algorithms.graph.field¶
Module: algorithms.graph.field¶
Inheritance diagram for nipy.algorithms.graph.field:
This module implements the Field class, which simply a WeightedGraph (see the graph.py) module, plus an arrray that yields (possibly multi-dimnesional) features associated with graph vertices. This allows some kinds of computations (all thoses relating to mathematical morphology, diffusion etc.)
Certain functions are provided to Instantiate Fields easily, given a WeightedGraph and feature data.
Author:Bertrand Thirion, 2006–2011
Class¶
Field¶
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class
nipy.algorithms.graph.field.Field(V, edges=None, weights=None, field=None)[source]¶ Bases:
nipy.algorithms.graph.graph.WeightedGraph- This is the basic field structure,
which contains the weighted graph structure plus an array of data (the ‘field’)
- field is an array of size(n, p)
where n is the number of vertices of the graph and p is the field dimension
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__init__(V, edges=None, weights=None, field=None)[source]¶ - Parameters
V (int > 0) the number of vertices of the graph
edges=None: the edge array of the graph
weights=None: the asociated weights array
field=None: the field data itself
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closing(nbiter=1)[source]¶ Morphological closing of the field data. self.field is changed inplace
- Parameters
nbiter=1 : the number of iterations required
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opening(nbiter=1)[source]¶ Morphological opening of the field data. self.field is changed inplace
- Parameters
nbiter: int, optional, the number of iterations required
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dilation(nbiter=1, fast=True)[source]¶ Morphological dilation of the field data, changed in place
- Parameters
nbiter: int, optional, the number of iterations required
Notes
When data dtype is not float64, a slow version of the code is used
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highest_neighbor(refdim=0)[source]¶ Computes the neighbor with highest field value along refdim
- Parameters
refdim: int, optional,
the dimension of the field under consideration
- Returns
hneighb: array of shape(self.V),
index of the neighbor with highest value
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erosion(nbiter=1)[source]¶ Morphological openeing of the field
- Parameters
nbiter: int, optional, the number of iterations required
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get_local_maxima(refdim=0, th=-inf)[source]¶ Look for the local maxima of one dimension (refdim) of self.field
- Parameters
refdim (int) the field dimension over which the maxima are looked after
th = float, optional
threshold so that only values above th are considered
- Returns
idx: array of shape (nmax)
indices of the vertices that are local maxima
depth: array of shape (nmax)
topological depth of the local maxima : depth[idx[i]] = q means that idx[i] is a q-order maximum
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local_maxima(refdim=0, th=-inf)[source]¶ Returns all the local maxima of a field
- Parameters
refdim (int) field dimension over which the maxima are looked after
th: float, optional
threshold so that only values above th are considered
- Returns
depth: array of shape (nmax)
a labelling of the vertices such that depth[v] = 0 if v is not a local maximum depth[v] = 1 if v is a first order maximum … depth[v] = q if v is a q-order maximum
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diffusion(nbiter=1)[source]¶ diffusion of the field data in the weighted graph structure self.field is changed inplace
- Parameters
nbiter: int, optional the number of iterations required
Notes
The process is run for all the dimensions of the field
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custom_watershed(refdim=0, th=-inf)[source]¶ customized watershed analysis of the field. Note that bassins are found around each maximum (and not minimum as conventionally)
- Parameters
refdim: int, optional
th: float optional, threshold of the field
- Returns
idx: array of shape (nbassins)
indices of the vertices that are local maxima
label : array of shape (self.V)
labelling of the vertices according to their bassin
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threshold_bifurcations(refdim=0, th=-inf)[source]¶ Analysis of the level sets of the field: Bifurcations are defined as changes in the topology in the level sets when the level (threshold) is varied This can been thought of as a kind of Morse analysis
- Parameters
th: float, optional,
threshold so that only values above th are considered
- Returns
idx: array of shape (nlsets)
indices of the vertices that are local maxima
height: array of shape (nlsets)
the depth of the local maxima depth[idx[i]] = q means that idx[i] is a q-order maximum Note that this is also the diameter of the basins associated with local maxima
parents: array of shape (nlsets)
the label of the maximum which dominates each local maximum i.e. it describes the hierarchy of the local maxima
label: array of shape (self.V)
a labelling of thevertices according to their bassin
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adjacency()¶ returns the adjacency matrix of the graph as a sparse coo matrix
- Returns
adj: scipy.sparse matrix instance,
that encodes the adjacency matrix of self
-
anti_symmeterize()¶ anti-symmeterize self, i.e. produces the graph whose adjacency matrix would be the antisymmetric part of its current adjacency matrix
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cc()¶ Compte the different connected components of the graph.
