import numpy as np
from scipy.sparse import csgraph, csr_matrix, triu
from . import PointCloud
class Graph(object):
r"""
Abstract class for Graph definitions and manipulation.
Parameters
----------
adjacency_matrix : ``(n_vertices, n_vertices)`` `ndarray` or `csr_matrix`
The adjacency matrix of the graph in which the rows represent source
vertices and columns represent destination vertices. The non-edges must
be represented with zeros and the edges can have a weight value.
The adjacency matrix of an undirected graph must be symmetric.
copy : `bool`, optional
If ``False``, the ``adjacency_matrix`` will not be copied on assignment.
skip_checks : `bool`, optional
If ``True``, no checks will be performed.
Raises
------
ValueError
adjacency_matrix must be either a numpy.ndarray or a
scipy.sparse.csr_matrix.
ValueError
Graph must have at least one vertex.
ValueError
adjacency_matrix must be square (n_vertices, n_vertices, ),
({adjacency_matrix.shape[0]}, {adjacency_matrix.shape[1]}) given
instead.
ValueError
The adjacency matrix of an undirected graph must be symmetric.
Examples
--------
The adjacency matrix of the following undirected graph ::
|---0---|
| |
| |
1-------2
| |
| |
3-------4
|
|
5
can be defined as ::
import numpy as np
adjacency_matrix = np.array([[0, 1, 1, 0, 0, 0],
[1, 0, 1, 1, 0, 0],
[1, 1, 0, 0, 1, 0],
[0, 1, 0, 0, 1, 1],
[0, 0, 1, 1, 0, 0],
[0, 0, 0, 1, 0, 0]])
or ::
from scipy.sparse import csr_matrix
adjacency_matrix = csr_matrix(
([1] * 14,
([0, 1, 0, 2, 1, 2, 1, 3, 2, 4, 3, 4, 3, 5],
[1, 0, 2, 0, 2, 1, 3, 1, 4, 2, 4, 3, 5, 3])),
shape=(6, 6))
The adjacency matrix of the following directed graph ::
|-->0<--|
| |
| |
1<----->2
| |
v v
3------>4
|
v
5
can be represented as ::
import numpy as np
adjacency_matrix = np.array([[0, 0, 0, 0, 0, 0],
[1, 0, 1, 1, 0, 0],
[1, 1, 0, 0, 1, 0],
[0, 0, 0, 0, 1, 1],
[0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0]])
or ::
from scipy.sparse import csr_matrix
adjacency_matrix = csr_matrix(([1] * 8, ([1, 2, 1, 2, 1, 2, 3, 3],
[0, 0, 2, 1, 3, 4, 4, 5])),
shape=(6, 6))
Finally, the adjacency matrix of the following graph with isolated
vertices ::
0---|
|
|
1 2
|
|
3-------4
5
can be defined as ::
import numpy as np
adjacency_matrix = np.array([[0, 0, 1, 0, 0, 0],
[0, 0, 0, 0, 0, 0],
[1, 0, 0, 0, 1, 0],
[0, 0, 0, 0, 1, 0],
[0, 0, 1, 1, 0, 0],
[0, 0, 0, 0, 0, 0]])
or ::
from scipy.sparse import csr_matrix
adjacency_matrix = csr_matrix(([1] * 6, ([0, 2, 2, 4, 3, 4],
[2, 0, 4, 2, 4, 3])),
shape=(6, 6))
"""
def __init__(self, adjacency_matrix, copy=True, skip_checks=False):
# check if adjacency_matrix is numpy.ndarray or scipy.sparse.csr_matrix
if isinstance(adjacency_matrix, np.ndarray):
# it is numpy.ndarray, thus convert it to scipy.sparse.csr_matrix
adjacency_matrix = csr_matrix(adjacency_matrix)
elif not (
isinstance(adjacency_matrix, np.ndarray)
or isinstance(adjacency_matrix, csr_matrix)
):
raise ValueError(
"adjacency_matrix must be either a numpy.ndarray"
"or a scipy.sparse.csr_matrix."
)
if not skip_checks:
# check that adjacency_matrix has expected shape
if adjacency_matrix.shape[0] == 0:
raise ValueError("Graph must have at least one vertex.")
elif adjacency_matrix.shape[0] != adjacency_matrix.shape[1]:
raise ValueError(
"adjacency_matrix must be square "
"(n_vertices, n_vertices, ), ({}, {}) given "
"instead".format(
adjacency_matrix.shape[0], adjacency_matrix.shape[1]
)
)
# check if adjacency matrix of undirected graph is symmetric
if not self._directed and not _is_symmetric(adjacency_matrix):
raise ValueError(
"The adjacency matrix of an undirected graph " "must be symmetric."
)
# store adjacency_matrix
if copy:
self.adjacency_matrix = adjacency_matrix.copy()
else:
self.adjacency_matrix = adjacency_matrix
@classmethod
def init_from_edges(cls, edges, n_vertices, skip_checks=False):
r"""
Initialize graph from edges array.
Parameters
----------
edges : ``(n_edges, 2, )`` `ndarray`
The `ndarray` of edges, i.e. all the pairs of vertices that are
connected with an edge.
n_vertices : `int`
The total number of vertices, assuming that the numbering of
vertices starts from ``0``. ``edges`` and ``n_vertices`` can be
defined in a way to set isolated vertices.
skip_checks : `bool`, optional
If ``True``, no checks will be performed.
Examples
--------
The following undirected graph ::
|---0---|
| |
| |
1-------2
| |
| |
3-------4
|
|
5
can be defined as ::
from menpo.shape import UndirectedGraph
import numpy as np
edges = np.array([[0, 1], [1, 0], [0, 2], [2, 0], [1, 2], [2, 1],
[1, 3], [3, 1], [2, 4], [4, 2], [3, 4], [4, 3],
[3, 5], [5, 3]])
graph = UndirectedGraph.init_from_edges(edges, n_vertices=6)
The following directed graph ::
|-->0<--|
| |
| |
1<----->2
| |
v v
3------>4
|
v
5
can be represented as ::
from menpo.shape import DirectedGraph
import numpy as np
edges = np.array([[1, 0], [2, 0], [1, 2], [2, 1], [1, 3], [2, 4],
[3, 4], [3, 5]])
graph = DirectedGraph.init_from_edges(edges, n_vertices=6)
Finally, the following graph with isolated vertices ::
0---|
|
|
1 2
|
|
3-------4
5
can be defined as ::
from menpo.shape import UndirectedGraph
import numpy as np
edges = np.array([[0, 2], [2, 0], [2, 4], [4, 2], [3, 4], [4, 3]])
graph = UndirectedGraph.init_from_edges(edges, n_vertices=6)
"""
adjacency_matrix = _convert_edges_to_adjacency_matrix(edges, n_vertices)
return cls(adjacency_matrix, copy=False, skip_checks=skip_checks)
@property
def vertices(self):
r"""
Returns the `list` of vertices.
:type: `list`
"""
return range(self.adjacency_matrix.shape[0])
@property
def n_vertices(self):
r"""
Returns the number of vertices.
:type: `int`
"""
return self.adjacency_matrix.shape[0]
@property
def edges(self):
r"""
Returns the `ndarray` of edges, i.e. all the pairs of vertices that are
connected with an edge.
:type: ``(n_edges, 2, )`` `ndarray`
"""
pass
@property
def n_edges(self):
r"""
Returns the number of edges.
:type: `int`
"""
return self.edges.shape[0]
def isolated_vertices(self):
r"""
Returns the isolated vertices of the graph (if any), i.e. the vertices
that have no edge connections.
Returns
-------
isolated_vertices : `list`
A `list` of the isolated vertices. If there aren't any, it returns
an empty `list`.
"""
return _isolated_vertices(self.adjacency_matrix)
def has_isolated_vertices(self):
r"""
Whether the graph has any isolated vertices, i.e. vertices with no edge
connections.
Returns
-------
has_isolated_vertices : `bool`
``True`` if the graph has at least one isolated vertex.
"""
return len(self.isolated_vertices()) > 0
def get_adjacency_list(self):
r"""
Returns the adjacency list of the graph, i.e. a `list` of length
``n_vertices`` that for each vertex has a `list` of the vertex
neighbours. If the graph is directed, the neighbours are children.
Returns
-------
adjacency_list : `list` of `list` of length ``n_vertices``
The adjacency list of the graph.
"""
# initialize list with empty lists
adjacency_list = [[] for _ in range(self.n_vertices)]
# get rows/columns of edges
rows, cols = self.adjacency_matrix.nonzero()
# store them accordingly
for i in range(rows.shape[0]):
from_v = rows[i]
to_v = cols[i]
adjacency_list[from_v].append(to_v)
return adjacency_list
def is_edge(self, vertex_1, vertex_2, skip_checks=False):
r"""
Whether there is an edge between the provided vertices.
Parameters
----------
vertex_1 : `int`
The first selected vertex. Parent if the graph is directed.
vertex_2 : `int`
The second selected vertex. Child if the graph is directed.
skip_checks : `bool`, optional
If ``False``, the given vertices will be checked.
Returns
-------
is_edge : `bool`
``True`` if there is an edge connecting ``vertex_1`` and
``vertex_2``.
Raises
------
ValueError
The vertex must be between 0 and {n_vertices-1}.
"""
if not skip_checks:
self._check_vertex(vertex_1)
self._check_vertex(vertex_2)
return self.adjacency_matrix[vertex_1, vertex_2] != 0
def find_path(self, start, end, method="bfs", skip_checks=False):
r"""
Returns a `list` with the first path (without cycles) found from the
``start`` vertex to the ``end`` vertex. It can employ either depth-first
search or breadth-first search.
Parameters
----------
start : `int`
The vertex from which the path starts.
end : `int`
The vertex to which the path ends.
method : {``bfs``, ``dfs``}, optional
The method to be used.
skip_checks : `bool`, optional
If ``True``, then input arguments won't pass through checks. Useful
for efficiency.
Returns
-------
path : `list`
The path's vertices.
Raises
------
ValueError
Method must be either bfs or dfs.
"""
# checks
if not skip_checks:
self._check_vertex(start)
self._check_vertex(end)
# search
if method == "bfs":
nodes, predecessors = csgraph.breadth_first_order(
self.adjacency_matrix,
start,
directed=self._directed,
return_predecessors=True,
)
elif method == "dfs":
nodes, predecessors = csgraph.depth_first_order(
self.adjacency_matrix,
start,
directed=self._directed,
return_predecessors=True,
)
else:
raise ValueError("Method must be either bfs or dfs.")
# get path
if predecessors[end] == -9999:
path = []
else:
path = [end]
i = None
while i != start:
i = predecessors[path[-1]]
path.append(i)
path.reverse()
return path
def find_all_paths(self, start, end, path=[]):
r"""
Returns a list of lists with all the paths (without cycles) found from
start vertex to end vertex.
Parameters
----------
start : `int`
The vertex from which the paths start.
end : `int`
The vertex from which the paths end.
path : `list`, optional
An existing path to append to.
Returns
-------
paths : `list` of `list`
The list containing all the paths from start to end.
"""
if path is None:
path = []
path = path + [start]
if start == end:
return [path]
if start > self.n_vertices - 1 or start < 0:
return []
paths = []
for v in list(self.adjacency_matrix[start, :].nonzero()[1]):
if v not in path:
newpaths = self.find_all_paths(v, end, path)
for newpath in newpaths:
paths.append(newpath)
return paths
def n_paths(self, start, end):
r"""
Returns the number of all the paths (without cycles) existing from
start vertex to end vertex.
Parameters
----------
start : `int`
The vertex from which the paths start.
end : `int`
The vertex from which the paths end.
Returns
-------
paths : `int`
The paths' numbers.
"""
return len(self.find_all_paths(start, end))
def find_all_shortest_paths(self, algorithm="auto", unweighted=False):
r"""
Returns the distances and predecessors arrays of the graph's shortest
paths.
Parameters
----------
algorithm : 'str', see below, optional
The algorithm to be used. Possible options are:
================ =========================================
'dijkstra' Dijkstra's algorithm with Fibonacci heaps
'bellman-ford' Bellman-Ford algorithm
'johnson' Johnson's algorithm
'floyd-warshall' Floyd-Warshall algorithm
'auto' Select the best among the above
================ =========================================
unweighted : `bool`, optional
If ``True``, then find unweighted distances. That is, rather than
finding the path between each vertex such that the sum of weights is
minimized, find the path such that the number of edges is minimized.
