# Source code for brainlit.algorithms.trace_analysis.fit_spline

```
from typing import Tuple
import numpy as np
from scipy.interpolate import splprep, splev, BSpline, CubicHermiteSpline, PPoly
from scipy.interpolate._cubic import prepare_input
import pandas as pd
import math
import warnings
import networkx as nx
import itertools
from brainlit.utils.util import (
check_type,
check_size,
check_precomputed,
check_iterable_type,
check_iterable_nonnegative,
)
from typing import Union, List
def compute_parameterization(positions: np.array) -> np.array:
"""Computes list of parameter values to be used for a list of positions using piecewise linear arclength.
Parameters
----------
positions : np.array
nxd array containing coordinates of the n points in d dimentional space.
Returns
-------
TotalDist : np.array
n array containing parameter values, first element is 0.
"""
NodeDist = np.linalg.norm(np.diff(positions, axis=0), axis=1)
TotalDist = np.concatenate(([0], np.cumsum(NodeDist)))
return TotalDist
[docs]class CubicHermiteChain(PPoly):
"""A third order spline class (continuous piecewise cubic representation), that is fit to a set of positions and one-sided derivatives. This is not a standard spline class (e.g. b-splines), because the derivatives are not necessarily continuous at the knots.
A subclass of PPoly, a piecewise polynomial class from scipy.
"""
def __init__(
self,
x: np.array,
y: np.array,
left_dydx: np.array,
right_dydx: np.array,
extrapolate=None,
):
"""Initialize object via:
Parameters
----------
x : np.array
Independent variable, shape n.
y : np.array
Dependent variable, shape n x d.
left_dydx : np.array
Derivatives on left sides of cubic segments (i.e. right hand derivatives of knots), shape n-1 x d.
right_dy_dx : np.array
Derivatives on right sides of cubic segments (i.e. left hand derivatives of knots), shape n-1 x d.
extrapolate : np.array
If bool, determines whether to extrapolate to out-of-bounds points based on first and last intervals, or to return NaNs. If â€˜periodicâ€™, periodic extrapolation is used. Default is True.
"""
if extrapolate is None:
extrapolate = True
x, dx, y, axis, _ = prepare_input(x, y, axis=0)
if not np.array_equal(left_dydx.shape, right_dydx.shape):
raise ValueError(
f"Left derivatives shape {left_dydx.shape} must be equal to right derivatives shape {right_dydx.shape}"
)
dxr = dx.reshape([dx.shape[0]] + [1] * (y.ndim - 1))
slope = np.diff(y, axis=0) / dxr
t = (left_dydx + right_dydx - 2 * slope) / dxr
c = np.empty((4, len(x) - 1) + y.shape[1:], dtype=t.dtype)
c[0] = t / dxr
c[1] = (slope - left_dydx) / dxr - t
c[2] = left_dydx
c[3] = y[:-1]
super().__init__(c, x, extrapolate=extrapolate)
self.axis = axis
"""
Geometric Graph class
"""
[docs]class GeometricGraph(nx.Graph):
r"""The shape of the neurons are expressed and fitted with splines in this undirected graph class.
The geometry of the neurons are projected on undirected graphs, based on which the trees of neurons consisted for splines is constructed.
It is required that each node has a loc attribute identifying that points location in space, and the location should be defined in 3-dimensional cartesian coordinates.
It extends `nx.Graph` and rejects duplicate node input.
"""
def __init__(self, df: pd.DataFrame = None, root=1) -> None:
super(GeometricGraph, self).__init__()
self.segments = None
self.cycle = None
self.root = root
self.spline_type = None
self.spline_tree = None
if df is not None:
self.__init_from_df(df)
def __init_from_df(self, df_neuron: pd.DataFrame) -> "GeometricGraph":
"""Converts dataframe of swc in voxel coordinates into a GeometricGraph
Parameters
----------
df_neuron : :class:`pandas.DataFrame`
Indicies, coordinates, and parents of each node in the swc.
Returns
-------
G : :class:`brainlit.algorithms.trace_analysis.fit_spline.GeometricGraph`
Neuron from swc represented as GeometricGraph. Coordinates `x,y,z`
are accessible in the `loc` attribute.
"""
# check that there are not duplicate nodes
dx = np.expand_dims(np.diff(df_neuron["x"].to_numpy()), axis=0).T
dy = np.expand_dims(np.diff(df_neuron["y"].to_numpy()), axis=0).T
dz = np.expand_dims(np.diff(df_neuron["z"].to_numpy()), axis=0).T
dr = np.concatenate((dx, dy, dz), axis=1)
if not all([any(du != 0) for du in dr]):
raise ValueError("cannot build GeometricGraph with duplicate nodes")
# build graph
for _, row in df_neuron.iterrows():
# extract id
id = int(row["sample"])
# add nodes
loc_x = row["x"]
loc_y = row["y"]
loc_z = row["z"]
loc = np.array([loc_x, loc_y, loc_z])
self.add_node(id, loc=loc)
# add edges
child = id
parent = int(row["parent"])
if parent > min(df_neuron["parent"]):
self.add_edge(parent, child)
def fit_spline_tree_invariant(
self, spline_type: Union[BSpline, CubicHermiteSpline] = BSpline, k=3
):
r"""Construct a spline tree based on the path lengths.
