def test_rotation_vector_rotation_matrix_regularize(self, n, point):
group = SpecialOrthogonal(n=n)
rot_mat = group.matrix_from_rotation_vector(gs.array(point))
self.assertAllClose(
group.regularize(gs.array(point)),
group.rotation_vector_from_matrix(rot_mat),
)
def test_squared_dist_is_less_than_squared_pi(self, point_1, point_2):
"""
This test only concerns the canonical metric.
For other metrics, the scaling factor can give
distances above pi.
"""
group = SpecialOrthogonal(3, "vector")
metric = self.metric(SpecialOrthogonal(3, "vector"))
point_1 = group.regularize(point_1)
point_2 = group.regularize(point_2)
sq_dist = metric.squared_dist(point_1, point_2)
diff = sq_dist - gs.pi**2
self.assertTrue(diff <= 0 or abs(diff) < EPSILON,
"sq_dist = {}".format(sq_dist))
def test_squared_dist_is_symmetric(
self, metric_mat_at_identity, left_or_right, point_1, point_2
):
group = SpecialOrthogonal(3, "vector")
metric = self.metric(
SpecialOrthogonal(n=3, point_type="vector"),
metric_mat_at_identity=metric_mat_at_identity,
left_or_right=left_or_right,
)
point_1 = group.regularize(point_1)
point_2 = group.regularize(point_2)
sq_dist_1_2 = gs.mod(metric.squared_dist(point_1, point_2) + 1e-4, gs.pi**2)
sq_dist_2_1 = gs.mod(metric.squared_dist(point_2, point_1) + 1e-4, gs.pi**2)
self.assertAllClose(sq_dist_1_2, sq_dist_2_1, atol=1e-4)
class _SpecialEuclideanVectors(LieGroup):
"""Base Class for the special Euclidean groups in 2d and 3d in vector form.
i.e. the Lie group of rigid transformations. Elements of SE(2), SE(3) can
either be represented as vectors (in 2d or 3d) or as matrices in general.
The matrix representation corresponds to homogeneous coordinates. This
class is specific to the vector representation of rotations. For the matrix
representation use the SpecialEuclidean class and set `n=2` or `n=3`.
Parameter
---------
epsilon : float
Precision to use for calculations involving potential
division by 0 in rotations.
Optional, default: 0.
"""
def __init__(self, n, epsilon=0.0):
dim = n * (n + 1) // 2
LieGroup.__init__(
self,
dim=dim,
shape=(dim, ),
default_point_type="vector",
lie_algebra=Euclidean(dim),
)
self.n = n
self.epsilon = epsilon
self.rotations = SpecialOrthogonal(n=n,
point_type="vector",
epsilon=epsilon)
self.translations = Euclidean(dim=n)
def get_identity(self, point_type=None):
"""Get the identity of the group.
Parameters
----------
point_type : str, {'vector', 'matrix'}
The point_type of the returned value.
Optional, default: self.default_point_type
Returns
-------
identity : array-like, shape={[dim], [n + 1, n + 1]}
"""
if point_type is None:
point_type = self.default_point_type
identity = gs.zeros(self.dim)
return identity
identity = property(get_identity)
def get_point_type_shape(self, point_type=None):
"""Get the shape of the instance given the default_point_style."""
return self.get_identity(point_type).shape
def belongs(self, point, atol=gs.atol):
"""Evaluate if a point belongs to SE(2) or SE(3).
Parameters
----------
point : array-like, shape=[..., dim]
Point to check.
Returns
-------
belongs : array-like, shape=[...,]
Boolean indicating whether point belongs to SE(2) or SE(3).
"""
point_dim = point.shape[-1]
point_ndim = point.ndim
belongs = gs.logical_and(point_dim == self.dim, point_ndim < 3)
belongs = gs.logical_and(
belongs,
self.rotations.belongs(point[..., :self.rotations.dim], atol=atol))
return belongs
def projection(self, point):
"""Project a point to the group.
The point is regularized, so that the norm of the rotation part lie in [0, pi).
Parameters
----------
point: array-like, shape[..., dim]
Point.
Returns
-------
projected: array-like, shape[..., dim]
Regularized point.
"""
return self.regularize(point)
def regularize(self, point):
"""Regularize a point to the default representation for SE(n).
Parameters
----------
point : array-like, shape=[..., dim]
Point to regularize.
Returns
-------
point : array-like, shape=[..., dim]
Regularized point.
"""
rotations = self.rotations
dim_rotations = rotations.dim
regularized_point = gs.copy(point)
rot_vec = regularized_point[..., :dim_rotations]
regularized_rot_vec = rotations.regularize(rot_vec)
translation = regularized_point[..., dim_rotations:]
return gs.concatenate([regularized_rot_vec, translation], axis=-1)
@geomstats.vectorization.decorator(["else", "vector", "else"])
def regularize_tangent_vec_at_identity(self, tangent_vec, metric=None):
"""Regularize a tangent vector at the identity.
Parameters
----------
tangent_vec: array-like, shape=[..., dim]
Tangent vector at base point.
metric : RiemannianMetric
Metric.
Optional, default: None.
Returns
-------
regularized_vec : array-like, shape=[..., dim]
Regularized vector.
"""
return self.regularize_tangent_vec(tangent_vec, self.identity, metric)
@geomstats.vectorization.decorator(["else", "vector"])
def matrix_from_vector(self, vec):
"""Convert point in vector point-type to matrix.
Parameters
----------
vec : array-like, shape=[..., dim]
Vector.
Returns
-------
mat : array-like, shape=[..., n+1, n+1]
Matrix.
"""
vec = self.regularize(vec)
output_shape = ((vec.shape[0], self.n + 1,
self.n + 1) if vec.ndim == 2 else (self.n + 1, ) * 2)
rot_vec = vec[..., :self.rotations.dim]
trans_vec = vec[..., self.rotations.dim:]
rot_mat = self.rotations.matrix_from_rotation_vector(rot_vec)
return homogeneous_representation(rot_mat, trans_vec, output_shape)
@geomstats.vectorization.decorator(["else", "vector", "vector"])
def compose(self, point_a, point_b):
r"""Compose two elements of SE(2) or SE(3).
Parameters
----------
point_a : array-like, shape=[..., dim]
Point of the group.
point_b : array-like, shape=[..., dim]
Point of the group.
