def hierarchy_to_optimal_MumfordShah_energy_cut_hierarchy(tree,
vertex_weights,
leaf_graph,
approximation_piecewise_linear_function=10):
"""
Transform the given hierarchy into an optimal energy cut hierarchy using the piecewise constant Mumford-Shah energy
(see function :func:`~higra.hierarchy_to_optimal_energy_cut_hierarchy`).
In this context:
- the data fidelity energy assumes a piecewise constant model in each node and is given by the variance of the vertex values inside the node (see function :func:`~higra.attribute_gaussian_region_weights_model`) multiplied by its area,
- the regularity energy is given by the length of the contour of the node (see function :func:`~higra.attribute_contour_length`).
:param tree: input tree (Concept :class:`~higra.CptHierarchy`)
:param vertex_weights: vertex weights of the leaf graph of the input tree
:param leaf_graph: leaf graph of the input tree (deduced from :class:`~higra.CptHierarchy`)
:param approximation_piecewise_linear_function: Maximum number of pieces used in the approximated piecewise linear model for the energy function (default 10).
:return: a tree (Concept :class:`~higra.CptHierarchy`) and its node altitudes
"""
area = hg.attribute_area(tree, leaf_graph=leaf_graph)
_, variance = hg.attribute_gaussian_region_weights_model(tree, vertex_weights, leaf_graph)
perimeter = hg.attribute_contour_length(tree, leaf_graph=leaf_graph)
if variance.ndim > 1:
variance = np.trace(variance, axis1=1, axis2=2)
return hierarchy_to_optimal_energy_cut_hierarchy(tree, variance * area, perimeter,
int(approximation_piecewise_linear_function))
def attribute_piecewise_constant_Mumford_Shah_energy(tree, vertex_weights,
gamma, leaf_graph):
"""
Piecewise constant Mumford-Shah energy of each node of the input tree.
The energy of a node is equal to its data fidelity energy plus gamma times its regularization energy.
For the piecewise constant Mumford-Shah model:
- the data fidelity energy assumes a piecewise constant model in each node and is given by the variance of the vertex values inside the node (see function :func:`~higra.attribute_gaussian_region_weights_model`) multiplied by its area,
- the regularity energy is given by the length of the contour of the node (see function :func:`~higra.attribute_contour_length`).
:param tree: input tree (Concept :class:`~higra.CptHierarchy`)
:param vertex_weights: vertex weights of the leaf graph of the input tree
:param gamma: weighting of the regularization term (should be a positive value)
:param leaf_graph: leaf graph of the input tree (deduced from :class:`~higra.CptHierarchy`)
:return: a 1d array measuring the energy of each node the input tree
"""
area = hg.attribute_area(tree, leaf_graph=leaf_graph)
_, variance = hg.attribute_gaussian_region_weights_model(
tree, vertex_weights, leaf_graph)
perimeter = hg.attribute_contour_length(tree, leaf_graph=leaf_graph)
if variance.ndim > 1:
variance = np.trace(variance, axis1=1, axis2=2)
return variance * area + gamma * perimeter
def test_attribute_gaussian_region_weights_model_scalar(self):
tree, altitudes = TestAttributes.get_test_tree()
vertex_list = hg.attribute_vertex_list(tree)
np.random.seed(42)
vertex_weights = np.random.rand(tree.num_leaves())
mean, variance = hg.attribute_gaussian_region_weights_model(tree, vertex_weights)
for i in tree.leaves_to_root_iterator():
m = np.mean(vertex_weights[vertex_list[i]])
v = np.var(vertex_weights[vertex_list[i]])
self.assertTrue(np.isclose(m, mean[i]))
self.assertTrue(np.isclose(v, variance[i]))
def test_attribute_gaussian_region_weights_model_vectorial(self):
tree, altitudes = TestAttributes.get_test_tree()
vertex_list = hg.attribute_vertex_list(tree)
np.random.seed(42)
vertex_weights = np.random.rand(tree.num_leaves(), 3)
mean, variance = hg.attribute_gaussian_region_weights_model(tree, vertex_weights)
for i in tree.leaves_to_root_iterator(include_leaves=True):
m = np.mean(vertex_weights[vertex_list[i]], 0)
self.assertTrue(np.allclose(m, mean[i, :]))
# numpy wrongly interprets a single observation with several variables as
# multiple observations of a single variables
if i >= tree.num_leaves():
v = np.cov(vertex_weights[vertex_list[i]], rowvar=False, bias=True)
self.assertTrue(np.allclose(v, variance[i, ...]))
else:
v = np.zeros_like(variance[i, ...])
self.assertTrue(np.allclose(v, variance[i, ...]))