# 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 array that yields (possibly multi-dimnesional) features associated with graph vertices. This allows some kinds of computations (all those 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

## `Field`¶

class nipy.algorithms.graph.field.Field(V, edges=None, weights=None, field=None)

Bases: `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

__init__(V, edges=None, weights=None, field=None)
Parameters:
V (int > 0) the number of vertices of the graph
edges=None: the edge array of the graph
weights=None: the associated weights array
field=None: the field data itself

returns the adjacency matrix of the graph as a sparse coo matrix

Returns:

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

cc()

Compte the different connected components of the graph.

Returns:
label: array of shape(self.V), labelling of the vertices
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

closing(nbiter=1)

Morphological closing of the field data. self.field is changed inplace

Parameters:
nbiter=1the number of iterations required
compact_neighb()

returns a compact representation of self

Returns:
idx: array of of shape(self.V + 1):

the positions where to find the neighbors 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
constrained_voronoi(seed)

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

copy()

copy function

custom_watershed(refdim=0, th=-inf)

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

labelarray of shape (self.V)

labelling of the vertices according to their bassin

cut_redundancies()

Returns a graph with redundant edges removed: ecah edge (ab) is present only 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
diffusion(nbiter=1)

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

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

dilation(nbiter=1, fast=True)

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

erosion(nbiter=1)

Morphological opening of the field

Parameters:
nbiter: int, optional, the number of iterations required
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…)

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
geodesic_kmeans(seeds=None, label=None, maxiter=100, eps=0.0001, verbose=0)

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
labelarray of shape (self.V), the resulting field label
J: float, inertia value
get_E()

To get the number of edges in the graph

get_V()

To get the number of vertices in the graph

get_edges()

To get the graph’s edges

get_field()
get_local_maxima(refdim=0, th=-inf)

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

get_vertices()

To get the graph’s vertices (as id)

get_weights()
highest_neighbor(refdim=0)

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

is_connected()

States whether self is connected or not

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

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

list_of_neighbors()

returns the set of neighbors of self as a list of arrays

local_maxima(refdim=0, th=-inf)

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

main_cc()

Returns the indexes of the vertices within the main cc

Returns:
idx: array of shape (sizeof main cc)
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

opening(nbiter=1)

Morphological opening of the field data. self.field is changed inplace

Parameters:
nbiter: int, optional, the number of iterations required
remove_edges(valid)

Removes all the edges for which valid==0

Parameters:
valid(self.E,) array
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
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

set_edges(edges)

Sets the graph’s edges

Preconditions:

• edges has a correct size

• edges take values in [1..V]

set_euclidian(X)

Compute the weights of the graph as the distances between the corresponding rows of X, which represents an embedding of self

Parameters:
X array of shape (self.V, edim),

the coordinate matrix of the embedding

set_field(field)
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 embedding 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))```

set_weights(weights)

Set edge weights

Parameters:
weights: array

array shape(self.V): edges weights

show(X=None, ax=None)

Plots the current graph in 2D

Parameters:
XNone 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.

subfield(valid)

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

subgraph(valid)

Creates a subgraph with the vertices for which valid>0 and with the corresponding 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

symmeterize()

Symmeterize self, modify edges and weights so that self.adjacency becomes the symmetric part of the current self.adjacency.

threshold_bifurcations(refdim=0, th=-inf)

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

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

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
ward(nbcluster)

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

## Functions¶

nipy.algorithms.graph.field.field_from_coo_matrix_and_data(x, data)

Instantiates a weighted graph from a (sparse) coo_matrix

Parameters:
x: (V, V) scipy.sparse.coo_matrix instance,

the input matrix

data: array of shape (V, dim),

the field data

Returns:
ifield: resulting Field instance
nipy.algorithms.graph.field.field_from_graph_and_data(g, data)

Instantiate a Fieldfrom a WeightedGraph plus some feature data Parameters ———- x: (V, V) scipy.sparse.coo_matrix instance,

the input matrix

data: array of shape (V, dim),

the field data

Returns:
ifield: resulting field instance