connectivity)]
n_compacts = 0
for node_idx in mask_idx:
#node_idx = 0
found = False
node = nodes_with_edges[node_idx]
for i, v in enumerate(node):
found = False
if weights is not None:
w = weights[i]
else:
w = 1.0
#print i, v
node1_idx, node2_idx = v
if mask_bool[node1_idx] and mask_bool[node2_idx]:
found = True
A[i][0] += [
map_full2masked[node1_idx], map_full2masked[node1_idx]
]
A[i][1] += [
map_full2masked[node1_idx], map_full2masked[node2_idx]
]
A[i][2] += [-w, w]
if found:
n_compacts += 1
p = mask.sum()
A = [
sparse.csr_matrix((A[i][2], (A[i][0], A[i][1])), shape=(p, p))
for i in xrange(len(A))
]
A = LinearOperator(*A)
A.n_compacts = n_compacts
return A
def linear_operator_from_shape(shape, weights=None):
"""Generates the linear operator for the total variation Nesterov function
from the shape of a 1D, 2D or 3D image.
Parameters
----------
shape : List or tuple with 1, 2 or 3 integers. The shape of the 1D, 2D or
3D image. shape has the form X, (X,), (Y, X) or (Z, Y, X), where Z
is the number of "layers", Y is the number of rows and X is the
number of columns. The shape does not involve any intercept
variables.
weights : Sequence, e.g. list or numpy (p-by-1) array. Weights put on the
groups. Default is weight 1 for each group, i.e. no weight.
"""
if not isinstance(shape, (list, tuple)):
shape = [shape]
while len(shape) < 3:
shape = tuple([1] + list(shape))
nz = shape[0]
ny = shape[1]
nx = shape[2]
p = nx * ny * nz
ind = np.arange(p).reshape((nz, ny, nx))
if weights is not None:
weights = np.array(weights)
weights = weights.ravel()
# w = sparse.spdiags(weights.ravel(), 0, p, p)
if nx > 1:
if weights is not None:
Ax = sparse.spdiags(weights, -1, p, p).T - \
sparse.spdiags(weights, 0, p, p)
Ax = Ax.tocsr()
else:
Ax = sparse.eye(p, p, 1, format='csr') - \
sparse.eye(p, p)
zind = ind[:, :, -1].ravel()
for i in zind:
Ax.data[Ax.indptr[i]: \
Ax.indptr[i + 1]] = 0
Ax.eliminate_zeros()
else:
Ax = sparse.csr_matrix((p, p), dtype=float)
if ny > 1:
if weights is not None:
Ay = sparse.spdiags(weights, -nx, p, p).T - \
sparse.spdiags(weights, 0, p, p)
Ay = Ay.tocsr()
else:
Ay = sparse.eye(p, p, nx, format='csr') - \
sparse.eye(p, p)
yind = ind[:, -1, :].ravel()
for i in yind:
Ay.data[Ay.indptr[i]: \
Ay.indptr[i + 1]] = 0
Ay.eliminate_zeros()
else:
Ay = sparse.csr_matrix((p, p), dtype=float)
if nz > 1:
if weights is not None:
Az = sparse.spdiags(weights, -(ny * nx), p, p).T - \
sparse.spdiags(weights, 0, p, p)
Az = Az.tocsr()
else:
Az = (sparse.eye(p, p, ny * nx, format='csr') - \
sparse.eye(p, p))
xind = ind[-1, :, :].ravel()
for i in xind:
Az.data[Az.indptr[i]: \
Az.indptr[i + 1]] = 0
Az.eliminate_zeros()
else:
Az = sparse.csr_matrix((p, p), dtype=float)
A = LinearOperator(Ax, Ay, Az)
A.n_compacts = (nz * ny * nx - 1)
return A
def linear_operator_from_subset_mask(mask, weights=None):
"""Generates the linear operator for the total variation Nesterov function
from a mask for a 3D image.
The binary mask marks a subset of the variables that are supposed to be
smoothed. The mask has the same size as the input and output image.
Parameters
----------
mask : Numpy array. The mask. The mask does not involve any intercept
variables.
weights : Numpy array. The weight put on the gradient of every point.
Default is weight 1 for each point, or equivalently, no weight. The
weights is a numpy array of the same shape as mask.
