class ProcrustesMetric(RiemannianMetric):
"""Procrustes metric on the pre-shape space.
Parameters
----------
k_landmarks : int
Number of landmarks
m_ambient : int
Number of coordinates of each landmark.
"""
def __init__(self, k_landmarks, m_ambient):
super(ProcrustesMetric, self).__init__(
dim=m_ambient * (k_landmarks - 1) - 1,
default_point_type='matrix')
self.embedding_metric = MatricesMetric(k_landmarks, m_ambient)
self.sphere_metric = Hypersphere(m_ambient * k_landmarks - 1).metric
self.k_landmarks = k_landmarks
self.m_ambient = m_ambient
def inner_product(self, tangent_vec_a, tangent_vec_b, base_point=None):
"""Compute the inner-product of two tangent vectors at a base point.
Parameters
----------
tangent_vec_a : array-like, shape=[..., k_landmarks, m_ambient]
First tangent vector at base point.
tangent_vec_b : array-like, shape=[..., k_landmarks, m_ambient]
Second tangent vector at base point.
base_point : array-like, shape=[..., dk_landmarks, m_ambient]
Point on the pre-shape space.
Returns
-------
inner_prod : array-like, shape=[...,]
Inner-product of the two tangent vectors.
"""
inner_prod = self.embedding_metric.inner_product(
tangent_vec_a, tangent_vec_b, base_point)
return inner_prod
def exp(self, tangent_vec, base_point):
"""Compute the Riemannian exponential of a tangent vector.
Parameters
----------
tangent_vec : array-like, shape=[..., k_landmarks, m_ambient]
Tangent vector at a base point.
base_point : array-like, shape=[..., k_landmarks, m_ambient]
Point on the pre-shape space.
Returns
-------
exp : array-like, shape=[..., k_landmarks, m_ambient]
Point on the pre-shape space equal to the Riemannian exponential
of tangent_vec at the base point.
"""
flat_bp = gs.reshape(base_point, (-1, self.sphere_metric.dim + 1))
flat_tan = gs.reshape(tangent_vec, (-1, self.sphere_metric.dim + 1))
flat_exp = self.sphere_metric.exp(flat_tan, flat_bp)
return gs.reshape(flat_exp, tangent_vec.shape)
def log(self, point, base_point):
"""Compute the Riemannian logarithm of a point.
Parameters
----------
point : array-like, shape=[..., k_landmarks, m_ambient]
Point on the pre-shape space.
base_point : array-like, shape=[..., k_landmarks, m_ambient]
Point on the pre-shape space.
Returns
-------
log : array-like, shape=[..., k_landmarks, m_ambient]
Tangent vector at the base point equal to the Riemannian logarithm
of point at the base point.
"""
flat_bp = gs.reshape(base_point, (-1, self.sphere_metric.dim + 1))
flat_pt = gs.reshape(point, (-1, self.sphere_metric.dim + 1))
flat_log = self.sphere_metric.log(flat_pt, flat_bp)
try:
log = gs.reshape(flat_log, base_point.shape)
except (RuntimeError,
check_tf_error(ValueError, 'InvalidArgumentError')):
log = gs.reshape(flat_log, point.shape)
return log
class PreShapeMetric(RiemannianMetric):
"""Procrustes metric on the pre-shape space.
Parameters
----------
k_landmarks : int
Number of landmarks
m_ambient : int
Number of coordinates of each landmark.
"""
def __init__(self, k_landmarks, m_ambient):
super(PreShapeMetric,
self).__init__(dim=m_ambient * (k_landmarks - 1) - 1,
default_point_type="matrix")
self.embedding_metric = MatricesMetric(k_landmarks, m_ambient)
self.sphere_metric = Hypersphere(m_ambient * k_landmarks - 1).metric
self.k_landmarks = k_landmarks
self.m_ambient = m_ambient
def inner_product(self, tangent_vec_a, tangent_vec_b, base_point=None):
"""Compute the inner-product of two tangent vectors at a base point.
Parameters
----------
tangent_vec_a : array-like, shape=[..., k_landmarks, m_ambient]
First tangent vector at base point.
tangent_vec_b : array-like, shape=[..., k_landmarks, m_ambient]
Second tangent vector at base point.
base_point : array-like, shape=[..., dk_landmarks, m_ambient]
Point on the pre-shape space.