Returns
-------
distances : ``(n_vertices, n_vertices,)`` `ndarray`
The matrix of distances between all graph vertices.
``distances[i,j]`` gives the shortest distance from vertex ``i`` to
vertex ``j`` along the graph.
predecessors : ``(n_vertices, n_vertices,)`` `ndarray`
The matrix of predecessors, which can be used to reconstruct the
shortest paths. Each entry ``predecessors[i, j]`` gives the index of
the previous vertex in the path from vertex ``i`` to vertex ``j``.
If no path exists between vertices ``i`` and ``j``, then
``predecessors[i, j] = -9999``.
"""
# find costs and predecessors of all shortest paths
return csgraph.shortest_path(
self.adjacency_matrix,
directed=self._directed,
method=algorithm,
unweighted=unweighted,
return_predecessors=True,
)
def find_shortest_path(
self, start, end, algorithm="auto", unweighted=False, skip_checks=False
):
r"""
Returns a `list` with the shortest path (without cycles) found from
``start`` vertex to ``end`` vertex.
Parameters
----------
start : `int`
The vertex from which the path starts.
end : `int`
The vertex to which the path ends.
algorithm : 'str', see below, optional
The algorithm to be used. Possible options are:
================ =========================================
'dijkstra' Dijkstra's algorithm with Fibonacci heaps
'bellman-ford' Bellman-Ford algorithm
'johnson' Johnson's algorithm
'floyd-warshall' Floyd-Warshall algorithm
'auto' Select the best among the above
================ =========================================
unweighted : `bool`, optional
If ``True``, then find unweighted distances. That is, rather than
finding the path such that the sum of weights is minimized, find
the path such that the number of edges is minimized.
skip_checks : `bool`, optional
If ``True``, then input arguments won't pass through checks. Useful
for efficiency.
Returns
-------
path : `list`
The shortest path's vertices, including ``start`` and ``end``. If
there was not path connecting the vertices, then an empty `list` is
returned.
distance : `int` or `float`
The distance (cost) of the path from ``start`` to ``end``.
"""
# checks
if not skip_checks:
self._check_vertex(start)
self._check_vertex(end)
# find distances and predecessors of all shortest paths
(distances, predecessors) = self.find_all_shortest_paths(
algorithm=algorithm, unweighted=unweighted
)
# retrieve shortest path and its distance
if predecessors[start, end] < 0:
path = []
distance = np.inf
else:
path = [end]
distance = 0
i = None
while i != start:
i = predecessors[start, path[-1]]
path.append(i)
distance += distances[start, path[-1]]
path.reverse()
return path, distance
def has_cycles(self):
r"""
Checks if the graph has at least one cycle.
Returns
-------
has_cycles : `bool`
``True`` if the graph has cycles.
"""
return _has_cycles(self.get_adjacency_list(), self._directed)
def is_tree(self):
r"""
Checks if the graph is tree.
Returns
-------
is_true : `bool`
If the graph is a tree.
"""
return not self.has_cycles() and self.n_edges == self.n_vertices - 1
def _check_vertex(self, vertex):
r"""
Checks that a given vertex is valid.
Parameters
----------
vertex : `int`
Index of a given vertex.
Raises
------
ValueError
The vertex must be between 0 and {n_vertices-1}.
"""
if vertex > self.n_vertices - 1 or vertex < 0:
raise ValueError(
"The vertex must be between " "0 and {}.".format(self.n_vertices - 1)
)
[docs]class UndirectedGraph(Graph):
r"""
Class for Undirected Graph definition and manipulation.
Parameters
----------
adjacency_matrix : ``(n_vertices, n_vertices, )`` `ndarray` or `csr_matrix`
The adjacency matrix of the graph. The non-edges must be represented
with zeros and the edges can have a weight value.
:Note: ``adjacency_matrix`` must be symmetric.
copy : `bool`, optional
If ``False``, the ``adjacency_matrix`` will not be copied on assignment.
skip_checks : `bool`, optional
If ``True``, no checks will be performed.
Raises
------
ValueError
adjacency_matrix must be either a numpy.ndarray or a
scipy.sparse.csr_matrix.
ValueError
Graph must have at least two vertices.
ValueError
adjacency_matrix must be square (n_vertices, n_vertices, ),
({adjacency_matrix.shape[0]}, {adjacency_matrix.shape[1]}) given
instead.
ValueError
The adjacency matrix of an undirected graph must be symmetric.
Examples
--------
The following undirected graph ::
|---0---|
| |
| |
1-------2
| |
| |
3-------4
|
|
5
can be defined as ::
import numpy as np
adjacency_matrix = np.array([[0, 1, 1, 0, 0, 0],
[1, 0, 1, 1, 0, 0],
[1, 1, 0, 0, 1, 0],
[0, 1, 0, 0, 1, 1],
[0, 0, 1, 1, 0, 0],
[0, 0, 0, 1, 0, 0]])
graph = UndirectedGraph(adjacency_matrix)
or ::
from scipy.sparse import csr_matrix
adjacency_matrix = csr_matrix(
([1] * 14,
([0, 1, 0, 2, 1, 2, 1, 3, 2, 4, 3, 4, 3, 5],
[1, 0, 2, 0, 2, 1, 3, 1, 4, 2, 4, 3, 5, 3])),
shape=(6, 6))
graph = UndirectedGraph(adjacency_matrix)
The adjacency matrix of the following graph with isolated vertices ::
0---|
|
|
1 2
|
|
3-------4
5
can be defined as ::
import numpy as np
adjacency_matrix = np.array([[0, 0, 1, 0, 0, 0],
[0, 0, 0, 0, 0, 0],
[1, 0, 0, 0, 1, 0],
[0, 0, 0, 0, 1, 0],
[0, 0, 1, 1, 0, 0],
[0, 0, 0, 0, 0, 0]])
graph = UndirectedGraph(adjacency_matrix)
or ::
from scipy.sparse import csr_matrix
adjacency_matrix = csr_matrix(([1] * 6, ([0, 2, 2, 4, 3, 4],
[2, 0, 4, 2, 4, 3])),
shape=(6, 6))
graph = UndirectedGraph(adjacency_matrix)
"""
def __init__(self, adjacency_matrix, copy=True, skip_checks=False):
self._directed = False
super(UndirectedGraph, self).__init__(
adjacency_matrix, copy=copy, skip_checks=skip_checks
)
[docs] @classmethod
def init_from_edges(cls, edges, n_vertices, skip_checks=False):
r"""
Initialize graph from edges array.
Parameters
----------
edges : ``(n_edges, 2, )`` `ndarray`
The `ndarray` of edges, i.e. all the pairs of vertices that are
connected with an edge.
n_vertices : `int`
The total number of vertices, assuming that the numbering of
vertices starts from ``0``. ``edges`` and ``n_vertices`` can be
defined in a way to set isolated vertices.
skip_checks : `bool`, optional
If ``True``, no checks will be performed.
Examples
--------
The following undirected graph ::
|---0---|
| |
| |
1-------2
| |
| |
3-------4
|
|
5
can be defined as ::
from menpo.shape import UndirectedGraph
import numpy as np
edges = np.array([[0, 1], [1, 0], [0, 2], [2, 0], [1, 2], [2, 1],
[1, 3], [3, 1], [2, 4], [4, 2], [3, 4], [4, 3],
[3, 5], [5, 3]])
graph = UndirectedGraph.init_from_edges(edges, n_vertices=6)
Finally, the following graph with isolated vertices ::
0---|
|
|
1 2
|
|
3-------4
5
can be defined as ::
from menpo.shape import UndirectedGraph
import numpy as np
edges = np.array([[0, 2], [2, 0], [2, 4], [4, 2], [3, 4], [4, 3]])
graph = UndirectedGraph.init_from_edges(edges, n_vertices=6)
"""
adjacency_matrix = _convert_edges_to_symmetric_adjacency_matrix(
edges, n_vertices
)
return cls(adjacency_matrix, copy=False, skip_checks=skip_checks)
@property
def edges(self):
return np.vstack(triu(self.adjacency_matrix).nonzero()).T
[docs] def neighbours(self, vertex, skip_checks=False):
r"""
Returns the neighbours of the selected vertex.
Parameters
----------
vertex : `int`
The selected vertex.
skip_checks : `bool`, optional
If ``False``, the given vertex will be checked.
Returns
-------
neighbours : `list`
The list of neighbours.
Raises
------
ValueError
The vertex must be between 0 and {n_vertices-1}.
"""
# check given vertex
if not skip_checks:
self._check_vertex(vertex)
return list(self.adjacency_matrix[vertex, :].nonzero()[1])
[docs] def n_neighbours(self, vertex, skip_checks=False):
r"""
Returns the number of neighbours of the selected vertex.
Parameters
----------
vertex : `int`
The selected vertex.
skip_checks : `bool`, optional
If ``False``, the given vertex will be checked.
Returns
-------
n_neighbours : `int`
The number of neighbours.
Raises
------
ValueError
The vertex must be between 0 and {n_vertices-1}.
"""
return len(self.neighbours(vertex, skip_checks=skip_checks))
[docs] def minimum_spanning_tree(self, root_vertex):
r"""
Returns the minimum spanning tree of the graph using Kruskal's
algorithm.
Parameters
----------
root_vertex : `int`
The vertex that will be set as root in the output MST.
Returns
-------
mst : :map:`Tree`
The computed minimum spanning tree.
Raises
------
ValueError
Cannot compute minimum spanning tree of a graph with isolated
vertices
"""
# check if graph has isolated vertices
if self.has_isolated_vertices():
raise ValueError(
"Cannot compute minimum spanning tree of a graph "
"with isolated vertices."
)
# Compute MST. It returns an undirected graph.
mst_adjacency = csgraph.minimum_spanning_tree(self.adjacency_matrix)
# Get directed tree from the above undirected graph using DFS.
mst_adjacency = csgraph.depth_first_tree(
mst_adjacency, root_vertex, directed=False
)
return Tree(mst_adjacency, root_vertex, skip_checks=True)
def __str__(self):
isolated = ""
if self.has_isolated_vertices():
isolated = " ({} isolated)".format(len(self.isolated_vertices()))
return "Undirected graph of {} vertices{} and {} " "edges.".format(
self.n_vertices, isolated, self.n_edges
)
[docs]class DirectedGraph(Graph):
r"""
Class for Directed Graph definition and manipulation.
Parameters
----------
adjacency_matrix : ``(n_vertices, n_vertices, )`` `ndarray` or `csr_matrix`
The adjacency matrix of the graph in which the rows represent source
vertices and columns represent destination vertices. The non-edges must
be represented with zeros and the edges can have a weight value.
copy : `bool`, optional
If ``False``, the ``adjacency_matrix`` will not be copied on assignment.
skip_checks : `bool`, optional
If ``True``, no checks will be performed.
Raises
------
ValueError
adjacency_matrix must be either a numpy.ndarray or a
scipy.sparse.csr_matrix.
ValueError
Graph must have at least two vertices.
ValueError
adjacency_matrix must be square (n_vertices, n_vertices, ),
({adjacency_matrix.shape[0]}, {adjacency_matrix.shape[1]}) given
instead.
Examples
--------
The following directed graph ::
|-->0<--|
| |
| |
1<----->2
| |
v v
3------>4
|
v
5
can be defined as ::
import numpy as np
adjacency_matrix = np.array([[0, 0, 0, 0, 0, 0],
[1, 0, 1, 1, 0, 0],
[1, 1, 0, 0, 1, 0],
[0, 0, 0, 0, 1, 1],
[0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0]])
graph = DirectedGraph(adjacency_matrix)
or ::
from scipy.sparse import csr_matrix
adjacency_matrix = csr_matrix(([1] * 8, ([1, 2, 1, 2, 1, 2, 3, 3],
[0, 0, 2, 1, 3, 4, 4, 5])),
shape=(6, 6))
graph = DirectedGraph(adjacency_matrix)
The following graph with isolated vertices ::
0<--|
|
|
1 2
|
v
3------>4
5
can be defined as ::
import numpy as np
adjacency_matrix = np.array([[0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0],
[1, 0, 0, 0, 1, 0],
[0, 0, 0, 0, 1, 0],
[0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0]])
graph = DirectedGraph(adjacency_matrix)
or ::
from scipy.sparse import csr_matrix
adjacency_matrix = csr_matrix(([1] * 3, ([2, 2, 3], [0, 4, 4])),
shape=(6, 6))
graph = DirectedGraph(adjacency_matrix)
"""
def __init__(self, adjacency_matrix, copy=True, skip_checks=False):
self._directed = True
super(DirectedGraph, self).__init__(
adjacency_matrix, copy=copy, skip_checks=skip_checks
)
@property
def edges(self):
return np.vstack(self.adjacency_matrix.nonzero()).T
[docs] def children(self, vertex, skip_checks=False):
r"""
Returns the children of the selected vertex.