Equation
--------
(:math: `(R_1, t_1) \\cdot (R_2, t_2) = (R_1 R_2, R_1 t_2 + t_1)`)
Returns
-------
composition : array-like, shape=[..., dim]
Composition of point_a and point_b.
"""
rotations = self.rotations
dim_rotations = rotations.dim
point_a = self.regularize(point_a)
point_b = self.regularize(point_b)
rot_vec_a = point_a[..., :dim_rotations]
rot_mat_a = rotations.matrix_from_rotation_vector(rot_vec_a)
rot_vec_b = point_b[..., :dim_rotations]
rot_mat_b = rotations.matrix_from_rotation_vector(rot_vec_b)
translation_a = point_a[..., dim_rotations:]
translation_b = point_b[..., dim_rotations:]
composition_rot_mat = gs.matmul(rot_mat_a, rot_mat_b)
composition_rot_vec = rotations.rotation_vector_from_matrix(
composition_rot_mat)
composition_translation = (
gs.einsum("...j,...kj->...k", translation_b, rot_mat_a) +
translation_a)
composition = gs.concatenate(
(composition_rot_vec, composition_translation), axis=-1)
return self.regularize(composition)
@geomstats.vectorization.decorator(["else", "vector"])
def inverse(self, point):
r"""Compute the group inverse in SE(n).
Parameters
----------
point: array-like, shape=[..., dim]
Point.
Returns
-------
inverse_point : array-like, shape=[..., dim]
Inverted point.
Notes
-----
:math:`(R, t)^{-1} = (R^{-1}, R^{-1}.(-t))`
"""
rotations = self.rotations
dim_rotations = rotations.dim
point = self.regularize(point)
rot_vec = point[:, :dim_rotations]
translation = point[:, dim_rotations:]
inverse_rotation = -rot_vec
inv_rot_mat = rotations.matrix_from_rotation_vector(inverse_rotation)
inverse_translation = gs.einsum(
"ni,nij->nj", -translation,
gs.transpose(inv_rot_mat, axes=(0, 2, 1)))
inverse_point = gs.concatenate([inverse_rotation, inverse_translation],
axis=-1)
return self.regularize(inverse_point)
@geomstats.vectorization.decorator(["else", "vector"])
def exp_from_identity(self, tangent_vec):
"""Compute group exponential of the tangent vector at the identity.
Parameters
----------
tangent_vec: array-like, shape=[..., 3]
Tangent vector at base point.
Returns
-------
group_exp: array-like, shape=[..., 3]
Group exponential of the tangent vectors computed
at the identity.
"""
rotations = self.rotations
dim_rotations = rotations.dim
rot_vec = tangent_vec[..., :dim_rotations]
rot_vec_regul = self.rotations.regularize(rot_vec)
rot_vec_regul = gs.to_ndarray(rot_vec_regul, to_ndim=2, axis=1)
transform = self._exp_translation_transform(rot_vec_regul)
translation = tangent_vec[..., dim_rotations:]
exp_translation = gs.einsum("ijk, ik -> ij", transform, translation)
group_exp = gs.concatenate([rot_vec, exp_translation], axis=1)
group_exp = self.regularize(group_exp)
return group_exp
@geomstats.vectorization.decorator(["else", "vector"])
def log_from_identity(self, point):
"""Compute the group logarithm of the point at the identity.
Parameters
----------
point: array-like, shape=[..., 3]
Point.
Returns
-------
group_log: array-like, shape=[..., 3]
Group logarithm in the Lie algebra.
"""
point = self.regularize(point)
rotations = self.rotations
dim_rotations = rotations.dim
rot_vec = point[:, :dim_rotations]
translation = point[:, dim_rotations:]
transform = self._log_translation_transform(rot_vec)
log_translation = gs.einsum("ijk, ik -> ij", transform, translation)
return gs.concatenate([rot_vec, log_translation], axis=1)
def random_point(self, n_samples=1, bound=1.0, **kwargs):
r"""Sample in SE(n) with the uniform distribution.
Parameters
----------
n_samples : int
Number of samples.
Optional, default: 1.
bound : float
Upper bound for the translation part of the sample.
Optional, default: 1.
Returns
-------
random_point : array-like, shape=[..., dim]
Sample.
"""
random_translation = self.translations.random_point(n_samples, bound)
random_rot_vec = self.rotations.random_uniform(n_samples)
return gs.concatenate([random_rot_vec, random_translation], axis=-1)
def test_regularize(self, point, expected):
group = SpecialOrthogonal(3, "vector")
result = group.regularize(point)
self.assertAllClose(result, expected)
def test_regularize(self, n, point_type, angle, expected):
group = SpecialOrthogonal(n, point_type)
result = group.regularize(angle)
self.assertAllClose(result, expected)
class _SpecialEuclidean3Vectors(LieGroup):
"""Class for the special euclidean group in 3d, SE(3).
i.e. the Lie group of rigid transformations. Elements of SE(3) can either
be represented as vectors (in 3d) or as matrices in general. The matrix
representation corresponds to homogeneous coordinates.This class is
specific to the vector representation of rotations. For the matrix
representation use the SpecialEuclidean class and set `n=3`.
Parameter
---------
epsilon : float, optional (defaults to 0)
Precision to use for calculations involving potential
division by 0 in rotations.
"""
def __init__(self, epsilon=0.):
super(_SpecialEuclidean3Vectors,
self).__init__(dim=6, default_point_type='vector')
self.n = 3
self.epsilon = epsilon
self.rotations = SpecialOrthogonal(n=3,
point_type='vector',
epsilon=epsilon)
self.translations = Euclidean(dim=3)
def get_identity(self, point_type=None):
"""Get the identity of the group.
Parameters
----------
point_type : str, {'vector', 'matrix'}, optional
The point_type of the returned value.
default: self.default_point_type
Returns
-------
identity : array-like, shape={[dim], [n + 1, n + 1]}
"""
if point_type is None:
point_type = self.default_point_type
identity = gs.zeros(self.dim)
if point_type == 'matrix':
identity = gs.eye(self.n + 1)
return identity
identity = property(get_identity)
def get_point_type_shape(self, point_type=None):
"""Get the shape of the instance given the default_point_style."""
return self.get_identity(point_type).shape
def belongs(self, point):
"""Evaluate if a point belongs to SE(3).
Parameters
----------
point : array-like, shape=[..., 3]
The point of which to check whether it belongs to SE(3).
