"""Module for the B-Spline model class."""
import warnings
import torch
from ..utils import check_positive_integer, check_consistency
[docs]
class Spline(torch.nn.Module):
r"""
The univariate B-Spline curve model class.
A univariate B-spline curve of order :math:`k` is a parametric curve defined
as a linear combination of B-spline basis functions and control points:
.. math::
S(x) = \sum_{i=1}^{n} B_{i,k}(x) C_i, \quad x \in [x_1, x_m]
where:
- :math:`C_i \in \mathbb{R}` are the control points. These fixed points
influence the shape of the curve but are not generally interpolated,
except at the boundaries under certain knot multiplicities.
- :math:`B_{i,k}(x)` are the B-spline basis functions of order :math:`k`,
i.e., piecewise polynomials of degree :math:`k-1` with support on the
interval :math:`[x_i, x_{i+k}]`.
- :math:`X = \{ x_1, x_2, \dots, x_m \}` is the non-decreasing knot vector.
If the first and last knots are repeated :math:`k` times, then the curve
interpolates the first and last control points.
.. note::
The curve is forced to be zero outside the interval defined by the
first and last knots.
:Example:
>>> from pina.model import Spline
>>> import torch
>>> knots1 = torch.tensor([0.0, 0.0, 0.0, 1.0, 2.0, 2.0, 2.0])
>>> spline1 = Spline(order=3, knots=knots1, control_points=None)
>>> knots2 = {"n": 7, "min": 0.0, "max": 2.0, "mode": "auto"}
>>> spline2 = Spline(order=3, knots=knots2, control_points=None)
>>> knots3 = torch.tensor([0.0, 0.0, 0.0, 1.0, 2.0, 2.0, 2.0])
>>> control_points3 = torch.tensor([0.0, 1.0, 3.0, 2.0])
>>> spline3 = Spline(order=3, knots=knots3, control_points=control_points3)
"""
def __init__(self, order=4, knots=None, control_points=None):
"""
Initialization of the :class:`Spline` class.
:param int order: The order of the spline. The corresponding basis
functions are polynomials of degree ``order - 1``. Default is 4.
:param knots: The knots of the spline. If a tensor is provided, knots
are set directly from the tensor. If a dictionary is provided, it
must contain the keys ``"n"``, ``"min"``, ``"max"``, and ``"mode"``.
Here, ``"n"`` specifies the number of knots, ``"min"`` and ``"max"``
define the interval, and ``"mode"`` selects the sampling strategy.
The supported modes are ``"uniform"``, where the knots are evenly
spaced over :math:`[min, max]`, and ``"auto"``, where knots are
constructed to ensure that the spline interpolates the first and
last control points. In this case, the number of knots is adjusted
if :math:`n < 2 * order`. If None is given, knots are initialized
automatically over :math:`[0, 1]` ensuring interpolation of the
first and last control points. Default is None.
:type knots: torch.Tensor | dict
:param torch.Tensor control_points: The control points of the spline.
If None, they are initialized as learnable parameters with an
initial value of zero. Default is None.
:raises AssertionError: If ``order`` is not a positive integer.
:raises ValueError: If ``knots`` is neither a torch.Tensor nor a
dictionary, when provided.
:raises ValueError: If ``control_points`` is not a torch.Tensor,
when provided.
:raises ValueError: If both ``knots`` and ``control_points`` are None.
:raises ValueError: If ``knots`` is not one-dimensional.
:raises ValueError: If ``control_points`` is not one-dimensional.
:raises ValueError: If the number of ``knots`` is not equal to the sum
of ``order`` and the number of ``control_points.``
:raises UserWarning: If the number of control points is lower than the
order, resulting in a degenerate spline.
"""
super().__init__()
# Check consistency
check_positive_integer(value=order, strict=True)
check_consistency(knots, (type(None), torch.Tensor, dict))
check_consistency(control_points, (type(None), torch.Tensor))
# Raise error if neither knots nor control points are provided
if knots is None and control_points is None:
raise ValueError("knots and control_points cannot both be None.")