Parameters
----------
vertex : `int`
The selected vertex.
skip_checks : `bool`, optional
If ``False``, the given vertex will be checked.
Returns
-------
children : `list`
The list of children.
Raises
------
ValueError
The vertex must be between 0 and {n_vertices-1}.
"""
if not skip_checks:
self._check_vertex(vertex)
return list(self.adjacency_matrix[vertex, :].nonzero()[1])
[docs] def n_children(self, vertex, skip_checks=False):
r"""
Returns the number of children of the selected vertex.
Parameters
----------
vertex : `int`
The selected vertex.
Returns
-------
n_children : `int`
The number of children.
skip_checks : `bool`, optional
If ``False``, the given vertex will be checked.
Raises
------
ValueError
The vertex must be in the range ``[0, n_vertices - 1]``.
"""
return len(self.children(vertex, skip_checks=skip_checks))
[docs] def parents(self, vertex, skip_checks=False):
r"""
Returns the parents of the selected vertex.
Parameters
----------
vertex : `int`
The selected vertex.
skip_checks : `bool`, optional
If ``False``, the given vertex will be checked.
Returns
-------
parents : `list`
The list of parents.
Raises
------
ValueError
The vertex must be in the range ``[0, n_vertices - 1]``.
"""
if not skip_checks:
self._check_vertex(vertex)
return list(self.adjacency_matrix[:, vertex].nonzero()[0])
[docs] def n_parents(self, vertex, skip_checks=False):
r"""
Returns the number of parents of the selected vertex.
Parameters
----------
vertex : `int`
The selected vertex.
skip_checks : `bool`, optional
If ``False``, the given vertex will be checked.
Returns
-------
n_parents : `int`
The number of parents.
Raises
------
ValueError
The vertex must be in the range ``[0, n_vertices - 1]``.
"""
return len(self.parents(vertex, skip_checks=skip_checks))
def __str__(self):
isolated = ""
if self.has_isolated_vertices():
isolated = " ({} isolated)".format(len(self.isolated_vertices()))
return "Directed graph of {} vertices{} and {} " "edges.".format(
self.n_vertices, isolated, self.n_edges
)
[docs]class Tree(DirectedGraph):
r"""
Class for Tree definitions and manipulation.
Parameters
----------
adjacency_matrix : ``(n_vertices, n_vertices, )`` `ndarray` or `csr_matrix`
The adjacency matrix of the tree in which the rows represent parents
and columns represent children. The non-edges must be represented with
zeros and the edges can have a weight value.
:Note: A tree must not have isolated vertices.
root_vertex : `int`
The vertex to be set as root.
copy : `bool`, optional
If ``False``, the ``adjacency_matrix`` will not be copied on assignment.
skip_checks : `bool`, optional
If ``True``, no checks will be performed.
Raises
------
ValueError
adjacency_matrix must be either a numpy.ndarray or a
scipy.sparse.csr_matrix.
ValueError
Graph must have at least two vertices.
ValueError
adjacency_matrix must be square (n_vertices, n_vertices, ),
({adjacency_matrix.shape[0]}, {adjacency_matrix.shape[1]}) given
instead.
ValueError
The provided edges do not represent a tree.
ValueError
The root_vertex must be in the range ``[0, n_vertices - 1]``.
ValueError
The combination of adjacency matrix and root vertex is not valid. BFS
returns a different tree.
Examples
--------
The following tree ::
0
|
___|___
1 2
| |
_|_ |
3 4 5
| | |
| | |
6 7 8
can be defined as ::
import numpy as np
adjacency_matrix = np.array([[0, 1, 1, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 1, 1, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 1, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 1, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 1, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 1],
[0, 0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 0]])
tree = Tree(adjacency_matrix, root_vertex=0)
or ::
from scipy.sparse import csr_matrix
adjacency_matrix = csr_matrix(([1] * 8, ([0, 0, 1, 1, 2, 3, 4, 5],
[1, 2, 3, 4, 5, 6, 7, 8])),
shape=(9, 9))
tree = Tree(adjacency_matrix, root_vertex=0)
"""
def __init__(self, adjacency_matrix, root_vertex, copy=True, skip_checks=False):
super(Tree, self).__init__(adjacency_matrix, copy=copy, skip_checks=skip_checks)
if not skip_checks:
# check if the provided tree has isolated vertices
if self.has_isolated_vertices():
raise ValueError("A tree cannot have isolated vertices.")
# check if provided adjacency_matrix represents a tree
if not self.is_tree():
raise ValueError("The provided edges do not represent a tree.")
# check if root_vertex is valid
self._check_vertex(root_vertex)
# check if the tree is properly defined given the root
if not np.allclose(
csgraph.breadth_first_tree(
self.adjacency_matrix, root_vertex, directed=True
).nonzero(),
self.adjacency_matrix.nonzero(),
):
raise ValueError(
"The combination of adjacency matrix and root "
"vertex is not valid. BFS returns a different "
"tree."
)
# store root and predecessors list
self.root_vertex = root_vertex
self.predecessors_list = self._get_predecessors_list()
[docs] @classmethod
def init_from_edges(
cls, edges, n_vertices, root_vertex, copy=True, skip_checks=False
):
r"""
Construct a :map:`Tree` from edges array.
Parameters
----------
edges : ``(n_edges, 2, )`` `ndarray`
The `ndarray` of edges, i.e. all the pairs of vertices that are
connected with an edge.
n_vertices : `int`
The total number of vertices, assuming that the numbering of
vertices starts from ``0``. ``edges`` and ``n_vertices`` can be
defined in a way to set isolated vertices.
root_vertex : `int`
That vertex that will be set as root.
copy : `bool`, optional
If ``False``, the ``adjacency_matrix`` will not be copied on
assignment.
skip_checks : `bool`, optional
If ``True``, no checks will be performed.
Examples
--------
The following tree ::
0
|
___|___
1 2
| |
_|_ |
3 4 5
| | |
| | |
6 7 8
can be defined as ::
from menpo.shape import PointTree
import numpy as np
points = np.array([[30, 30], [10, 20], [50, 20], [0, 10], [20, 10],
[50, 10], [0, 0], [20, 0], [50, 0]])
edges = np.array([[0, 1], [0, 2], [1, 3], [1, 4], [2, 5], [3, 6],
[4, 7], [5, 8]])
tree = PointTree.init_from_edges(points, edges, root_vertex=0)
"""
adjacency_matrix = _convert_edges_to_adjacency_matrix(edges, n_vertices)
return cls(
adjacency_matrix,
root_vertex=root_vertex,
copy=copy,
skip_checks=skip_checks,
)
def _get_predecessors_list(self):
r"""
Returns the predecessors list of the tree, i.e. a `list` of length
``n_vertices`` that stores the parent for each vertex. The value of the
root vertex is ``None``.
:type: `list` of length ``n_vertices``
"""
# initialize list with None
predecessors_list = [None] * self.n_vertices
# get rows/columns of edges
parents, children = self.adjacency_matrix.nonzero()
# store them accordingly
for i in range(children.shape[0]):
parent = parents[i]
child = children[i]
predecessors_list[child] = parent
return predecessors_list
[docs] def depth_of_vertex(self, vertex, skip_checks=False):
r"""
Returns the depth of the specified vertex.
Parameters
----------
vertex : `int`
The selected vertex.
skip_checks : `bool`, optional
If ``False``, the given vertex will be checked.
Returns
-------
depth : `int`
The depth of the selected vertex.
Raises
------
ValueError
The vertex must be in the range ``[0, n_vertices - 1]``.
"""
if not skip_checks:
self._check_vertex(vertex)
parent = vertex
depth = 0
while not parent == self.root_vertex:
current = parent
parent = self.predecessors_list[current]
depth += 1
return depth
@property
def maximum_depth(self):
r"""
Returns the maximum depth of the tree.
:type: `int`
"""
all_depths = [self.depth_of_vertex(v) for v in range(self.n_vertices)]
return np.max(all_depths)
[docs] def vertices_at_depth(self, depth):
r"""
Returns a list of vertices at the specified depth.
Parameters
----------
depth : `int`
The selected depth.
Returns
-------
vertices : `list`
The vertices that lie in the specified depth.
"""
ver = []
for v in range(self.n_vertices):
if self.depth_of_vertex(v) == depth:
ver.append(v)
return ver
[docs] def n_vertices_at_depth(self, depth):
r"""
Returns the number of vertices at the specified depth.
Parameters
----------
depth : `int`
The selected depth.
Returns
-------
n_vertices : `int`
The number of vertices that lie in the specified depth.
"""
n_ver = 0
for v in range(self.n_vertices):
if self.depth_of_vertex(v) == depth:
n_ver += 1
return n_ver
[docs] def is_leaf(self, vertex, skip_checks=False):
r"""
Whether the vertex is a leaf.
Parameters
----------
vertex : `int`
The selected vertex.
skip_checks : `bool`, optional
If ``False``, the given vertex will be checked.
Returns
-------
is_leaf : `bool`
If ``True``, then selected vertex is a leaf.
Raises
------
ValueError
The vertex must be in the range ``[0, n_vertices - 1]``.
"""
if not skip_checks:
self._check_vertex(vertex)
return len(self.children(vertex)) == 0
@property
def leaves(self):
r"""
Returns a `list` with the all leaves of the tree.
:type: `list`
"""
leaves = []
for v in range(self.n_vertices):
if self.is_leaf(v):
leaves.append(v)
return leaves
@property
def n_leaves(self):
r"""
Returns the number of leaves of the tree.
:type: `int`
"""
return len(self.leaves)
[docs] def parent(self, vertex, skip_checks=False):
r"""
Returns the parent of the selected vertex.
Parameters
----------
vertex : `int`
The selected vertex.
skip_checks : `bool`, optional
If ``False``, the given vertex will be checked.
Returns
-------
parent : `int`
The parent vertex.
Raises
------
ValueError
The vertex must be in the range ``[0, n_vertices - 1]``.
"""
if not skip_checks:
self._check_vertex(vertex)
return self.predecessors_list[vertex]
def __str__(self):
return "Tree of depth {} with {} vertices and {} leaves.".format(
self.maximum_depth, self.n_vertices, self.n_leaves
)
class PointGraph(Graph, PointCloud):
r"""
Class for defining a Graph with geometry.
Parameters
----------
points : ``(n_vertices, n_dims, )`` `ndarray`
The array of point locations.
adjacency_matrix : ``(n_vertices, n_vertices)`` `ndarray` or `csr_matrix`
The adjacency matrix of the graph in which the rows represent source
vertices and columns represent destination vertices. The non-edges must
be represented with zeros and the edges can have a weight value.
The adjacency matrix of an undirected graph must be symmetric.
copy : `bool`, optional
If ``False``, the ``adjacency_matrix`` will not be copied on assignment.
skip_checks : `bool`, optional
If ``True``, no checks will be performed.
Raises
------
ValueError
adjacency_matrix must be either a numpy.ndarray or a
scipy.sparse.csr_matrix.
ValueError
Graph must have at least two vertices.
ValueError
adjacency_matrix must be square (n_vertices, n_vertices, ),
({adjacency_matrix.shape[0]}, {adjacency_matrix.shape[1]}) given
instead.
ValueError
The adjacency matrix of an undirected graph must be symmetric.
