Namespace hg

Warning

The cpp documentation is in progress, so here is a raw dump of the whole hg namespace: enjoy!

enum class hg::accumulators

Values:

enumerator first

enumerator last

enumerator mean

enumerator min

enumerator max

enumerator counter

enumerator sum

enumerator prod

enumerator argmin

enumerator argmax

enum class hg::weight_functions

Predefined edge-weighting functions (see weight_graph function)

Values:

enumerator mean

enumerator min

enumerator max

enumerator L0

enumerator L1

enumerator L2

enumerator L_infinity

enumerator L2_squared

enumerator source

enumerator target

enum class hg::optimal_cut_measure

Values:

enumerator BCE

enumerator DHamming

enumerator DCovering

enum class hg::partition_measure

Values:

enumerator BCE

enumerator DHamming

enumerator DCovering

enum hg::tos_padding

Padding mode for the function component_tree_tree_of_shapes.

Values:

enumerator none

enumerator mean

enumerator zero

enum class hg::tree_category

Values:

enumerator component_tree

enumerator partition_tree

enum class hg::leaves_it

Enum used in tree node iterator (leaves_to_root_iterator and root_to_leaves_iterator) to include or exclude leaves from the iterator.

Values:

enumerator include

enumerator exclude

enum class hg::root_it

Enum used in tree node iterator (leaves_to_root_iterator and root_to_leaves_iterator) to include or exclude the root from the iterator.

Values:

enumerator include

enumerator exclude

using hg::contour_2d = contour_2d_internal::contour_2d<point_2d_f>

using hg::polyline_contour_2d = contour_2d_internal::polyline_contour_2d<point_2d_f>

using hg::contour_segment_2d = contour_2d_internal::contour_segment_2d<point_2d_f>

template<typename value_t>
using hg::array_1d = xt::xtensor<value_t, 1>

template<typename value_t>
using hg::array_2d = xt::xtensor<value_t, 2>

template<typename value_t>
using hg::array_3d = xt::xtensor<value_t, 3>

template<typename value_t>
using hg::array_4d = xt::xtensor<value_t, 4>

template<typename value_t>
using hg::array_nd = xt::xarray<value_t>

template<int dim>
using hg::embedding_grid = typename embedding_internal::embedding_grid<dim, index_t>

using hg::embedding_grid_1d = typename embedding_internal::embedding_grid<1, index_t>

using hg::embedding_grid_2d = typename embedding_internal::embedding_grid<2, index_t>

using hg::embedding_grid_3d = typename embedding_internal::embedding_grid<3, index_t>

using hg::embedding_grid_4d = typename embedding_internal::embedding_grid<4, index_t>

template<typename value_type>
using hg::fibonacci_heap = fibonacci_heap_internal::fibonacci_heap<value_type>

using hg::lca_sparse_table_block = lca_internal::lca_rmq<tree, range_minimum_query_internal::rmq_sparse_table_block<index_t>>

using hg::lca_sparse_table = lca_internal::lca_rmq<tree, range_minimum_query_internal::rmq_sparse_table<index_t>>

using hg::lca_fast = lca_sparse_table_block

template<typename value_t, unsigned int dim>
using hg::point = xt::xtensor_fixed<value_t, xt::xshape<dim>>

using hg::point_1d_f = point<double, 1>

using hg::point_1d_i = point<index_t, 1>

using hg::point_2d_f = point<double, 2>

using hg::point_2d_i = point<index_t, 2>

using hg::point_3d_f = point<double, 3>

using hg::point_3d_i = point<index_t, 3>

using hg::point_4d_f = point<double, 4>

using hg::point_4d_i = point<index_t, 4>

template<typename embedding_t>
using hg::regular_graph = regular_graph_internal::regular_graph<embedding_t>

using hg::regular_grid_graph_1d = regular_graph<hg::embedding_grid_1d>

using hg::regular_grid_graph_2d = regular_graph<hg::embedding_grid_2d>

using hg::regular_grid_graph_3d = regular_graph<hg::embedding_grid_3d>

using hg::regular_grid_graph_4d = regular_graph<hg::embedding_grid_4d>

template<typename embedding_t>
using hg::regular_graph_out_edge_iterator = typename regular_graph_internal::regular_graph<embedding_t>::out_edge_iterator

template<typename embedding_t>
using hg::regular_graph_adjacent_vertex_iterator = typename regular_graph_internal::regular_graph<embedding_t>::adjacency_iterator

using hg::tree = tree_internal::tree

using hg::vecS = undirected_graph_internal::vecS

using hg::hash_setS = undirected_graph_internal::hash_setS

template<typename storage_type = vecS>
using hg::undirected_graph = undirected_graph_internal::undirected_graph<storage_type>

using hg::ugraph = undirected_graph_internal::undirected_graph<>

using hg::union_find = union_find_internal::union_find<>

using hg::index_t = int64_t

Preferred type to represent an index.

using hg::size_t = std::size_t

Preferred type to represent a size.

template<typename T>
using hg::stackv = std::stack<T, std::vector<T>>

const index_t hg::invalid_index = -1

Constant used to represent an invalid index (eg.

not initialized)

template<typename T, typename accumulator_t, typename output_t = typename T::value_type>
auto hg::accumulate_at(const array_1d<index_t> &indices, const xt::xexpression<T> &xweights, const accumulator_t &accumulator)

Accumulate the given weights located at given indices.

Let :math:M = max(indices). For all :math:i \in \{0, \ldots, M\}

.. math::

 result[i] = accumulator(\{weights[j, :] \mid indices[j] = i  \})

Template Parameters:

• T –

• accumulator_t –

• output_t –

Parameters:

• indices – a 1d array of indices (entry equals to :math:-1 are ignored)

• xweights – a nd-array of shape :math:(s_1, \ldots, s_n) such that :math:s_1=indices.size()

• accumulator –

Returns:

a nd-array of size :math:(M, s_2, \ldots, s_n)

template<typename graph_t, typename T, typename accumulator_t, typename output_t = typename T::value_type>
auto hg::accumulate_graph_edges(const graph_t &graph, const xt::xexpression<T> &xedge_weights, const accumulator_t &accumulator)

template<typename graph_t, typename T, typename accumulator_t, typename output_t = typename T::value_type>
auto hg::accumulate_graph_vertices(const graph_t &graph, const xt::xexpression<T> &xvertex_weights, const accumulator_t &accumulator)

template<typename tree_t, typename T, typename accumulator_t, typename output_t = typename T::value_type>
auto hg::accumulate_parallel(const tree_t &tree, const xt::xexpression<T> &xinput, const accumulator_t &accumulator)

template<typename tree_t, typename T, typename accumulator_t, typename output_t = typename T::value_type>
auto hg::accumulate_sequential(const tree_t &tree, const xt::xexpression<T> &xvertex_data, const accumulator_t &accumulator)

template<typename tree_t, typename T1, typename T2, typename accumulator_t, typename combination_fun_t, typename output_t = typename T1::value_type>
auto hg::accumulate_and_combine_sequential(const tree_t &tree, const xt::xexpression<T1> &xinput, const xt::xexpression<T2> &xvertex_data, const accumulator_t &accumulator, const combination_fun_t &combine)

template<typename tree_t, typename T1>
auto hg::propagate_parallel(const tree_t &tree, const xt::xexpression<T1> &xinput)

template<typename tree_t, typename T1, typename T2>
auto hg::propagate_parallel(const tree_t &tree, const xt::xexpression<T1> &xinput, const xt::xexpression<T2> &xcondition)

template<typename tree_t, typename T1, typename T2>
auto hg::propagate_sequential(const tree_t &tree, const xt::xexpression<T1> &xinput, const xt::xexpression<T2> &xcondition)

template<typename tree_t, typename T, typename accumulator_t>
auto hg::propagate_sequential_and_accumulate(const tree_t &tree, const xt::xexpression<T> &xinput, const accumulator_t &accumulator)

template<typename tree_t, typename T1, typename T2, typename accumulator_t>
auto hg::propagate_sequential_and_accumulate(const tree_t &tree, const xt::xexpression<T1> &xinput, const accumulator_t &accumulator, const xt::xexpression<T2> &xcondition)

template<typename graph_t, typename tree_t, typename T, typename T1, typename accumulator_t, typename output_t = typename T::value_type>
auto hg::accumulate_on_contours(const graph_t &graph, const tree_t &tree, const xt::xexpression<T> &xinput, const xt::xexpression<T1> &xdepth, const accumulator_t &accumulator)

template<typename T1, typename T2>
auto hg::project_fine_to_coarse_labelisation(const xt::xexpression<T1> &xlabelisation_fine, const xt::xexpression<T2> &xlabelisation_coarse, size_t num_regions_fine = 0, size_t num_regions_coarse = 0)

Given two labelisations, a fine and a coarse one, of a same set of elements.

Find for each label (ie. region) of the fine labelisation, the label of the region in the coarse labelisation that maximises the intersection with the “fine” region.

Pre-condition: range(xlabelisation_fine) = [0..num_regions_fine[ range(xlabelisation_coarse) = [0..num_regions_coarse[

If num_regions_fine or num_regions_coarse are not provided, they will be determined as max(xlabelisation_fine) + 1 and max(xlabelisation_coarse) + 1

Template Parameters:

• T1 –

• T2 –

Parameters:

• xlabelisation_fine –

• num_regions_fine –

• xlabelisation_coarse –

• num_regions_coarse –

Returns:

a 1d array of size num_regions_fine

inline auto hg::project_fine_to_coarse_rag(const region_adjacency_graph &fine_rag, const region_adjacency_graph &coarse_rag)

Given two region adjacency graphs, a fine and a coarse one, of a same set of elements.

Find for each region of the fine rag, the region of the coarse rag that maximises the intersection with the “fine” region.

Parameters:

• fine_rag –

• coarse_rag –

Returns:

a 1d array of size num_vertices(fine_rag.rag)

template<typename graph_t, typename T>
auto hg::make_hierarchy_aligner_from_graph_cut(const graph_t &graph, const xt::xexpression<T> &saliency_map)

template<typename graph_t, typename T>
auto hg::make_hierarchy_aligner_from_labelisation(const graph_t &graph, const xt::xexpression<T> &vertex_labels)

template<typename graph_t, typename tree_t, typename T>
auto hg::make_hierarchy_aligner_from_hierarchy(const graph_t &graph, const tree_t &tree, const xt::xexpression<T> &altitudes)

template<typename graph_t>
std::pair<bool, array_1d<unsigned char>> hg::is_bipartite_graph(const graph_t &g)

Check if a graph is bipartite.

A bipartite graph is a graph whose vertices can be divided into two disjoint and independent sets X and Y such that every edge connects a vertex in X to one in Y.

This function is implemented using a depth first search. Its runtime complexity is O(|V| + |E|).

If the graph is bipartite, the function returns a pair of a boolean set to true and an array of colors. For any vertex v, color[v] == 0 if v belongs to X and color[v] == 1 if v belongs to Y. Note that the coloring is not unique, the algorithm returns any valid coloring.

If the graph is not bipartite, the function returns a pair of a boolean set to false and an empty array.

Template Parameters:

graph_t – type of the input graph

Parameters:

g – input graph

Returns:

a pair of a boolean and an array of colors. The boolean is true if the graph is bipartite and false otherwise. If the graph is bipartite, the array contains the color of each vertex. If the graph is not bipartite, the array is empty.

template<typename T>
std::pair<bool, array_1d<unsigned char>> hg::is_bipartite_graph(const xt::xexpression<T> &xsources, const xt::xexpression<T> &xtargets, index_t num_vertices)

Check if a graph is bipartite.

A bipartite graph is a graph whose vertices can be divided into two disjoint and independent sets X and Y such that every edge connects a vertex in X to one in Y.

This function is implemented using a union find approach. Its runtime complexity is O(|E| α(|V|)) time, where α is the extremely slowly growing inverse of the single-valued Ackermann function.

If the graph is bipartite, the function returns a pair of a boolean set to true and an array of colors. For any vertex v, color[v] == 0 if v belongs to X and color[v] == 1 if v belongs to Y. Note that the coloring is not unique, the algorithm returns any valid coloring.

If the graph is not bipartite, the function returns a pair of a boolean set to false and an empty array.

Template Parameters:

T – type of the input arrays

Parameters:

• xsources – source vertices of the edges

• xtargets – target vertices of the edges

• num_vertices – number of vertices in the graph

Returns:

a pair of a boolean and an array of colors. The boolean is true if the graph is bipartite and false otherwise. If the graph is bipartite, the array contains the color of each vertex. If the graph is not bipartite, the array is empty.

template<typename graph_t, typename T>
auto hg::bipartite_graph_matching(const graph_t &graph, const xt::xexpression<T> &xedge_weights)

Bipartite graph matching.

The bipartite graph matching problem is to find a maximum cardinality matching with minimum weight in a bipartite graph.

The input graph must be a balanced bipartite graph, i.e. the number of vertices in each side must be equal. Moreover, the left hand side must be composed of the vertices 0 to num_vertices(graph)/2 - 1 and the right hand side of the vertices num_vertices(graph)/2 to num_vertices(graph) - 1. A perfect match must exist in the graph or this function might never terminate or return an incorrect result.

Edge weights must be of integral type.

This function is implemented using the CSA library by Andrew V. Goldberg and Robert Kennedy. It is based on a push-relabel method.

Template Parameters:

• graph_t – type of the input graph

• T – type of the weights

Parameters:

• graph – input graph

• xedge_weights – edge weights

Returns:

the indices of the edges in the matching

template<typename graph_t, typename T>
auto hg::graph_cut_2_labelisation(const graph_t &graph, const xt::xexpression<T> &xedge_weights)

Labelize graph vertices according to the given graph cut.

Each edge having a non zero value in the given edge_weights are assumed to be part of the cut.

Template Parameters:

• graph_t –

• T –

• label_type –

Parameters:

• graph –

• edge_weights –

Returns:

template<typename graph_t, typename T>
auto hg::labelisation_2_graph_cut(const graph_t &graph, const xt::xexpression<T> &xvertex_labels)

Determine the graph cut that corresponds to a given labeling of the graph vertices.

The result is a weighting of the graph edges where edges with a non zero weight are part of the cut.