# Initialize knots if not provided
if knots is None and control_points is not None:
knots = {
"n": len(control_points) + order,
"min": 0,
"max": 1,
"mode": "auto",
}
# Initialization - knots and control points managed by their setters
self.order = order
self.knots = knots
self.control_points = control_points
# Check dimensionality of knots
if self.knots.ndim > 1:
raise ValueError("knots must be one-dimensional.")
# Check dimensionality of control points
if self.control_points.ndim > 1:
raise ValueError("control_points must be one-dimensional.")
# Raise error if #knots != order + #control_points
if len(self.knots) != self.order + len(self.control_points):
raise ValueError(
f" The number of knots must be equal to order + number of"
f" control points. Got {len(self.knots)} knots, {self.order}"
f" order and {len(self.control_points)} control points."
)
# Raise warning if spline is degenerate
if len(self.control_points) < self.order:
warnings.warn(
"The number of control points is smaller than the spline order."
" This creates a degenerate spline with limited flexibility.",
UserWarning,
)
# Precompute boundary interval index
self._boundary_interval_idx = self._compute_boundary_interval()
def _compute_boundary_interval(self):
"""
Precompute the index of the rightmost non-degenerate interval to improve
performance, eliminating the need to perform a search loop in the basis
function on each call.
:return: The index of the rightmost non-degenerate interval.
:rtype: int
"""
# Return 0 if there is a single interval
if len(self.knots) < 2:
return 0
# Find all indices where knots are strictly increasing
diffs = self.knots[1:] - self.knots[:-1]
valid = torch.nonzero(diffs > 0, as_tuple=False)
# If all knots are equal, return 0 for degenerate spline
if valid.numel() == 0:
return 0
# Otherwise, return the last valid index
return int(valid[-1])
[docs]
def basis(self, x):
"""
Compute the basis functions for the spline using an iterative approach.
This is a vectorized implementation based on the Cox-de Boor recursion.
:param torch.Tensor x: The points to be evaluated.
:return: The basis functions evaluated at x.
:rtype: torch.Tensor
"""
# Add a final dimension to x
x = x.unsqueeze(-1)
# Add an initial dimension to knots
knots = self.knots.unsqueeze(0)
# Base case of recursion: indicator functions for the intervals
basis = (x >= knots[..., :-1]) & (x < knots[..., 1:])
basis = basis.to(x.dtype)
# One-dimensional knots case: ensure rightmost boundary inclusion
if self._boundary_interval_idx is not None:
# Extract left and right knots of the rightmost interval
knot_left = knots[..., self._boundary_interval_idx]
knot_right = knots[..., self._boundary_interval_idx + 1]
# Identify points at the rightmost boundary
at_rightmost_boundary = (
x.squeeze(-1) >= knot_left
) & torch.isclose(x.squeeze(-1), knot_right, rtol=1e-8, atol=1e-10)
# Ensure the correct value is set at the rightmost boundary
if torch.any(at_rightmost_boundary):
basis[..., self._boundary_interval_idx] = torch.logical_or(
basis[..., self._boundary_interval_idx].bool(),
at_rightmost_boundary,
).to(basis.dtype)
# Iterative case of recursion
for i in range(1, self.order):
# Compute the denominators for both terms
denom1 = knots[..., i:-1] - knots[..., : -(i + 1)]
denom2 = knots[..., i + 1 :] - knots[..., 1:-i]
# Ensure no division by zero
denom1 = torch.where(
torch.abs(denom1) < 1e-8, torch.ones_like(denom1), denom1
)
denom2 = torch.where(
torch.abs(denom2) < 1e-8, torch.ones_like(denom2), denom2
)
# Compute the two terms of the recursion
term1 = ((x - knots[..., : -(i + 1)]) / denom1) * basis[..., :-1]
term2 = ((knots[..., i + 1 :] - x) / denom2) * basis[..., 1:]
# Combine terms to get the new basis
basis = term1 + term2
return basis
[docs]
def forward(self, x):
"""
Forward pass for the :class:`Spline` model.