ValueError
A point for each graph vertex needs to be passed. Got {} points instead
of {}
Examples
--------
The adjacency matrix of the following undirected graph ::
|---0---|
| |
| |
1-------2
| |
| |
3-------4
|
|
5
can be defined as ::
import numpy as np
adjacency_matrix = np.array([[0, 1, 1, 0, 0, 0],
[1, 0, 1, 1, 0, 0],
[1, 1, 0, 0, 1, 0],
[0, 1, 0, 0, 1, 1],
[0, 0, 1, 1, 0, 0],
[0, 0, 0, 1, 0, 0]])
or ::
from scipy.sparse import csr_matrix
adjacency_matrix = csr_matrix(
([1] * 14,
([0, 1, 0, 2, 1, 2, 1, 3, 2, 4, 3, 4, 3, 5],
[1, 0, 2, 0, 2, 1, 3, 1, 4, 2, 4, 3, 5, 3])),
shape=(6, 6))
The adjacency matrix of the following directed graph ::
|-->0<--|
| |
| |
1<----->2
| |
v v
3------>4
|
v
5
can be represented as ::
import numpy as np
adjacency_matrix = np.array([[0, 0, 0, 0, 0, 0],
[1, 0, 1, 1, 0, 0],
[1, 1, 0, 0, 1, 0],
[0, 0, 0, 0, 1, 1],
[0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0]])
or ::
from scipy.sparse import csr_matrix
adjacency_matrix = csr_matrix(([1] * 8, ([1, 2, 1, 2, 1, 2, 3, 3],
[0, 0, 2, 1, 3, 4, 4, 5])),
shape=(6, 6))
Finally, the adjacency matrix of the following graph with isolated
vertices ::
0---|
|
|
1 2
|
|
3-------4
5
can be defined as ::
import numpy as np
adjacency_matrix = np.array([[0, 0, 1, 0, 0, 0],
[0, 0, 0, 0, 0, 0],
[1, 0, 0, 0, 1, 0],
[0, 0, 0, 0, 1, 0],
[0, 0, 1, 1, 0, 0],
[0, 0, 0, 0, 0, 0]])
or ::
from scipy.sparse import csr_matrix
adjacency_matrix = csr_matrix(([1] * 6, ([0, 2, 2, 4, 3, 4],
[2, 0, 4, 2, 4, 3])),
shape=(6, 6))
"""
def __init__(self, points, adjacency_matrix, copy=True, skip_checks=False):
if not skip_checks:
# check the number of points
_check_n_points(points, adjacency_matrix)
Graph.__init__(self, adjacency_matrix, copy=copy, skip_checks=skip_checks)
PointCloud.__init__(self, points, copy=copy)
@classmethod
def init_from_edges(cls, points, edges, copy=True, skip_checks=False):
r"""
Construct a PointGraph from edges array.
Parameters
----------
points : ``(n_vertices, n_dims, )`` `ndarray`
The array of point locations.
edges : ``(n_edges, 2, )`` `ndarray`
The `ndarray` of edges, i.e. all the pairs of vertices that are
connected with an edge.
copy : `bool`, optional
If ``False``, the ``adjacency_matrix`` will not be copied on
assignment.
skip_checks : `bool`, optional
If ``True``, no checks will be performed.
Examples
--------
The following undirected graph ::
|---0---|
| |
| |
1-------2
| |
| |
3-------4
|
|
5
can be defined as ::
from menpo.shape import PointUndirectedGraph
import numpy as np
points = np.array([[10, 30], [0, 20], [20, 20], [0, 10], [20, 10],
[0, 0]])
edges = np.array([[0, 1], [1, 0], [0, 2], [2, 0], [1, 2], [2, 1],
[1, 3], [3, 1], [2, 4], [4, 2], [3, 4], [4, 3],
[3, 5], [5, 3]])
graph = PointUndirectedGraph.init_from_edges(points, edges)
The following directed graph ::
|-->0<--|
| |
| |
1<----->2
| |
v v
3------>4
|
v
5
can be represented as ::
from menpo.shape import PointDirectedGraph
import numpy as np
points = np.array([[10, 30], [0, 20], [20, 20], [0, 10], [20, 10],
[0, 0]])
edges = np.array([[1, 0], [2, 0], [1, 2], [2, 1], [1, 3], [2, 4],
[3, 4], [3, 5]])
graph = PointDirectedGraph.init_from_edges(points, edges)
Finally, the following graph with isolated vertices ::
0---|
|
|
1 2
|
|
3-------4
5
can be defined as ::
from menpo.shape import PointUndirectedGraph
import numpy as np
points = np.array([[10, 30], [0, 20], [20, 20], [0, 10], [20, 10],
[0, 0]])
edges = np.array([[0, 2], [2, 0], [2, 4], [4, 2], [3, 4], [4, 3]])
graph = PointUndirectedGraph.init_from_edges(points, edges)
"""
adjacency_matrix = _convert_edges_to_adjacency_matrix(edges, points.shape[0])
return cls(points, adjacency_matrix, copy=copy, skip_checks=skip_checks)
@classmethod
def init_2d_grid(
cls, shape, spacing=None, adjacency_matrix=None, skip_checks=False
):
r"""
Create a PointGraph that exists on a regular 2D grid. The first
dimension is the number of rows in the grid and the second dimension
of the shape is the number of columns. ``spacing`` optionally allows
the definition of the distance between points (uniform over points).
The spacing may be different for rows and columns.
If no adjacency matrix is provided, the default connectivity will
be a 4-connected lattice.
Parameters
----------
shape : `tuple` of 2 `int`
The size of the grid to create, this defines the number of points
across each dimension in the grid. The first element is the number
of rows and the second is the number of columns.
spacing : `int` or `tuple` of 2 `int`, optional
The spacing between points. If a single `int` is provided, this
is applied uniformly across each dimension. If a `tuple` is
provided, the spacing is applied non-uniformly as defined e.g.
``(2, 3)`` gives a spacing of 2 for the rows and 3 for the
columns.
adjacency_matrix : ``(n_vertices, n_vertices)`` `ndarray` or `csr_matrix`, optional
The adjacency matrix of the graph in which the rows represent source
vertices and columns represent destination vertices. The non-edges must
be represented with zeros and the edges can have a weight value.
The adjacency matrix of an undirected graph must be symmetric.
skip_checks : `bool`, optional
If ``True``, no checks will be performed. Only considered if no
adjacency matrix is provided.
Returns
-------
pgraph : PointGraph
A pointgraph arranged in a grid.
"""
from .graph_predefined import stencil_grid
pc = PointCloud.init_2d_grid(shape, spacing=spacing)
points = pc.points
if adjacency_matrix is None:
stencil = np.array([[0, 1, 0], [1, 0, 1], [0, 1, 0]])
adjacency_matrix = stencil_grid(stencil, shape, format="csr")
# Skip checks if we construct the adjacency.
skip_checks = True
else:
adjacency_matrix = adjacency_matrix.copy()
return cls(points, adjacency_matrix, copy=False, skip_checks=skip_checks)
@classmethod
def init_from_depth_image(
cls, depth_image, spacing=None, adjacency_matrix=None, skip_checks=False
):
r"""
Return a 3D point graph from the given depth image. The depth image
is assumed to represent height/depth values and the XY coordinates
are assumed to unit spaced and represent image coordinates. This is
particularly useful for visualising depth values that have been
recovered from images.
If no adjacency matrix is provided, the default connectivity will
be a 4-connected lattice.
Parameters
----------
depth_image : :map:`Image` or subclass
A single channel image that contains depth values - as commonly
returned by RGBD cameras, for example.
spacing : `int` or `tuple` of 2 `int`, optional
The spacing between points. If a single `int` is provided, this
is applied uniformly across each dimension. If a `tuple` is
provided, the spacing is applied non-uniformly as defined e.g.
``(2, 3)`` gives a spacing of 2 for the rows and 3 for the
columns.
adjacency_matrix : ``(n_vertices, n_vertices)`` `ndarray` or `csr_matrix`, optional
The adjacency matrix of the graph in which the rows represent source
vertices and columns represent destination vertices. The non-edges must
be represented with zeros and the edges can have a weight value.
The adjacency matrix of an undirected graph must be symmetric.
skip_checks : `bool`, optional
If ``True``, no checks will be performed. Only considered if no
adjacency matrix is provided.
Returns
-------
depth_cloud : ``type(cls)``
A new 3D PointGraph with unit XY coordinates and the given depth
values as Z coordinates.
"""
from menpo.image import MaskedImage
new_pcloud = cls.init_2d_grid(
depth_image.shape,
spacing=spacing,
adjacency_matrix=adjacency_matrix,
skip_checks=skip_checks,
)
if isinstance(depth_image, MaskedImage):
new_pcloud = new_pcloud.from_mask(depth_image.mask.as_vector())
return cls(
np.hstack([new_pcloud.points, depth_image.as_vector(keep_channels=True).T]),
new_pcloud.adjacency_matrix,
copy=False,
skip_checks=True,
)
def tojson(self):
r"""
Convert this PointGraph to a dictionary representation suitable for
inclusion in the LJSON landmark format.
Returns
-------
json : `dict`
Dictionary with ``points`` and ``connectivity`` keys.
"""
json_dict = PointCloud.tojson(self)
json_dict["landmarks"]["connectivity"] = self.edges.tolist()
return json_dict
def _view_2d(
self,
figure_id=None,
new_figure=False,
image_view=True,
render_lines=True,
line_colour="r",
line_style="-",
line_width=1.0,
render_markers=True,
marker_style="o",
marker_size=5,
marker_face_colour="k",
marker_edge_colour="k",
marker_edge_width=1.0,
render_numbering=False,
numbers_horizontal_align="center",
numbers_vertical_align="bottom",
numbers_font_name="sans-serif",
numbers_font_size=10,
numbers_font_style="normal",
numbers_font_weight="normal",
numbers_font_colour="k",
render_axes=True,
axes_font_name="sans-serif",
axes_font_size=10,
axes_font_style="normal",
axes_font_weight="normal",
axes_x_limits=None,
axes_y_limits=None,
axes_x_ticks=None,
axes_y_ticks=None,
figure_size=(7, 7),
label=None,
**kwargs,
):
r"""
Visualization of the PointGraph in 2D.
Returns
-------
figure_id : `object`, optional
The id of the figure to be used.
new_figure : `bool`, optional
If ``True``, a new figure is created.
image_view : `bool`, optional
If ``True`` the PointGraph will be viewed as if it is in the image
coordinate system.
render_lines : `bool`, optional
If ``True``, the edges will be rendered.
line_colour : See Below, optional
The colour of the lines.
Example options::
{r, g, b, c, m, k, w}
or
(3, ) ndarray
line_style : ``{'-', '--', '-.', ':'}``, optional
The style of the lines.
line_width : `float`, optional
The width of the lines.
render_markers : `bool`, optional
If ``True``, the markers will be rendered.
marker_style : See Below, optional
The style of the markers. Example options ::
{., ,, o, v, ^, <, >, +, x, D, d, s, p, *, h, H, 1, 2, 3, 4, 8}
marker_size : `int`, optional
The size of the markers in points.
marker_face_colour : See Below, optional
The face (filling) colour of the markers.
Example options ::
{r, g, b, c, m, k, w}
or
(3, ) ndarray
marker_edge_colour : See Below, optional
The edge colour of the markers.
Example options ::
{r, g, b, c, m, k, w}
or
(3, ) ndarray
marker_edge_width : `float`, optional
The width of the markers' edge.
render_numbering : `bool`, optional
If ``True``, the landmarks will be numbered.
numbers_horizontal_align : ``{center, right, left}``, optional
The horizontal alignment of the numbers' texts.
numbers_vertical_align : ``{center, top, bottom, baseline}``, optional
The vertical alignment of the numbers' texts.
numbers_font_name : See Below, optional
The font of the numbers. Example options ::
{serif, sans-serif, cursive, fantasy, monospace}
numbers_font_size : `int`, optional
The font size of the numbers.
numbers_font_style : ``{normal, italic, oblique}``, optional
The font style of the numbers.
numbers_font_weight : See Below, optional
The font weight of the numbers.
Example options ::
{ultralight, light, normal, regular, book, medium, roman,
semibold, demibold, demi, bold, heavy, extra bold, black}
numbers_font_colour : See Below, optional
The font colour of the numbers.