Template Parameters:

• graph_t –

• T –

Parameters:

• graph –

• xvertex_labels –

Returns:

template<typename graph_t, typename T>
auto hg::minimum_spanning_tree(const graph_t &graph, const xt::xexpression<T> &xedge_weights)

Computes a minimum spanning tree of the given edge weighted graph using Kruskal’s algorithm.

The returned structure contains two elements:

• the minimum spanning tree (mst)

• a map (mst_edge_map) that indicates for each edge of the mst, the corresponding edge index in the input graph

If the input graph is not connected, the result is indeed a minimum spanning forest.

Template Parameters:

• graph_t – Input graph type

• T – Input edge weights type

Parameters:

• graph – Input graph

• xedge_weights – Input edge weights

Returns:

a mst structure

template<typename graph_t, typename T>
auto hg::subgraph_spanning(const graph_t &graph, const xt::xexpression<T> &xedge_indices)

Compute a spanning subgraph of the given graph composed of the edges of the input graph indicated in the edge_indices array.

The edges of the subgraph will be in the order given in edge_indices array.

Template Parameters:

• graph_t –

• T –

Parameters:

• graph – input graph

• xedges_indices – list of edges of the input graph to include in the subgraph

Returns:

a spanning subgraph

inline ugraph hg::line_graph(const ugraph &graph)

Compute the line graph of an undirected graph.

The line graph :math:LG of an undirected graph :math:G is a graph such that:

• each vertex of :math:LG represents an edge of :math:G: the :math:i-th vertex of :math:LG corresponds to the :math:i-th edge of :math:G; and

• two vertices :math:x and :math:y of :math:LG are adjacent if their corresponding edges in :math:G share a common extremity. Formally, if :math:x represents the edge :math:\{i, j \} and if :math:y represents the edge :math:\{k, j \}, then the edge :math:\{x, y\} belongs to :math:LG if :math:\{i, j \} \\cap \{k, j \} \\neq \emptyset.

The line graph is also known as: the covering graph, the derivative, the edge-to-vertex dual, the conjugate, the representative graph, the edge graph, the interchange graph, the adjoint graph, or the derived graph.

Parameters:

graph –

Returns:

template<typename graph_t>
ugraph hg::line_graph(const graph_t &graph)

See description of the function above.

Template Parameters:

graph_t –

Parameters:

graph –

Returns:

template<typename result_value_t = double, typename graph_t>
auto hg::weight_graph(const graph_t &graph, const std::function<result_value_t(typename graph_t::vertex_descriptor, typename graph_t::vertex_descriptor)> &fun)

Compute edge-weights of a graph based on a weighting function.

A weighting function is a function that associates a weights to a pair of vertices.

Template Parameters:

• graph_t –

• result_value_t –

Parameters:

• graph –

• fun –

Returns:

an array of weights

template<typename result_value_t = double, typename promoted_type = double, typename graph_t, typename T>
auto hg::weight_graph(const graph_t &graph, const xt::xexpression<T> &xvertex_weights, weight_functions weight)

Compute edge-weights of a graph based from the vertex-weights and a predefined weighting function (see weight_functions enum).

Each edge is weighted with a combination of its extremities weights.

Template Parameters:

• result_value_t – The value type of the result

• promoted_type – The value type used for internal computation

• graph_t –

• T –

Parameters:

• graph –

• xvertex_weights –

• weight –

Returns:

template<typename value_t>
decltype(auto) hg::make_horizontal_cut_nodes(array_1d<index_t> &&nodes, value_t altitude)

template<typename tree_t, typename T>
decltype(auto) hg::make_horizontal_cut_explorer(tree_t &&tree, T &&altitudes)

template<typename graph_t, typename T>
auto hg::make_region_adjacency_graph_from_labelisation(const graph_t &graph, const xt::xexpression<T> &xvertex_labels)

Construct a region adjacency graph from a vertex labeled graph in linear time.

Template Parameters:

• graph_t –

• T –

Parameters:

• graph –

• xvertex_labels –

Returns:

see struct region_adjacency_graph

template<typename graph_t, typename T>
auto hg::make_region_adjacency_graph_from_graph_cut(const graph_t &graph, const xt::xexpression<T> &xedge_weights)

Construct a region adjacency graph from a graph cut in linear time.

Any edge with weight different from 0 belongs to the cut.

TODO factorize method with make_region_adjacency_graph_from_labelisation

Template Parameters:

• graph_t –

• T –

Parameters:

• graph –

• xedge_weights –

Returns:

see struct region_adjacency_graph

template<typename T>
auto hg::rag_back_project_weights(const array_1d<index_t> &rag_map, const xt::xexpression<T> &xrag_weights)

Projects weights on the rag (vertices or edges) onto the original graph.

Template Parameters:

T –

Parameters:

• rag_map – rag vertex_map or rag edge_map (see struct region_adjacency_graph)

• xrag_weights – node or edge weights of the rag (depending of the provided rag_map)

Returns:

original graph (vertices or edges) weights

template<typename T, typename accumulator_t, typename output_t = typename T::value_type>
auto hg::rag_accumulate(const array_1d<index_t> &rag_map, const xt::xexpression<T> &xweights, const accumulator_t &accumulator)

Accumulate original graph (vertices or edges) weights onto the rag (vertices or edges)

Template Parameters:

• T –

• accumulator_t –

• output_t –

Parameters:

• rag_map – rag vertex_map or rag edge_map (see struct region_adjacency_graph)

• xweights – node or edge weights of the original graph (depending of the provided rag_map)

• accumulator –

Returns:

rag (vertices or edges) weights

template<typename tree_t, typename T1, typename T2>
auto hg::reconstruct_leaf_data(const tree_t &tree, const xt::xexpression<T1> &altitudes, const xt::xexpression<T2> &deleted_nodes)

Each leaf of the tree takes the weight of its closest non deleted ancestor.

Template Parameters:

• tree_t –

• T1 –

• T2 –

Parameters:

• tree –

• altitudes –

• deleted_nodes –

Returns:

template<typename tree_t, typename T, typename value_t>
auto hg::labelisation_horizontal_cut_from_threshold(const tree_t &tree, const xt::xexpression<T> &xaltitudes, const value_t threshold)

Labelize tree leaves according to an horizontal cut in the tree.

Two leaves are in the same region (ie. have the same label) if the altitude of their lowest common ancestor is smaller or equal than the specified threshold.

The label of a leave l is equal to the index of the smallest node containing l and whose altitude is strictly greater than the specified threshold.

Template Parameters:

• tree_t –

• T –

• value_t –

Parameters:

• tree –

• xaltitudes –

• threshold –

Returns:

template<typename tree_t, typename T>
auto hg::labelisation_hierarchy_supervertices(const tree_t &tree, const xt::xexpression<T> &xaltitudes)

Labelize the tree leaves into supervertices.

Two leaves are in the same supervertex if they have a common ancestor of altitude 0.

This functions guaranties that the labels are in the range [0, num_supervertices-1].

Template Parameters:

• tree_t –

• T –

Parameters:

• tree –

• xaltitudes –

Returns:

template<typename tree_t, typename T>
auto hg::supervertices_hierarchy(const tree_t &tree, const xt::xexpression<T> &xaltitudes)

Extract the supervertices associated to the given tree and create the equivalent tree on this supervertices.

Two leaves are in the same supervertex if they have a common ancestor of altitude 0.

The equivalent tree is obtained by removing all nodes of the given tree which does not contain any of the supervertices. Its leaves are thus the supervertices.

Also returns an array that maps any node index i of the new tree, to the index of this node in the original tree.

Template Parameters:

• tree_t –

• T –

Parameters:

• tree –

• xaltitudes –

Returns:

template<typename tree1_t, typename tree2_t>
bool hg::test_tree_isomorphism(const tree1_t &t1, const tree2_t &t2)

Test if 2 trees are isomorph assuming that they share the same leaves.

By this definition t1 is isomorph to t2 if there exist a bijection f from vertices(t1) to vertices(t2) such that: 1) for any leaf node n of t1, f(n) = n and n 2) for any node n of t1, f(t1.parent(n)) = t2.parent(f(n))

Note that the root r node of a tree t is defined by t.parent(r) = r, thus 2) becomes for the root node r1 of t1, f(r1) = t2.parent(f(r1)), i.e. f(r1) is the root node of t2

Template Parameters:

• tree1_t –

• tree2_t –

Parameters:

• t1 –

• t2 –

Returns:

template<typename tree_t, typename T1, typename T2>
auto hg::binary_labelisation_from_markers(const tree_t &tree, const xt::xexpression<T1> &xobject_marker, const xt::xexpression<T2> &xbackground_marker)

Given two binary markers o (object) and b (background) (given by their indicator functions) on the leaves of a tree t, the corresponding binary labelization of the leaves of t is defined as the union of all the nodes intersecting o but not b.

final_object = union {R in T | R cap o neq emptyset and R cap b = emptyset}

Template Parameters:

• tree_t – tree type

• T1 – xtensor type, value_type must be castable to bool

• T2 – xtensor type, value_type must be castable to bool

Parameters:

• tree – input tree

• xobject_marker – indicator function of the object marker

• xbackground_marker – indicator function of the background marker

Returns:

indicator function of the final_object

template<typename tree_t, typename T>
auto hg::sort_hierarchy_with_altitudes(const tree_t &tree, const xt::xexpression<T> &xaltitudes)

Sort the nodes of a tree according to their altitudes.

The altitudes must be increasing, i.e. for any nodes i, j such that j is an ancestor of j, then altitudes[i] <= altitudes[j].

The result is a new tree and a node map, isomorph to the input tree such that for any nodes i and j, i<j => altitudes[node_map[i]] <= altitudes[node_map[j]]

The latter condition is stronger than the original condition on the altitudes as j is an ancestor of i implies i<j while the converse is not true.

Note that the altitudes of the new tree can be obtained with:

auto res = sort_hierarchy_with_altitudes(tree, altitudes); auto & new_tree = res.tree; auto new_altitudes = xt::index_view(altitudes, res.node_map);

Template Parameters:

• tree_t –

• T –

Parameters:

• tree –

• xaltitudes –

Returns:

template<typename tree_t>
auto hg::sub_tree(const tree_t &tree, index_t root)

Extract the sub tree rooted in the given node from the given tree.

The result is a new tree :math:st and a node map :math:nm such that

• the node map associates each node of the sub tree :math:st to its corresponding node in the original tree

• the order of the nodes of the original tree is preserved in the sub tree: for any vertices :math:x and :math:y of :math:st such that :math:x < y then :math:nm[x] < nm[y]

:Complexity:

The tree is constructed in linearithmic time :math:\mathcal{O}(n\log(n)) with :math:n the number of vertices in the sub tree.

Template Parameters:

tree_t –

Parameters:

• tree –

• root –

Returns:

template<typename tree_type, typename T, typename accumulator_type = hg::accumulator_sum>
auto hg::labelisation_optimal_cut_from_energy(const tree_type &tree, const xt::xexpression<T> &xenergy_attribute, const accumulator_type accumulator = hg::accumulator_sum())

Computes the labelisation of the input tree leaves corresponding to the optimal cut according to the given energy attribute.

Given a node i, the value energy_attribute(i) represents the energy fo the partial partition composed of the single region i. Given a node i, the energy of the partial partition composed of the children of i is given by accumulator(energy_attribute(children(i))).

This function computes the partition (ie. a set of node forming a cut of the tree) that has a minimal energy according to the definition above.

The algorithm used is based on dynamic programming and runs in linear time w.r.t. to the number of nodes in the tree.

See:

Laurent Guigues, Jean Pierre Cocquerez, H ervé Le Men. Scale-sets Image Analysis. International Journal of Computer Vision, Springer Verlag, 2006, 68 (3), pp.289-317

and

Bangalore Ravi Kiran, Jean Serra. Global-local optimizations by hierarchical cuts and climbing energies. Pattern Recognition Letters, Elsevier, 2014, 47 (1), pp.12-24.

Template Parameters:

• tree_type – input tree type

• T – energy attribute type

• accumulator_type – accumulator type

Parameters:

• tree – input tree

• xenergy_attribute – 1d array of energy attribute for the input tree

• accumulator – accumulator used to define how children energies are combined in order to obtain the energy of the corresponding partial partition

Returns:

a 1d integer array with num_leaves(tree) elements representing the minimal energy partition

template<typename tree_type, typename T>
auto hg::hierarchy_to_optimal_energy_cut_hierarchy(const tree_type &tree, const xt::xexpression<T> &xdata_fidelity_attribute, const xt::xexpression<T> &xregularization_attribute, const int approximation_piecewise_linear_function = 10)

Transforms the given hierarchy into its optimal energy cut hierarchy for the given energy terms.

In the optimal energy cut hierarchy, any horizontal cut corresponds to an optimal energy cut in the original hierarchy.

Each node i of the tree is associated to a data fidelity energy D(i) and a regularization energy R(i). The algorithm construct a new hierarchy with associated altitudes such that the horizontal cut of level lambda is the optimal cut for the energy attribute D + lambda * R of the input tree (see function labelisation_optimal_cut_from_energy). In other words, the horizontal cut of level lambda in the result is the cut of the input composed of the nodes N such that sum_{r in N} D(r) + lambda * R(r) is minimal.

PRECONDITION: the regularization energy R must be sub additive: for each node i: R(i) <= sum_{c in children(i)} R(c)

The algorithm runs in linear time O(n)

See:

Laurent Guigues, Jean Pierre Cocquerez, Hervé Le Men. Scale-sets Image Analysis. International Journal of Computer Vision, Springer Verlag, 2006, 68 (3), pp.289-317

Template Parameters:

• tree_type –

• T –

Parameters:

• tree – Input tree

• xdata_fidelity_attribute – Data fidelity energy (1d array)

• xregularization_attribute – Regularization energy (1d array)

• approximation_piecewise_linear_function – Maximum number of pieces used in the approximated piecewise linear model for the energy.

Returns:

a node_weighted_tree

template<typename graph_t, typename T1, typename T2, typename T3, typename T4, typename T5>
auto hg::binary_partition_tree_MumfordShah_energy(const graph_t &graph, const xt::xexpression<T1> &xvertex_perimeter, const xt::xexpression<T2> &xvertex_area, const xt::xexpression<T3> &xvertex_values, const xt::xexpression<T4> &xsquared_vertex_values, const xt::xexpression<T5> &xedge_length)

Compute the binary partition tree, i.e.

the agglomerative clustering, according to the Mumford-Shah energy with a constant piecewise model.