Returns
-------
belongs : array-like, shape=[..., 1]
Boolean indicating whether point belongs to SE(3).
"""
point_dim = point.shape[-1]
point_ndim = point.ndim
belongs = gs.logical_and(point_dim == self.dim, point_ndim < 3)
belongs = gs.logical_and(belongs,
self.rotations.belongs(point[..., :self.n]))
return belongs
def regularize(self, point):
"""Regularize a point to the default representation for SE(n).
Parameters
----------
point : array-like, shape=[..., 3]
The point to regularize.
Returns
-------
point : array-like, shape=[..., 3]
"""
rotations = self.rotations
dim_rotations = rotations.dim
rot_vec = point[..., :dim_rotations]
regularized_rot_vec = rotations.regularize(rot_vec)
translation = point[..., dim_rotations:]
return gs.concatenate([regularized_rot_vec, translation], axis=-1)
@geomstats.vectorization.decorator(['else', 'vector', 'else'])
def regularize_tangent_vec_at_identity(self, tangent_vec, metric=None):
"""Regularize a tangent vector at the identity.
Parameters
----------
tangent_vec: array-like, shape=[..., 3]
metric : RiemannianMetric, optional
Returns
-------
regularized_vec : the regularized tangent vector
"""
return self.regularize_tangent_vec(tangent_vec, self.identity, metric)
def regularize_tangent_vec(self, tangent_vec, base_point, metric=None):
"""Regularize a tangent vector at a base point.
Parameters
----------
tangent_vec: array-like, shape=[..., 3]
base_point : array-like, shape=[..., 3]
metric : RiemannianMetric, optional
default: self.left_canonical_metric
Returns
-------
regularized_vec : the regularized tangent vector
"""
if metric is None:
metric = self.left_canonical_metric
rotations = self.rotations
dim_rotations = rotations.dim
rot_tangent_vec = tangent_vec[..., :dim_rotations]
rot_base_point = base_point[..., :dim_rotations]
metric_mat = metric.inner_product_mat_at_identity
rot_metric_mat = metric_mat[:dim_rotations, :dim_rotations]
rot_metric = InvariantMetric(
group=rotations,
inner_product_mat_at_identity=rot_metric_mat,
left_or_right=metric.left_or_right)
rotations_vec = rotations.regularize_tangent_vec(
tangent_vec=rot_tangent_vec,
base_point=rot_base_point,
metric=rot_metric)
return gs.concatenate(
[rotations_vec, tangent_vec[..., dim_rotations:]], axis=-1)
@geomstats.vectorization.decorator(['else', 'vector'])
def matrix_from_vector(self, vec):
"""Convert point in vector point-type to matrix.
Parameters
----------
vec: array-like, shape=[..., 3]
Returns
-------
mat: array-like, shape=[..., n+1, n+1]
"""
vec = self.regularize(vec)
n_vecs, _ = vec.shape
rot_vec = vec[:, :self.rotations.dim]
trans_vec = vec[:, self.rotations.dim:]
rot_mat = self.rotations.matrix_from_rotation_vector(rot_vec)
trans_vec = gs.reshape(trans_vec, (n_vecs, self.n, 1))
mat = gs.concatenate((rot_mat, trans_vec), axis=2)
last_lines = gs.array(gs.get_mask_i_float(self.n, self.n + 1))
last_lines = gs.to_ndarray(last_lines, to_ndim=2)
last_lines = gs.to_ndarray(last_lines, to_ndim=3)
mat = gs.concatenate((mat, last_lines), axis=1)
return mat
@geomstats.vectorization.decorator(['else', 'vector', 'vector'])
def compose(self, point_a, point_b):
r"""Compose two elements of SE(3).
Parameters
----------
point_a : array-like, shape=[..., 3]
Point of the group.
point_b : array-like, shape=[..., 3]
Point of the group.
Equation
---------
(:math: `(R_1, t_1) \\cdot (R_2, t_2) = (R_1 R_2, R_1 t_2 + t_1)`)
Returns
-------
composition :
The composition of point_a and point_b.
"""
rotations = self.rotations
dim_rotations = rotations.dim
point_a = self.regularize(point_a)
point_b = self.regularize(point_b)
rot_vec_a = point_a[..., :dim_rotations]
rot_mat_a = rotations.matrix_from_rotation_vector(rot_vec_a)
rot_vec_b = point_b[..., :dim_rotations]
rot_mat_b = rotations.matrix_from_rotation_vector(rot_vec_b)
translation_a = point_a[..., dim_rotations:]
translation_b = point_b[..., dim_rotations:]
composition_rot_mat = gs.matmul(rot_mat_a, rot_mat_b)
composition_rot_vec = rotations.rotation_vector_from_matrix(
composition_rot_mat)
composition_translation = gs.einsum('...j,...kj->...k', translation_b,
rot_mat_a) + translation_a
composition = gs.concatenate(
(composition_rot_vec, composition_translation), axis=-1)
return self.regularize(composition)
@geomstats.vectorization.decorator(['else', 'vector'])
def inverse(self, point):
r"""Compute the group inverse in SE(n).
Parameters
----------
point: array-like, shape=[..., 3]
Returns
-------
inverse_point : array-like, shape=[..., 3]
The inverted point.
Notes
-----
:math:`(R, t)^{-1} = (R^{-1}, R^{-1}.(-t))`
"""
rotations = self.rotations
dim_rotations = rotations.dim
point = self.regularize(point)
rot_vec = point[:, :dim_rotations]
translation = point[:, dim_rotations:]
inverse_rotation = -rot_vec
inv_rot_mat = rotations.matrix_from_rotation_vector(inverse_rotation)
inverse_translation = gs.einsum(
'ni,nij->nj', -translation,
gs.transpose(inv_rot_mat, axes=(0, 2, 1)))
inverse_point = gs.concatenate([inverse_rotation, inverse_translation],
axis=-1)
return self.regularize(inverse_point)
@geomstats.vectorization.decorator(['else', 'vector', 'else'])
def jacobian_translation(self, point, left_or_right='left'):
"""Compute the Jacobian matrix resulting from translation.
Compute the matrix of the differential of the left/right translations
from the identity to point in SE(3).
Parameters
----------
point: array-like, shape=[..., 3]
left_or_right: str, {'left', 'right'}, optional
Whether to compute the jacobian of the left or right translation.