Returns
-------
inner_prod : array-like, shape=[...,]
Inner-product of the two tangent vectors.
"""
inner_prod = self.embedding_metric.inner_product(
tangent_vec_a, tangent_vec_b, base_point)
return inner_prod
def exp(self, tangent_vec, base_point, **kwargs):
"""Compute the Riemannian exponential of a tangent vector.
Parameters
----------
tangent_vec : array-like, shape=[..., k_landmarks, m_ambient]
Tangent vector at a base point.
base_point : array-like, shape=[..., k_landmarks, m_ambient]
Point on the pre-shape space.
Returns
-------
exp : array-like, shape=[..., k_landmarks, m_ambient]
Point on the pre-shape space equal to the Riemannian exponential
of tangent_vec at the base point.
"""
flat_bp = gs.reshape(base_point, (-1, self.sphere_metric.dim + 1))
flat_tan = gs.reshape(tangent_vec, (-1, self.sphere_metric.dim + 1))
flat_exp = self.sphere_metric.exp(flat_tan, flat_bp)
return gs.reshape(flat_exp, tangent_vec.shape)
def log(self, point, base_point, **kwargs):
"""Compute the Riemannian logarithm of a point.
Parameters
----------
point : array-like, shape=[..., k_landmarks, m_ambient]
Point on the pre-shape space.
base_point : array-like, shape=[..., k_landmarks, m_ambient]
Point on the pre-shape space.
Returns
-------
log : array-like, shape=[..., k_landmarks, m_ambient]
Tangent vector at the base point equal to the Riemannian logarithm
of point at the base point.
"""
flat_bp = gs.reshape(base_point, (-1, self.sphere_metric.dim + 1))
flat_pt = gs.reshape(point, (-1, self.sphere_metric.dim + 1))
flat_log = self.sphere_metric.log(flat_pt, flat_bp)
try:
log = gs.reshape(flat_log, base_point.shape)
except (RuntimeError, check_tf_error(ValueError,
"InvalidArgumentError")):
log = gs.reshape(flat_log, point.shape)
return log
def curvature(self, tangent_vec_a, tangent_vec_b, tangent_vec_c,
base_point):
r"""Compute the curvature.
For three tangent vectors at a base point :math:`x,y,z`,
the curvature is defined by
:math:`R(X, Y)Z = \nabla_{[X,Y]}Z
- \nabla_X\nabla_Y Z + - \nabla_Y\nabla_X Z`, where :math:`\nabla`
is the Levi-Civita connection. In the case of the hypersphere,
we have the closed formula
:math:`R(X,Y)Z = \langle X, Z \rangle Y - \langle Y,Z \rangle X`.
Parameters
----------
tangent_vec_a : array-like, shape=[..., k_landmarks, m_ambient]
Tangent vector at `base_point`.
tangent_vec_b : array-like, shape=[..., k_landmarks, m_ambient]
Tangent vector at `base_point`.
tangent_vec_c : array-like, shape=[..., k_landmarks, m_ambient]
Tangent vector at `base_point`.
base_point : array-like, shape=[..., k_landmarks, m_ambient]
Point on the group. Optional, default is the identity.
Returns
-------
curvature : array-like, shape=[..., k_landmarks, m_ambient]
Tangent vector at `base_point`.
"""
max_shape = base_point.shape
for arg in [tangent_vec_a, tangent_vec_b, tangent_vec_c]:
if arg.ndim >= 3:
max_shape = arg.shape
flat_shape = (-1, self.sphere_metric.dim + 1)
flat_a = gs.reshape(tangent_vec_a, flat_shape)
flat_b = gs.reshape(tangent_vec_b, flat_shape)
flat_c = gs.reshape(tangent_vec_c, flat_shape)
flat_bp = gs.reshape(base_point, flat_shape)
curvature = self.sphere_metric.curvature(flat_a, flat_b, flat_c,
flat_bp)
curvature = gs.reshape(curvature, max_shape)
return curvature
def curvature_derivative(
self,
tangent_vec_a,
tangent_vec_b=None,
tangent_vec_c=None,
tangent_vec_d=None,
base_point=None,
):
r"""Compute the covariant derivative of the curvature.