- Returns
label: array of shape(self.V), labelling of the vertices
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cliques()¶ Extraction of the graphe cliques these are defined using replicator dynamics equations
- Returns
cliques: array of shape (self.V), type (np.int)
labelling of the vertices according to the clique they belong to
-
compact_neighb()¶ returns a compact representation of self
- Returns
idx: array of of shape(self.V + 1):
the positions where to find the neighors of each node within neighb and weights
neighb: array of shape(self.E), concatenated list of neighbors
weights: array of shape(self.E), concatenated list of weights
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constrained_voronoi(seed)[source]¶ Voronoi parcellation of the field starting from the input seed
- Parameters
seed: int array of shape(p), the input seeds
- Returns
label: The resulting labelling of the data
Notes
FIXME: deal with graphs with several ccs
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cut_redundancies()¶ Returns a graph with redundant edges removed: ecah edge (ab) is present ony once in the edge matrix: the correspondng weights are added.
- Returns
the resulting WeightedGraph
-
degrees()¶ Returns the degree of the graph vertices.
- Returns
rdegree: (array, type=int, shape=(self.V,)), the right degrees
ldegree: (array, type=int, shape=(self.V,)), the left degrees
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dijkstra(seed=0)¶ Returns all the [graph] geodesic distances starting from seed x
- seed (int, >-1, <self.V) or array of shape(p)
edge(s) from which the distances are computed
- Returns
dg: array of shape (self.V),
the graph distance dg from ant vertex to the nearest seed
Notes
It is mandatory that the graph weights are non-negative
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floyd(seed=None)¶ Compute all the geodesic distances starting from seeds
- Parameters
seed= None: array of shape (nbseed), type np.int
vertex indexes from which the distances are computed if seed==None, then every edge is a seed point
- Returns
dg array of shape (nbseed, self.V)
the graph distance dg from each seed to any vertex
Notes
It is mandatory that the graph weights are non-negative. The algorithm proceeds by repeating Dijkstra’s algo for each seed. Floyd’s algo is not used (O(self.V)^3 complexity…)
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from_3d_grid(xyz, k=18)¶ Sets the graph to be the topological neighbours graph of the three-dimensional coordinates set xyz, in the k-connectivity scheme
- Parameters
xyz: array of shape (self.V, 3) and type np.int,
k = 18: the number of neighbours considered. (6, 18 or 26)
- Returns
E(int): the number of edges of self
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get_E()¶ To get the number of edges in the graph
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get_V()¶ To get the number of vertices in the graph
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get_edges()¶ To get the graph’s edges
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get_vertices()¶ To get the graph’s vertices (as id)
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get_weights()¶
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is_connected()¶ States whether self is connected or not
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kruskal()¶ Creates the Minimum Spanning Tree of self using Kruskal’s algo. efficient is self is sparse
- Returns
K, WeightedGraph instance: the resulting MST
Notes
If self contains several connected components, will have the same number k of connected components
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left_incidence()¶ Return left incidence matrix
- Returns
left_incid: list
the left incidence matrix of self as a list of lists: i.e. the list[[e.0.0, .., e.0.i(0)], .., [e.V.0, E.V.i(V)]] where e.i.j is the set of edge indexes so that e.i.j[0] = i
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list_of_neighbors()¶ returns the set of neighbors of self as a list of arrays
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main_cc()¶ Returns the indexes of the vertices within the main cc
- Returns
idx: array of shape (sizeof main cc)
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normalize(c=0)¶ Normalize the graph according to the index c Normalization means that the sum of the edges values that go into or out each vertex must sum to 1
- Parameters
c=0 in {0, 1, 2}, optional: index that designates the way
according to which D is normalized c == 0 => for each vertex a, sum{edge[e, 0]=a} D[e]=1 c == 1 => for each vertex b, sum{edge[e, 1]=b} D[e]=1 c == 2 => symmetric (‘l2’) normalization
Notes
Note that when sum_{edge[e, .] == a } D[e] = 0, nothing is performed
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remove_edges(valid)¶ Removes all the edges for which valid==0
- Parameters
valid : (self.E,) array