Returns
-------
jacobian : array-like, shape=[..., 3]
The jacobian of the left / right translation.
"""
if left_or_right not in ('left', 'right'):
raise ValueError('`left_or_right` must be `left` or `right`.')
rotations = self.rotations
translations = self.translations
dim_rotations = rotations.dim
dim_translations = translations.dim
point = self.regularize(point)
n_points, _ = point.shape
rot_vec = point[:, :dim_rotations]
jacobian_rot = self.rotations.jacobian_translation(
point=rot_vec, left_or_right=left_or_right)
block_zeros_1 = gs.zeros((n_points, dim_rotations, dim_translations))
jacobian_block_line_1 = gs.concatenate([jacobian_rot, block_zeros_1],
axis=2)
if left_or_right == 'left':
rot_mat = self.rotations.matrix_from_rotation_vector(rot_vec)
jacobian_trans = rot_mat
block_zeros_2 = gs.zeros(
(n_points, dim_translations, dim_rotations))
jacobian_block_line_2 = gs.concatenate(
[block_zeros_2, jacobian_trans], axis=2)
else:
inv_skew_mat = -self.rotations.skew_matrix_from_vector(rot_vec)
eye = gs.to_ndarray(gs.eye(self.n), to_ndim=3)
eye = gs.tile(eye, [n_points, 1, 1])
jacobian_block_line_2 = gs.concatenate([inv_skew_mat, eye], axis=2)
return gs.concatenate([jacobian_block_line_1, jacobian_block_line_2],
axis=1)
@geomstats.vectorization.decorator(['else', 'vector'])
def exp_from_identity(self, tangent_vec):
"""Compute group exponential of the tangent vector at the identity.
Parameters
----------
tangent_vec: array-like, shape=[..., 3]
Returns
-------
group_exp: array-like, shape=[..., 3]
The group exponential of the tangent vectors calculated
at the identity.
"""
rotations = self.rotations
dim_rotations = rotations.dim
rot_vec = tangent_vec[..., :dim_rotations]
rot_vec = self.rotations.regularize(rot_vec)
translation = tangent_vec[..., dim_rotations:]
angle = gs.linalg.norm(rot_vec, axis=-1)
angle = gs.to_ndarray(angle, to_ndim=2, axis=1)
skew_mat = self.rotations.skew_matrix_from_vector(rot_vec)
sq_skew_mat = gs.matmul(skew_mat, skew_mat)
mask_0 = gs.equal(angle, 0.)
mask_close_0 = gs.isclose(angle, 0.) & ~mask_0
mask_else = ~mask_0 & ~mask_close_0
mask_0_float = gs.cast(mask_0, gs.float32)
mask_close_0_float = gs.cast(mask_close_0, gs.float32)
mask_else_float = gs.cast(mask_else, gs.float32)
angle += mask_0_float * gs.ones_like(angle)
coef_1 = gs.zeros_like(angle)
coef_2 = gs.zeros_like(angle)
coef_1 += mask_0_float * 1. / 2. * gs.ones_like(angle)
coef_2 += mask_0_float * 1. / 6. * gs.ones_like(angle)
coef_1 += mask_close_0_float * (TAYLOR_COEFFS_1_AT_0[0] +
TAYLOR_COEFFS_1_AT_0[2] * angle**2 +
TAYLOR_COEFFS_1_AT_0[4] * angle**4 +
TAYLOR_COEFFS_1_AT_0[6] * angle**6)
coef_2 += mask_close_0_float * (TAYLOR_COEFFS_2_AT_0[0] +
TAYLOR_COEFFS_2_AT_0[2] * angle**2 +
TAYLOR_COEFFS_2_AT_0[4] * angle**4 +
TAYLOR_COEFFS_2_AT_0[6] * angle**6)
coef_1 += mask_else_float * ((1. - gs.cos(angle)) / angle**2)
coef_2 += mask_else_float * ((angle - gs.sin(angle)) / angle**3)
n_tangent_vecs, _ = tangent_vec.shape
exp_translation = gs.zeros((n_tangent_vecs, self.n))
for i in range(n_tangent_vecs):
translation_i = translation[i]
term_1_i = coef_1[i] * gs.dot(translation_i,
gs.transpose(skew_mat[i]))
term_2_i = coef_2[i] * gs.dot(translation_i,
gs.transpose(sq_skew_mat[i]))
mask_i_float = gs.get_mask_i_float(i, n_tangent_vecs)
exp_translation += gs.outer(mask_i_float,
translation_i + term_1_i + term_2_i)
group_exp = gs.concatenate([rot_vec, exp_translation], axis=1)
group_exp = self.regularize(group_exp)
return group_exp
@geomstats.vectorization.decorator(['else', 'vector'])
def log_from_identity(self, point):
"""Compute the group logarithm of the point at the identity.
Parameters
----------
point: array-like, shape=[..., 3]
Returns
-------
group_log: array-like, shape=[..., 3]
the group logarithm in the Lie algbra
"""
point = self.regularize(point)
rotations = self.rotations
dim_rotations = rotations.dim
rot_vec = point[:, :dim_rotations]
angle = gs.linalg.norm(rot_vec, axis=1)
angle = gs.to_ndarray(angle, to_ndim=2, axis=1)
translation = point[:, dim_rotations:]
skew_rot_vec = rotations.skew_matrix_from_vector(rot_vec)
sq_skew_rot_vec = gs.matmul(skew_rot_vec, skew_rot_vec)
mask_close_0 = gs.isclose(angle, 0.)
mask_close_pi = gs.isclose(angle, gs.pi)
mask_else = ~mask_close_0 & ~mask_close_pi
mask_close_0_float = gs.cast(mask_close_0, gs.float32)
mask_close_pi_float = gs.cast(mask_close_pi, gs.float32)
mask_else_float = gs.cast(mask_else, gs.float32)
mask_0 = gs.isclose(angle, 0., atol=1e-6)
mask_0_float = gs.cast(mask_0, gs.float32)
angle += mask_0_float * gs.ones_like(angle)
coef_1 = -0.5 * gs.ones_like(angle)
coef_2 = gs.zeros_like(angle)
coef_2 += mask_close_0_float * (1. / 12. + angle**2 / 720. + angle**4 /
30240. + angle**6 / 1209600.)