:param x: The input tensor.
:type x: torch.Tensor | LabelTensor
:return: The output tensor.
:rtype: torch.Tensor
"""
return torch.einsum(
"...bi, i -> ...b",
self.basis(x.as_subclass(torch.Tensor)).squeeze(-1),
self.control_points,
)
@property
def control_points(self):
"""
The control points of the spline.
:return: The control points.
:rtype: torch.Tensor
"""
return self._control_points
@control_points.setter
def control_points(self, control_points):
"""
Set the control points of the spline.
:param torch.Tensor control_points: The control points tensor. If None,
control points are initialized to learnable parameters with zero
initial value. Default is None.
:raises ValueError: If there are not enough knots to define the control
points, due to the relation: #knots = order + #control_points.
"""
# If control points are not provided, initialize them
if control_points is None:
# Check that there are enough knots to define control points
if len(self.knots) < self.order + 1:
raise ValueError(
f"Not enough knots to define control points. Got "
f"{len(self.knots)} knots, but need at least "
f"{self.order + 1}."
)
# Initialize control points to zero
control_points = torch.zeros(len(self.knots) - self.order)
# Set control points
self._control_points = torch.nn.Parameter(
control_points, requires_grad=True
)
@property
def knots(self):
"""
The knots of the spline.
:return: The knots.
:rtype: torch.Tensor
"""
return self._knots
@knots.setter
def knots(self, value):
"""
Set the knots of the spline.
:param value: The knots of the spline. If a tensor is provided, knots
are set directly from the tensor. If a dictionary is provided, it
must contain the keys ``"n"``, ``"min"``, ``"max"``, and ``"mode"``.
Here, ``"n"`` specifies the number of knots, ``"min"`` and ``"max"``
define the interval, and ``"mode"`` selects the sampling strategy.
The supported modes are ``"uniform"``, where the knots are evenly
spaced over :math:`[min, max]`, and ``"auto"``, where knots are
constructed to ensure that the spline interpolates the first and
last control points. In this case, the number of knots is inferred
and the ``"n"`` key is ignored.
:type value: torch.Tensor | dict
:raises ValueError: If a dictionary is provided but does not contain
the required keys.
:raises ValueError: If the mode specified in the dictionary is invalid.
"""
# If a dictionary is provided, initialize knots accordingly
if isinstance(value, dict):
# Check that required keys are present
required_keys = {"n", "min", "max", "mode"}
if not required_keys.issubset(value.keys()):
raise ValueError(
f"When providing knots as a dictionary, the following "
f"keys must be present: {required_keys}. Got "
f"{value.keys()}."
)
# Uniform sampling of knots
if value["mode"] == "uniform":
value = torch.linspace(value["min"], value["max"], value["n"])
# Automatic sampling of interpolating knots
elif value["mode"] == "auto":
# Repeat the first and last knots 'order' times
initial_knots = torch.ones(self.order) * value["min"]
final_knots = torch.ones(self.order) * value["max"]
# Number of internal knots
n_internal = value["n"] - 2 * self.order
# If no internal knots are needed, just concatenate boundaries
if n_internal <= 0:
value = torch.cat((initial_knots, final_knots))
# Else, sample internal knots uniformly and exclude boundaries
# Recover the correct number of internal knots when slicing by
# adding 2 to n_internal
else:
internal_knots = torch.linspace(
value["min"], value["max"], n_internal + 2
)[1:-1]
value = torch.cat(
(initial_knots, internal_knots, final_knots)
)
# Raise error if mode is invalid
else:
raise ValueError(
f"Invalid mode for knots initialization. Got "
f"{value['mode']}, but expected 'uniform' or 'auto'."
)
# Set knots
self.register_buffer("_knots", value.sort(dim=0).values)
# Recompute boundary interval when knots change
if hasattr(self, "_boundary_interval_idx"):
self._boundary_interval_idx = self._compute_boundary_interval()