Example options ::
{r, g, b, c, m, k, w}
or
(3, ) ndarray
render_axes : `bool`, optional
If ``True``, the axes will be rendered.
axes_font_name : See Below, optional
The font of the axes.
Example options ::
{serif, sans-serif, cursive, fantasy, monospace}
axes_font_size : `int`, optional
The font size of the axes.
axes_font_style : {``normal``, ``italic``, ``oblique``}, optional
The font style of the axes.
axes_font_weight : See Below, optional
The font weight of the axes.
Example options ::
{ultralight, light, normal, regular, book, medium, roman,
semibold, demibold, demi, bold, heavy, extra bold, black}
axes_x_limits : `float` or (`float`, `float`) or ``None``, optional
The limits of the x axis. If `float`, then it sets padding on the
right and left of the PointGraph as a percentage of the PointGraph's
width. If `tuple` or `list`, then it defines the axis limits. If
``None``, then the limits are set automatically.
axes_y_limits : (`float`, `float`) `tuple` or ``None``, optional
The limits of the y axis. If `float`, then it sets padding on the
top and bottom of the PointGraph as a percentage of the PointGraph's
height. If `tuple` or `list`, then it defines the axis limits. If
``None``, then the limits are set automatically.
axes_x_ticks : `list` or `tuple` or ``None``, optional
The ticks of the x axis.
axes_y_ticks : `list` or `tuple` or ``None``, optional
The ticks of the y axis.
figure_size : (`float`, `float`) `tuple` or ``None``, optional
The size of the figure in inches.
label : `str`, optional
The name entry in case of a legend.
Returns
-------
viewer : :map:`PointGraphViewer2d`
The viewer object.
"""
from menpo.visualize import PointGraphViewer2d
renderer = PointGraphViewer2d(figure_id, new_figure, self.points, self.edges)
renderer.render(
image_view=image_view,
render_lines=render_lines,
line_colour=line_colour,
line_style=line_style,
line_width=line_width,
render_markers=render_markers,
marker_style=marker_style,
marker_size=marker_size,
marker_face_colour=marker_face_colour,
marker_edge_colour=marker_edge_colour,
marker_edge_width=marker_edge_width,
render_numbering=render_numbering,
numbers_horizontal_align=numbers_horizontal_align,
numbers_vertical_align=numbers_vertical_align,
numbers_font_name=numbers_font_name,
numbers_font_size=numbers_font_size,
numbers_font_style=numbers_font_style,
numbers_font_weight=numbers_font_weight,
numbers_font_colour=numbers_font_colour,
render_axes=render_axes,
axes_font_name=axes_font_name,
axes_font_size=axes_font_size,
axes_font_style=axes_font_style,
axes_font_weight=axes_font_weight,
axes_x_limits=axes_x_limits,
axes_y_limits=axes_y_limits,
axes_x_ticks=axes_x_ticks,
axes_y_ticks=axes_y_ticks,
figure_size=figure_size,
label=label,
)
return renderer
def _view_landmarks_2d(
self,
group=None,
with_labels=None,
without_labels=None,
figure_id=None,
new_figure=False,
image_view=True,
render_lines=True,
line_colour="k",
line_style="-",
line_width=2,
render_markers=True,
marker_style="s",
marker_size=7,
marker_face_colour="k",
marker_edge_colour="k",
marker_edge_width=1.0,
render_lines_lms=True,
line_colour_lms=None,
line_style_lms="-",
line_width_lms=1,
render_markers_lms=True,
marker_style_lms="o",
marker_size_lms=5,
marker_face_colour_lms=None,
marker_edge_colour_lms=None,
marker_edge_width_lms=1.0,
render_numbering=False,
numbers_horizontal_align="center",
numbers_vertical_align="bottom",
numbers_font_name="sans-serif",
numbers_font_size=10,
numbers_font_style="normal",
numbers_font_weight="normal",
numbers_font_colour="k",
render_legend=False,
legend_title="",
legend_font_name="sans-serif",
legend_font_style="normal",
legend_font_size=10,
legend_font_weight="normal",
legend_marker_scale=None,
legend_location=2,
legend_bbox_to_anchor=(1.05, 1.0),
legend_border_axes_pad=None,
legend_n_columns=1,
legend_horizontal_spacing=None,
legend_vertical_spacing=None,
legend_border=True,
legend_border_padding=None,
legend_shadow=False,
legend_rounded_corners=False,
render_axes=False,
axes_font_name="sans-serif",
axes_font_size=10,
axes_font_style="normal",
axes_font_weight="normal",
axes_x_limits=None,
axes_y_limits=None,
axes_x_ticks=None,
axes_y_ticks=None,
figure_size=(7, 7),
):
"""
Visualize the landmarks. This method will appear on the `PointGraph` as
``view_landmarks``.
Parameters
----------
group : `str` or``None`` optional
The landmark group to be visualized. If ``None`` and there are more
than one landmark groups, an error is raised.
with_labels : ``None`` or `str` or `list` of `str`, optional
If not ``None``, only show the given label(s). Should **not** be
used with the ``without_labels`` kwarg.
without_labels : ``None`` or `str` or `list` of `str`, optional
If not ``None``, show all except the given label(s). Should **not**
be used with the ``with_labels`` kwarg.
figure_id : `object`, optional
The id of the figure to be used.
new_figure : `bool`, optional
If ``True``, a new figure is created.
image_view : `bool`, optional
If ``True`` the PointCloud will be viewed as if it is in the image
coordinate system.
render_lines : `bool`, optional
If ``True``, the edges will be rendered.
line_colour : See Below, optional
The colour of the lines.
Example options::
{r, g, b, c, m, k, w}
or
(3, ) ndarray
line_style : ``{-, --, -., :}``, optional
The style of the lines.
line_width : `float`, optional
The width of the lines.
render_markers : `bool`, optional
If ``True``, the markers will be rendered.
marker_style : See Below, optional
The style of the markers. Example options ::
{., ,, o, v, ^, <, >, +, x, D, d, s, p, *, h, H, 1, 2, 3, 4, 8}
marker_size : `int`, optional
The size of the markers in points.
marker_face_colour : See Below, optional
The face (filling) colour of the markers.
Example options ::
{r, g, b, c, m, k, w}
or
(3, ) ndarray
marker_edge_colour : See Below, optional
The edge colour of the markers.
Example options ::
{r, g, b, c, m, k, w}
or
(3, ) ndarray
marker_edge_width : `float`, optional
The width of the markers' edge.
render_lines_lms : `bool`, optional
If ``True``, the edges of the landmarks will be rendered.
line_colour_lms : See Below, optional
The colour of the lines of the landmarks.
Example options::
{r, g, b, c, m, k, w}
or
(3, ) ndarray
line_style_lms : ``{-, --, -., :}``, optional
The style of the lines of the landmarks.
line_width_lms : `float`, optional
The width of the lines of the landmarks.
render_markers : `bool`, optional
If ``True``, the markers of the landmarks will be rendered.
marker_style : See Below, optional
The style of the markers of the landmarks. Example options ::
{., ,, o, v, ^, <, >, +, x, D, d, s, p, *, h, H, 1, 2, 3, 4, 8}
marker_size : `int`, optional
The size of the markers of the landmarks in points.
marker_face_colour : See Below, optional
The face (filling) colour of the markers of the landmarks.
Example options ::
{r, g, b, c, m, k, w}
or
(3, ) ndarray
marker_edge_colour : See Below, optional
The edge colour of the markers of the landmarks.
Example options ::
{r, g, b, c, m, k, w}
or
(3, ) ndarray
marker_edge_width : `float`, optional
The width of the markers' edge of the landmarks.
render_numbering : `bool`, optional
If ``True``, the landmarks will be numbered.
numbers_horizontal_align : ``{center, right, left}``, optional
The horizontal alignment of the numbers' texts.
numbers_vertical_align : ``{center, top, bottom, baseline}``, optional
The vertical alignment of the numbers' texts.
numbers_font_name : See Below, optional
The font of the numbers. Example options ::
{serif, sans-serif, cursive, fantasy, monospace}
numbers_font_size : `int`, optional
The font size of the numbers.
numbers_font_style : ``{normal, italic, oblique}``, optional
The font style of the numbers.
numbers_font_weight : See Below, optional
The font weight of the numbers.
Example options ::
{ultralight, light, normal, regular, book, medium, roman,
semibold, demibold, demi, bold, heavy, extra bold, black}
numbers_font_colour : See Below, optional
The font colour of the numbers.
Example options ::
{r, g, b, c, m, k, w}
or
(3, ) ndarray
render_legend : `bool`, optional
If ``True``, the legend will be rendered.
legend_title : `str`, optional
The title of the legend.
legend_font_name : See below, optional
The font of the legend. Example options ::
{serif, sans-serif, cursive, fantasy, monospace}
legend_font_style : ``{normal, italic, oblique}``, optional
The font style of the legend.
legend_font_size : `int`, optional
The font size of the legend.
legend_font_weight : See Below, optional
The font weight of the legend.
Example options ::
{ultralight, light, normal, regular, book, medium, roman,
semibold, demibold, demi, bold, heavy, extra bold, black}
legend_marker_scale : `float`, optional
The relative size of the legend markers with respect to the original
legend_location : `int`, optional
The location of the legend. The predefined values are:
=============== ==
'best' 0
'upper right' 1
'upper left' 2
'lower left' 3
'lower right' 4
'right' 5
'center left' 6
'center right' 7
'lower center' 8
'upper center' 9
'center' 10
=============== ==
legend_bbox_to_anchor : (`float`, `float`) `tuple`, optional
The bbox that the legend will be anchored.
legend_border_axes_pad : `float`, optional
The pad between the axes and legend border.
legend_n_columns : `int`, optional
The number of the legend's columns.
legend_horizontal_spacing : `float`, optional
The spacing between the columns.
legend_vertical_spacing : `float`, optional
The vertical space between the legend entries.
legend_border : `bool`, optional
If ``True``, a frame will be drawn around the legend.
legend_border_padding : `float`, optional
The fractional whitespace inside the legend border.
legend_shadow : `bool`, optional
If ``True``, a shadow will be drawn behind legend.
legend_rounded_corners : `bool`, optional
If ``True``, the frame's corners will be rounded (fancybox).
render_axes : `bool`, optional
If ``True``, the axes will be rendered.
axes_font_name : See Below, optional
The font of the axes. Example options ::
{serif, sans-serif, cursive, fantasy, monospace}
axes_font_size : `int`, optional
The font size of the axes.
axes_font_style : ``{normal, italic, oblique}``, optional
The font style of the axes.
axes_font_weight : See Below, optional
The font weight of the axes.
Example options ::
{ultralight, light, normal, regular, book, medium, roman,
semibold,demibold, demi, bold, heavy, extra bold, black}
axes_x_limits : `float` or (`float`, `float`) or ``None``, optional
The limits of the x axis. If `float`, then it sets padding on the
right and left of the PointCloud as a percentage of the PointCloud's
width. If `tuple` or `list`, then it defines the axis limits. If
``None``, then the limits are set automatically.
axes_y_limits : (`float`, `float`) `tuple` or ``None``, optional
The limits of the y axis. If `float`, then it sets padding on the
top and bottom of the PointCloud as a percentage of the PointCloud's
height. If `tuple` or `list`, then it defines the axis limits. If
``None``, then the limits are set automatically.
axes_x_ticks : `list` or `tuple` or ``None``, optional
The ticks of the x axis.
axes_y_ticks : `list` or `tuple` or ``None``, optional
The ticks of the y axis.
figure_size : (`float`, `float`) `tuple` or ``None`` optional
The size of the figure in inches.
Raises
------
ValueError
If both ``with_labels`` and ``without_labels`` are passed.
ValueError
If the landmark manager doesn't contain the provided group label.
"""
if not self.has_landmarks:
raise ValueError(
"PointGraph does not have landmarks attached, "
"unable to view landmarks."