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remove_trivial_edges()¶ Removes trivial edges, i.e. edges that are (vv)-like self.weights and self.E are corrected accordingly
- Returns
self.E (int): The number of edges
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right_incidence()¶ Return right incidence matrix
- Returns
right_incid: list
the right incidence matrix of self as a list of lists: i.e. the list[[e.0.0, .., e.0.i(0)], .., [e.V.0, E.V.i(V)]] where e.i.j is the set of edge indexes so that e.i.j[1] = i
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set_edges(edges)¶ Sets the graph’s edges
Preconditions:
edges has a correct size
edges take values in [1..V]
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set_euclidian(X)¶ Compute the weights of the graph as the distances between the corresponding rows of X, which represents an embdedding of self
- Parameters
X array of shape (self.V, edim),
the coordinate matrix of the embedding
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set_gaussian(X, sigma=0)¶ Compute the weights of the graph as a gaussian function of the distance between the corresponding rows of X, which represents an embdedding of self
- Parameters
X array of shape (self.V, dim)
the coordinate matrix of the embedding
sigma=0, float: the parameter of the gaussian function
Notes
When sigma == 0, the following value is used:
sigma = sqrt(mean(||X[self.edges[:, 0], :]-X[self.edges[:, 1], :]||^2))
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set_weights(weights)¶ Set edge weights
- Parameters
weights: array
array shape(self.V): edges weights
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show(X=None, ax=None)¶ Plots the current graph in 2D
- Parameters
X : None or array of shape (self.V, 2)
a set of coordinates that can be used to embed the vertices in 2D. If X.shape[1]>2, a svd reduces X for display. By default, the graph is presented on a circle
ax: None or int, optional
ax handle
- Returns
ax: axis handle
Notes
This should be used only for small graphs.
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subgraph(valid)¶ Creates a subgraph with the vertices for which valid>0 and with the correponding set of edges
- Parameters
valid, array of shape (self.V): nonzero for vertices to be retained
- Returns
G, WeightedGraph instance, the desired subgraph of self
Notes
The vertices are renumbered as [1..p] where p = sum(valid>0) when sum(valid==0) then None is returned
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symmeterize()¶ Symmeterize self, modify edges and weights so that self.adjacency becomes the symmetric part of the current self.adjacency.
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to_coo_matrix()¶ Return adjacency matrix as coo sparse
- Returns
sp: scipy.sparse matrix instance
that encodes the adjacency matrix of self
-
voronoi_diagram(seeds, samples)¶ Defines the graph as the Voronoi diagram (VD) that links the seeds. The VD is defined using the sample points.
- Parameters
seeds: array of shape (self.V, dim)
samples: array of shape (nsamples, dim)
Notes
By default, the weights are a Gaussian function of the distance The implementation is not optimal
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voronoi_labelling(seed)¶ Performs a voronoi labelling of the graph
- Parameters
seed: array of shape (nseeds), type (np.int),
vertices from which the cells are built
- Returns
labels: array of shape (self.V) the labelling of the vertices
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geodesic_kmeans(seeds=None, label=None, maxiter=100, eps=0.0001, verbose=0)[source]¶ Geodesic k-means algorithm i.e. obtention of clusters that are topologically connected and minimally variable concerning the information of self.field
- Parameters
seeds: array of shape(p), optional,
initial indices of the seeds within the field if seeds==None the labels are used as initialization
labels: array of shape(self.V) initial labels, optional,
it is expected that labels take their values in a certain range (0..lmax) if Labels==None, this is not used if seeds==None and labels==None, an ewxception is raised
maxiter: int, optional,
maximal number of iterations
eps: float, optional,
increase of inertia at which convergence is declared
- Returns
seeds: array of shape (p), the final seeds
label : array of shape (self.V), the resulting field label
J: float, inertia value
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ward(nbcluster)[source]¶ Ward’s clustering of self
- Parameters
nbcluster: int,
the number of desired clusters
- Returns
label: array of shape (self.V)
the resulting field label
J (float): the resulting inertia
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subfield(valid)[source]¶ Returns a subfield of self, with only vertices such that valid > 0
- Parameters
valid: array of shape (self.V),
nonzero for vertices to be retained
- Returns
F: Field instance,
the desired subfield of self
Notes
The vertices are renumbered as [1..p] where p = sum(valid>0) when sum(valid) == 0 then None is returned