"""Module for the bivariate B-Spline surface model class."""
import torch
from .spline import Spline
from ..label_tensor import LabelTensor
from ..utils import check_consistency, check_positive_integer
[docs]
class SplineSurface(torch.nn.Module):
r"""
The bivariate B-Spline surface model class.
A bivariate B-spline surface is a parametric surface defined as the tensor
product of two univariate B-spline curves:
.. math::
S(x, y) = \sum_{i=1}^{n_x} \sum_{j=1}^{n_y} B_{i,k}(x) B_{j,s}(y)
C_{i,j}, \quad x \in [x_1, x_m], y \in [y_1, y_l]
where:
- :math:`C \in \mathbb{R}^{n_x \times n_y}` is the matrix of learnable
control coefficients. Its entries :math:`C_{i,j}` influence the shape of
the surface but are not generally interpolated, except under certain knot
multiplicities.
- :math:`B_{i,k}(x)` and :math:`B_{j,s}(y)` are the B-spline basis functions
defined over two orthogonal directions, with orders :math:`k` and
:math:`s`, respectively.
- :math:`X = \{ x_1, x_2, \dots, x_m \}` and
:math:`Y = \{ y_1, y_2, \dots, y_l \}` are the non-decreasing knot
vectors along the two directions.
"""
def __init__(self, orders, knots_u=None, knots_v=None, control_points=None):
"""
Initialization of the :class:`SplineSurface` class.
:param list[int] orders: The orders of the spline along each parametric
direction. Each order defines the degree of the corresponding basis
as ``degree = order - 1``.
:param knots_u: The knots of the spline along the first direction.
For details on valid formats and initialization modes, see the
:class:`Spline` class. Default is None.
:type knots_u: torch.Tensor | dict
:param knots_v: The knots of the spline along the second direction.
For details on valid formats and initialization modes, see the
:class:`Spline` class. Default is None.
:type knots_v: torch.Tensor | dict
:param torch.Tensor control_points: The control points defining the
surface geometry. It must be a two-dimensional tensor of shape
``[len(knots_u) - orders[0], len(knots_v) - orders[1]]``.
If None, they are initialized as learnable parameters with zero
values. Default is None.
:raises ValueError: If ``orders`` is not a list of integers.
:raises ValueError: If ``knots_u`` is neither a torch.Tensor nor a
dictionary, when provided.
:raises ValueError: If ``knots_v`` 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 ``orders`` is not a list of two elements.
:raises ValueError: If ``knots_u``, ``knots_v``, and ``control_points``
are all None.
"""
super().__init__()
# Check consistency
check_consistency(orders, int)
check_consistency(control_points, (type(None), torch.Tensor))
check_consistency(knots_u, (type(None), torch.Tensor, dict))
check_consistency(knots_v, (type(None), torch.Tensor, dict))
# Check orders is a list of two elements
if len(orders) != 2:
raise ValueError("orders must be a list of two elements.")
# Raise error if neither knots nor control points are provided
if (knots_u is None or knots_v is None) and control_points is None:
raise ValueError(
"control_points cannot be None if knots_u or knots_v is None."
)
# Initialize knots_u if not provided
if knots_u is None and control_points is not None:
knots_u = {
"n": control_points.shape[0] + orders[0],
"min": 0,
"max": 1,
"mode": "auto",
}
# Initialize knots_v if not provided
if knots_v is None and control_points is not None:
knots_v = {
"n": control_points.shape[1] + orders[1],
"min": 0,
"max": 1,
"mode": "auto",
}
# Create two univariate b-splines
self.spline_u = Spline(order=orders[0], knots=knots_u)
self.spline_v = Spline(order=orders[1], knots=knots_v)
self.control_points = control_points
# Delete unneeded parameters
delattr(self.spline_u, "_control_points")
delattr(self.spline_v, "_control_points")
[docs]
def forward(self, x):
"""
Forward pass for the :class:`SplineSurface` model.
:param x: The input tensor.
:type x: torch.Tensor | LabelTensor
:return: The output tensor.
:rtype: torch.Tensor
"""
return torch.einsum(
"...bi, ...bj, ij -> ...b",
self.spline_u.basis(x.as_subclass(torch.Tensor)[..., 0]),
self.spline_v.basis(x.as_subclass(torch.Tensor)[..., 1]),
self.control_points,
).unsqueeze(-1)
[docs]
def derivative(self, x, degree_u, degree_v):
"""
Compute the partial derivatives of the spline at the given points.