The distance between two regions is equal to the apparition scale of the merged region.

See:

Laurent Guigues, Jean Pierre Cocquerez, Hervé Le Men. Scale-sets Image Analysis. International Journal of Computer Vision, Springer Verlag, 2006, 68 (3), pp.289-317

Template Parameters:

• graph_t –

• T1 –

• T2 –

• T3 –

• T4 –

• T5 –

Parameters:

• graph – Input graph

• xvertex_perimeter – Perimeter of each vertex of the input graph

• xvertex_area – Area of each vertex of the input graph

• xvertex_values – Sum of values inside the region represented by each vertex of the input graph.

• xsquared_vertex_values – Sum of the squared values inside the region represented by each vertex of the input graph.

• xedge_length – Length of the frontier represented by each edge.

Returns:

a node_weighted_tree

template<typename tree_iterator>
auto hg::tree_fusion_depth_map(const tree_iterator first, const tree_iterator last)

The result is a directed acyclic graph with a single root (corresponding to the roots of the input trees).

The depth of a node in this graph is defined as the length of the longest path from the root this node.

This function returns the depth of the leaves of this graph (which are the same as the leaves of the input trees).

Template Parameters:

tree_iterator – Iterator on tree pointers

Parameters:

• first –

• last –

Returns:

an 1d array of size num_leaves(**first)

template<typename range_tree_t>
auto hg::tree_fusion_depth_map(const range_tree_t &range)

template<typename tree_t, typename T, typename Tw>
auto hg::tree_monotonic_regression(const tree_t &tree, const xt::xexpression<T> &xaltitudes, const xt::xexpression<Tw> &xweights, const std::string &mode)

Monotonic regression on the given tree altitudes.

Computes new altitudes naltitudes that are close to the given :attr:altitudes and that are increasing for the given :attr:tree: i.e. for any nodes :math:i, j such that :math:j is an ancestor of :math:i, then :math:naltitudes[i] \leq naltitudes[j].

The definition of close depends of the value of :attr:mode:

• If :attr:mode is equal to "min" then naltitudes is the largest increasing function below :attr:altitudes.

• If :attr:mode is equal to "max" then naltitudes is the smallest increasing function above :attr:altitudes.

• If :attr:mode is equal to "least_square" then naltitudes minizes the following minization problem:

.. math::

naltitudes = \\arg \min_x \sum_i (weights[i] * (altitudes[i] - x[i])^2)

such that naltitudes is increasing for :attr:tree.

:Complexity:

With :math:n the number of nodes in the :attr:tree:

• For the modes "min" and "max", the runtime complexity is linear :math:\mathcal{O}(n).

• For the mode "least_square", the runtime complexity is linearithmic :math:\mathcal{O}(n\log(n)) and the space complexity is linear :math:\mathcal{O}(n). The algorithm used is described in:

  P. Pardalos and G. Xue ‘Algorithms for a Class of Isotonic Regression Problems. https://link.springer.com/article/10.1007/PL00009258`_ Algorithmica (1999) 23: 211. doi:10.1007/PL00009258

Template Parameters:

• tree_t –

• T –

• Tw –

Parameters:

• tree – input tree

• xaltitudes – tree node altitudes

• xweights – tree node weights

• mode – “min”, “max”, or “ least_square”

Returns:

template<typename tree_t, typename T>
auto hg::tree_monotonic_regression(const tree_t &tree, const xt::xexpression<T> &xaltitudes, const std::string &mode)

template<typename graph_t, typename T>
auto hg::labelisation_watershed(const graph_t &graph, const xt::xexpression<T> &xedge_weights)

Linear time watershed cut algorithm.

Jean Cousty, Gilles Bertrand, Laurent Najman, Michel Couprie. Watershed Cuts: Minimum Spanning Forests and the Drop of Water Principle. IEEE Transactions on Pattern Analysis and Machine Intelligence, Institute of Electrical and Electronics Engineers, 2009, 31 (8), pp.1362-1374.

Template Parameters:

• graph_t –

• T –

Parameters:

• graph –

• xedge_weights –

Returns:

array of labels on graph vertices, numbered from 1 to n with n the number of minima

template<typename graph_t, typename T1, typename T2>
auto hg::labelisation_seeded_watershed(const graph_t &graph, const xt::xexpression<T1> &xedge_weights, const xt::xexpression<T2> &xvertex_seeds, const typename T2::value_type background_label = 0)

template<typename tree_t, typename T>
auto hg::dendrogram_purity(const tree_t &tree, const xt::xexpression<T> &xleaf_labels)

Weighted average of the purity of each node of the tree with respect to a ground truth labelization of the tree leaves.

Let :math:T be a tree with leaves :math:V=\{1, \ldots, n\}. Let :math:C=\{C_1, \ldots, C_K\} be a partition of :math:V into :math:k (label) sets.

The purity of a subset :math:X of :math:V with respect to class :math:C_\ell\in C is the fraction of elements of :math:X that belongs to class :math:C_\ell:

.. math::

 pur(X, C_\ell) = \\frac{| X \cap C_\ell |}{| X |}.

The purity of :math:T is the defined as:

.. math::

 pur(T) = \\frac{1}{Z}\sum_{k=1}^{K}\sum_{x,y\in C_k, x\\neq y} pur(lca_T(x,y), C_k)

with :math:Z=| \{\{x,y\} \subseteq V \mid x\\neq y, \exists k, \{x,y\}\subseteq C_k\} |.

:See:

 Heller, Katherine A., and Zoubin Ghahramani. "`Bayesian hierarchical clustering <https://www2.stat.duke.edu/~kheller/bhcnew.pdf>`_ ."
 Proc. ICML. ACM, 2005.

:Complexity:

The dendrogram purity is computed in :math:\mathcal{O}(N\\times K \\times C^2) with :math:N the number of nodes in the tree, :math:K the number of classes, and :math:C the maximal number of children of a node in the tree.

Let :maht:T be a tree with leaves :maht:V=\{1, \ldots, n\}. Let :maht:C=\{C_1, \ldots, C_K\} be a partition of :maht:V into :maht:k (label) sets.

The purity of a subset :maht:X of :maht:V with respect to class :maht:C_\ell\in C is the fraction of elements of :maht:X that belongs to class :maht:C_\ell:

.. math::

 pur(X, C_\ell) = \\frac{| X \cap C_\ell |}{| X |}.

The purity of :maht:T is the defined as:

.. math::

 pur(T) = \\frac{1}{Z}\sum_{k=1}^{K}\sum_{x,y\in C_k, x\neq y} pur(lca_T(x,y), C_k)

with :math:Z=| \{\{x,y\} \subseteq V \mid x\neq y, \exists k, \{x,y\}\subseteq C_k\} |.

:See:

 Heller, Katherine A., and Zoubin Ghahramani. "Bayesian hierarchical clustering."
 Proc. ICML. ACM, 2005.

:Complexity:

The dendrogram purity is computed in :math:\mathcal{O}(N\times K \times C^2) with :math:N the number of nodes in the tree, :math:K the number of classes, and :math:C the maximal number of children of a node in the tree.

Template Parameters:

• tree_t –

• T –

• tree_t –

• T –

Parameters:

• tree – input tree

• xleaf_labels – must be a 1d array with values in [0, max_label]

• tree – input tree

• xleaf_labels – must be a 1d array with values in [0, max_label]

Returns:

a score between 0 and 1 (higher is better)

Returns:

a score between 0 and 1 (higher is better)

template<typename tree_t, typename T1, typename T2, typename scorer_t>
auto hg::assess_fragmentation_horizontal_cut(const tree_t &tree, const xt::xexpression<T1> &xaltitudes, const xt::xexpression<T2> &xground_truth, const scorer_t &partition_scorer, const array_1d<index_t> &vertex_map = {}, size_t max_regions = 200)

template<typename value_type = index_t, typename T1, typename T2>
auto hg::card_intersections(const xt::xexpression<T1> &xcandidate, const xt::xexpression<T2> &xground_truths)

template<typename T, typename scorer_t>
auto hg::assess_partition(const std::vector<T> &card_intersections, const scorer_t &scorer)

template<typename T1, typename T2, typename scorer_t>
auto hg::assess_partition(const xt::xexpression<T1> &xcandidate, const xt::xexpression<T2> &xground_truths, const scorer_t &scorer)

template<typename tree_t, typename T>
auto hg::attribute_area(const tree_t &tree, const xt::xexpression<T> &xleaf_area)

The area of a node n of the tree t is equal to the sum of the area of the leaves in the subtree rooted in n.

area(n) = sum_{l in leaves(t), l is a descendant of n} area(n)

Template Parameters:

• tree_t – tree type

• T – xexpression derived type of xleaf_area

Parameters:

• tree – input tree

• xleaf_area – area of the leaves of the input tree

Returns:

an array with the area of each node of the tree

template<typename tree_t>
auto hg::attribute_area(const tree_t &tree)

The area of a node n of the tree t is equal to the number of leaves in the subtree rooted in n.

area(n) = |{l in leaves(t), l is a descendant of n}|

Template Parameters:

tree_t – tree type

Parameters:

tree – input tree

Returns:

an array with the area of each node of the tree

template<typename tree_t, typename T1, typename T2>
auto hg::attribute_volume(const tree_t &tree, const xt::xexpression<T1> &xnode_altitude, const xt::xexpression<T2> &xnode_area)

The volume of a node n of the tree t is defined recursively as: volume(n) = abs(altitude(n) - altitude(parent(n)) * area(n) + sum_{c in children(n, t)} volume(c)

Template Parameters:

• tree_t – tree type

• T1 – xexpression derived type of xnode_altitude

• T2 – xexpression derived type of xnode_area

Parameters:

• tree – input tree

• xnode_altitude – altitude of the nodes of the input tree

• xnode_area – area of the nodes of the input tree

Returns:

an array with the volume of each node of the tree

template<typename tree_t>
auto hg::attribute_depth(const tree_t &tree)

The depth of a node n of the tree t is equal to the number of ancestors of n.

Template Parameters:

tree_t – tree type

Parameters:

tree – input tree

Returns:

an array with the depth of each node of the tree

template<typename tree_t, typename T>
auto hg::attribute_height(const tree_t &tree, const xt::xexpression<T> &xaltitudes, bool increasing_altitudes)

In a tree :math:t, given that the altitudes of the nodes vary monotically from the leaves to the root, the height of a node :math:n of :math:t is equal to the difference between the altitude of the parent of :math:n and the altitude of the deepest non-leaf node in the subtree of :math:t rooted in :math:n.

If :attr:increasing_altitude is true, this means that altitudes are increasing from the leaves to the root (ie. for any node :math:n, :math:altitudes(n) \leq altitudes(parent(n)). Else, if :attr:increasing_altitude is false, this means that altitudes are decreasing from the leaves to the root (ie. for any node :math:n, :math:altitude(n) \geq altitude(parent(n)).

Template Parameters:

• tree_t – tree type

• T – xexpression derived type of xnode_altitude

Parameters:

• tree – input tree

• xnode_altitude – altitude of the nodes of the input tree

• increasing_altitudes – must be true if altitude is increasing, false if it is decreasing

Returns:

an array with the height of each node of the tree

template<typename tree_t, typename T>
auto hg::attribute_extrema(const tree_t &tree, const xt::xexpression<T> &xaltitudes)

Identify nodes in a hierarchy that represent extrema.

An extremum of the hierarchy :math:T with altitudes :math:alt is a node :math:n of :math:T such that the altitude of any non leaf node included in :math:n is equal to the altitude of :math:n and the altitude of the parent of :math:n is different from the altitude of :math:n.

The result is a boolean array such that :math:result(n)=true if the node :math:n is an extremum.

Template Parameters:

• tree_t –

• T –

Parameters:

• tree – Input tree

• xaltitudes – Tree node altitudes

Returns:

template<typename tree_t, typename T1, typename T2>
auto hg::attribute_extinction_value(const tree_t &tree, const xt::xexpression<T1> &xaltitudes, const xt::xexpression<T2> &xattribute, bool increasing_altitudes)

The extinction value of a node :math:n of the input tree :math:t with increasing altitudes :math:alt for the increasing attribute :math:att is the equal to the threshold :math:k such that the node :math:n is still in an minima of :math:t when all nodes having an attribute value smaller than :math:k are removed.

Formally, let :math:\{M_i\} be the set of minima of :math:(t, altitudes). Let :math:prec be a total ordering of :math:\{M_i\} such that :math:M_i \prec M_j \Rightarrow alt(M_i) \leq alt(M_j). Let :math:r(M_i) be the smallest node of :math:t containing :math:M_i and another minima :mathM_j such that :math:M_j \prec M_i. The extinction value of :mathM_i is then defined as :math:alt(r(M_i)) - alt(M_i).

Extinction values of minima are then extended to other nodes in the tree with the following rules:

• the extinction value of a non-leaf node :math:n which is not a minimum is defined as the largest extinction values among all the minima contained in :math:n (and 0 if :math:n does not contain any minima); and

• the extinction value of a leaf node :math:n belonging to a minima :math:M_i is equal to the extinction value of :math:M_i. I :math:n does not belong to any minima its extinction value is 0.

If :attr:increasing_altitude is true, this means that altitudes are increasing from the leaves to the root (ie. for any node :math:n, :math:altitudes(n) \leq altitudes(parent(n)). Else, if :attr:increasing_altitude is false, this means that altitudes are decreasing from the leaves to the root (ie. for any node :math:n, :math:altitude(n) \geq altitude(parent(n)): you should then replace minima by maxima in the description above.

Template Parameters:

• tree_t – tree type

• T – xexpression derived type of xaltitude

Parameters:

• tree – input tree

• xaltitudes – altitude of the nodes of the input tree

• xattribute – attribute used for filtering

• increasing_altitudes – must be true if altitude is increasing, false if it is decreasing

Returns:

an array with the dynamics of each node of the tree

template<typename tree_t, typename T>
auto hg::attribute_dynamics(const tree_t &tree, const xt::xexpression<T> &xaltitudes, bool increasing_altitudes)

Given a node :math:n of the tree :math:t, the dynamics of :math:n is the difference between the altitude of the deepest minima of the subtree rooted in :math:n and the altitude of the closest ancestor of :math:n that has a deeper minima in its subtree.