Arguments:
spline_type: BSpline or CubicHermiteSpline, spline type that will be fit to the data. BSplines are typically used to fit position data only, and CubicHermiteSplines can only be used if derivative, and independent variable information is also known.
Raises:
ValueError: check if every node is unigue in location
ValueError: check if every node is assigned to at least one edge
ValueError: check if the graph contains undirected cycle(s)
ValueErorr: check if the graph has disconnected segment(s)
Returns:
spline_tree: nx.DiGraph a parent tree with the longest path in the directed graph
"""
# check integrity of 'loc' attributes in the neuron
if any([self.nodes[node].get("loc") is None for node in self.nodes]):
raise KeyError("some nodes are missing the 'loc' attribute")
for node in self.nodes:
check_type(self.nodes[node].get("loc"), np.ndarray)
if any([self.nodes[node].get("loc").ndim != 1 for node in self.nodes]):
raise ValueError("nodes must be flat arrays")
if any([len(self.nodes[node].get("loc")) == 0 for node in self.nodes]):
raise ValueError("nodes cannot have empty 'loc' attributes")
for node in self.nodes:
check_iterable_type(self.nodes[node].get("loc"), (np.integer, float))
if any([len(self.nodes[node].get("loc")) != 3 for node in self.nodes]):
raise ValueError("'loc' attributes must contain 3 coordinates")
# check there are no duplicate nodes
LOCs = [np.ndarray.tolist(self.nodes[node]["loc"]) for node in self.nodes]
LOCs.sort()
unique_LOCs = list(LOC for LOC, _ in itertools.groupby(LOCs))
if len(LOCs) != len(unique_LOCs):
raise ValueError("there are duplicate nodes")
# check the graph is edge-covering
if not nx.algorithms.is_edge_cover(self, self.edges):
raise ValueError("the edges are not a valid cover of the graph")
# check there are no undirected cycles in the graph
if not nx.algorithms.tree.recognition.is_forest(self):
raise ValueError("the graph contains undirected cycles")
# check there are no disconnected segments
if not nx.algorithms.tree.recognition.is_tree(self):
raise ValueError("the graph contains disconnected segments")
# Identify paths
spline_tree = nx.DiGraph()
curr_spline_num = 0
stack = []
root = self.root
tree = nx.algorithms.traversal.depth_first_search.dfs_tree(self, source=root)
main_branch, collateral_branches = self.__find_main_branch(tree)
spline_tree.add_node(curr_spline_num, path=main_branch, starting_length=0)
for tree in collateral_branches:
stack.append((tree, curr_spline_num))
while len(stack) > 0:
curr_spline_num = curr_spline_num + 1
treenum = stack.pop()
tree = treenum[0]
parent_num = treenum[1]
main_branch, collateral_branches = self.__find_main_branch(
tree[0], starting_length=tree[2]
)
main_branch.insert(0, tree[1])
spline_tree.add_node(
curr_spline_num, path=main_branch, starting_length=tree[2]
)
spline_tree.add_edge(parent_num, curr_spline_num)
for tree in collateral_branches:
stack.append((tree, curr_spline_num))
# Fit splines
self.spline_type = spline_type
for node in spline_tree.nodes:
main_branch = spline_tree.nodes[node]["path"]
if spline_type == BSpline: # each node must have "loc" attribute
spline_tree.nodes[node]["spline"] = self.__fit_bspline_path(
main_branch, k=k
)
elif (
spline_type == CubicHermiteSpline
): # each node must have "u," "loc," and "deriv" attributes
spline_tree.nodes[node]["spline"] = self.__fit_chspline_path(
main_branch
)
self.spline_tree = spline_tree
return spline_tree
def __fit_bspline_path(self, path, k):
r"""Fit a B-Spline to a path.
Compute the knots, coefficients, and the degree of the
B-Spline fitting the path
Arguments:
path: list, a list of nodes.
Raises:
ValueError: Nodes should be defined under loc attribute
TypeError: loc should be of numpy.ndarray class
ValueError: loc should be 3-dimensional
Returns:
tck: tuple, contains the vector of knots, the coefficients, and the degree of the B-Spline.
u: list, contains the values of the parameters where the B-Spline is evaluated.
"""
x = np.zeros((len(path), 3))
for row, node in enumerate(path):
x[row, :] = self.nodes[node]["loc"]
path_length = x.shape[0]
TotalDist = compute_parameterization(x)
# old
# if path_length != 5:
# k = np.amin([path_length - 1, 5])
# else:
# k = 3
k = np.amin([k, path_length - 1]) # change
tck, u = splprep([x[:, 0], x[:, 1], x[:, 2]], s=0, u=TotalDist, k=k)
return tck, u
def __fit_chspline_path(self, path: List):
"""Fit cubic hermite spline to path of nodes that has independent variable (u), position (loc) and derivative (deriv) attributes.
