def canonize_hierarchy(tree, altitudes, return_node_map=False):
"""
Removes consecutive tree nodes with equal altitudes.
The new tree is composed of the inner nodes :math:`n` of the input tree such that
:math:`altitudes[n] \\neq altitudes[tree.parent(n)]` or :math:`n = tree.root(n)`.
For example, applying this function to the result of :func:`~higra.bpt_canonical` on an edge weighted graph
is the same as computing the :func:`~higra.quasi_flat_zone_hierarchy` of the same edge weighted graph.
If :attr:`return_node_map` is ``True``, an extra array that maps any vertex index :math:`i` of the new tree,
to the index of the corresponding vertex in the original tree is returned.
:param tree: input tree
:param altitudes: altitudes of the vertices of the tree
:param return_node_map: if ``True``, also return the node map.
:return: a tree (Concept :class:`~higra.CptHierarchy` if input tree already satisfied this concept)
its node altitudes, and, if requested, its node map.
"""
tree, node_map = hg.simplify_tree(tree,
altitudes == altitudes[tree.parents()])
new_altitudes = altitudes[node_map]
if return_node_map:
return tree, new_altitudes, node_map
else:
return tree, new_altitudes
def num_parents(bpt_tree, altitudes):
# construct quasi flat zone hierarchy from input bpt
tree, node_map = hg.simplify_tree(
bpt_tree, altitudes == altitudes[bpt_tree.parents()])
# determine inner nodes of the min tree, i.e. nodes of qfz having at least one node that is not a leaf
num_children = tree.num_children(np.arange(tree.num_vertices()))
num_children_leaf = np.zeros((tree.num_vertices(), ), dtype=np.int64)
np.add.at(num_children_leaf, tree.parents()[:tree.num_leaves()], 1)
inner_nodes = num_children != num_children_leaf
# go back into bpt space
inner_nodes_bpt = np.zeros((bpt_tree.num_vertices(), ), dtype=np.int64)
inner_nodes_bpt[node_map] = inner_nodes
inner_nodes = inner_nodes_bpt
# count number of min tree inner nodes in the subtree rooted in the given node
res = hg.accumulate_and_add_sequential(bpt_tree, inner_nodes,
inner_nodes[:tree.num_leaves()],
hg.Accumulators.sum)
# add 1 to avoid having a zero measure in a minima
res[bpt_tree.num_leaves():] = res[bpt_tree.num_leaves():] + 1
return res
def constrained_connectivity_hierarchy_strong_connection(graph, edge_weights):
"""
Strongly constrained connectivity hierarchy based on the given edge weighted graph.
Let :math:`X` be a set of vertices, the range of :math:`X` is the maximal weight of the edges linking two vertices inside :math:`X`.
Let :math:`\\alpha` be a positive real number, a set of vertices :math:`X` is :math:`\\alpha`-connected, if for any two vertices
:math:`i` and :math:`j` in :math:`X`, there exists a path from :math:`i` to :math:`j` in :math:`X` composed of edges of weights
lower than or equal to :math:`\\alpha`.
Let :math:`\\alpha` be a positive real numbers, the :math:`\\alpha`-strongly connected components of the graph are
the maximal :math:`\\alpha'`-connected sets of vertices with a range lower than or equal to :math:`\\alpha` with :math:`\\alpha'\leq\\alpha`.
Finally, the strongly constrained connectivity hierarchy is defined as the hierarchy composed of all the
:math:`\\alpha`- strongly connected components for all positive :math:`\\alpha`.
The definition used follows the one given in:
P. Soille,
"Constrained connectivity for hierarchical image partitioning and simplification,"
in IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 30, no. 7, pp. 1132-1145, July 2008.