If no such ancestor exists then, the dynamics of :math:n is equal to the difference between the altitude of the highest node of the tree (the root) and the depth of the deepest minima.

The dynamics is the extinction values for the attribute height.

If :attr:increasing_altitude is true, this means that altitudes are increasing from the leaves to the root (ie. for any node :math:n, :math:altitudes(n) \leq altitudes(parent(n)). Else, if :attr:increasing_altitude is false, this means that altitudes are decreasing from the leaves to the root (ie. for any node :math:n, :math:altitude(n) \geq altitude(parent(n)).

Template Parameters:

• tree_t – tree type

• T – xexpression derived type of xaltitudes

Parameters:

• tree – input tree

• xaltitudes – altitude of the nodes of the input tree

• increasing_altitudes – must be true if altitude is increasing, false if it is decreasing

Returns:

an array with the dynamics of each node of the tree

template<typename tree_t>
auto hg::attribute_sibling(const tree_t &tree, index_t skip = 1)

For each node n which is the k-th child of its parent node p among N children, the attribute sibling of n is the index of the (k + skip) % N-th child of p.

The sibling of the root node is itself.

The sibling attribute enables to easily emulates a (doubly) linked list among brothers.

In a binary tree, the sibling attribute of a node is effectively its only brother (with skip equals to 1).

Template Parameters:

tree_t –

Parameters:

• tree – Input tree

• skip – Number of skipped element in the children list (including yourself)

Returns:

an array with the sibling index of each node of the tree

template<typename tree_t, typename graph_t, typename T1, typename T2>
auto hg::attribute_contour_length_component_tree(const tree_t &tree, const graph_t &base_graph, const xt::xexpression<T1> &xvertex_perimeter, const xt::xexpression<T2> &xedge_length)

Computes the contour length (perimeter) of each node of the input component tree.

Warning: does not work for tree of shapes left in original space (the problem is that two children of a node may become adjacent when the interpolated pixels are removed).

Template Parameters:

• tree_t –

• graph_t –

• T1 –

• T2 –

Parameters:

• tree – input tree

• base_graph – graph on the leaves of tree

• xvertex_perimeter – perimeter of each vertex of the base graph

• xedge_length – length of each edge of the base graph (length of the frontier between the two adjacent vertices)

Returns:

template<typename tree_t>
auto hg::attribute_child_number(const tree_t &tree)

Given a node :math:n whose parent is :math:p, the attribute value of :math:n is the rank of :math:n in the list of children of :math:p.

In other :math:attribute(n)=i means that :math:n is the :math:i-th child of :math:p.

The root of the tree, who has no parent, take the value -1.

Template Parameters:

tree_t –

Parameters:

tree –

Returns:

template<typename tree_t>
auto hg::attribute_smallest_enclosing_shape(const tree_t &t1, const tree_t &t2)

Given two trees :math:t_1 and :math:t_2 defined over the same domain, ie sharing the same set of leaves.

For each node :math:n of :math:t1, computes the index of the smallest node of :math:t2 containing :math:n.

Template Parameters:

tree_t –

Parameters:

• t1 –

• t2 –

Returns:

template<typename tree_t, typename T, typename value_type = typename T::value_type>
auto hg::attribute_children_pair_sum_product(const tree_t &tree, const xt::xexpression<T> &xnode_weights)

Given a tree :math:T with node weights :math:w: the children pair sum product for a node :math:n sums for every pairs :math:(c_i, c_j) of children of :math:n, the product of the node weights of :math:c_i and :math:c_j.

Formally:

.. math::

 res(n) = \sum_{i=0}^{i<numc(n)} \sum_{j=0}^{j<i} w(child(i, n)) * w(child(j, n))

where :math:numc(n) is the number of children of :math:n and :math:child(i, n) is the :math:i-th child of the node :math:n.

The result is thus an array of the same shape of :attr:node_weights

Template Parameters:

• tree_t –

• T –

• value_type –

Parameters:

• tree –

• xnode_weights –

Returns:

template<typename graph_t>
auto hg::source(const std::pair<typename graph::graph_traits<graph_t>::vertex_descriptor, typename graph::graph_traits<graph_t>::vertex_descriptor> &e, const graph_t&)

Source vertex of an edge.

Template Parameters:

graph_t –

Parameters:

e –

Returns:

template<typename graph_t>
auto hg::sources(const graph_t &t)

Source vertex of all edges oin the given graph.

Template Parameters:

graph_t –

Parameters:

t –

Returns:

1d expression with num_edges(t) element

template<typename graph_t>
auto hg::target(const std::pair<typename graph::graph_traits<graph_t>::vertex_descriptor, typename graph::graph_traits<graph_t>::vertex_descriptor> &e, const graph_t&)

Target vertex of an edge.

Template Parameters:

graph_t –

Parameters:

e –

Returns:

template<typename graph_t>
auto hg::targets(const graph_t &t)

Target vertex of all edges oin the given graph.

Template Parameters:

graph_t –

Parameters:

t –

Returns:

1d expression with num_edges(t) element

template<typename graph_t>
auto hg::vertex_iterator(const graph_t &g)

Range over all vertices of the given graph.

Template Parameters:

graph_t –

Parameters:

g –

Returns:

template<typename graph_t>
auto hg::edge_iterator(const graph_t &g)

Range over all edges of the given graph.

Template Parameters:

graph_t –

Parameters:

g –

Returns:

template<typename graph_t>
auto hg::out_edge_iterator(typename graph::graph_traits<graph_t>::vertex_descriptor v, const graph_t &g)

Range over all edges whose source is the given vertex in the given graph.

Template Parameters:

graph_t –

Parameters:

• v –

• g –

Returns:

template<typename graph_t>
auto hg::in_edge_iterator(typename graph::graph_traits<graph_t>::vertex_descriptor v, const graph_t &g)

Range over all edges whose target is the given vertex in the given graph.

Template Parameters:

graph_t –

Parameters:

• v –

• g –

Returns:

template<typename graph_t>
auto hg::adjacent_vertex_iterator(typename graph::graph_traits<graph_t>::vertex_descriptor v, const graph_t &g)

Range over all vertices adjacent to the given vertex.

Template Parameters:

graph_t –

Parameters:

• v –

• g –

Returns:

template<typename graph_t>
auto hg::children_iterator(typename graph_t::vertex_descriptor v, const graph_t &g)

Range over the children vertices of the given node in the given tree.

Template Parameters:

graph_t –

Parameters:

• v –

• g –

Returns:

template<typename graph_t>
auto hg::ancestors_iterator(typename graph_t::vertex_descriptor v, const graph_t &g)

Range over the ancestors of v in topological order (starting from v included)

Template Parameters:

graph_t –

Parameters:

• v –

• g –

Returns:

template<typename T, typename graph_t>
auto hg::degree(const xt::xexpression<T> &xindex, const graph_t &g)

Degrees of all the given vertices in the given graph.

Template Parameters:

• T – type of indices (must be integral, preferably index_t)

• graph_t –

Parameters:

• xindex – array of vertex indices

• g –

Returns:

array of the same size as xindex containing the degree of each vertex indicated in xindex

template<typename T, typename graph_t>
auto hg::in_degree(const xt::xexpression<T> &xindex, const graph_t &g)

In-degrees of all the given vertices in the given graph.

Template Parameters:

• T – type of indices (must be integral, preferably index_t)

• graph_t –

Parameters:

• xindex – array of vertex indices

• g –

Returns:

array of the same size as xindex containing the in-degree of each vertex indicated in xindex

template<typename T, typename graph_t>
auto hg::out_degree(const xt::xexpression<T> &xindex, const graph_t &g)

Out-degrees of all the given vertices in the given graph.

Template Parameters:

• T – type of indices (must be integral, preferably index_t)

• graph_t –

Parameters:

• xindex – array of vertex indices

• g –

Returns:

array of the same size as xindex containing the out-degree of each vertex indicated in xindex

template<typename T, typename graph_t>
void hg::add_edges(const xt::xexpression<T> &xsources, const xt::xexpression<T> &xtargets, graph_t &g)

Add all edges given as a pair of arrays (sources, targets) to the graph.

Template Parameters:

• T – xexpression type

• graph_t – Mutable graph type

Parameters:

• xsources – Must be a 1d array of integral values

• xtargets – Must have the same shape as xsources

• g – A mutable graph

template<typename output_graph_type = ugraph, typename T>
output_graph_type hg::copy_graph(const T &graph)

Create a new graph as a copy of the given graph.

Template Parameters:

• T – input graph type

• output_graph_type – return type (default = ugraph)

Parameters:

graph –

Returns:

template<typename output_graph_type>
output_graph_type hg::copy_graph(const ugraph &graph)

template<>
inline ugraph hg::copy_graph(const ugraph &graph)

template<typename graph_t>
auto hg::other_vertex(const typename graph::graph_traits<graph_t>::edge_descriptor &edge, typename graph::graph_traits<graph_t>::vertex_descriptor vertex, const graph_t &graph)

Given an edge and one of the two extremities of this edge, return the other extremity (if the source is given it returns the target and vice versa).

Template Parameters:

graph_t –

Parameters:

• edge –

• vertex –

• graph –

Returns:

template<typename undirected_graph, typename T, typename value_type = typename T::value_type>
auto hg::undirected_graph_2_adjacency_matrix(const undirected_graph &graph, const xt::xexpression<T> &xedge_weights, const value_type &non_edge_value = 0)

Create an adjacency matrix from an undirected edge-weighted graph (the result is thus symmetric).

As the given graph is not necessarily complete, non-existing edges will receive the value non_edge_value in the adjacency matrix.

Template Parameters:

• undirected_graph –

• T –

• value_type –

Parameters:

• graph – Input undirected graph

• xedge_weights – Input edge-weights

• non_edge_value – Value used to represent non existing edges

Returns:

A 2d square array

template<typename T, typename value_type = typename T::value_type>
auto hg::adjacency_matrix_2_undirected_graph(const xt::xexpression<T> &xadjacency_matrix, const value_type &non_edge_value = 0)

Creates an undirected edge-weighted graph from an adjacency matrix.

Adjacency matrix entries which are equal to non_edge_value are not considered to be part of the graph.

Template Parameters:

• T –

• value_type –

Parameters:

• xadjacency_matrix – Input adjacency matrix

• non_edge_value – Value used to represent non existing edges

Returns:

a pair of types (ugraph, array_1d) representing the graph and its edge-weights

template<typename graph_t, typename weighter, typename T>
auto hg::binary_partition_tree(const graph_t &graph, const xt::xexpression<T> &xedge_weights, weighter weight_function)

Compute the binary partition tree of the graph.

At each step: 1 - the algorithm finds the edge of smallest weight. 2 - the two vertices linked by this edge are merged: the new vertex is the parent of the two merged vertices 3 - the weight of the edges linking the new vertex to the remaining vertices of the graph are updated according to the user provided function (weight_function) 4 - repeat until a single edge remain

The initial weight of the edges (xedge_weights) and the callback (weight_function) determine the shape of the hierarchy.

The weight_function callback can be anything that defining the operator() and should follow the following pattern:

struct my_weighter { …

template<typename graph_t, typename neighbours_t> void operator()(const graph_t &g, // the current state of the graph index_t fusion_edge_index, // the edge between the two vertices being merged index_t new_region, // the new vertex in the graph index_t merged_region1, // the first vertex merged index_t merged_region2, // the second vertex merged neighbours_t &new_neighbours){ // list of edges to be weighted (see below) … for (auto &n: new_neighbours) { … n.new_edge_weight() = new_edge_value; // define the weight of this edge } }

Each element in the parameter new_neighbours represent an edge between the new vertex and another vertex of the graph. For each element of the list, the following methods are available:

• neighbour_vertex(): the other vertex

• num_edges(): returns 2 if both the two merged vertices add an edge linking themselves with neighbour_vertex() and 1 otherwise

• first_edge_index(): the index of the edge linking one of the merged region to neighbour_vertex()

• second_edge_index(): the index of the edge linking the other merged region to neighbour_vertex() (only if num_edges()==2)

• new_edge_weight(): weight of the new edge (THIS HAS TO BE DEFINED IN THE WEIGHTING FUNCTION)

• new_edge_index(): the index of the new edge: the weighting function will probably have to track new weight values

Example of weighting function: binary_partition_tree_min_linkage

Template Parameters:

• graph_t –

• weighter –

• T –

Parameters:

• graph –

• xedge_weights –

• weight_function –

Returns:

a node weighted tree

template<typename graph_t, typename T>
auto hg::binary_partition_tree_min_linkage(const graph_t &graph, const xt::xexpression<T> &xedge_weights)

Binary partition tree, i.e.

the agglomerative clustering, with the minimum/single linkage rule.

Given a graph :math:G=(V, E), with initial edge weights :math:w, the distance :math:d(X,Y) between any two clusters :math:X and :math:Y is

.. math::

 d(X,Y) = \min \{w(\{x,y\}) | x \in X, y \in Y, \{x,y\} \in E \}

Regions are then iteratively merged following the above distance (closest first) until a single region remains.

Template Parameters:

• graph_t –

• T –

Parameters:

• graph –

• xedge_weights –

Returns:

a node weighted tree

template<typename graph_t, typename T>
auto hg::binary_partition_tree_complete_linkage(const graph_t &graph, const xt::xexpression<T> &xedge_weights)

Binary partition tree, i.e.

the agglomerative clustering, with the maximum/complete linkage rule.

Given a graph :math:G=(V, E), with initial edge weights :math:w, the distance :math:d(X,Y) between any two clusters :math:X and :math:Y is

.. math::

 d(X,Y) = \max \{w(\{x,y\}) | x \in X, y \in Y, \{x,y\} \in E \}

Regions are then iteratively merged following the above distance (closest first) until a single region remains

Template Parameters:

• graph_t –

• T –

Parameters:

• graph –

• xedge_weights –

Returns:

a node weighted tree

template<typename graph_t, typename T>
auto hg::binary_partition_tree_average_linkage(const graph_t &graph, const xt::xexpression<T> &xedge_weights, const xt::xexpression<T> &xedge_weight_weights)

Binary partition tree, i.e.

the agglomerative clustering, with the average linkage rule.