For four vectors fields :math:`H|_P = tangent\_vec\_a, X|_P =
tangent\_vec\_b, Y|_P = tangent\_vec\_c, Z|_P = tangent\_vec\_d` with
tangent vector value specified in argument at the base point `P`,
the covariant derivative of the curvature
:math:`(\nabla_H R)(X, Y) Z |_P` is computed at the base point P.
Since the sphere is a constant curvature space this
vanishes identically.
Parameters
----------
tangent_vec_a : array-like, shape=[..., k_landmarks, m_ambient]
Tangent vector at `base_point` along which the curvature is
derived.
tangent_vec_b : array-like, shape=[..., k_landmarks, m_ambient]
Unused tangent vector at `base_point` (since curvature derivative
vanishes).
tangent_vec_c : array-like, shape=[..., k_landmarks, m_ambient]
Unused tangent vector at `base_point` (since curvature derivative
vanishes).
tangent_vec_d : array-like, shape=[..., k_landmarks, m_ambient]
Unused tangent vector at `base_point` (since curvature derivative
vanishes).
base_point : array-like, shape=[..., k_landmarks, m_ambient]
Unused point on the group.
Returns
-------
curvature_derivative : array-like, shape=[..., k_landmarks, m_ambient]
Tangent vector at base point.
"""
return gs.zeros_like(tangent_vec_a)
def parallel_transport(self,
tangent_vec,
base_point,
direction=None,
end_point=None):
"""Compute the Riemannian parallel transport of a tangent vector.
Parameters
----------
tangent_vec : array-like, shape=[..., k_landmarks, m_ambient]
Tangent vector at a base point.
base_point : array-like, shape=[..., k_landmarks, m_ambient]
Point on the pre-shape space.
direction : array-like, shape=[..., k_landmarks, m_ambient]
Tangent vector at a base point.
Optional, default : None.
end_point : array-like, shape=[..., k_landmarks, m_ambient]
Point on the pre-shape space, to transport to. Unused if `tangent_vec_b`
is given.
Optional, default : None.
Returns
-------
transported : array-like, shape=[..., k_landmarks, m_ambient]
Point on the pre-shape space equal to the Riemannian exponential
of tangent_vec at the base point.
"""
if direction is None:
if end_point is not None:
direction = self.log(end_point, base_point)
else:
raise ValueError(
"Either an end_point or a tangent_vec_b must be given to define the"
" geodesic along which to transport.")
max_shape = tangent_vec.shape if tangent_vec.ndim == 3 else direction.shape
flat_bp = gs.reshape(base_point, (-1, self.sphere_metric.dim + 1))
flat_tan_a = gs.reshape(tangent_vec, (-1, self.sphere_metric.dim + 1))
flat_tan_b = gs.reshape(direction, (-1, self.sphere_metric.dim + 1))
flat_transport = self.sphere_metric.parallel_transport(
flat_tan_a, flat_bp, flat_tan_b)
return gs.reshape(flat_transport, max_shape)
def injectivity_radius(self, base_point):
"""Compute the radius of the injectivity domain.
This is is the supremum of radii r for which the exponential map is a
diffeomorphism from the open ball of radius r centered at the base
point onto its image.
In the case of the sphere, it does not depend on the base point and is
Pi everywhere.
Parameters
----------
base_point : array-like, shape=[..., k_landmarks, m_ambient]
Point on the manifold.
Returns
-------
radius : float
Injectivity radius.
"""
return gs.pi
class PreShapeMetric(RiemannianMetric):
"""Procrustes metric on the pre-shape space.
Parameters
----------
k_landmarks : int
Number of landmarks
m_ambient : int
Number of coordinates of each landmark.
"""
def __init__(self, k_landmarks, m_ambient):
super(PreShapeMetric,
self).__init__(dim=m_ambient * (k_landmarks - 1) - 1,
default_point_type='matrix')
self.embedding_metric = MatricesMetric(k_landmarks, m_ambient)
self.sphere_metric = Hypersphere(m_ambient * k_landmarks - 1).metric
self.k_landmarks = k_landmarks
self.m_ambient = m_ambient
def inner_product(self, tangent_vec_a, tangent_vec_b, base_point=None):
"""Compute the inner-product of two tangent vectors at a base point.