doi: 10.1109/TPAMI.2007.70817
The algorithm runs in time :math:`\mathcal{O}(n\log(n))` and proceeds by filtering a quasi-flat zone hierarchy (see :func:`~higra.quasi_flat_zones_hierarchy`)
:param graph: input graph
:param edge_weights: edge_weights: edge weights of the input graph
:return: a tree (Concept :class:`~higra.CptHierarchy`) and its node altitudes
"""
tree, altitudes = hg.quasi_flat_zone_hierarchy(graph, edge_weights)
altitude_parents = altitudes[tree.parents()]
# max edge weights inside each region
lca_map = hg.attribute_lca_map(tree)
max_edge_weights = np.zeros((tree.num_vertices(),), dtype=edge_weights.dtype)
np.maximum.at(max_edge_weights, lca_map, edge_weights)
max_edge_weights = hg.accumulate_and_max_sequential(tree,
max_edge_weights,
max_edge_weights[:tree.num_leaves()],
hg.Accumulators.max)
# parent node can't be deleted
altitude_parents[tree.root()] = max(altitudes[tree.root()], max_edge_weights[tree.root()])
# nodes whith a range greater than the altitudes of their parent have to be deleted
violated_constraints = max_edge_weights >= altitude_parents
# the altitude of nodes with a range greater than their altitude but lower than the one of their parent must be changed
reparable_node_indices = np.nonzero(
np.logical_and(max_edge_weights > altitudes, max_edge_weights < altitude_parents))
altitudes[reparable_node_indices] = max_edge_weights[reparable_node_indices]
# final result construction
tree, node_map = hg.simplify_tree(tree, violated_constraints)
altitudes = altitudes[node_map]
hg.CptHierarchy.link(tree, graph)
return tree, altitudes
def test_simplify_tree_with_leaves3(self):
t = hg.Tree((7, 7, 8, 8, 8, 9, 9, 11, 10, 10, 11, 11))
criterion = np.zeros(t.num_vertices(), dtype=np.bool)
criterion[:9] = True
new_tree, node_map = hg.simplify_tree(t,
criterion,
process_leaves=True)
ref_tree = hg.Tree((1, 2, 2))
self.assertTrue(hg.test_tree_isomorphism(new_tree, ref_tree))
self.assertFalse(np.max(criterion[node_map]))
def test_simplify_tree_with_leaves(self):
t = hg.Tree((7, 7, 8, 8, 8, 9, 9, 11, 10, 10, 11, 11))
criterion = np.asarray((False, False, False, True, True, True, True,
False, True, False, True, False),
dtype=np.bool)
new_tree, node_map = hg.simplify_tree(t,
criterion,
process_leaves=True)
ref_tree = hg.Tree((4, 4, 5, 5, 5, 5))
self.assertTrue(hg.test_tree_isomorphism(new_tree, ref_tree))
self.assertFalse(np.max(criterion[node_map]))
def test_simplify_tree_propagate_category(self):
g = hg.get_4_adjacency_implicit_graph((1, 6))
vertex_values = np.asarray((1, 5, 4, 3, 3, 6), dtype=np.int32)
tree, altitudes = hg.component_tree_max_tree(g, vertex_values)
condition = np.asarray((False, False, False, False, False, False,
False, True, False, True, False), np.bool)
new_tree, node_map = hg.simplify_tree(tree, condition)
self.assertTrue(
np.all(new_tree.parents() == (8, 7, 7, 8, 8, 6, 8, 8, 8)))
self.assertTrue(np.all(node_map == (0, 1, 2, 3, 4, 5, 6, 8, 10)))
self.assertTrue(new_tree.category() == hg.TreeCategory.ComponentTree)
rec = hg.reconstruct_leaf_data(new_tree, altitudes[node_map])
self.assertTrue(np.all(rec == (1, 4, 4, 1, 1, 6)))
def canonize_hierarchy(tree, altitudes):
"""
Removes consecutive tree nodes with equal altitudes.
The new tree is composed of the inner nodes :math:`n` of the input tree such that
:math:`altitudes[n] \\neq altitudes[tree.parent(n)]` or :math:`n = tree.root(n)`.
For example, applying this function to the result of :func:`~higra.bpt_canonical` on an edge weighted graph
is the same as computing the :func:`~higra.quasi_flat_zone_hierarchy` of the same edge weighted graph.