Given a graph :math:G=(V, E), with initial edge weights :math:w with associated weights :math:w, the distance :math:d(X,Y)between any two clusters :math:Xand :math:Y` is

.. math::

 d(X,Y) = \\frac{1}{Z} sum_{x \in X, y \in Y, \{x,y\} in E} w(\{x,y\}) \\times w'(\{x,y\})

with :math:Z = \sum_{x \in X, y \in Y, \{x,y\} \in E} w({x,y})`.

Regions are then iteratively merged following the above distance (closest first) until a single region remains

Template Parameters:

• graph_t –

• T –

Parameters:

• graph –

• xedge_weights –

• xedge_weight_weights –

Returns:

a node weighted tree

template<typename graph_t, typename T>
auto hg::binary_partition_tree_exponential_linkage(const graph_t &graph, const xt::xexpression<T> &xedge_weights, const typename T::value_type &alpha, const xt::xexpression<T> &xedge_weight_weights)

))`.

Regions are then iteratively merged following the above distance (closest first) until a single region remains

Note that:

• :math:\\apha=0 is equivalent to average linkage clustering

• :math:\\alpha=-\infty is equivalent to single linkage clustering

• :math:\\alpha=+\infty is equivalent to complete linkage clustering

See:

 Nishant Yadav, Ari Kobren, Nicholas Monath, Andrew Mccallum ;
 Supervised Hierarchical Clustering with Exponential Linkage
 Proceedings of the 36th International Conference on Machine Learning, PMLR 97:6973-6983, 2019.

Template Parameters:

• graph_t –

• T –

Parameters:

• graph –

• xedge_weights –

• alpha –

• xedge_weight_weights –

Returns:

a node weighted tree

template<typename graph_t, typename T1, typename T2>
auto hg::binary_partition_tree_ward_linkage(const graph_t &graph, const xt::xexpression<T1> &xvertex_centroids, const xt::xexpression<T2> &xvertex_sizes, const std::string &altitude_correction = "max")

Binary partition tree, i.e.

the agglomerative clustering, with the Ward linkage rule.

Given a graph :math:G=(V, E), with initial edge weights :math:w with associated weights :math:w, the distance :math:d(X,Y)between any two clusters :math:Xand :math:Y` is

.. math::

 d(X,Y) = \\frac{| X |\\times| Y |}{| X |+| Y |} \| \\vec{X} - \\vec{Y} \|^2

where :math:\\vec{X} and :math:\\vec{Y} are the centroids of :math:X and :math:Y.

Regions are then iteratively merged following the above distance (closest first) until a single region remains

Note that the Ward distance is not necessarily strictly increasing when processing a non complete graph. This can be corrected afterward with an altitude correction strategy. Valid values for altitude correction are:

 - ``"none"``: nothing is done and the altitude of a node is equal to the Ward distance between its 2 children;
     this may not be non-decreasing
 - ``"max"``: the altitude of a node :math:`n` is defined as the maximum of the the Ward distance associated
     to each node in the subtree rooted in :math:`n`.

Template Parameters:

• graph_t –

• T1 –

• T2 –

Parameters:

• graph –

• xvertex_centroids – Centroids of the graph vertices (must be a 2d array)

• xvertex_sizes – Size (number of elements) of the graph vertices

• altitude_correction – can be "none" or "max" (default)

Returns:

a node weighted tree

template<typename tree_t, typename altitude_t>
decltype(auto) hg::make_node_weighted_tree(tree_t &&tree, altitude_t &&node_altitude)

template<typename tree_t, typename node_map_t>
decltype(auto) hg::make_remapped_tree(tree_t &&tree, node_map_t &&node_map)

template<typename graph_t, typename T>
auto hg::component_tree_max_tree(const graph_t &graph, const xt::xexpression<T> &xvertex_weights)

Construct the Max Tree of the vertex weighted graph.

The Min/Max Tree structure were proposed in [1], [2]. The algorithm used in this implementation was first described in [3].

[1] Ph. Salembier, A. Oliveras, and L. Garrido, “Anti-extensive connected operators for image

and sequence processing,” IEEE Trans. Image Process., vol. 7, no. 4, pp. 555-570, Apr. 1998.

[2] Ro. Jones, “Connected filtering and segmentation using component trees,” Comput. Vis. Image Understand., vol. 75, no. 3, pp. 215-228, Sep. 1999.

[3] Ch. Berger, T. Geraud, R. Levillain, N. Widynski, A. Baillard, and E. Bertin, “Effective

Component Tree Computation with Application to Pattern Recognition in Astronomical Imaging,” IEEE ICIP 2007.

Template Parameters:

• graph_t –

• T –

Parameters:

• graph – input graph

• vertex_weights – graph vertex weights

Returns:

a node weighted tree

template<typename graph_t, typename T>
auto hg::component_tree_min_tree(const graph_t &graph, const xt::xexpression<T> &xvertex_weights)

Construct the Min Tree of the vertex weighted graph.

The Min/Max Tree structure were proposed in [1], [2]. The algorithm used in this implementation was first described in [3].

[1] Ph. Salembier, A. Oliveras, and L. Garrido, “Anti-extensive connected operators for image

and sequence processing,” IEEE Trans. Image Process., vol. 7, no. 4, pp. 555-570, Apr. 1998.

[2] Ro. Jones, “Connected filtering and segmentation using component trees,” Comput. Vis. Image Understand., vol. 75, no. 3, pp. 215-228, Sep. 1999.

[3] Ch. Berger, T. Geraud, R. Levillain, N. Widynski, A. Baillard, and E. Bertin, “Effective

Component Tree Computation with Application to Pattern Recognition in Astronomical Imaging,” IEEE ICIP 2007.

Template Parameters:

• graph_t –

• T –

Parameters:

• graph – input graph

• vertex_weights – graph vertex weights

Returns:

a node weighted tree

template<typename tree_t, typename altitude_t>
decltype(auto) hg::make_node_weighted_tree_and_mst(tree_t &&tree, altitude_t &&node_altitude, array_1d<index_t> &&mst_edge_map)

template<typename graph_t, typename T>
auto hg::bpt_canonical(const graph_t &graph, const xt::xexpression<T> &xedge_weights)

Compute the canonical binary partition tree (or binary partition tree by altitude ordering) of the given edge weighted graph.

The algorithm returns a tuple composed of:

• the binary partition tree,

• the levels of the vertices of the tree,

• the minimum spanning tree of the given graph that corresponds to this tree.

L. Najman, J. Cousty, B. Perret. Playing with Kruskal: algorithms for morphological trees in edge-weighted graphs. In, 11th International Symposium on Mathematical Morphology, ISMM 2013, Uppsala, Sweden, Mai 2013.

Template Parameters:

• graph_t –

• T –

Parameters:

• graph –

• xedge_weights –

Returns:

template<typename criterion_t>
auto hg::simplify_tree(const tree &t, const criterion_t &criterion, bool process_leaves = false)

Creates a copy of the current Tree and deletes the nodes such that the criterion function is true.

Also returns an array that maps any node index i of the new tree, to the index of this node in the original tree.

The criterion function is a predicate that associates true (this node must be deleted) or false (do not delete this node) to a node index (with operator ()).

Template Parameters:

criterion_t –

Parameters:

• t – input tree

• criterion – For any vertex n of the tree, n has to be removed if criterion(n) == true

• process_leaves – If false, a leaf vertex will never be removed disregarding the value of criterion.

Returns:

a remapped_tree

template<typename graph_t, typename T>
auto hg::quasi_flat_zone_hierarchy(const graph_t &graph, const xt::xexpression<T> &xedge_weights)

Compute the quasi-flat zone hierarchy of an edge weighted graph.

For a given positive real value lamba:

• a set of vertices X is lambda-connected if, for any two vertices x, y in X there exists an xy-path in X composed of edges of weights smaller of equal than lambda;

• a lambda-connected component is a lambda-connected set of maximal extent;

• the set of lambda-connected components forms a partition, called lambda-partition, of the graph vertices.

The quasi-flat zone hierarchy is composed of the sequence of lambda-partitions obtained for all lambda in edge_weights.

Template Parameters:

• graph_t – Input graph type

• T – xepression derived type of input edge weights

Parameters:

• graph – Input graph

• xedge_weights – Input graph weights

Returns:

A node weighted tree

template<typename graph_t, typename tree_t, typename T>
auto hg::saliency_map(const graph_t &graph, const tree_t &tree, const xt::xexpression<T> &xaltitudes)

Compute the saliency map of the given hierarchy for the given graph.

The saliency map is a weighting of the graph edges. The weight of an edge {x, y} is the altitude of the lowest common ancestor of x and y in the hierarchy.

Template Parameters:

• graph_t – Input graph type

• tree_t – Input tree type

• T – xepression derived type of input altitudes

Parameters:

• graph – Input graph

• tree – Input tree

• xaltitudes – Input node altitudes of the given tree

Returns:

An array of shape (num_edges(graph)) and with the same value type as T.

template<typename tree_t>
auto hg::tree_2_binary_tree(const tree_t &tree)

Transforms a tree into a binary tree.

Each non-leaf node of the input tree must have at least 2 children!

Whenever a non-leaf node :math:n with :math:k > 2 children is found:

• an extra node :math:m is created;

• the first 2 children of :math:n become children of the new node :math:m; and

• the new node :math:m becomes the first child of :math:n.

The number of children of :math:n is thus reduced by 1. This operation is repeated :math:k-2 times, i.e. until :math:n has only 2 children.

Template Parameters:

tree_t – Input tree type

Parameters:

tree – Input tree

Returns:

a remapped_tree

template<typename graph_t, typename T, typename F>
auto hg::watershed_hierarchy_by_attribute(const graph_t &graph, const xt::xexpression<T> &xedge_weights, const F &attribute_functor)

Computes a hierarchical watershed for the given regional attribute.

The algorithm used is described in:

Laurent Najman, Jean Cousty, Benjamin Perret: Playing with Kruskal: Algorithms for Morphological Trees in Edge-Weighted Graphs. ISMM 2013: 135-146

The regional attribute is specified by an attribute functor, that is a function that

• takes 2 input parameter: a binary partition tree, the altitudes of its nodes;

• returns an 1d array giving the attribute value for each node of the input tree. The computed regional attribute must be scalar, positive and increasing (the attribute value of a node is smaller than or equal to the attribute value of its parent).

Template Parameters:

• graph_t –

• T –

• F –

Parameters:

• graph – input graph

• xedge_weights – input graph edge weights

• attribute_functor – function that computes the attribute value from a tree and its node altitudes

Returns:

a node_weighted_tree

template<typename graph_t, typename T1, typename T2>
auto hg::watershed_hierarchy_by_minima_ordering(const graph_t &graph, const xt::xexpression<T1> &xedge_weights, const xt::xexpression<T2> &xminima_ranks)

Computes a hierarchical watershed for the given minima ordering.

The algorithm used is described in:

Laurent Najman, Jean Cousty, Benjamin Perret: Playing with Kruskal: Algorithms for Morphological Trees in Edge-Weighted Graphs. ISMM 2013: 135-146

The ranking ranking of the minima of the given edge weighted graph (G,w) is given as vertex weights with values in {0..n} with n the number of minima of (G,w). It must satisfy the following pre-conditions:

• each minimum of (G,w) contains at least one non zero vertex,

• all non zero vertices in a minimum have the same weight,

• there is no non zero value vertex outside minima, and

• no two minima contain non zero vertices with the same weight.

The altitude associated to each minimum is a non decreasing 1d array of size n + 1 with non negative values. Note that the first entry of the minima altitudes array, ie. the value at index 0, does not represent a minimum and its value should be 0.

Template Parameters:

• graph_t –

• T1 –

• T2 –

Parameters:

• graph – input graph

• xedge_weights – input graph edge weights

• xminima_ranks – input graph vertex weights containing the rank of each minima of the input edge weighted graph

Returns:

a node_weighted_tree

template<typename graph_t, typename T1, typename T2>
auto hg::watershed_hierarchy_by_area(const graph_t &graph, const xt::xexpression<T1> &edge_weights, const xt::xexpression<T2> &vertex_area)

template<typename graph_t, typename T1>
auto hg::watershed_hierarchy_by_area(const graph_t &graph, const xt::xexpression<T1> &xedge_weights)

template<typename graph_t, typename T1, typename T2>
auto hg::watershed_hierarchy_by_volume(const graph_t &graph, const xt::xexpression<T1> &edge_weights, const xt::xexpression<T2> &vertex_area)

template<typename graph_t, typename T1>
auto hg::watershed_hierarchy_by_volume(const graph_t &graph, const xt::xexpression<T1> &xedge_weights)

template<typename graph_t, typename T>
auto hg::watershed_hierarchy_by_dynamics(const graph_t &graph, const xt::xexpression<T> &edge_weights)

template<typename graph_t, typename T>
auto hg::fit_contour_2d(const graph_t &graph, const embedding_grid_2d &embedding, const xt::xexpression<T> &xedge_weights)

Construct a contour_2d object from a graph cut of a 2d image with a 4 adjacency (non zero edges are part of the cut).

Template Parameters:

• graph_t –

• T –

Parameters:

• graph –

• embedding –

• xedge_weights –

Returns:

template<typename rag_graph_t, typename graph_t, typename T>
auto hg::rag_2d_vertex_perimeter_and_edge_length(const rag_graph_t &rag_graph, const xt::xexpression<T> &xvertex_map, const xt::xexpression<T> &xedge_map, const embedding_grid_2d &embedding, const graph_t &graph, double epsilon = 0.1, bool relative_epsilon = true, int min_size = 2)

Estimate the vertex perimeter and the length of the frontier associated to the edges of a region adjacency graph constructed on a 2d 4 adjacency graph.

The region boundaries are simplified with Ramer–Douglas–Peucker algorithm and is controlled by the parameters epsilon, relative_epsilon, min_size. See function subdivide of the class contour_2d for more information.