Args:
path (List): sequence of nodes to which the spline will be fit.
Returns:
CubicHermiteSpline: spline that fits the nodes in path.
"""
x = np.zeros((len(path)))
y = np.zeros((len(path), 3))
dy = np.zeros((len(path), 3))
for row, node in enumerate(path):
x[row] = self.nodes[node]["u"]
y[row, :] = self.nodes[node]["loc"]
dy[row, :] = self.nodes[node]["deriv"]
chspline = CubicHermiteSpline(x, y, dy)
return chspline
def __find_main_branch(self, tree: nx.DiGraph, starting_length: float = 0):
r"""Find the main branch in a directed graph.
It is used in `fit_spline_tree_invariant` to identify the main branch
in a neuron and group the collateral branches for later analysis.
The main branch is defined as the longest possible path connecting the
neuron's nodes, in terms of spatial distance. An example is provided in
the following figure:
.. figure:: https://raw.githubusercontent.com/neurodata/brainlit/develop/docs/images/find_main_branch.png
:scale: 25%
:alt: find_main_branch example
Graphic example of `find_main_branch()` functionality.
Arguments:
tree: nx.DiGraph, a directed graph.
It is the result of nx.algorithms.traversal.depth_first_search.dfs_tree()
which returns an oriented tree constructed from a depth-first search of
the neuron.
starting_length: float, optional.
It is the spatial distance between the root of the neuron (i.e `self.root`) and
the root of the current main branch. It must be real-valued, non-negative.
It is defaulted to `0` for the first main branch, that starts from the root of
the neuron.
Returns:
main_branch: list, a list of nodes.
collateral_branches: list, directed graphs of children trees.
"""
# Initialize the list of collateral branches
collateral_branches = []
# If there is only one node in the tree, that is the main branch
if len(tree.nodes) == 1:
main_branch = tree.nodes
else:
# Find the root of the tree.
# A node is a candidate to be the root if it does not
# have any edges pointing to it (i.e. in_degree == 0)
roots = [node for node, degree in tree.in_degree() if degree == 0]
root = roots[0]
# Find the leaves of the tree.
# A node is a leaf if it has only one edge pointing
# to it (i.e. in_degree == 1), and no edges pointing
# out of it (i.e. out_degree == 0)
leaves = [
node
for node in tree.nodes()
if tree.out_degree(node) == 0 and tree.in_degree(node) == 1
]
# For each leaf, compute the shortest path to reach it
shortest_paths = [
nx.algorithms.shortest_paths.generic.shortest_path(
tree, source=root, target=l
)
for l in leaves
]
# Compute the lengths of the paths
lengths = [self.__path_length(path) for path in shortest_paths]
# Find the longest path
longest_path_idx = np.argmax(lengths)
furthest_leaf = leaves[longest_path_idx]
# Find the main branch
main_branch = nx.algorithms.shortest_paths.generic.shortest_path(
tree, source=root, target=furthest_leaf
)
# Here, we walk on the main branch to find
# the collateral branches
for i, node in enumerate(main_branch):
# Increase starting_length by the size of
# the step on the main branch
if i > 0:
loc1 = self.nodes[node]["loc"]
loc2 = self.nodes[main_branch[i - 1]]["loc"]
starting_length += np.linalg.norm(loc2 - loc1)
# Find all successors of the current node on
# the main branch. A node m is a successor of the node
# n if there is a directed edge that goes from n to m
children = tree.successors(node)
for child in children:
# If the successor is not on the main branch, then
# we found a branching point of the neuron
if child != main_branch[i + 1]:
# Explore the newly-found branch and
# append it to the list of collateral branches
collateral_branches.append(
(
nx.algorithms.traversal.depth_first_search.dfs_tree(
tree, source=child
),
node,
starting_length,
)
)
return list(main_branch), collateral_branches
def __path_length(self, path: list) -> float:
r"""Compute the length of a path.
Given a path ::math::`p = (r_1, \dots, r_N)`, where
::math::`r_k = [x_k, y_k, z_k], k = 1, \dots, N`, the length
`l` of a path is computed as the sum of the lengths of the
edges of the path. We can write:
.. math::
l = \sum_{k=2}^N \lVert r_k - r_{k-1} \rVert
Arguments:
path: a list of nodes.
The integrity of the nodes is checked for at the beginning of
`fit_spline_tree_invariant`.
Returns:
length: float.
It is the length of the path.
"""
length = sum(
[
np.linalg.norm(self.nodes[node]["loc"] - self.nodes[path[i - 1]]["loc"])
if i >= 1
else 0
for i, node in enumerate(path)
]
)
return length
```