delta_angle = angle - gs.pi
coef_2 += mask_close_pi_float * (
1. / PI2 + (PI2 - 8.) * delta_angle / (4. * PI3) -
((PI2 - 12.) * delta_angle**2 / (4. * PI4)) +
((-192. + 12. * PI2 + PI4) * delta_angle**3 / (48. * PI5)) -
((-240. + 12. * PI2 + PI4) * delta_angle**4 / (48. * PI6)) +
((-2880. + 120. * PI2 + 10. * PI4 + PI6) * delta_angle**5 /
(480. * PI7)) -
((-3360 + 120. * PI2 + 10. * PI4 + PI6) * delta_angle**6 /
(480. * PI8)))
psi = 0.5 * angle * gs.sin(angle) / (1 - gs.cos(angle))
coef_2 += mask_else_float * (1 - psi) / (angle**2)
n_points, _ = point.shape
log_translation = gs.zeros((n_points, self.n))
for i in range(n_points):
translation_i = translation[i]
term_1_i = coef_1[i] * gs.dot(translation_i,
gs.transpose(skew_rot_vec[i]))
term_2_i = coef_2[i] * gs.dot(translation_i,
gs.transpose(sq_skew_rot_vec[i]))
mask_i_float = gs.get_mask_i_float(i, n_points)
log_translation += gs.outer(mask_i_float,
translation_i + term_1_i + term_2_i)
return gs.concatenate([rot_vec, log_translation], axis=1)
def random_uniform(self, n_samples=1):
"""Sample in SE(3) with the uniform distribution.
Parameters
----------
n_samples : int, optional
default : 1
Returns
-------
random_point : array-like, shape=[..., 3]
An array of random elements in SE(3) having the given.
"""
random_translation = self.translations.random_uniform(n_samples)
random_rot_vec = self.rotations.random_uniform(n_samples)
return gs.concatenate([random_rot_vec, random_translation], axis=-1)
def _exponential_matrix(self, rot_vec):
"""Compute exponential of rotation matrix represented by rot_vec.
Parameters
----------
rot_vec : array-like, shape=[..., 3]
Returns
-------
exponential_mat : The matrix exponential of rot_vec
"""
# TODO(nguigs): find usecase for this method
rot_vec = self.rotations.regularize(rot_vec)
n_rot_vecs, _ = rot_vec.shape
angle = gs.linalg.norm(rot_vec, axis=1)
angle = gs.to_ndarray(angle, to_ndim=2, axis=1)
skew_rot_vec = self.rotations.skew_matrix_from_vector(rot_vec)
coef_1 = gs.empty_like(angle)
coef_2 = gs.empty_like(coef_1)
mask_0 = gs.equal(angle, 0)
mask_0 = gs.squeeze(mask_0, axis=1)
mask_close_to_0 = gs.isclose(angle, 0)
mask_close_to_0 = gs.squeeze(mask_close_to_0, axis=1)
mask_else = ~mask_0 & ~mask_close_to_0
coef_1[mask_close_to_0] = (1. / 2. - angle[mask_close_to_0]**2 / 24.)
coef_2[mask_close_to_0] = (1. / 6. - angle[mask_close_to_0]**3 / 120.)
# TODO(nina): Check if the discontinuity at 0 is expected.
coef_1[mask_0] = 0
coef_2[mask_0] = 0
coef_1[mask_else] = (angle[mask_else]**(-2) *
(1. - gs.cos(angle[mask_else])))
coef_2[mask_else] = (angle[mask_else]**(-2) *
(1. -
(gs.sin(angle[mask_else]) / angle[mask_else])))
term_1 = gs.zeros((n_rot_vecs, self.n, self.n))
term_2 = gs.zeros_like(term_1)
for i in range(n_rot_vecs):
term_1[i] = gs.eye(self.n) + skew_rot_vec[i] * coef_1[i]
term_2[i] = gs.matmul(skew_rot_vec[i], skew_rot_vec[i]) * coef_2[i]
exponential_mat = term_1 + term_2
return exponential_mat
class TestSpecialOrthogonal2(geomstats.tests.TestCase):
def setup_method(self):
warnings.simplefilter("ignore", category=ImportWarning)
warnings.simplefilter("ignore", category=UserWarning)
gs.random.seed(1234)
self.group = SpecialOrthogonal(n=2, point_type="vector")
# -- Set attributes
self.n_samples = 4
def test_projection(self):
# Test 2D and nD cases
rot_mat = gs.eye(2)
delta = 1e-12 * gs.ones((2, 2))
rot_mat_plus_delta = rot_mat + delta
result = self.group.projection(rot_mat_plus_delta)
expected = rot_mat
self.assertAllClose(result, expected)
def test_projection_vectorization(self):
n_samples = self.n_samples
mats = gs.ones((n_samples, 2, 2))
result = self.group.projection(mats)
self.assertAllClose(gs.shape(result), (n_samples, 2, 2))
def test_skew_matrix_from_vector(self):
rot_vec = gs.array([0.9])
skew_matrix = self.group.skew_matrix_from_vector(rot_vec)
result = gs.matmul(skew_matrix, skew_matrix)
diag = gs.array([-0.81, -0.81])
expected = algebra_utils.from_vector_to_diagonal_matrix(diag)
self.assertAllClose(result, expected)
def test_skew_matrix_and_vector(self):
rot_vec = gs.array([0.8])
skew_mat = self.group.skew_matrix_from_vector(rot_vec)
result = self.group.vector_from_skew_matrix(skew_mat)
expected = rot_vec
self.assertAllClose(result, expected)
def test_skew_matrix_from_vector_vectorization(self):
n_samples = self.n_samples
rot_vecs = self.group.random_uniform(n_samples=n_samples)
result = self.group.skew_matrix_from_vector(rot_vecs)
self.assertAllClose(gs.shape(result), (n_samples, 2, 2))
def test_random_uniform_shape(self):
result = self.group.random_uniform()
self.assertAllClose(gs.shape(result), (self.group.dim, ))
def test_random_and_belongs(self):
point = self.group.random_uniform()
result = self.group.belongs(point)
expected = True
self.assertAllClose(result, expected)
def test_random_and_belongs_vectorization(self):
n_samples = self.n_samples
points = self.group.random_uniform(n_samples=n_samples)
result = self.group.belongs(points)
expected = gs.array([True] * n_samples)
self.assertAllClose(result, expected)
def test_regularize(self):
angle = 2 * gs.pi + 1
result = self.group.regularize(gs.array([angle]))
expected = gs.array([1.0])
self.assertAllClose(result, expected)
def test_regularize_vectorization(self):
n_samples = self.n_samples
rot_vecs = self.group.random_uniform(n_samples=n_samples)
result = self.group.regularize(rot_vecs)
self.assertAllClose(gs.shape(result), (n_samples, self.group.dim))
def test_matrix_from_rotation_vector(self):
angle = gs.pi / 3
expected = gs.array([[1.0 / 2, -gs.sqrt(3.0) / 2],
[gs.sqrt(3.0) / 2, 1.0 / 2]])
result = self.group.matrix_from_rotation_vector(gs.array([angle]))
self.assertAllClose(result, expected)
def test_matrix_from_rotation_vector_vectorization(self):
n_samples = self.n_samples
rot_vecs = self.group.random_uniform(n_samples=n_samples)
rot_mats = self.group.matrix_from_rotation_vector(rot_vecs)
self.assertAllClose(gs.shape(rot_mats),
(n_samples, self.group.n, self.group.n))
def test_rotation_vector_from_matrix(self):
angle = 0.12
rot_mat = gs.array([[gs.cos(angle), -gs.sin(angle)],
[gs.sin(angle), gs.cos(angle)]])
result = self.group.rotation_vector_from_matrix(rot_mat)
expected = gs.array([0.12])
self.assertAllClose(result, expected)
def test_rotation_vector_and_rotation_matrix(self):
"""
This tests that the composition of
rotation_vector_from_matrix
and
matrix_from_rotation_vector
is the identity.