)
self_view = self.view(
figure_id=figure_id,
new_figure=new_figure,
image_view=image_view,
figure_size=figure_size,
render_markers=render_markers,
marker_style=marker_style,
marker_size=marker_size,
marker_face_colour=marker_face_colour,
marker_edge_colour=marker_edge_colour,
marker_edge_width=marker_edge_width,
render_lines=render_lines,
line_colour=line_colour,
line_style=line_style,
line_width=line_width,
)
# correct group label in legend
if group is None:
group = self.landmarks.group_labels[0]
landmark_view = self.landmarks[group].view(
with_labels=with_labels,
without_labels=without_labels,
figure_id=self_view.figure_id,
new_figure=False,
group=group,
image_view=image_view,
render_lines=render_lines_lms,
line_colour=line_colour_lms,
line_style=line_style_lms,
line_width=line_width_lms,
render_markers=render_markers_lms,
marker_style=marker_style_lms,
marker_size=marker_size_lms,
marker_face_colour=marker_face_colour_lms,
marker_edge_colour=marker_edge_colour_lms,
marker_edge_width=marker_edge_width_lms,
render_numbering=render_numbering,
numbers_horizontal_align=numbers_horizontal_align,
numbers_vertical_align=numbers_vertical_align,
numbers_font_name=numbers_font_name,
numbers_font_size=numbers_font_size,
numbers_font_style=numbers_font_style,
numbers_font_weight=numbers_font_weight,
numbers_font_colour=numbers_font_colour,
render_legend=render_legend,
legend_title=legend_title,
legend_font_name=legend_font_name,
legend_font_style=legend_font_style,
legend_font_size=legend_font_size,
legend_font_weight=legend_font_weight,
legend_marker_scale=legend_marker_scale,
legend_location=legend_location,
legend_bbox_to_anchor=legend_bbox_to_anchor,
legend_border_axes_pad=legend_border_axes_pad,
legend_n_columns=legend_n_columns,
legend_horizontal_spacing=legend_horizontal_spacing,
legend_vertical_spacing=legend_vertical_spacing,
legend_border=legend_border,
legend_border_padding=legend_border_padding,
legend_shadow=legend_shadow,
legend_rounded_corners=legend_rounded_corners,
render_axes=render_axes,
axes_font_name=axes_font_name,
axes_font_size=axes_font_size,
axes_font_style=axes_font_style,
axes_font_weight=axes_font_weight,
axes_x_limits=axes_x_limits,
axes_y_limits=axes_y_limits,
axes_x_ticks=axes_x_ticks,
axes_y_ticks=axes_y_ticks,
figure_size=figure_size,
)
return landmark_view
def _view_3d(
self,
figure_id=None,
new_figure=True,
render_lines=True,
line_colour="r",
line_width=2,
render_markers=True,
marker_style="sphere",
marker_size=None,
marker_colour="k",
marker_resolution=8,
step=None,
alpha=1.0,
render_numbering=False,
numbers_colour="k",
numbers_size=None,
):
r"""
Visualization of the PointGraph in 3D.
Parameters
----------
figure_id : `object`, optional
The id of the figure to be used.
new_figure : `bool`, optional
If ``True``, a new figure is created.
render_lines : `bool`, optional
If ``True``, then the lines will be rendered.
line_colour : See Below, optional
The colour of the lines.
Example options ::
{r, g, b, c, m, k, w}
or
(3, ) ndarray
line_width : `float`, optional
The width of the lines.
render_markers : `bool`, optional
If ``True``, then the markers will be rendered.
marker_style : `str`, optional
The style of the markers.
Example options ::
{2darrow, 2dcircle, 2dcross, 2ddash, 2ddiamond, 2dhooked_arrow,
2dsquare, 2dthick_arrow, 2dthick_cross, 2dtriangle, 2dvertex,
arrow, axes, cone, cube, cylinder, point, sphere}
marker_size : `float` or ``None``, optional
The size of the markers. This size can be seen as a scale factor
applied to the size markers, which is by default calculated from
the inter-marker spacing. If ``None``, then an optimal marker size
value will be set automatically.
marker_colour : See Below, optional
The colour of the markers.
Example options ::
{r, g, b, c, m, k, w}
or
(3, ) ndarray
marker_resolution : `int`, optional
The resolution of the markers. For spheres, for instance, this is
the number of divisions along theta and phi.
step : `int` or ``None``, optional
If `int`, then one every `step` vertexes will be rendered.
If ``None``, then all vertexes will be rendered.
alpha : `float`, optional
Defines the transparency (opacity) of the object.
render_numbering : `bool`, optional
If ``True``, the points will be numbered.
numbers_colour : See Below, optional
The colour of the numbers.
Example options ::
{r, g, b, c, m, k, w}
or
(3, ) ndarray
numbers_size : `float` or ``None``, optional
The size of the numbers. This size can be seen as a scale factor
applied to the numbers, which is by default calculated from
the inter-marker spacing. If ``None``, then an optimal numbers size
value will be set automatically.
Returns
-------
renderer : `menpo3d.visualize.PointGraphViewer3d`
The Menpo3D rendering object.
"""
try:
from menpo3d.visualize import PointGraphViewer3d
renderer = PointGraphViewer3d(
figure_id, new_figure, self.points, self.edges
)
renderer.render(
render_lines=render_lines,
line_colour=line_colour,
line_width=line_width,
render_markers=render_markers,
marker_style=marker_style,
marker_size=marker_size,
marker_colour=marker_colour,
marker_resolution=marker_resolution,
step=step,
alpha=alpha,
render_numbering=render_numbering,
numbers_colour=numbers_colour,
numbers_size=numbers_size,
)
return renderer
except ImportError as e:
from menpo.visualize import Menpo3dMissingError
raise Menpo3dMissingError(e)
[docs]class PointUndirectedGraph(PointGraph, UndirectedGraph):
r"""
Class for defining an Undirected Graph with geometry.
Parameters
----------
points : ``(n_vertices, n_dims, )`` `ndarray`
The array of point locations.
adjacency_matrix : ``(n_vertices, n_vertices, )`` `ndarray` or `csr_matrix`
The adjacency matrix of the graph. The non-edges must be represented
with zeros and the edges can have a weight value.
:Note: ``adjacency_matrix`` must be symmetric.
copy : `bool`, optional
If ``False``, the ``adjacency_matrix`` will not be copied on assignment.
skip_checks : `bool`, optional
If ``True``, no checks will be performed.
Raises
------
ValueError
A point for each graph vertex needs to be passed. Got ``n_points``
points instead of ``n_vertices``.
ValueError
adjacency_matrix must be either a numpy.ndarray or a
scipy.sparse.csr_matrix.
ValueError
Graph must have at least two vertices.
ValueError
adjacency_matrix must be square (n_vertices, n_vertices, ),
({adjacency_matrix.shape[0]}, {adjacency_matrix.shape[1]}) given
instead.
ValueError
The adjacency matrix of an undirected graph must be symmetric.
Examples
--------
The following undirected graph ::
|---0---|
| |
| |
1-------2
| |
| |
3-------4
|
|
5
can be defined as ::
import numpy as np
adjacency_matrix = np.array([[0, 1, 1, 0, 0, 0],
[1, 0, 1, 1, 0, 0],
[1, 1, 0, 0, 1, 0],
[0, 1, 0, 0, 1, 1],
[0, 0, 1, 1, 0, 0],
[0, 0, 0, 1, 0, 0]])
points = np.array([[10, 30], [0, 20], [20, 20], [0, 10], [20, 10],
[0, 0]])
graph = PointUndirectedGraph(points, adjacency_matrix)
or ::
from scipy.sparse import csr_matrix
adjacency_matrix = csr_matrix(
([1] * 14,
([0, 1, 0, 2, 1, 2, 1, 3, 2, 4, 3, 4, 3, 5],
[1, 0, 2, 0, 2, 1, 3, 1, 4, 2, 4, 3, 5, 3])),
shape=(6, 6))
points = np.array([[10, 30], [0, 20], [20, 20], [0, 10], [20, 10],
[0, 0]])
graph = PointUndirectedGraph(points, adjacency_matrix)
The adjacency matrix of the following graph with isolated vertices ::
0---|
|
|
1 2
|
|
3-------4
5
can be defined as ::
import numpy as np
adjacency_matrix = np.array([[0, 0, 1, 0, 0, 0],
[0, 0, 0, 0, 0, 0],
[1, 0, 0, 0, 1, 0],
[0, 0, 0, 0, 1, 0],
[0, 0, 1, 1, 0, 0],
[0, 0, 0, 0, 0, 0]])
points = np.array([[10, 30], [0, 20], [20, 20], [0, 10], [20, 10],
[0, 0]])
graph = PointUndirectedGraph(points, adjacency_matrix)
or ::
from scipy.sparse import csr_matrix
adjacency_matrix = csr_matrix(([1] * 6, ([0, 2, 2, 4, 3, 4],
[2, 0, 4, 2, 4, 3])),
shape=(6, 6))
points = np.array([[10, 30], [0, 20], [20, 20], [0, 10], [20, 10],
[0, 0]])
graph = PointUndirectedGraph(points, adjacency_matrix)
"""
def __init__(self, points, adjacency_matrix, copy=True, skip_checks=False):
self._directed = False
super(PointUndirectedGraph, self).__init__(
points, adjacency_matrix, copy=copy, skip_checks=skip_checks
)
[docs] @classmethod
def init_from_edges(cls, points, edges, copy=True, skip_checks=False):
r"""
Construct a :map:`PointUndirectedGraph` from edges array.
Parameters
----------
points : ``(n_vertices, n_dims, )`` `ndarray`
The array of point locations.
edges : ``(n_edges, 2, )`` `ndarray`
The `ndarray` of edges, i.e. all the pairs of vertices that are
connected with an edge.
copy : `bool`, optional
If ``False``, the ``adjacency_matrix`` will not be copied on
assignment.
skip_checks : `bool`, optional
If ``True``, no checks will be performed.
Examples
--------
The following undirected graph ::
|---0---|
| |
| |
1-------2
| |
| |
3-------4
|
|
5
can be defined as ::
from menpo.shape import PointUndirectedGraph
import numpy as np
points = np.array([[10, 30], [0, 20], [20, 20], [0, 10], [20, 10],
[0, 0]])
edges = np.array([[0, 1], [1, 0], [0, 2], [2, 0], [1, 2], [2, 1],
[1, 3], [3, 1], [2, 4], [4, 2], [3, 4], [4, 3],
[3, 5], [5, 3]])
graph = PointUndirectedGraph.init_from_edges(points, edges)
Finally, the following graph with isolated vertices ::
0---|
|
|
1 2
|
|
3-------4
5
can be defined as ::
from menpo.shape import PointUndirectedGraph
import numpy as np
points = np.array([[10, 30], [0, 20], [20, 20], [0, 10], [20, 10],
[0, 0]])
edges = np.array([[0, 2], [2, 0], [2, 4], [4, 2], [3, 4], [4, 3]])
graph = PointUndirectedGraph.init_from_edges(points, edges)
"""
adjacency_matrix = _convert_edges_to_symmetric_adjacency_matrix(
edges, points.shape[0]
)
return cls(points, adjacency_matrix, copy=copy, skip_checks=skip_checks)
[docs] def from_mask(self, mask):
"""
A 1D boolean array with the same number of elements as the number of
points in the `PointUndirectedGraph`. This is then broadcast across
the dimensions of the `PointUndirectedGraph` and returns a new
`PointUndirectedGraph` containing only those points that were ``True``
in the mask.
Parameters
----------
mask : ``(n_vertices,)`` `ndarray`
1D array of booleans
Returns
-------
pointgraph : :map:`PointUndirectedGraph`
A new pointgraph that has been masked.
Raises
------
ValueError
Mask must be a 1D boolean array of the same number of entries as
points in this PointUndirectedGraph.
"""
if mask.shape[0] != self.n_points:
raise ValueError(
"Mask must be a 1D boolean array of the same "
"number of entries as points in this "
"PointUndirectedGraph."
)
if np.all(mask): # Shortcut for all true masks
return PointUndirectedGraph(
self.points, self.adjacency_matrix, copy=True, skip_checks=True
)
else:
# Get new adjacency_matrix and points
(adjacency_matrix, points) = _mask_adjacency_matrix_and_points(
mask, self.adjacency_matrix, self.points
)
return PointUndirectedGraph(
points, adjacency_matrix, copy=True, skip_checks=False
)
[docs] def minimum_spanning_tree(self, root_vertex):
r"""
Returns the minimum spanning tree of the graph using Kruskal's
algorithm.