"""
while len(mask.shape) < 3:
mask = mask[np.newaxis, :]
if weights is not None:
while len(weights.shape) < 3:
weights = weights[np.newaxis, :]
nz, ny, nx = mask.shape
mask = mask.astype(bool)
zyx_mask = np.where(mask)
Ax_i = list()
Ax_j = list()
Ax_v = list()
Ay_i = list()
Ay_j = list()
Ay_v = list()
Az_i = list()
Az_j = list()
Az_v = list()
num_compacts = 0
# p = np.sum(mask)
# Mapping from image coordinate to flat masked array.
def im2flat(sub, dims):
return sub[0] * dims[2] * dims[1] + \
sub[1] * dims[2] + \
sub[2]
# im2flat = np.zeros(mask.shape, dtype=int)
# im2flat[:] = -1
# im2flat[mask] = np.arange(p)
# im2flat[np.arange(p)] = np.arange(p)
for pt in xrange(len(zyx_mask[0])):
found = False
z, y, x = zyx_mask[0][pt], zyx_mask[1][pt], zyx_mask[2][pt]
i_pt = im2flat((z, y, x), mask.shape)
if weights is not None:
w = weights[z, y, x]
else:
w = 1.0
if z + 1 < nz and mask[z + 1, y, x]:
found = True
Az_i += [i_pt, i_pt]
Az_j += [i_pt, im2flat((z + 1, y, x), mask.shape)]
Az_v += [-w, w]
if y + 1 < ny and mask[z, y + 1, x]:
found = True
Ay_i += [i_pt, i_pt]
Ay_j += [i_pt, im2flat((z, y + 1, x), mask.shape)]
Ay_v += [-w, w]
if x + 1 < nx and mask[z, y, x + 1]:
found = True
Ax_i += [i_pt, i_pt]
Ax_j += [i_pt, im2flat((z, y, x + 1), mask.shape)]
Ax_v += [-w, w]
if found:
num_compacts += 1
p = np.prod(mask.shape)
Az = sparse.csr_matrix((Az_v, (Az_i, Az_j)), shape=(p, p))
Ay = sparse.csr_matrix((Ay_v, (Ay_i, Ay_j)), shape=(p, p))
Ax = sparse.csr_matrix((Ax_v, (Ax_i, Ax_j)), shape=(p, p))
A = LinearOperator(Ax, Ay, Az)
A.n_compacts = num_compacts
return A
def linear_operator_from_mask(mask, offset=0, weights=None):
"""Generates the linear operator for the total variation Nesterov function
from a mask for a 3D image.
Parameters
----------
mask : Numpy array of integers. The mask has the same shape as the original
data. Non-null values correspond to columns of X. Groups may be
defined using different values in the mask. TV will be applied
within groups of the same value in the mask.
offset: Non-negative integer. The index of the first column, variable,
where TV applies. This is different from penalty_start which
define where the penalty applies. The offset defines where TV
applies within the penalised variables.
Example: X := [Intercept, Age, Weight, Image]. Intercept is
not penalized, TV does not apply on Age and Weight but only on
Image. Thus: penalty_start = 1, offset = 2 (skip Age and
Weight).
weights : Numpy array. The weight put on the gradient of every point.
Default is weight 1 for each point, or equivalently, no weight. The
weights is a numpy array of the same shape as mask.
"""
while len(mask.shape) < 3:
mask = mask[..., np.newaxis]
if weights is not None:
while len(weights.shape) < 3:
weights = weights[..., np.newaxis]
nx, ny, nz = mask.shape
mask_bool = mask != 0
xyz_mask = np.where(mask_bool)
Ax_i = list()
Ax_j = list()
Ax_v = list()
Ay_i = list()
Ay_j = list()
Ay_v = list()
Az_i = list()
Az_j = list()
Az_v = list()
n_compacts = 0
p = np.sum(mask_bool) + offset
# Mapping from image coordinate to flat masked array.
im2flat = np.zeros(mask.shape, dtype=int)
im2flat[:] = -1
im2flat[mask_bool] = np.arange(np.sum(mask_bool)) + offset
for pt in xrange(len(xyz_mask[0])):
found = False
x, y, z = xyz_mask[0][pt], xyz_mask[1][pt], xyz_mask[2][pt]
i_pt = im2flat[x, y, z]
val = mask[x, y, z]
if weights is not None:
w = weights[x, y, z]
else:
w = 1.0
if x + 1 < nx and (mask[x + 1, y, z] == val):
found = True
Ax_i += [i_pt, i_pt]
Ax_j += [i_pt, im2flat[x + 1, y, z]]
Ax_v += [-w, w]
if y + 1 < ny and (mask[x, y + 1, z] == val):
found = True
Ay_i += [i_pt, i_pt]
Ay_j += [i_pt, im2flat[x, y + 1, z]]
Ay_v += [-w, w]
if z + 1 < nz and (mask[x, y, z + 1] == val):
found = True
Az_i += [i_pt, i_pt]
Az_j += [i_pt, im2flat[x, y, z + 1]]
Az_v += [-w, w]
if found:
n_compacts += 1
Ax = sparse.csr_matrix((Ax_v, (Ax_i, Ax_j)), shape=(p, p))
Ay = sparse.csr_matrix((Ay_v, (Ay_i, Ay_j)), shape=(p, p))
Az = sparse.csr_matrix((Az_v, (Az_i, Az_j)), shape=(p, p))
A = LinearOperator(Ax, Ay, Az)
A.n_compacts = n_compacts
return A