"""
# TODO(nguigs): bring back a 1d representation of SO2
point = gs.array([0.78])
rot_mat = self.group.matrix_from_rotation_vector(point)
result = self.group.rotation_vector_from_matrix(rot_mat)
expected = point
self.assertAllClose(result, expected)
def test_rotation_vector_and_rotation_matrix_vectorization(self):
rot_vecs = gs.array([[2.0], [1.3], [0.8], [0.03]])
rot_mats = self.group.matrix_from_rotation_vector(rot_vecs)
result = self.group.rotation_vector_from_matrix(rot_mats)
expected = self.group.regularize(rot_vecs)
self.assertAllClose(result, expected)
def test_compose(self):
point_a = gs.array([0.12])
point_b = gs.array([-0.15])
result = self.group.compose(point_a, point_b)
expected = self.group.regularize(gs.array([-0.03]))
self.assertAllClose(result, expected)
def test_compose_and_inverse(self):
angle = 0.986
point = gs.array([angle])
inv_point = self.group.inverse(point)
result = self.group.compose(point, inv_point)
expected = self.group.identity
self.assertAllClose(result, expected)
result = self.group.compose(inv_point, point)
expected = self.group.identity
self.assertAllClose(result, expected)
def test_compose_vectorization(self):
point_type = "vector"
self.group.default_point_type = point_type
n_samples = self.n_samples
n_points_a = self.group.random_uniform(n_samples=n_samples)
n_points_b = self.group.random_uniform(n_samples=n_samples)
one_point = self.group.random_uniform(n_samples=1)
result = self.group.compose(one_point, n_points_a)
self.assertAllClose(gs.shape(result), (n_samples, self.group.dim))
result = self.group.compose(n_points_a, one_point)
self.assertAllClose(gs.shape(result), (n_samples, self.group.dim))
result = self.group.compose(n_points_a, n_points_b)
self.assertAllClose(gs.shape(result), (n_samples, self.group.dim))
def test_inverse_vectorization(self):
n_samples = self.n_samples
points = self.group.random_uniform(n_samples=n_samples)
result = self.group.inverse(points)
self.assertAllClose(gs.shape(result), (n_samples, self.group.dim))
def test_group_exp(self):
"""
The Riemannian exp and log are inverse functions of each other.
This test is the inverse of test_log's.
"""
rot_vec_base_point = gs.array([gs.pi / 5])
rot_vec = gs.array([2 * gs.pi / 5])
expected = gs.array([3 * gs.pi / 5])
result = self.group.exp(base_point=rot_vec_base_point,
tangent_vec=rot_vec)
self.assertAllClose(result, expected)
def test_group_exp_vectorization(self):
n_samples = self.n_samples
one_tangent_vec = self.group.random_uniform(n_samples=1)
one_base_point = self.group.random_uniform(n_samples=1)
n_tangent_vec = self.group.random_uniform(n_samples=n_samples)
n_base_point = self.group.random_uniform(n_samples=n_samples)
# Test with the 1 base point, and n tangent vecs
result = self.group.exp(n_tangent_vec, one_base_point)
self.assertAllClose(gs.shape(result), (n_samples, self.group.dim))
# Test with the several base point, and one tangent vec
result = self.group.exp(one_tangent_vec, n_base_point)
self.assertAllClose(gs.shape(result), (n_samples, self.group.dim))
# Test with the same number n of base point and n tangent vec
result = self.group.exp(n_tangent_vec, n_base_point)
self.assertAllClose(gs.shape(result), (n_samples, self.group.dim))
def test_group_log(self):
"""
The Riemannian exp and log are inverse functions of each other.
This test is the inverse of test_exp's.
"""
rot_vec_base_point = gs.array([gs.pi / 5])
rot_vec = gs.array([2 * gs.pi / 5])
expected = gs.array([1 * gs.pi / 5])
result = self.group.log(point=rot_vec, base_point=rot_vec_base_point)
self.assertAllClose(result, expected)
def test_group_log_vectorization(self):
n_samples = self.n_samples
one_point = self.group.random_uniform(n_samples=1)
one_base_point = self.group.random_uniform(n_samples=1)
n_point = self.group.random_uniform(n_samples=n_samples)
n_base_point = self.group.random_uniform(n_samples=n_samples)
# Test with the 1 base point, and several different points
result = self.group.log(n_point, one_base_point)
self.assertAllClose(gs.shape(result), (n_samples, self.group.dim))
# Test with the several base point, and 1 point
result = self.group.log(one_point, n_base_point)
self.assertAllClose(gs.shape(result), (n_samples, self.group.dim))
# Test with the same number n of base point and point
result = self.group.log(n_point, n_base_point)
self.assertAllClose(gs.shape(result), (n_samples, self.group.dim))
def test_group_exp_then_log_from_identity(self):
"""
Test that the group exponential
and the group logarithm are inverse.