:param x: The input tensor.
:type x: torch.Tensor | LabelTensor
:param int degree_u: The degree of the derivative along the first
parameter direction.
:param int degree_v: The degree of the derivative along the second
parameter direction.
:raise ValueError: If ``degree_u`` is not an integer.
:raise ValueError: If ``degree_v`` is not an integer.
:return: The derivative tensor.
:rtype: torch.Tensor
"""
# Check consistency
check_positive_integer(degree_u, strict=False)
check_positive_integer(degree_v, strict=False)
# Split input into u and v components
if isinstance(x, LabelTensor):
u = x[x.labels[0]].as_subclass(torch.Tensor)
v = x[x.labels[1]].as_subclass(torch.Tensor)
else:
u = x[..., 0]
v = x[..., 1]
# Compute basis derivatives
der_u = self.spline_u._basis_derivative(u, degree=degree_u)
der_v = self.spline_v._basis_derivative(v, degree=degree_v)
return torch.einsum(
"...bi, ...bj, ij -> ...b", der_u, der_v, self.control_points
)
[docs]
def gradient(self, x):
"""
Convenience method to compute the gradient of the spline surface.
:param x: The input tensor.
:type x: torch.Tensor | LabelTensor
:return: The gradient tensor.
:rtype: torch.Tensor
"""
# Compute partial derivatives
du = self.derivative(x, degree_u=1, degree_v=0)
dv = self.derivative(x, degree_u=0, degree_v=1)
return torch.cat((du, dv), dim=-1)
[docs]
def laplacian(self, x):
"""
Convenience method to compute the laplacian of the spline surface.
:param x: The input tensor.
:type x: torch.Tensor | LabelTensor
:return: The laplacian tensor.
:rtype: torch.Tensor
"""
# Compute second partial derivatives
ddu = self.derivative(x, degree_u=2, degree_v=0)
ddv = self.derivative(x, degree_u=0, degree_v=2)
return ddu + ddv
@property
def knots(self):
"""
The knots of the univariate splines defining the spline surface.
:return: The knots.
:rtype: tuple(torch.Tensor, torch.Tensor)
"""
return self.spline_u.knots, self.spline_v.knots
@knots.setter
def knots(self, value):
"""
Set the knots of the spline surface.
:param value: A tuple (knots_u, knots_v) containing the knots for both
parametric directions.
:type value: tuple(torch.Tensor | dict, torch.Tensor | dict)
:raises ValueError: If value is not a tuple of two elements.
"""
# Check value is a tuple of two elements
if not (isinstance(value, tuple) and len(value) == 2):
raise ValueError("Knots must be a tuple of two elements.")
knots_u, knots_v = value
self.spline_u.knots = knots_u
self.spline_v.knots = knots_v
@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 surface.
:param torch.Tensor control_points: The bidimensional control points
tensor, where each dimension refers to a direction in the parameter
space. If None, control points are initialized to learnable
parameters with zero initial value. Default is None.
:raises ValueError: If in any direction there are not enough knots to
define the control points, due to the relation:
#knots = order + #control_points.
:raises ValueError: If ``control_points`` is not of the correct shape.
"""
# Save correct shape of control points
__valid_shape = (
len(self.spline_u.knots) - self.spline_u.order,
len(self.spline_v.knots) - self.spline_v.order,
)
# 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.spline_u.knots) < self.spline_u.order + 1
or len(self.spline_v.knots) < self.spline_v.order + 1
):
raise ValueError(
f"Not enough knots to define control points. Got "
f"{len(self.spline_u.knots)} knots along u and "
f"{len(self.spline_v.knots)} knots along v, but need at "
f"least {self.spline_u.order + 1} and "
f"{self.spline_v.order + 1}, respectively."
)
# Initialize control points to zero
control_points = torch.zeros(__valid_shape)
# Check control points
if control_points.shape != __valid_shape:
raise ValueError(
f"control_points must be of the correct shape. "
f"Expected {__valid_shape}, got {control_points.shape}."
)
# Register control points as a learnable parameter
self._control_points = torch.nn.Parameter(
control_points, requires_grad=True
)