Template Parameters:

graph_t –

Parameters:

• rag_graph – Region Adjacency Graph

• xvertex_map – Vertex map of the rag_graph

• xedge_map – Edge map of the rag_graph

• embedding – 2d shape of the input graph

• graph – input graph on which the region adjacency graph has been build: must be a 4 adjacency graph whose shape correspond to the given embedding

• epsilon – larger epsilon values will provide stronger contour shapes simplification

• relative_epsilon – Is epsilon given in relative or absolute units

• min_size – Boundaries elements smaller than min_size will be deleted

Returns:

a pair composed of two 1d arrays: vertex_perimeter and edge_length.

template<typename graph_t>
auto hg::rag_2d_vertex_perimeter_and_edge_length(const region_adjacency_graph &rag, const embedding_grid_2d &embedding, const graph_t &graph, double epsilon = 0.1, bool relative_epsilon = true, int min_size = 2)

Estimate the vertex perimeter and the length of the frontier associated to the edges of a region adjacency graph constructed on a 2d 4 adjacency graph.

The region boundaries are simplified with Ramer–Douglas–Peucker algorithm and is controlled by the parameters epsilon, relative_epsilon, min_size. See function subdivide of the class contour_2d for more information.

Template Parameters:

graph_t –

Parameters:

• rag – Region Adjacency Graph

• embedding – 2d shape of the input graph

• graph – input graph on which the region adjacency graph has been build: must be a 4 adjacency graph whose shape correspond to the given embedding

• epsilon – larger epsilon values will provide stronger contour shapes simplification

• relative_epsilon – Is epsilon given in relative or absolute units

• min_size – Boundaries elements smaller than min_size will be deleted

Returns:

a pair composed of two 1d arrays: vertex_perimeter and edge_length.

inline auto hg::get_4_adjacency_implicit_graph(const embedding_grid_2d &embedding)

Create a 4 adjacency implicit regular graph for the given embedding.

Parameters:

embedding –

Returns:

inline auto hg::get_8_adjacency_implicit_graph(const embedding_grid_2d &embedding)

Create of 4 adjacency implicit regular graph for the given embedding.

Parameters:

embedding –

Returns:

inline auto hg::get_6_adjacency_implicit_graph(const embedding_grid_3d &embedding)

Create of 6 adjacency implicit regular graph for the given embedding.

Parameters:

embedding –

Returns:

inline auto hg::get_4_adjacency_graph(const embedding_grid_2d &embedding)

Create of 4 adjacency explicit regular graph for the given embedding.

Parameters:

embedding –

Returns:

inline auto hg::get_8_adjacency_graph(const embedding_grid_2d &embedding)

Create of 8 adjacency explicit regular graph for the given embedding.

Parameters:

embedding –

Returns:

template<typename graph_t, typename T, typename result_type = typename T::value_type>
auto hg::graph_4_adjacency_2_khalimsky(const graph_t &graph, const embedding_grid_2d &embedding, const xt::xexpression<T> &xedge_weights, bool add_extra_border = false, result_type extra_border_value = 0)

Represents a 4 adjacency edge weighted regular graph in 2d Khalimsky space.

Parameters:

embedding –

Returns:

template<typename T, typename result_type = typename T::value_type>
auto hg::khalimsky_2_graph_4_adjacency(const xt::xexpression<T> &xkhalimsky, const ugraph &g, const embedding_grid_2d &embedding, bool extra_border = false)

Transforms a contour map represented in 2d Khalimsky space into a weighted 4 adjacency edge weighted regular graph (0-face and 2-face of the Khalimsky space are ignored).

Parameters:

• xkhalimsky – 2d array representing the khalimsky space

• g – 4 adjacency graph, with the same number of vertices as the embedding

• embedding – embedding of the graph, dimension is half the size of the khalimsky space

• extra_border – if true, the khalimsky space will have an extra border of 0-faces and 1-faces

Returns:

2d array representing the weights of the edges in the khalimsky space

template<typename T, typename result_type = typename T::value_type>
auto hg::khalimsky_2_graph_4_adjacency(const xt::xexpression<T> &xkhalimsky, bool extra_border = false)

Transforms a contour map represented in 2d Khalimsky space into a weighted 4 adjacency edge weighted regular graph (0-face and 2-face of the Khalimsky space are ignored).

Parameters:

• xkhalimsky – 2d array representing the khalimsky space

• extra_border – if true, the khalimsky space will have an extra border of 0-faces and 1-faces

Returns:

A tuple composed of the 4 adjacency graph, the embedding of the graph, and the weights of the edges in the khalimsky space

template<typename T>
auto hg::get_bipartite_matching_graph_contour_image_2d(const embedding_grid_2d &embedding, const xt::xexpression<T> &xcontour_image1, const xt::xexpression<T> &xcontour_image2, double max_distance)

Creates a bipartite graph, linking each pixel of the first contour image to any pixel of the second contour image that is within a given distance.

Template Parameters:

T –

Parameters:

• embedding –

• xcontour_image1 –

• xcontour_image2 –

• max_distance –

Returns:

template<typename graph_t, typename T1, typename T2>
auto hg::oriented_watershed(const graph_t &graph, const embedding_grid_2d &embedding, const xt::xexpression<T1> &xedge_weights, const xt::xexpression<T2> &xedge_orientations = array_nd<int>())

Compute the oriented watershed as described in [ArbelaezPAMI2011]_ .

Given a 4 adjacency graph with edge boundary probabilities and estimated boundary orientations, the algorithms computes:

• a region adjacency graph of the watershed regions of the edge boundary probabilities

• the boundaries between watershed regions are vectorized and simplified (see contour_2d class)

• the orientation of each boundary element is estimated

• the edge boundary probabilities are reweighted according to the concordance between user provided boundary orientations and estimated orientation of boundary elements

• the weight of the region adjacency graph edges as the mean value of reweighted edge boundary probabilities on the frontier between the 2 regions

The algorithm returns the region adjacency graph of watershed pixels and its edge weights.

.. [ArbelaezPAMI2011] Arbelaez, P., Maire, M., Fowlkes, C., & Malik, J.. Contour detection and hierarchical image segmentation. IEEE transactions on pattern analysis and machine intelligence, 33(5), 898-916.

Template Parameters:

• graph_t –

• T1 –

• T2 –

Parameters:

• graph –

• embedding –

• xedge_weights –

• xedge_orientations –

Returns:

template<typename graph_t, typename T1, typename T2>
auto hg::mean_pb_hierarchy(const graph_t &graph, const embedding_grid_2d &embedding, const xt::xexpression<T1> &xedge_weights, const xt::xexpression<T2> &xedge_orientations = array_nd<int>())

Compute the mean probability boundary hierarchy as described in [ArbelaezPAMI2011]_ .

Given a 4 adjacency graph with edge boundary probabilities and estimated boundary orientations, the algorithms computes:

• the oriented watershed of the given graph

• the average linkage clustering ot the oriented watershed

The algorithm returns the region adjacency graph of watershed pixels and teh valued tree computed on this graph.

.. [ArbelaezPAMI2011] Arbelaez, P., Maire, M., Fowlkes, C., & Malik, J.. Contour detection and hierarchical image segmentation. IEEE transactions on pattern analysis and machine intelligence, 33(5), 898-916.

Template Parameters:

• graph_t –

• T1 –

• T2 –

Parameters:

• graph –

• embedding –

• xedge_weights –

• xedge_orientations –

Returns:

template<typename T>
auto hg::component_tree_tree_of_shapes_image(const xt::xexpression<T> &ximage, tos_padding padding = tos_padding::mean, bool original_size = true, bool immersion = true, index_t exterior_vertex = 0)

Computes the tree of shapes of an image.

The Tree of Shapes was described in [1].

The algorithm used in this implementation was first described in [2].

The tree is computed in the interpolated multivalued Khalimsky space to provide a continuous and autodual representation of input image.

If padding is different from tos_padding::none, an extra border of pixels is added to the input image before anything else. This will ensure the existence of a shape encompassing all the shapes inside the input image (if exterior_vertex is inside the extra border): this shape will be the root of the tree. The padding value can be:

• 0 is padding == tos_padding::zero

• the mean value of the boundary pixels of the input image if padding == tos_padding::mean

If original_size is true, all the nodes corresponding to pixels not belonging to the input image are removed (except for the root node). If original_size is false, the returned tree is the tree constructed in the interpolated/padded space. In practice if the size of the input image is (h, w, d), the leaves of the returned tree will correspond to an image of size:

• (h, w, d) if original_size is true;

• (h * 2 - 1, w * 2 - 1, d * 2 - 1) is original_size is false and padding is tos_padding::none; and

• ((h + 2) * 2 - 1, (w + 2) * 2 - 1, (d + 2) * 2 - 1) otherwise.

:Advanced options:

Use with care the following options may lead to unexpected results:

Immersion defines if the initial image should be first converted as an equivalent continuous representation called a plain map. If the immersion is deactivated the level lines of the shapes of the image may intersect (if the image is not well composed) and the result of the algorithm is undefined. If immersion is deactivated, the factor :math:*2 - 1 has to be removed in the result sizes given above.

Exterior_vertex defines the linear coordinates of the pixel corresponding to the exterior (interior and exterior of a shape is defined with respect to this point). The coordinate of this point must be given in the padded/interpolated space.

[1] Pa. Monasse, and F. Guichard, “Fast computation of a contrast-invariant image representation,” Image Processing, IEEE Transactions on, vol.9, no.5, pp.860-872, May 2000

[2] Th. Géraud, E. Carlinet, S. Crozet, and L. Najman, “A Quasi-linear Algorithm to Compute the Tree

of Shapes of nD Images”, ISMM 2013.

Template Parameters:

T –

Parameters:

• ximage – Must be a 2d or 3d array

• padding – Defines if an extra boundary of pixels is added to the original image (see enum tos_padding).

• original_size – remove all nodes corresponding to interpolated/padded pixels

• exterior_vertex – linear coordinate of the exterior point

Returns:

a node weighted tree

template<typename T>
auto hg::component_tree_tree_of_shapes_image2d(const xt::xexpression<T> &ximage, tos_padding padding = tos_padding::mean, bool original_size = true, bool immersion = true, index_t exterior_vertex = 0)

Deprecated:

use component_tree_tree_of_shapes_image instead

See also

description of component_tree_tree_of_shapes_image

template<typename T>
auto hg::component_tree_tree_of_shapes_image3d(const xt::xexpression<T> &ximage, tos_padding padding = tos_padding::mean, bool original_size = true, bool immersion = true, index_t exterior_vertex = 0)

Deprecated:

use component_tree_tree_of_shapes_image instead

See also

description of component_tree_tree_of_shapes_image

inline auto hg::read_pink_graph(std::istream &in)

inline auto hg::read_pink_graph(const std::string &filename)

template<typename graph_t, typename T1, typename T2, typename S>
auto hg::save_pink_graph(std::ostream &out, const graph_t &graph, const xt::xexpression<T1> &xvertex_values = xt::xscalar<char>(0), const xt::xexpression<T2> &xedge_values = xt::xscalar<char>(0), S &shape = std::vector<std::size_t>())

template<typename graph_t, typename T1, typename T2, typename S>
void hg::save_pink_graph(const std::string &filename, const graph_t &graph, const xt::xexpression<T1> &xvertex_values = xt::xscalar<char>(0), const xt::xexpression<T2> &xedge_values = xt::xscalar<char>(0), S &shape = std::vector<std::size_t>())

inline array_nd<unsigned char> hg::read_image_pnm(const char *filename)

Read the given pnm image (pbm, pgm or ppm formats).

Current the following pnm specification are supported P1 binary ascii: supported P2 byte ascii: supported (max value <= 255) P3 RGB ascii: supported (max value <= 255) P4 binary raw: NOT supported P5 byte raw: supported (max value <= 255) P6 RGB rax: supported (max value <= 255)

Parameters:

filename – Path to the file to read

Returns:

an array with pixel data

template<typename T>
void hg::save_image_pnm(const char *filename, const xt::xexpression<T> &ximage)

Save an array as a pnm file (pgm or ppm).

The array value_type MUST be unsigned char. If the array has 2 dimensions it is saved as a pgm raw file (format P5). If the array has 3 dimensions, the size of the third dimension must be 3 and it is saved as a ppm raw file (format P6).

If the provided filename already exists, it will be overwritten !

Template Parameters:

T – xexpression derived type for ximage

Parameters:

• filename – Path to the file to write.

• ximage – The array containing the data to save.

inline auto hg::save_tree(std::ostream &out, const tree &t)

inline auto hg::read_tree(std::istream &in)

template<typename RandomAccessIterator, typename Compare>
void hg::stable_sort(RandomAccessIterator xs, RandomAccessIterator xe, Compare comp)

template<typename RandomAccessIterator, typename Compare>
void hg::sort(RandomAccessIterator xs, RandomAccessIterator xe, Compare comp)

template<typename RandomAccessIterator>
void hg::stable_sort(RandomAccessIterator xs, RandomAccessIterator xe)

template<typename T, typename Compare>
void hg::stable_sort(xt::xexpression<T> &arrayx, Compare comp)

template<typename T>
void hg::stable_sort(xt::xexpression<T> &arrayx)

template<typename RandomAccessIterator>
void hg::sort(RandomAccessIterator xs, RandomAccessIterator xe)

template<typename T, typename Compare>
void hg::sort(xt::xexpression<T> &arrayx, Compare comp)

template<typename T>
void hg::sort(xt::xexpression<T> &arrayx)

template<typename T, typename Compare>
auto hg::arg_sort(const xt::xexpression<T> &arrayx, Compare comp)

template<typename T>
auto hg::arg_sort(const xt::xexpression<T> &arrayx)

template<typename T, typename Compare>
auto hg::stable_arg_sort(const xt::xexpression<T> &arrayx, Compare comp)

template<typename T>
auto hg::stable_arg_sort(const xt::xexpression<T> &arrayx)

template<typename graph_t>
auto &hg::source(const indexed_edge<typename graph::graph_traits<graph_t>::vertex_descriptor, typename graph::graph_traits<graph_t>::edge_index> &e, const graph_t&)

Source vertex of an edge.