Expect their composition to give the identity function.
"""
tangent_vec = gs.array([0.12])
result = helper.group_exp_then_log_from_identity(
group=self.group, tangent_vec=tangent_vec)
expected = self.group.regularize(tangent_vec)
self.assertAllClose(result, expected)
def test_group_log_then_exp_from_identity(self):
"""
Test that the group exponential
and the group logarithm are inverse.
Expect their composition to give the identity function.
"""
point = gs.array([0.12])
result = helper.group_log_then_exp_from_identity(group=self.group,
point=point)
expected = self.group.regularize(point)
self.assertAllClose(result, expected)
def test_group_exp_then_log(self):
"""
This tests that the composition of
log and exp gives identity.
"""
base_point = gs.array([0.12])
tangent_vec = gs.array([0.35])
result = helper.group_exp_then_log(group=self.group,
tangent_vec=tangent_vec,
base_point=base_point)
expected = self.group.regularize_tangent_vec(tangent_vec=tangent_vec,
base_point=base_point)
self.assertAllClose(result, expected)
def test_group_log_then_exp(self):
"""
This tests that the composition of
log and exp gives identity.
"""
base_point = gs.array([0.12])
point = gs.array([0.35])
result = helper.group_log_then_exp(group=self.group,
point=point,
base_point=base_point)
expected = self.group.regularize(point)
self.assertAllClose(result, expected)
class SpecialEuclidean(LieGroup):
"""Class for the special euclidean group SE(n).
i.e. the Lie group of rigid transformations.
"""
def __init__(self, n, point_type=None, epsilon=0.):
"""Initiate an object of class SpecialEuclidean.
Parameter
---------
n : int
the dimension of the euclidean space that SE(n) acts upon
point_type : str, {'vector', 'matrix'}, optional
whether to represent elmenents of SE(n) by vectors or matrices
if None is given, point_type is set to 'vector' for dimension 3
and 'matrix' otherwise
epsilon : float, optional
precision to use for calculations involving potential division by
rotations
default: 0
"""
assert isinstance(n, int) and n > 1
self.n = n
self.dimension = int((n * (n - 1)) / 2 + n)
self.epsilon = epsilon
self.default_point_type = point_type
if point_type is None:
self.default_point_type = 'vector' if n == 3 else 'matrix'
super(SpecialEuclidean, self).__init__(dimension=self.dimension)
self.rotations = SpecialOrthogonal(n=n, epsilon=epsilon)
self.translations = Euclidean(dimension=n)
def get_identity(self, point_type=None):
"""Get the identity of the group.
Parameters
----------
point_type : str, {'vector', 'matrix'}, optional
the point_type of the returned value
default: self.default_point_type
Returns
-------
identity : array-like, shape={[dimension], [n + 1, n + 1]}
"""
if point_type is None:
point_type = self.default_point_type
identity = gs.zeros(self.dimension)
if self.default_point_type == 'matrix':
identity = gs.eye(self.n)
return identity
identity = property(get_identity)
def belongs(self, point, point_type=None):
"""Evaluate if a point belongs to SE(n).
Parameters
----------
point : array-like, shape=[n_samples, {dimension, [n + 1, n + 1]}]
the point of which to check whether it belongs to SE(n)
point_type : str, {'vector', 'matrix'}, optional
default: self.default_point_type
Returns
-------
belongs : array-like, shape=[n_samples, 1]
array of booleans indicating whether point belongs to SE(n)
"""
if point_type is None:
point_type = self.default_point_type
if point_type == 'vector':
point = gs.to_ndarray(point, to_ndim=2)
n_points, point_dim = point.shape
belongs = point_dim == self.dimension
belongs = gs.to_ndarray(belongs, to_ndim=1)
belongs = gs.to_ndarray(belongs, to_ndim=2, axis=1)
belongs = gs.tile(belongs, (n_points, 1))
elif point_type == 'matrix':
point = gs.to_ndarray(point, to_ndim=3)
raise NotImplementedError()
return belongs
def regularize(self, point, point_type=None):
"""Regularize a point to the default representation for SE(n).
Parameters
----------
point : array-like, shape=[n_samples, {dimension, [n + 1, n + 1]}]
the point which should be regularized
point_type : str, {'vector', 'matrix'}, optional
default: self.default_point_type
Returns
-------
point : array-like, shape=[n_samples, {dimension, [n + 1, n + 1]}]
"""
if point_type is None:
point_type = self.default_point_type
if point_type == 'vector':
point = gs.to_ndarray(point, to_ndim=2)
rotations = self.rotations
dim_rotations = rotations.dimension
rot_vec = point[:, :dim_rotations]
regularized_rot_vec = rotations.regularize(rot_vec,
point_type=point_type)
translation = point[:, dim_rotations:]
regularized_point = gs.concatenate(
[regularized_rot_vec, translation], axis=1)
elif point_type == 'matrix':
point = gs.to_ndarray(point, to_ndim=3)
regularized_point = gs.copy(point)
return regularized_point
def regularize_tangent_vec_at_identity(self,
tangent_vec,
metric=None,
point_type=None):
"""Regularize a tangent vector at the identity.
Parameters
----------
tangent_vec: array-like, shape=[n_samples, {dimension, [n + 1, n + 1]}]
metric : RiemannianMetric, optional
point_type : str, {'vector', 'matrix'}, optional
default: self.default_point_type
Returns
-------
regularized_vec : the regularized tangent vector
"""
if point_type is None:
point_type = self.default_point_type
return self.regularize_tangent_vec(tangent_vec,
self.identity,
metric,
point_type=point_type)
def regularize_tangent_vec(self,
tangent_vec,
base_point,
metric=None,
point_type=None):
"""Regularize a tangent vector at a base point.