:param tree: input tree
:param altitudes: altitudes of the vertices of the tree
:return: a tree (Concept :class:`~higra.CptHierarchy` if input tree already satisfied this concept)
and its node altitudes
"""
ctree, node_map = hg.simplify_tree(tree,
altitudes == altitudes[tree.parents()])
return ctree, altitudes[node_map]
def test_simplifyTreeWithLeaves(self):
t = hg.Tree((8, 8, 9, 7, 7, 11, 11, 9, 10, 10, 12, 12, 12))
criterion = np.asarray((False, True, True, False, False, False, False,
False, True, True, False, False),
dtype=np.bool)
new_tree, node_map = hg.simplify_tree(t,
criterion,
process_leaves=True)
self.assertTrue(new_tree.num_vertices() == 9)
refp = np.asarray((6, 5, 5, 7, 7, 6, 8, 8, 8))
self.assertTrue(np.all(refp == new_tree.parents()))
refnm = np.asarray((0, 3, 4, 5, 6, 7, 10, 11, 12))
self.assertTrue(np.all(refnm == node_map))
def test_simplify_tree(self):
t = TestHierarchyCore.getTree()
altitudes = np.asarray((0, 0, 0, 0, 0, 1, 2, 2))
criterion = np.equal(altitudes, altitudes[t.parents()])
new_tree, node_map = hg.simplify_tree(t, criterion)
# for reference
new_altitudes = altitudes[node_map]
self.assertTrue(new_tree.num_vertices() == 7)
refp = np.asarray((5, 5, 6, 6, 6, 6, 6))
self.assertTrue(np.all(refp == new_tree.parents()))
refnm = np.asarray((0, 1, 2, 3, 4, 5, 7))
self.assertTrue(np.all(refnm == node_map))
def constrained_connectivity_hierarchy_alpha_omega(graph, vertex_weights):
"""
Alpha-omega constrained connectivity hierarchy based on the given vertex weighted graph.
For :math:`(i,j)` be an edge of the graph, we define :math:`w(i,j)=|w(i) - w(j)|`, the weight of this edge.
Let :math:`X` be a set of vertices, the range of :math:`X` is the maximal absolute difference between the weights of any two vertices in :math:`X`:
:math:`R(X) = \max\{|w(i) - w(j)|, (i,j)\in X^2\}`
Let :math:`\\alpha` be a positive real number, a set of vertices :math:`X` is :math:`\\alpha`-connected, if for any two vertices
:math:`i` and :math:`j` in :math:`X`, there exists a path from :math:`i` to :math:`j` in :math:`X` composed of edges of weights
lower than or equal to :math:`\\alpha`.
Let :math:`\\alpha` and :math:`\omega` be a two positive real numbers, the :math:`\\alpha-\omega`-connected components of the graph are
the maximal :math:`\\alpha'`-connected sets of vertices with a range lower than or equal to :math:`\omega`, with :math:`\\alpha'\leq\\alpha`.
Finally, the alpha-omega constrained connectivity hierarchy is defined as the hierarchy composed of all the :math:`k-k`-connected components for all positive :math:`k`.
The definition used follows the one given in:
P. Soille,
"Constrained connectivity for hierarchical image partitioning and simplification,"
in IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 30, no. 7, pp. 1132-1145, July 2008.
doi: 10.1109/TPAMI.2007.70817
The algorithm runs in time :math:`\mathcal{O}(n\log(n))` and proceeds by filtering a quasi-flat zone hierarchy (see :func:`~higra.quasi_flat_zones_hierarchy`)
:param graph: input graph
:param vertex_weights: edge_weights: edge weights of the input graph
:return: a tree (Concept :class:`~higra.CptHierarchy`) and its node altitudes
"""
vertex_weights = hg.linearize_vertex_weights(vertex_weights, graph)
if vertex_weights.ndim != 1:
raise ValueError("constrainted_connectivity_hierarchy_alpha_omega only works for scalar vertex weights.")