Template Parameters:

graph_t –

Parameters:

e –

Returns:

template<typename graph_t>
auto &hg::target(const indexed_edge<typename graph::graph_traits<graph_t>::vertex_descriptor, typename graph::graph_traits<graph_t>::edge_index> &e, const graph_t&)

Target vertex of an edge.

Template Parameters:

graph_t –

Parameters:

e –

Returns:

template<typename graph_t>
auto &hg::index(const indexed_edge<typename graph::graph_traits<graph_t>::vertex_descriptor, typename graph::graph_traits<graph_t>::edge_index> &e, const graph_t&)

Index of an edge.

Template Parameters:

graph_t –

Parameters:

e –

Returns:

template<bool vectorial = true, typename T>
auto hg::make_light_axis_view(T &e, index_t position = 0)

template<typename embedding_t>
std::pair<typename hg::regular_graph<embedding_t>::out_edge_iterator, typename hg::regular_graph<embedding_t>::out_edge_iterator> hg::out_edges(typename hg::regular_graph<embedding_t>::vertex_descriptor u, const hg::regular_graph<embedding_t> &g)

template<typename embedding_t>
std::pair<typename hg::regular_graph<embedding_t>::out_edge_iterator, typename hg::regular_graph<embedding_t>::out_edge_iterator> hg::in_edges(typename hg::regular_graph<embedding_t>::vertex_descriptor u, const hg::regular_graph<embedding_t> &g)

template<typename embedding_t>
hg::regular_graph<embedding_t>::vertices_size_type hg::num_vertices(const hg::regular_graph<embedding_t> &g)

template<typename embedding_t>
std::pair<typename hg::regular_graph<embedding_t>::vertex_iterator, typename hg::regular_graph<embedding_t>::vertex_iterator> hg::vertices(const hg::regular_graph<embedding_t> &g)

template<typename embedding_t>
std::pair<typename hg::regular_graph<embedding_t>::adjacency_iterator, typename hg::regular_graph<embedding_t>::adjacency_iterator> hg::adjacent_vertices(typename hg::regular_graph<embedding_t>::vertex_descriptor u, const hg::regular_graph<embedding_t> &g)

template<typename embedding_t>
hg::regular_graph<embedding_t>::degree_size_type hg::out_degree(const typename hg::regular_graph<embedding_t>::vertex_descriptor v, const hg::regular_graph<embedding_t> &g)

template<typename embedding_t>
hg::regular_graph<embedding_t>::degree_size_type hg::in_degree(const typename hg::regular_graph<embedding_t>::vertex_descriptor v, const hg::regular_graph<embedding_t> &g)

template<typename embedding_t>
hg::regular_graph<embedding_t>::degree_size_type hg::degree(const typename hg::regular_graph<embedding_t>::vertex_descriptor v, const hg::regular_graph<embedding_t> &g)

inline auto hg::num_leaves(const tree &t)

inline auto hg::num_children(tree::vertex_descriptor v, const tree &t)

template<typename T>
auto hg::num_children(const xt::xexpression<T> &xvertices, const tree &t)

inline const auto &hg::category(const tree &t)

inline auto hg::root(const tree &t)

inline auto hg::parent(tree::vertex_descriptor v, const tree &t)

template<typename T>
auto hg::parent(const xt::xexpression<T> &xvertices, const tree &t)

inline auto hg::is_leaf(tree::vertex_descriptor v, const tree &t)

template<typename T>
auto hg::is_leaf(const xt::xexpression<T> &xvertices, const tree &t)

inline const auto &hg::parents(const tree &t)

inline auto hg::leaves_to_root_iterator(const tree &t, leaves_it leaves_opt = leaves_it::include, root_it root_opt = root_it::include)

inline auto hg::root_to_leaves_iterator(const tree &t, leaves_it leaves_opt = leaves_it::include, root_it root_opt = root_it::include)

inline auto hg::leaves_iterator(const tree &t)

inline auto hg::ancestors(tree::vertex_descriptor v, const tree &t)

inline tree::edge_descriptor hg::edge_from_index(tree::edge_index_t ei, const tree &g)

inline std::pair<tree::children_iterator, tree::children_iterator> hg::children(const tree::vertex_descriptor v, const tree &g)

inline tree::vertex_descriptor hg::child(index_t i, tree::vertex_descriptor v, const tree &t)

template<typename T>
auto hg::child(index_t i, const xt::xexpression<T> &xvertices, const tree &t)

inline hg::tree::vertices_size_type hg::num_vertices(const hg::tree &g)

inline hg::tree::edges_size_type hg::num_edges(const hg::tree &g)

inline hg::tree::degree_size_type hg::degree(const typename hg::tree::vertex_descriptor v, const hg::tree &g)

inline hg::tree::degree_size_type hg::in_degree(const typename hg::tree::vertex_descriptor v, const hg::tree &g)

inline hg::tree::degree_size_type hg::out_degree(const typename hg::tree::vertex_descriptor v, const hg::tree &g)

inline std::pair<typename hg::tree::vertex_iterator, typename hg::tree::vertex_iterator> hg::vertices(const hg::tree &g)

inline std::pair<typename hg::tree::edge_iterator, typename hg::tree::edge_iterator> hg::edges(const hg::tree &g)

inline std::pair<typename hg::tree::adjacency_iterator, typename hg::tree::adjacency_iterator> hg::adjacent_vertices(typename hg::tree::vertex_descriptor v, const hg::tree &g)

inline std::pair<hg::tree::out_edge_iterator, hg::tree::out_edge_iterator> hg::out_edges(hg::tree::vertex_descriptor v, const hg::tree &g)

inline std::pair<hg::tree::out_edge_iterator, hg::tree::out_edge_iterator> hg::in_edges(hg::tree::vertex_descriptor v, const hg::tree &g)

template<typename T>
auto hg::find_region(tree::vertex_descriptor v, const typename T::value_type &lambda, const T &altitudes, const tree &tree)

template<typename T1, typename T2, typename T3>
auto hg::find_region(const xt::xexpression<T1> &xvertices, const xt::xexpression<T2> &xlambdas, const xt::xexpression<T3> &xaltitudes, const tree &t)

inline auto hg::lowest_common_ancestor(tree::vertex_descriptor v1, tree::vertex_descriptor v2, const tree &t)

template<typename T>
auto hg::lowest_common_ancestor(const xt::xexpression<T> &xvertices_1, const xt::xexpression<T> &xvertices_2, const tree &t)

template<typename T>
const auto &hg::edge_from_index(const typename undirected_graph<T>::vertex_descriptor v, const undirected_graph<T> &g)

template<typename T>
hg::undirected_graph<T>::vertices_size_type hg::num_vertices(const hg::undirected_graph<T> &g)

template<typename T>
hg::undirected_graph<T>::edges_size_type hg::num_edges(const hg::undirected_graph<T> &g)

template<typename T>
hg::undirected_graph<T>::degree_size_type hg::degree(typename hg::undirected_graph<T>::vertex_descriptor v, const hg::undirected_graph<T> &g)

template<typename T>
hg::undirected_graph<T>::degree_size_type hg::in_degree(typename hg::undirected_graph<T>::vertex_descriptor v, const hg::undirected_graph<T> &g)

template<typename T>
hg::undirected_graph<T>::degree_size_type hg::out_degree(typename hg::undirected_graph<T>::vertex_descriptor v, const hg::undirected_graph<T> &g)

template<typename T>
hg::undirected_graph<T>::vertex_descriptor hg::add_vertex(hg::undirected_graph<T> &g)

template<typename T>
void hg::add_vertices(size_t num, hg::undirected_graph<T> &g)

template<typename T>
hg::undirected_graph<T>::edge_descriptor hg::add_edge(typename hg::undirected_graph<T>::vertex_descriptor v1, typename hg::undirected_graph<T>::vertex_descriptor v2, hg::undirected_graph<T> &g)

template<typename T>
void hg::remove_edge(typename hg::undirected_graph<T>::edge_index_t ei, hg::undirected_graph<T> &g)

template<typename T>
void hg::set_edge(typename hg::undirected_graph<T>::edge_index_t ei, typename hg::undirected_graph<T>::vertex_descriptor v1, typename hg::undirected_graph<T>::vertex_descriptor v2, hg::undirected_graph<T> &g)

template<typename T>
std::pair<typename hg::undirected_graph<T>::vertex_iterator, typename hg::undirected_graph<T>::vertex_iterator> hg::vertices(const hg::undirected_graph<T> &g)

template<typename T>
std::pair<typename hg::undirected_graph<T>::edge_iterator, typename hg::undirected_graph<T>::edge_iterator> hg::edges(const hg::undirected_graph<T> &g)

template<typename T>
std::pair<typename hg::undirected_graph<T>::out_edge_iterator, typename hg::undirected_graph<T>::out_edge_iterator> hg::out_edges(typename hg::undirected_graph<T>::vertex_descriptor v, const hg::undirected_graph<T> &g)

template<typename T>
std::pair<typename hg::undirected_graph<T>::out_edge_iterator, typename hg::undirected_graph<T>::out_edge_iterator> hg::in_edges(typename hg::undirected_graph<T>::vertex_descriptor v, const hg::undirected_graph<T> &g)

template<typename T>
std::pair<typename hg::undirected_graph<T>::adjacency_iterator, typename hg::undirected_graph<T>::adjacency_iterator> hg::adjacent_vertices(typename hg::undirected_graph<T>::vertex_descriptor v, const hg::undirected_graph<T> &g)

template<typename lambda_t>
void hg::parfor(index_t start_index, index_t end_index, lambda_t fun, index_t step_size = 1)

template<typename T1, typename T2>
void hg::extend(T1 &a, const T2 &b)

Insert all elements of collection b at the end of collection a.

Template Parameters:

• T1 – must have an insert method (STL like) and a range interface (begin, end)

• T2 – must have a range interface (begin, end)

Parameters:

• a –

• b –

inline void hg::unreachable()

struct accumulator_argmax

Public Functions

template<bool vectorial = true, typename S>
inline auto make_accumulator(S &storage) const

Public Static Functions

template<typename shape_t>
static inline auto get_output_shape(const shape_t &input_shape)

struct accumulator_argmin

Public Functions

template<bool vectorial = true, typename S>
inline auto make_accumulator(S &storage) const

Public Static Functions

template<typename shape_t>
static inline auto get_output_shape(const shape_t &input_shape)

struct accumulator_counter

Public Functions

template<bool vectorial = true, typename S>
inline auto make_accumulator(S &storage) const

Public Static Functions

template<typename shape_t>
static inline auto get_output_shape(const shape_t&)

struct accumulator_first

Public Functions

template<bool vectorial = true, typename S>
inline auto make_accumulator(S &storage) const

Public Static Functions

template<typename shape_t>
static inline auto get_output_shape(const shape_t &input_shape)

struct accumulator_last

Public Functions

template<bool vectorial = true, typename S>
inline auto make_accumulator(S &storage) const

Public Static Functions

template<typename shape_t>
static inline auto get_output_shape(const shape_t &input_shape)

struct accumulator_max

Public Functions

template<bool vectorial = true, typename S>
inline auto make_accumulator(S &storage) const

Public Static Functions

template<typename shape_t>
static inline auto get_output_shape(const shape_t &input_shape)

template<typename value_type>
static inline value_type init_value()

template<typename value_type>
static inline value_type reduce(const value_type &v1, const value_type &v2)

struct accumulator_mean

Public Functions

template<bool vectorial = true, typename S>
inline auto make_accumulator(S &storage) const

Public Static Functions

template<typename shape_t>
static inline auto get_output_shape(const shape_t &input_shape)

struct accumulator_min

Public Functions

template<bool vectorial = true, typename S>
inline auto make_accumulator(S &storage) const

Public Static Functions

template<typename shape_t>
static inline auto get_output_shape(const shape_t &input_shape)

template<typename value_type>
static inline value_type init_value()

template<typename value_type>
static inline value_type reduce(const value_type &v1, const value_type &v2)

struct accumulator_prod

Public Functions

template<bool vectorial = true, typename S>
inline auto make_accumulator(S &storage) const

Public Static Functions

template<typename shape_t>
static inline auto get_output_shape(const shape_t &input_shape)

template<typename value_type>
static inline value_type init_value()

template<typename value_type>
static inline value_type reduce(const value_type &v1, const value_type &v2)

struct accumulator_sum

Public Functions

template<bool vectorial = true, typename S>
inline auto make_accumulator(S &storage) const

Public Static Functions

template<typename shape_t>
static inline auto get_output_shape(const shape_t &input_shape)

template<typename value_type>
static inline constexpr value_type init_value()

template<typename value_type>
static inline value_type reduce(const value_type &v1, const value_type &v2)

class assesser_fragmentation_optimal_cut

#include <fragmentation_curve.hpp>

This class is used to assess the optimal cuts of a hierarchy of partitions with respect to a given ground-truth labelisation of its base graph and the BCE measure.

Public Functions

template<typename tree_t, typename T>
inline assesser_fragmentation_optimal_cut(const tree_t &tree, const xt::xexpression<T> &xground_truth, optimal_cut_measure measure, const array_1d<index_t> &vertex_map = {}, size_t max_regions = 200)

Create an assesser for hierarchy optimal cuts w.r.t.

a given ground-truth partition of hierarchy leaves and the BCE quality measure. The algorithms will explore optimal cuts containing at most max_regions regions.

The ground truth labelisation must be normalized (i.e. its labels must be positive integers in the interval [0, num_regions[).

Template Parameters:

• tree_t – tree type

• T – type of labels

Parameters:

• tree – input hierarchy

• xground_truth – ground truth labelisation of the tree leaves

• vertex_map – super-vertices map (if tree is built on a rag, leave empty otherwise)

• max_regions – maximum number of regions in the considered cuts.

inline auto fragmentation_curve() const

Fragmentation curve, i.e.

for each number of region k between 1 and max_regions, the BCE score of the optimal cut with k regions.

Returns:

a fragmentation_curve

inline auto optimal_number_of_regions() const

Number of regions in the optimal cut.

Returns:

inline auto optimal_score() const

Score of the optimal cut.

Returns:

inline auto optimal_partition(size_t num_regions = 0) const

Labelisation of the base graph that corresponds to the optimal cut with the given number of regions.

If the number of regions is equal to 0 (default), the global optimal cut it returned (it will contain get_optimal_number_of_regions regions).