Parameters
----------
tangent_vec_a : array-like, shape=[..., k_landmarks, m_ambient]
First tangent vector at base point.
tangent_vec_b : array-like, shape=[..., k_landmarks, m_ambient]
Second tangent vector at base point.
base_point : array-like, shape=[..., dk_landmarks, m_ambient]
Point on the pre-shape space.
Returns
-------
inner_prod : array-like, shape=[...,]
Inner-product of the two tangent vectors.
"""
inner_prod = self.embedding_metric.inner_product(
tangent_vec_a, tangent_vec_b, base_point)
return inner_prod
def exp(self, tangent_vec, base_point):
"""Compute the Riemannian exponential of a tangent vector.
Parameters
----------
tangent_vec : array-like, shape=[..., k_landmarks, m_ambient]
Tangent vector at a base point.
base_point : array-like, shape=[..., k_landmarks, m_ambient]
Point on the pre-shape space.
Returns
-------
exp : array-like, shape=[..., k_landmarks, m_ambient]
Point on the pre-shape space equal to the Riemannian exponential
of tangent_vec at the base point.
"""
flat_bp = gs.reshape(base_point, (-1, self.sphere_metric.dim + 1))
flat_tan = gs.reshape(tangent_vec, (-1, self.sphere_metric.dim + 1))
flat_exp = self.sphere_metric.exp(flat_tan, flat_bp)
return gs.reshape(flat_exp, tangent_vec.shape)
def log(self, point, base_point):
"""Compute the Riemannian logarithm of a point.
Parameters
----------
point : array-like, shape=[..., k_landmarks, m_ambient]
Point on the pre-shape space.
base_point : array-like, shape=[..., k_landmarks, m_ambient]
Point on the pre-shape space.
Returns
-------
log : array-like, shape=[..., k_landmarks, m_ambient]
Tangent vector at the base point equal to the Riemannian logarithm
of point at the base point.
"""
flat_bp = gs.reshape(base_point, (-1, self.sphere_metric.dim + 1))
flat_pt = gs.reshape(point, (-1, self.sphere_metric.dim + 1))
flat_log = self.sphere_metric.log(flat_pt, flat_bp)
try:
log = gs.reshape(flat_log, base_point.shape)
except (RuntimeError, check_tf_error(ValueError,
'InvalidArgumentError')):
log = gs.reshape(flat_log, point.shape)
return log
def curvature(self, tangent_vec_a, tangent_vec_b, tangent_vec_c,
base_point):
r"""Compute the curvature.
For three tangent vectors at a base point :math: `x,y,z`,
the curvature is defined by
:math: `R(X, Y)Z = \nabla_{[X,Y]}Z
- \nabla_X\nabla_Y Z + - \nabla_Y\nabla_X Z`, where :math: `\nabla`
is the Levi-Civita connection. In the case of the hypersphere,
we have the closed formula
:math: `R(X,Y)Z = \langle X, Z \rangle Y - \langle Y,Z \rangle X`.
Parameters
----------
tangent_vec_a : array-like, shape=[..., n, n]
Tangent vector at `base_point`.
tangent_vec_b : array-like, shape=[..., n, n]
Tangent vector at `base_point`.
tangent_vec_c : array-like, shape=[..., n, n]
Tangent vector at `base_point`.
base_point : array-like, shape=[..., n, n]
Point on the group. Optional, default is the identity.
Returns
-------
curvature : array-like, shape=[..., n, n]
Tangent vector at `base_point`.
"""
max_shape = base_point.shape
for arg in [tangent_vec_a, tangent_vec_b, tangent_vec_c]:
if arg.ndim >= 3:
max_shape = arg.shape
flat_shape = (-1, self.sphere_metric.dim + 1)
flat_a = gs.reshape(tangent_vec_a, flat_shape)
flat_b = gs.reshape(tangent_vec_b, flat_shape)
flat_c = gs.reshape(tangent_vec_c, flat_shape)
flat_bp = gs.reshape(base_point, flat_shape)
curvature = self.sphere_metric.curvature(flat_a, flat_b, flat_c,
flat_bp)
curvature = gs.reshape(curvature, max_shape)
return curvature