Parameters
----------
root_vertex : `int`
The vertex that will be set as root in the output MST.
Returns
-------
mst : :map:`PointTree`
The computed minimum spanning tree with the `points` of `self`.
Raises
------
ValueError
Cannot compute minimum spanning tree of a graph with isolated
vertices
"""
# check if graph has isolated vertices
if self.has_isolated_vertices():
raise ValueError(
"Cannot compute minimum spanning tree of a graph "
"with isolated vertices."
)
# Compute MST. It returns an undirected graph.
mst_adjacency = csgraph.minimum_spanning_tree(self.adjacency_matrix)
# Get directed tree from the above undirected graph using DFS.
mst_adjacency = csgraph.depth_first_tree(
mst_adjacency, root_vertex, directed=False
)
# remove isolated vertices from the points
return PointTree(
self.points, mst_adjacency, root_vertex, copy=True, skip_checks=True
)
def __str__(self):
isolated = ""
if self.has_isolated_vertices():
isolated = " ({} isolated)".format(len(self.isolated_vertices()))
return "{}D undirected graph of {} vertices{} and {} " "edges.".format(
self.n_dims, self.n_vertices, isolated, self.n_edges
)
[docs]class PointDirectedGraph(PointGraph, DirectedGraph):
r"""
Class for defining a directed graph with geometry.
Parameters
----------
points : ``(n_vertices, n_dims)`` `ndarray`
The array representing the points.
adjacency_matrix : ``(n_vertices, n_vertices, )`` `ndarray` or `csr_matrix`
The adjacency matrix of the graph in which the rows represent source
vertices and columns represent destination vertices. The non-edges must
be represented with zeros and the edges can have a weight value.
copy : `bool`, optional
If ``False``, the ``adjacency_matrix`` will not be copied on assignment.
skip_checks : `bool`, optional
If ``True``, no checks will be performed.
Raises
------
ValueError
A point for each graph vertex needs to be passed. Got {n_points} points
instead of {n_vertices}.
ValueError
adjacency_matrix must be either a numpy.ndarray or a
scipy.sparse.csr_matrix.
ValueError
Graph must have at least two vertices.
ValueError
adjacency_matrix must be square (n_vertices, n_vertices, ),
({adjacency_matrix.shape[0]}, {adjacency_matrix.shape[1]}) given
instead.
Examples
--------
The following directed graph ::
|-->0<--|
| |
| |
1<----->2
| |
v v
3------>4
|
v
5
can be defined as ::
import numpy as np
adjacency_matrix = np.array([[0, 0, 0, 0, 0, 0],
[1, 0, 1, 1, 0, 0],
[1, 1, 0, 0, 1, 0],
[0, 0, 0, 0, 1, 1],
[0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0]])
points = np.array([[10, 30], [0, 20], [20, 20], [0, 10], [20, 10],
[0, 0]])
graph = PointDirectedGraph(points, adjacency_matrix)
or ::
from scipy.sparse import csr_matrix
adjacency_matrix = csr_matrix(([1] * 8, ([1, 2, 1, 2, 1, 2, 3, 3],
[0, 0, 2, 1, 3, 4, 4, 5])),
shape=(6, 6))
points = np.array([[10, 30], [0, 20], [20, 20], [0, 10], [20, 10],
[0, 0]])
graph = PointDirectedGraph(points, adjacency_matrix)
The following graph with isolated vertices ::
0<--|
|
|
1 2
|
v
3------>4
5
can be defined as ::
import numpy as np
adjacency_matrix = np.array([[0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0],
[1, 0, 0, 0, 1, 0],
[0, 0, 0, 0, 1, 0],
[0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0]])
points = np.array([[10, 30], [0, 20], [20, 20], [0, 10], [20, 10],
[0, 0]])
graph = PointDirectedGraph(points, adjacency_matrix)
or ::
from scipy.sparse import csr_matrix
adjacency_matrix = csr_matrix(([1] * 3, ([2, 2, 3], [0, 4, 4])),
shape=(6, 6))
points = np.array([[10, 30], [0, 20], [20, 20], [0, 10], [20, 10],
[0, 0]])
graph = PointDirectedGraph(points, adjacency_matrix)
"""
def __init__(self, points, adjacency_matrix, copy=True, skip_checks=False):
self._directed = True
super(PointDirectedGraph, self).__init__(
points, adjacency_matrix, copy=copy, skip_checks=skip_checks
)
[docs] def relative_location_edge(self, parent, child):
r"""
Returns the relative location between the provided vertices. That is
if vertex j is the parent and vertex i is its child and vector l
denotes the coordinates of a vertex, then
::
l_i - l_j = [[x_i], [y_i]] - [[x_j], [y_j]] =
= [[x_i - x_j], [y_i - y_j]]
Parameters
----------
parent : `int`
The first selected vertex which is considered as the parent.
child : `int`
The second selected vertex which is considered as the child.
Returns
-------
relative_location : ``(2,)`` `ndarray`
The relative location vector.
Raises
------
ValueError
Vertices ``parent`` and ``child`` are not connected with an edge.
"""
if not self.is_edge(parent, child):
raise ValueError(
"Vertices {} and {} are not connected "
"with an edge.".format(parent, child)
)
return self.points[child, ...] - self.points[parent, ...]
[docs] def relative_locations(self):
r"""
Returns the relative location between the vertices of each edge. If
vertex j is the parent and vertex i is its child and vector l denotes
the coordinates of a vertex, then:
::
l_i - l_j = [[x_i], [y_i]] - [[x_j], [y_j]] =
= [[x_i - x_j], [y_i - y_j]]
Returns
-------
relative_locations : ``(n_vertexes, 2)`` `ndarray`
The relative locations vector.
"""
parents = list(self.adjacency_matrix.nonzero()[0])
children = list(self.adjacency_matrix.nonzero()[1])
return self.points[children] - self.points[parents]
[docs] def from_mask(self, mask):
"""
A 1D boolean array with the same number of elements as the number of
points in the `PointDirectedGraph`. This is then broadcast across the
dimensions of the `PointDirectedGraph` and returns a new
`PointDirectedGraph` containing only those points that were ``True`` in
the mask.
Parameters
----------
mask : ``(n_points,)`` `ndarray`
1D array of booleans
Returns
-------
pointgraph : :map:`PointDirectedGraph`
A new pointgraph that has been masked.
Raises
------
ValueError
Mask must be a 1D boolean array of the same number of entries as
points in this PointDirectedGraph.
"""
if mask.shape[0] != self.n_points:
raise ValueError(
"Mask must be a 1D boolean array of the same "
"number of entries as points in this "
"PointDirectedGraph."
)
if np.all(mask): # Shortcut for all true masks
return self.copy()
else:
# Get new adjacency_matrix and points
(adjacency_matrix, points) = _mask_adjacency_matrix_and_points(
mask, self.adjacency_matrix, self.points
)
return PointDirectedGraph(
points, adjacency_matrix, copy=True, skip_checks=False
)
def __str__(self):
isolated = ""
if self.has_isolated_vertices():
isolated = " ({} isolated)".format(len(self.isolated_vertices()))
return "{}D directed graph of {} vertices{} and {} " "edges.".format(
self.n_dims, self.n_vertices, isolated, self.n_edges
)
[docs]class PointTree(PointDirectedGraph, Tree):
r"""
Class for defining a Tree with geometry.
Parameters
----------
points : ``(n_vertices, n_dims)`` `ndarray`
The array representing the points.
adjacency_matrix : ``(n_vertices, n_vertices)`` `ndarray` or `csr_matrix`
The adjacency matrix of the tree in which the rows represent parents
and columns represent children. The non-edges must be represented with
zeros and the edges can have a weight value.
:Note: A tree must not have isolated vertices.
root_vertex : `int`
The vertex to be set as root.
copy : `bool`, optional
If ``False``, the ``adjacency_matrix`` will not be copied on assignment.
skip_checks : `bool`, optional
If ``True``, no checks will be performed.
Raises
------
ValueError
A point for each graph vertex needs to be passed. Got {n_points} points
instead of {n_vertices}.
ValueError
adjacency_matrix must be either a numpy.ndarray or a
scipy.sparse.csr_matrix.
ValueError
Graph must have at least two vertices.
ValueError
adjacency_matrix must be square (n_vertices, n_vertices, ),
({adjacency_matrix.shape[0]}, {adjacency_matrix.shape[1]}) given
instead.
ValueError
The provided edges do not represent a tree.
ValueError
The root_vertex must be in the range ``[0, n_vertices - 1]``.
ValueError
The combination of adjacency matrix and root vertex is not valid. BFS
returns a different tree.
Examples
--------
The following tree ::
0
|
___|___
1 2
| |
_|_ |
3 4 5
| | |
| | |
6 7 8
can be defined as ::
import numpy as np
adjacency_matrix = np.array([[0, 1, 1, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 1, 1, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 1, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 1, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 1, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 1],
[0, 0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 0]])
points = np.array([[30, 30], [10, 20], [50, 20], [0, 10], [20, 10],
[50, 10], [0, 0], [20, 0], [50, 0]])
tree = PointTree(points, adjacency_matrix, root_vertex=0)
or ::
from scipy.sparse import csr_matrix
adjacency_matrix = csr_matrix(([1] * 8, ([0, 0, 1, 1, 2, 3, 4, 5],
[1, 2, 3, 4, 5, 6, 7, 8])),
shape=(9, 9))
points = np.array([[30, 30], [10, 20], [50, 20], [0, 10], [20, 10],
[50, 10], [0, 0], [20, 0], [50, 0]])
tree = PointTree(points, adjacency_matrix, root_vertex=0)
"""
def __init__(
self, points, adjacency_matrix, root_vertex, copy=True, skip_checks=False
):
super(PointTree, self).__init__(
points, adjacency_matrix, copy=copy, skip_checks=skip_checks
)
Tree.__init__(
self, adjacency_matrix, root_vertex, copy=copy, skip_checks=skip_checks
)
[docs] @classmethod
def init_from_edges(cls, points, edges, root_vertex, copy=True, skip_checks=False):
r"""
Construct a :map:`PointTree` from edges array.
Parameters
----------
points : ``(n_vertices, n_dims, )`` `ndarray`
The array of point locations.
edges : ``(n_edges, 2, )`` `ndarray`
The `ndarray` of edges, i.e. all the pairs of vertices that are
connected with an edge.
root_vertex : `int`
That vertex that will be set as root.
copy : `bool`, optional
If ``False``, the ``adjacency_matrix`` will not be copied on
assignment.
skip_checks : `bool`, optional
If ``True``, no checks will be performed.
Examples
--------
The following tree ::
0
|
___|___
1 2
| |
_|_ |
3 4 5
| | |
| | |
6 7 8
can be defined as ::
from menpo.shape import PointTree
import numpy as np
points = np.array([[30, 30], [10, 20], [50, 20], [0, 10], [20, 10],
[50, 10], [0, 0], [20, 0], [50, 0]])
edges = np.array([[0, 1], [0, 2], [1, 3], [1, 4], [2, 5], [3, 6],
[4, 7], [5, 8]])
tree = PointTree.init_from_edges(points, edges, root_vertex=0)
"""
adjacency_matrix = _convert_edges_to_adjacency_matrix(edges, points.shape[0])
return cls(
points, adjacency_matrix, root_vertex, copy=copy, skip_checks=skip_checks
)
[docs] @classmethod
def init_2d_grid(
cls,
shape,
spacing=None,
adjacency_matrix=None,
root_vertex=None,
skip_checks=False,
):
r"""
Create a pointtree that exists on a regular 2D grid. The first
dimension is the number of rows in the grid and the second dimension
of the shape is the number of columns. ``spacing`` optionally allows
the definition of the distance between points (uniform over points).
The spacing may be different for rows and columns.
The default connectivity is the minimum spanning tree formed from
a triangulation of the grid. The default root will be the centre
of the grid.
Parameters
----------
shape : `tuple` of 2 `int`
The size of the grid to create, this defines the number of points
across each dimension in the grid. The first element is the number
of rows and the second is the number of columns.
spacing : `int` or `tuple` of 2 `int`, optional
The spacing between points. If a single `int` is provided, this
is applied uniformly across each dimension. If a `tuple` is
provided, the spacing is applied non-uniformly as defined e.g.