Parameters:

num_regions –

Returns:

inline auto straightened_altitudes(bool gain_only = false, bool normalize_result = true) const

Compute tree node altitudes such that the horizontal cut of the resulting vertex valued hierarchy corresponds to the optimal cut of the tree.

Parameters:

• gain_only – If true, optimal cuts with a number of regions greater than optimal_number_of_regions() are ignored

• normalize_result – Ensure that altitudes[tree(root)] == optimal_score

Returns:

template<typename T>
struct COMPILE_ERROR

Do not use except if you want a compile error showing the type of the provided template parameter !

Template Parameters:

T –

template<typename value_type = index_t>
struct counting_iterator : public hg::random_iterator_facade<counting_iterator<index_t>, index_t>

#include <iterators.hpp>

Quick implementation of a counting iterator.

Counting iterator has an integer value and a step size. Incrementing the iterator simply add the step size to the current value of the iterator.

Template Parameters:

value_type –

Public Types

using self_type = counting_iterator<value_type>

Public Functions

inline counting_iterator()

inline counting_iterator(value_type position, value_type step = 1)

inline void increment()

inline void decrement()

inline void advance(size_t n)

inline auto distance_to(const self_type &rhs) const

inline bool equal(const counting_iterator<value_type> &other) const

inline value_type dereference() const

template<typename derived_type, typename value_t, typename reference_t = value_t>
struct forward_iterator_facade

#include <iterators.hpp>

Facade to ease the declaration of new forward iterators (quick reimplementation of boost facade iterator).

The iterator must inherit of the facade and declare the following 3 functions:

• deference: obtain the element at the current position of the iterator

• increment: move the iterator to next element

• equal: test if two iterators are equal

Template Parameters:

• derived_type –

• value_t –

• reference_t –

Subclassed by hg::contour_2d_internal::contour_segment_2d_iterator< point_type >, hg::contour_2d_internal::polyline_contour_2d_iterator< point_type >, hg::transform_forward_iterator< transform_fun_type, base_iterator_t, value_t, reference_t >, hg::tree_internal::tree_graph_adjacent_vertex_iterator< edge_index_iterator >

Public Types

using self_type = forward_iterator_facade<derived_type, value_t, reference_t>

using value_type = value_t

using difference_type = std::ptrdiff_t

using reference = reference_t

using pointer = value_t*

using iterator_category = std::forward_iterator_tag

Friends

inline friend reference operator*(const derived_type &lhs)

inline friend const reference operator*(derived_type &lhs)

inline friend derived_type operator++(derived_type &lhs)

inline friend derived_type operator++(derived_type &lhs, int)

inline friend bool operator==(const derived_type &lhs, const derived_type &rhs)

inline friend bool operator!=(const derived_type &lhs, const derived_type &rhs)

template<typename value_t = double>
struct fragmentation_curve

Public Functions

inline fragmentation_curve(array_1d<value_t> &&num_regions, array_1d<value_t> &&scores, size_t num_regions_ground_truth)

inline auto num_regions_ground_truth() const

Number of regions in the ground truth labelisation of the hierarchy base graph.

Returns:

inline auto optimal_number_of_regions() const

Number of regions in the optimal cut.

Returns:

inline auto optimal_score() const

Score of the optimal cut.

Returns:

inline const auto &scores() const

inline const auto &num_regions() const

inline auto num_regions_normalized() const

class hierarchy_aligner

#include <alignment.hpp>

This class allows to project hierarchies build from coarse supervertices onto fine supervertices.

The class is contructed by providing a fine supervertices decomposition of a graph. Then the functions align_hierarchy allows to project a hierarchy, given as a tree or as a saliency map, onto the fine supervertices.

Given:

• a graph g

• a fine labelisation l1 of the vertices of g;

• a tree t on g whose supervertices corresponds to the coarse labelisation l2 of the vertices of g; and

• the altitudes a of the nodes of t. Let us denote:

• given a vertex x of g and a labelisation l, l(x) is the region of l that contains x

• given a region r of l1, s(r, l2) is the region R of l2 that has the largest intersection with r: s(r, l2) = arg_max(R in l2) |R \cap r| The projection of t onto l1 is a hierarchy given by the saliency map sm on g defined by: for all {x,y} in edges(g), sm({x,y}) = a(lca_t(s(l1(x), l2), s(l1(y), l2)))

See the following helper functions for instanciation

• make_hierarchy_aligner_from_graph_cut

• make_hierarchy_aligner_from_labelisation

• make_hierarchy_aligner_from_hierarchy

Public Functions

inline hierarchy_aligner(region_adjacency_graph &&rag)

template<typename T>
inline auto align_hierarchy(const hg::tree &tree, const xt::xexpression<T> &xaltitudes) const

template<typename graph_t, typename T>
inline auto align_hierarchy(const graph_t &graph, const xt::xexpression<T> &xsaliency_map) const

template<typename T1, typename T2>
inline auto align_hierarchy(const xt::xexpression<T1> &xcoarse_supervertices, const hg::tree &tree, const xt::xexpression<T2> &xaltitudes) const

template<typename tree_t, typename value_t>
class horizontal_cut_explorer

Public Types

using tree_type = tree_t

using value_type = value_t

Public Functions

template<typename T>
inline horizontal_cut_explorer(const tree_t &tree, const xt::xexpression<T> &xaltitudes)

inline auto num_cuts() const

inline auto num_regions_cut(index_t i) const

inline const auto &num_regions_cuts() const

inline auto altitude_cut(index_t i) const

inline const auto &altitude_cuts() const

inline auto horizontal_cut_from_index(index_t cut_index) const

inline auto horizontal_cut_from_altitude(value_t threshold) const

inline auto horizontal_cut_from_num_regions(index_t num_regions, bool at_least = true) const

template<typename value_t>
struct horizontal_cut_nodes

Public Functions

inline horizontal_cut_nodes(array_1d<index_t> &&_nodes, value_t _altitude)

template<typename tree_t>
inline auto labelisation_leaves(const tree_t &tree) const

template<typename tree_t, typename T>
inline auto reconstruct_leaf_data(const tree_t &tree, const xt::xexpression<T> &altitudes) const

template<typename tree_t, typename graph_t>
inline auto graph_cut(const tree_t &tree, const graph_t &leaf_graph) const

Public Members

array_1d<index_t> nodes

value_t altitude

template<typename vertex_descriptor, typename edge_index_t>
class indexed_edge

#include <indexed_edge.hpp>

An edge with a source vertex, a target vertex, and an index.

Template Parameters:

• vertex_descriptor –

• edge_index_t –

Public Functions

inline indexed_edge(vertex_descriptor _source, vertex_descriptor _target, edge_index_t _index)

inline indexed_edge(const std::pair<vertex_descriptor, vertex_descriptor> &edge, edge_index_t _index)

inline operator edge_index_t() const

inline operator std::pair<vertex_descriptor, vertex_descriptor>() const

Public Members

vertex_descriptor source

vertex_descriptor first

union hg::indexed_edge

vertex_descriptor target

vertex_descriptor second

union hg::indexed_edge

edge_index_t index

template<typename value_type = index_t>
struct irange

#include <iterators.hpp>

A range of integer value with a start value, a step size, and a maximum value (excluded).

irange(start, stop, step) is the equivalent to

• python range(start, stop, step)

• matlab start:step:stop

Template Parameters:

value_type –

Public Functions

inline irange()

inline irange(value_type start, value_type stop, value_type step = 1)

inline auto begin() const

inline auto end() const

template<typename iterator_t>
struct iterator_wrapper

#include <graph.hpp>

Simple wrapper over two iterators to create a “range” object usable in foreach loops.

Template Parameters:

iterator_t –

Public Functions

inline iterator_wrapper(iterator_t &_first, iterator_t &_last)

inline iterator_wrapper(const std::pair<iterator_t, iterator_t> &p)

inline iterator_t begin()

inline iterator_t end()

inline iterator_t begin() const

inline iterator_t end() const

Public Members

iterator_t const first

iterator_t const last

struct logger

Public Types

using callback_list = std::vector<std::function<void(const std::string&)>>

Public Static Functions

static inline bool &trace_enabled()

static inline callback_list &callbacks()

template<typename ...Args>
static inline void emit(const char *format, Args&&... args)

Public Static Attributes

static const std::size_t MAX_MSG_SIZE = 8096

template<typename mst_t>
struct minimum_spanning_tree_result

#include <graph_core.hpp>

A simple structure to hold the result of minimum_spanning_tree function.

The structures holds 2 elements:

• the minimum spanning tree (mst)

• a map (mst_edge_map) that indicates for each edge of the mst, the corresponding edge index in the original graph

Template Parameters:

mst_t –

Public Members

mst_t mst

array_1d<index_t> mst_edge_map

template<typename tree_t, typename altitude_t>
struct node_weighted_tree

#include <common.hpp>

A simple structure to hold the result of hierarchy construction algorithms.

See make_node_weighted_tree for construction

Template Parameters:

• tree_t –

• altitude_t –

Public Members

tree_t tree

altitude_t altitudes

template<typename tree_t, typename altitude_t>
struct node_weighted_tree_and_mst

#include <hierarchy_core.hpp>

A simple structure to hold the result of canonical bpt function.

See make_node_weighted_tree_and_mst for construction

Template Parameters:

• tree_t –

• altitude_t –

• mst_t –

Public Members

tree_t tree

altitude_t altitudes

array_1d<index_t> mst_edge_map

template<typename A = array_1d<double>, typename B = A>
struct pink_graph

Public Members

ugraph graph

std::vector<std::size_t> shape

A vertex_weights

B edge_weights

template<typename derived_type, typename value_t, typename reference_t = value_t>
struct random_iterator_facade

#include <iterators.hpp>

Facade to ease the declaration of new random iterators (quick reimplementation of boost facade iterator).

The iterator must inherit of the facade and declare the following 3 functions:

• deference: obtain the element at the current position of the iterator

• increment: move the iterator to next element

• decrement: move the iterator to previous element

• advance(n): Advance by n positions

• distance_to(j) Measure the distance to j

• equal: test if two iterators are equal

Template Parameters:

• derived_type –

• value_t –

• reference_t –

Subclassed by hg::counting_iterator< value_type >

Public Types

using self_type = random_iterator_facade<derived_type, value_t, reference_t>

using value_type = value_t

using difference_type = std::ptrdiff_t

using reference = reference_t

using pointer = value_t*

using iterator_category = std::random_access_iterator_tag

Public Functions

inline auto operator[](const difference_type &n)

Friends

inline friend reference operator*(const derived_type &lhs)

inline friend const reference operator*(derived_type &lhs)

inline friend derived_type operator++(derived_type &lhs)

inline friend derived_type operator++(derived_type &lhs, int)

inline friend derived_type operator--(derived_type &lhs)

inline friend derived_type operator--(derived_type &lhs, int)

inline friend bool operator==(const derived_type &lhs, const derived_type &rhs)

inline friend bool operator!=(const derived_type &lhs, const derived_type &rhs)

inline friend bool operator<=(const derived_type &lhs, const derived_type &rhs)

inline friend bool operator<(const derived_type &lhs, const derived_type &rhs)

inline friend bool operator>=(const derived_type &lhs, const derived_type &rhs)

inline friend bool operator>(const derived_type &lhs, const derived_type &rhs)

inline friend auto operator-(const derived_type &lhs, const derived_type &rhs)

inline friend derived_type operator+(const derived_type &lhs, const difference_type &n)

inline friend derived_type operator+(const difference_type &n, const derived_type &rhs)

struct region_adjacency_graph

#include <rag.hpp>

Result of the region adjacency graph (rag) construction algorithm.

Public Members

ugraph rag

The region adjacency graph.

array_1d<index_t> vertex_map

An array indicating for each vertex of the original graph, the corresponding vertex of the rag.

array_1d<index_t> edge_map

An array indicating for each edge of the original graph, the corresponding edge of the rag.

An edge with no corresponding edge in the rag (edge within a region) is indicated with the value invalid_index.

template<typename tree_t, typename node_map_t>
struct remapped_tree

#include <common.hpp>

A simple structure to hold the result of a remapping operation on the nodes of a tree.

When algorithm transforms a tree into a new tree by removing or duplicating or reordering some of its nodes, it is useful to know the relation between the nodes of the new tree and the nodes of the original one.

For each node i of the new tree, node_map[i] gives the corresponding node in the original tree.

Template Parameters:

• tree_t –

• node_map_t –

Public Members

tree_t tree

node_map_t node_map

struct scorer_partition_BCE

Public Static Functions

template<typename T>
static inline auto score(const xt::xexpression<T> &xcard_intersection)

struct scorer_partition_DCovering

Public Static Functions

template<typename T>
static inline auto score(const xt::xexpression<T> &xcard_intersection)

struct scorer_partition_DHamming

Public Static Functions

template<typename T>
static inline auto score(const xt::xexpression<T> &xcard_intersection)

template<typename supervertex_labels_t, typename tree_t, typename node_map_t>
struct supervertex_hierarchy

#include <tree.hpp>

A simple structure to hold the result of supervertices_hierarchy algorithm.

Template Parameters:

• node_map_t –

• tree_t –

• node_map_t –

Public Members

supervertex_labels_t supervertex_labelisation

tree_t tree

node_map_t node_map

template<typename transform_fun_type, typename base_iterator_t, typename value_t, typename reference_t = value_t>
struct transform_forward_iterator : public hg::forward_iterator_facade<transform_forward_iterator<transform_fun_type, base_iterator_t, value_t, value_t>, value_t, value_t>

#include <iterators.hpp>

Facade to ease the declaration of new forward iterators as a transformation of an existing forward iterator (quick reimplementation of boost transform iterator).

Template Parameters:

• transform_fun_type –

• base_iterator_t –

• value_t –

• reference_t –

Public Types

using self_type = transform_forward_iterator<transform_fun_type, base_iterator_t, value_t, reference_t>

Public Functions

inline void increment()

inline bool equal(const self_type &other) const

inline value_t dereference() const

inline transform_forward_iterator(base_iterator_t &&base, transform_fun_type &&fun)

inline transform_forward_iterator(const base_iterator_t &base, const transform_fun_type &fun)

inline transform_forward_iterator()