# QFZ on the L1 distance weighted graph
edge_weights = hg.weight_graph(graph, vertex_weights, hg.WeightFunction.L1)
tree, altitudes = hg.quasi_flat_zone_hierarchy(graph, edge_weights)
altitude_parents = altitudes[tree.parents()]
# vertex value range inside each region
min_value = hg.accumulate_sequential(tree, vertex_weights, hg.Accumulators.min)
max_value = hg.accumulate_sequential(tree, vertex_weights, hg.Accumulators.max)
value_range = max_value - min_value
# parent node can't be deleted
altitude_parents[tree.root()] = max(altitudes[tree.root()], value_range[tree.root()])
# nodes whith a range greater than the altitudes of their parent have to be deleted
violated_constraints = value_range >= altitude_parents
# the altitude of nodes with a range greater than their altitude but lower than the one of their parent must be changed
reparable_node_indices = np.nonzero(np.logical_and(value_range > altitudes, value_range < altitude_parents))
altitudes[reparable_node_indices] = value_range[reparable_node_indices]
# final result construction
tree, node_map = hg.simplify_tree(tree, violated_constraints)
altitudes = altitudes[node_map]
hg.CptHierarchy.link(tree, graph)
return tree, altitudes
def component_tree_multivariate_tree_of_shapes_image2d(image,
padding='mean',
original_size=True,
immersion=True):
"""
Multivariate tree of shapes for a 2d multi-band image. This tree is defined as a fusion of the marginal
trees of shapes. The method is described in:
E. Carlinet.
`A Tree of shapes for multivariate images <https://pastel.archives-ouvertes.fr/tel-01280131/file/TH2015PESC1118.pdf>`_.
PhD Thesis, Université Paris-Est, 2015.
The input :attr:`image` must be a 3d array of shape :math:`(height, width, channel)`.
Note that the constructed hierarchy doesn't have natural altitudes associated to its node: as a node is generally
a fusion of several marginal nodes, we can't associate a single canonical value from the original image to this node.
The parameters :attr:`padding`, :attr:`original_size`, and :attr:`immersion` are forwarded to the function
:func:`~higra.component_tree_tree_of_shapes_image2d`: please look at this function documentation for more details.
:Complexity:
The worst case runtime complexity of this method is :math:`\mathcal{O}(N^2D^2)` with :math:`N` the number of pixels
and :math:`D` the number of bands. If the image pixel values are quantized on :math:`K<<N` different values
(eg. with a color image in :math:`[0..255]^D`), then the worst case runtime complexity can be tightened to
:math:`\mathcal{O}(NKD^2)`.
:See:
This function relies on :func:`~higra.tree_fusion_depth_map` to compute the fusion of the marinal trees.
:param image: input *color* 2d image
:param padding: possible values are `'none'`, `'zero'`, and `'mean'` (default = `'mean'`)
:param original_size: remove all nodes corresponding to interpolated/padded pixels (default = `True`)
:param immersion: performs a plain map continuous immersion fo the original image (default = `True`)
:return: a tree (Concept :class:`~higra.CptHierarchy`)
"""
assert len(
image.shape
) == 3, "This multivariate tree of shapes implementation only supports multichannel 2d images."
ndim = image.shape[2]
trees = tuple(
hg.component_tree_tree_of_shapes_image2d(
image[:, :,
k], padding, original_size=False, immersion=immersion)[0]
for k in range(ndim))
g = hg.CptHierarchy.get_leaf_graph(trees[0])
shape = hg.CptGridGraph.get_shape(g)
depth_map = hg.tree_fusion_depth_map(trees)
tree, altitudes = hg.component_tree_tree_of_shapes_image2d(
np.reshape(depth_map, shape),
padding="none",
original_size=False,
immersion=False)
if original_size and (immersion or padding != "none"):
deleted_vertices = np.ones((tree.num_leaves(), ), dtype=np.bool)
deleted = np.reshape(deleted_vertices, shape)
if immersion:
if padding != "none":
deleted[2:-2:2, 2:-2:2] = False
else:
deleted[0::2, 0::2] = False
else:
if padding != "none":
deleted[1:-1, 1::-1] = False
all_deleted = hg.accumulate_sequential(tree, deleted_vertices,
hg.Accumulators.min)
shape = (image.shape[0], image.shape[1])
else:
all_deleted = np.zeros((tree.num_vertices(), ), dtype=np.bool)
holes = altitudes < altitudes[tree.parents()]
all_deleted = np.logical_or(all_deleted, holes)
tree, _ = hg.simplify_tree(tree, all_deleted, process_leaves=True)
g = hg.get_4_adjacency_graph(shape)
hg.CptHierarchy.link(tree, g)
return tree