``(2, 3)`` gives a spacing of 2 for the rows and 3 for the
columns.
adjacency_matrix : ``(n_vertices, n_vertices)`` `ndarray` or `csr_matrix`, optional
The adjacency matrix of the tree in which the rows represent parents
and columns represent children. The non-edges must be represented with
zeros and the edges can have a weight value.
:Note: A tree must not have isolated vertices.
root_vertex : `int`
The vertex to be set as root.
skip_checks : `bool`, optional
If ``True``, no checks will be performed. Only considered if an
adjacency matrix is provided.
Returns
-------
shape_cls : `type(cls)`
A PointCloud or subclass arranged in a grid.
"""
if root_vertex is None:
# Centre of the grid
root_vertex = np.ravel_multi_index(np.array(shape) // 2, shape)
if adjacency_matrix is None:
# Default tree is a spanning tree. Create a triangular mesh
# because it has a low average degree and is a connected graph.
from .mesh.base import TriMesh
tmesh = TriMesh.init_2d_grid(shape, spacing=spacing).as_pointgraph(
copy=False, skip_checks=True
)
return tmesh.minimum_spanning_tree(root_vertex)
else:
return cls(
PointCloud.init_2d_grid(shape, spacing=spacing).points,
adjacency_matrix.copy(),
root_vertex,
copy=False,
skip_checks=skip_checks,
)
[docs] @classmethod
def init_from_depth_image(
cls,
depth_image,
spacing=None,
adjacency_matrix=None,
root_vertex=None,
skip_checks=False,
):
r"""
Return a 3D point cloud from the given depth image. The depth image
is assumed to represent height/depth values and the XY coordinates
are assumed to unit spaced and represent image coordinates. This is
particularly useful for visualising depth values that have been
recovered from images.
The default connectivity is the minimum spanning tree formed from
a triangulation of the grid. The default root will be the centre
of the grid (for an unmasked image), otherwise it will be the
first pixel in the masked are of the image.
Parameters
----------
depth_image : :map:`Image` or subclass
A single channel image that contains depth values - as commonly
returned by RGBD cameras, for example.
spacing : `int` or `tuple` of 2 `int`, optional
The spacing between points. If a single `int` is provided, this
is applied uniformly across each dimension. If a `tuple` is
provided, the spacing is applied non-uniformly as defined e.g.
``(2, 3)`` gives a spacing of 2 for the rows and 3 for the
columns.
adjacency_matrix : ``(n_vertices, n_vertices)`` `ndarray` or `csr_matrix`, optional
The adjacency matrix of the tree in which the rows represent parents
and columns represent children. The non-edges must be represented with
zeros and the edges can have a weight value.
:Note: A tree must not have isolated vertices.
root_vertex : `int`
The vertex to be set as root.
skip_checks : `bool`, optional
If ``True``, no checks will be performed. Only considered if an
adjacency matrix is provided.
Returns
-------
depth_cloud : ``type(cls)``
A new 3D PointCloud with unit XY coordinates and the given depth
values as Z coordinates.
"""
from menpo.image import MaskedImage
if root_vertex is None and isinstance(depth_image, MaskedImage):
# If the image is masked then the masked area may not contain the
# default 'centre' root vertex, so we choose the first pixel
# in the masked area.
root_vertex = np.ravel_multi_index(
depth_image.indices()[0], depth_image.shape
)
elif root_vertex is None:
# Otherwise the default root is the centre of the image
root_vertex = np.ravel_multi_index(
np.array(depth_image.shape) // 2, depth_image.shape
)
if adjacency_matrix is None:
# Default tree is a spanning tree. Create a triangular mesh
# because it has a low average degree and is a connected graph.
from .mesh.base import TriMesh
tmesh = TriMesh.init_2d_grid(depth_image.shape, spacing=spacing)
tmesh = tmesh.as_pointgraph(copy=False, skip_checks=True)
# Performing masking before spanning tree to ensure that the
# spanning tree is valid
if isinstance(depth_image, MaskedImage):
tmesh = tmesh.from_mask(depth_image.mask.as_vector())
# Reindex root vertex according to mask
mask = depth_image.mask.mask.ravel()
root_vertex = root_vertex - np.sum(~mask[:root_vertex])
tree_2d = tmesh.minimum_spanning_tree(root_vertex)
else:
points = PointCloud.init_2d_grid(depth_image.shape, spacing=spacing).points
tree_2d = cls(
points,
adjacency_matrix.copy(),
root_vertex,
copy=False,
skip_checks=skip_checks,
)
return cls(
np.hstack([tree_2d.points, depth_image.as_vector(keep_channels=True).T]),
tree_2d.adjacency_matrix,
tree_2d.root_vertex,
copy=False,
skip_checks=True,
)
[docs] def from_mask(self, mask):
"""
A 1D boolean array with the same number of elements as the number of
points in the `PointTree`. This is then broadcast across the dimensions
of the `PointTree` and returns a new `PointTree` containing only those
points that were ``True`` in the mask.
Parameters
----------
mask : ``(n_points,)`` `ndarray`
1D array of booleans
Returns
-------
pointtree : :map:`PointTree`
A new pointtree that has been masked.
Raises
------
ValueError
Mask must be a 1D boolean array of the same number of entries as
points in this PointTree.
ValueError
Cannot remove root vertex.
"""
if mask.shape[0] != self.n_points:
raise ValueError(
"Mask must be a 1D boolean array of the same "
"number of entries as points in this PointTree."
)
if np.all(mask): # Shortcut for all true masks
return self.copy()
else:
# Impossible to remove root vertex
if not mask[self.root_vertex]:
raise ValueError("Cannot remove root vertex.")
# Get new adjacency_matrix and points
(adjacency_matrix, points) = _mask_adjacency_matrix_and_points(
mask, self.adjacency_matrix, self.points
)
root_vertex = self.root_vertex - np.sum(~mask[: self.root_vertex])
# iteratively find isolated vertices and remove them
n_components, labels = csgraph.connected_components(
adjacency_matrix, directed=True
)
while n_components > 1:
label_to_keep = labels[root_vertex]
mask = labels == label_to_keep
(adjacency_matrix, points) = _mask_adjacency_matrix_and_points(
mask, adjacency_matrix, points
)
root_vertex = root_vertex - np.sum(~mask[:root_vertex])
n_components, labels = csgraph.connected_components(
adjacency_matrix, directed=True
)
return PointTree(
points,
adjacency_matrix,
root_vertex=root_vertex,
copy=True,
skip_checks=False,
)
def __str__(self):
return "{}D tree of depth {} with {} vertices and {} leaves.".format(
self.n_dims, self.maximum_depth, self.n_vertices, self.n_leaves
)
def _is_symmetric(array):
r"""
Check if an array is symmetric.
Parameters
----------
array : `ndarray` or `scipy.sparse.csr_matrix`
The array to check.
Returns
-------
is_symmetric : `bool`
``True`` if the array is symmetric.
"""
return np.allclose(array.transpose().nonzero(), array.nonzero())
def _check_n_points(points, adjacency_matrix):
r"""
Checks whether the ``points`` array and the ``adjacency_matrix`` have the
same number of points.
Parameters
----------
points : ``(n_vertices, n_dims,)`` `ndarray`
Points array.
adjacency_matrix : ``(n_vertices, n_vertices,)`` `ndarray`
The adjacency matrix.
Raises
------
ValueError
A point for each graph vertex needs to be passed. Got {} points instead
of {}
"""
if not points.shape[0] == adjacency_matrix.shape[0]:
raise ValueError(
"A point for each graph vertex needs to be passed. "
"Got {} points instead "
"of {}".format(points.shape[0], adjacency_matrix.shape[0])
)
def _has_cycles(adjacency_list, directed):
r"""
Function that checks if the provided directed graph has cycles using a Depth
First Search (DFS).
Parameters
----------
adjacency_list : `list` of `list` of length ``n_vertices``
The adjacency list of the graph.
directed : `bool`
Defines if the provided graph is directed or not.
Returns
-------
has_cycles : `bool`
Whether the graph has cycles.
"""
def dfs(node, entered, exited, tree_edges, back_edges):
if node not in entered:
entered.add(node)
for y in adjacency_list[node]:
if y not in entered:
tree_edges[y] = node
elif (
not directed
and tree_edges.get(node, None) != y
or directed
and y not in exited
):
back_edges.setdefault(y, set()).add(node)
dfs(y, entered, exited, tree_edges, back_edges)
exited.add(node)
return tree_edges, back_edges
for x in range(len(adjacency_list)):
if dfs(x, entered=set(), exited=set(), tree_edges={}, back_edges={})[1]:
return True
else:
return False
def _mask_adjacency_matrix_and_points(mask, adjacency_matrix, points):
r"""
Function that masks a provided adjacency matrix and points array.
Parameters
----------
mask : ``(n_vertices,)`` `ndarray`
1D array of booleans
adjacency_matrix : ``(n_vertices, n_vertices,)`` `ndarray`
The adjacency matrix.
points : ``(n_vertices, n_dims)`` `ndarray`
The array representing the points.
Returns
-------
adjacency_matrix : `ndarray`
The masked adjacency matrix.
points : `ndarray`
The masked points array.
"""
# Find the indices that have been asked to be removed
indices_to_keep = np.nonzero(mask)[0]
# Remove rows and columns from adjacency matrix
adjacency_matrix = adjacency_matrix[indices_to_keep, :]
adjacency_matrix = adjacency_matrix[:, indices_to_keep]
# remove rows from points
points = points[mask, :]
return adjacency_matrix, points
def _isolated_vertices(adjacency_matrix):
all_vertices = set(range(adjacency_matrix.shape[0]))
# find the set difference between {0, 1, ..., n_vertices} and the set
# of rows (columns) that have at least one non-zero element.
rows = all_vertices.difference(set(adjacency_matrix.nonzero()[0]))
cols = all_vertices.difference(set(adjacency_matrix.nonzero()[1]))
return list(rows.intersection(cols))
def _convert_edges_to_adjacency_matrix(edges, n_vertices):
r"""
Converts an edges array to an adjacency matrix.
Parameters
----------
edges : ``(n_edges, 2, )`` `ndarray` or ``None``
The `ndarray` of edges, i.e. all the pairs of vertices that are
connected with an edge.
n_vertices : `int`
The total number of vertices, assuming that the numbering of
vertices starts from ``0``. ``edges`` and ``n_vertices`` can be
defined in a way to set isolated vertices.
Returns
-------
adjacency_matrix : ``(n_vertices, n_vertices, )`` `csr_matrix`
The adjacency matrix of the graph in which the rows represent source
vertices and columns represent destination vertices.
"""
if isinstance(edges, list):
edges = np.array(edges)
if edges is None or edges.shape[0] == 0:
# create adjacency with zeros
return csr_matrix((n_vertices, n_vertices), dtype=int)
else:
# create sparse adjacency
return csr_matrix(
([1] * edges.shape[0], (edges[:, 0], edges[:, 1])),
shape=(n_vertices, n_vertices),
)
def _convert_edges_to_symmetric_adjacency_matrix(edges, n_vertices):
r"""
Converts an edges array to an adjacency matrix.
Parameters
----------
edges : ``(n_edges, 2, )`` `ndarray` or ``None``
The `ndarray` of edges, i.e. all the pairs of vertices that are
connected with an edge.
n_vertices : `int`
The total number of vertices, assuming that the numbering of
vertices starts from ``0``. ``edges`` and ``n_vertices`` can be
defined in a way to set isolated vertices.
Returns
-------
adjacency_matrix : ``(n_vertices, n_vertices, )`` `csr_matrix`
The adjacency matrix of the graph in which the rows represent source
vertices and columns represent destination vertices.
"""
if isinstance(edges, list):
edges = np.array(edges)
if edges is None or edges.shape[0] == 0:
# create adjacency with zeros
adjacency_matrix = csr_matrix((n_vertices, n_vertices), dtype=int)
else:
rows = np.hstack((edges[:, 0], edges[:, 1]))
cols = np.hstack((edges[:, 1], edges[:, 0]))
adjacency_matrix = csr_matrix(
([1] * rows.shape[0], (rows, cols)), shape=(n_vertices, n_vertices)
)
adjacency_matrix[adjacency_matrix.nonzero()] = 1
return adjacency_matrix