Parameters
----------
tangent_vec: array-like, shape=[n_samples, {dimension, [n + 1, n + 1]}]
base_point : array-like, shape=[n_samples, {dimension, [n + 1, n + 1]}]
metric : RiemannianMetric, optional
default: self.left_canonical_metric
point_type: str, {'vector', 'matrix'}, optional
default: self.default_point_type
Returns
-------
regularized_vec : the regularized tangent vector
"""
if point_type is None:
point_type = self.default_point_type
if metric is None:
metric = self.left_canonical_metric
if point_type == 'vector':
tangent_vec = gs.to_ndarray(tangent_vec, to_ndim=2)
base_point = gs.to_ndarray(base_point, to_ndim=2)
rotations = self.rotations
dim_rotations = rotations.dimension
rot_tangent_vec = tangent_vec[:, :dim_rotations]
rot_base_point = base_point[:, :dim_rotations]
metric_mat = metric.inner_product_mat_at_identity
rot_metric_mat = metric_mat[:, :dim_rotations, :dim_rotations]
rot_metric = InvariantMetric(
group=rotations,
inner_product_mat_at_identity=rot_metric_mat,
left_or_right=metric.left_or_right)
regularized_vec = gs.zeros_like(tangent_vec)
rotations_vec = rotations.regularize_tangent_vec(
tangent_vec=rot_tangent_vec,
base_point=rot_base_point,
metric=rot_metric,
point_type=point_type)
regularized_vec = gs.concatenate(
[rotations_vec, tangent_vec[:, dim_rotations:]], axis=1)
elif point_type == 'matrix':
regularized_vec = tangent_vec
return regularized_vec
def compose(self, point_1, point_2, point_type=None):
r"""Compose two elements of SE(n).
Parameters
----------
point_1 : array-like, shape=[n_samples, {dimension, [n + 1, n + 1]}]
point_2 : array-like, shape=[n_samples, {dimension, [n + 1, n + 1]}]
point_type: str, {'vector', 'matrix'}, optional
default: self.default_point_type
Equation
---------
(:math: `(R_1, t_1) \\cdot (R_2, t_2) = (R_1 R_2, R_1 t_2 + t_1)`)
Returns
-------
composition : the composition of point_1 and point_2
"""
if point_type is None:
point_type = self.default_point_type
rotations = self.rotations
dim_rotations = rotations.dimension
point_1 = self.regularize(point_1, point_type=point_type)
point_2 = self.regularize(point_2, point_type=point_type)
if point_type == 'vector':
n_points_1, _ = point_1.shape
n_points_2, _ = point_2.shape
assert (point_1.shape == point_2.shape or n_points_1 == 1
or n_points_2 == 1)
if n_points_1 == 1:
point_1 = gs.stack([point_1[0]] * n_points_2)
if n_points_2 == 1:
point_2 = gs.stack([point_2[0]] * n_points_1)
rot_vec_1 = point_1[:, :dim_rotations]
rot_mat_1 = rotations.matrix_from_rotation_vector(rot_vec_1)
rot_vec_2 = point_2[:, :dim_rotations]
rot_mat_2 = rotations.matrix_from_rotation_vector(rot_vec_2)
translation_1 = point_1[:, dim_rotations:]
translation_2 = point_2[:, dim_rotations:]
composition_rot_mat = gs.matmul(rot_mat_1, rot_mat_2)
composition_rot_vec = rotations.rotation_vector_from_matrix(
composition_rot_mat)
composition_translation = gs.einsum('ij,ikj->ik', translation_2,
rot_mat_1) + translation_1
composition = gs.concatenate(
(composition_rot_vec, composition_translation), axis=1)
elif point_type == 'matrix':
raise NotImplementedError()
composition = self.regularize(composition, point_type=point_type)
return composition
def inverse(self, point, point_type=None):
r"""Compute the group inverse in SE(n).
Parameters
----------
point: array-like, shape=[n_samples, {dimension, [n + 1, n + 1]}]
Formula
-------
:math:`(R, t)^{-1} = (R^{-1}, R^{-1}.(-t))`
Returns
-------
inverse_point : array-like,
shape=[n_samples, {dimension, [n + 1, n + 1]}]
the inverted point
"""
if point_type is None:
point_type = self.default_point_type
rotations = self.rotations
dim_rotations = rotations.dimension
point = self.regularize(point)
if point_type == 'vector':
n_points, _ = point.shape
rot_vec = point[:, :dim_rotations]
translation = point[:, dim_rotations:]
inverse_point = gs.zeros_like(point)
inverse_rotation = -rot_vec
inv_rot_mat = rotations.matrix_from_rotation_vector(
inverse_rotation)
inverse_translation = gs.einsum(
'ni,nij->nj', -translation,
gs.transpose(inv_rot_mat, axes=(0, 2, 1)))
inverse_point = gs.concatenate(
[inverse_rotation, inverse_translation], axis=1)
elif point_type == 'matrix':
raise NotImplementedError()
inverse_point = self.regularize(inverse_point, point_type=point_type)
return inverse_point
def jacobian_translation(self,
point,
left_or_right='left',
point_type=None):
"""Compute the Jacobian matrix resulting from translation.
Compute the jacobian matrix of the differential
of the left/right translations from the identity to point in SE(n).
Currently only implemented for point_type == 'vector'.
Parameters
----------
point: array-like, shape=[n_samples, {dimension, [n + 1, n + 1]}]
left_or_right: str, {'left', 'right'}, optional
default: 'left'
whether to compute the jacobian of the left or right translation
point_type : str, {'vector', 'matrix'}, optional
default: self.default_point_type
Returns
-------
jacobian : array-like, shape=[n_samples, dimension]
The jacobian of the left / right translation
"""
if point_type is None:
point_type = self.default_point_type
assert left_or_right in ('left', 'right')
dim = self.dimension
rotations = self.rotations
translations = self.translations
dim_rotations = rotations.dimension
dim_translations = translations.dimension
point = self.regularize(point, point_type=point_type)
if point_type == 'vector':
n_points, _ = point.shape
rot_vec = point[:, :dim_rotations]
jacobian = gs.zeros((n_points, ) + (dim, ) * 2)
jacobian_rot = self.rotations.jacobian_translation(
point=rot_vec,
left_or_right=left_or_right,
point_type=point_type)
block_zeros_1 = gs.zeros(
(n_points, dim_rotations, dim_translations))
jacobian_block_line_1 = gs.concatenate(
[jacobian_rot, block_zeros_1], axis=2)
if left_or_right == 'left':
rot_mat = self.rotations.matrix_from_rotation_vector(rot_vec)
jacobian_trans = rot_mat
block_zeros_2 = gs.zeros(
(n_points, dim_translations, dim_rotations))
jacobian_block_line_2 = gs.concatenate(
[block_zeros_2, jacobian_trans], axis=2)
else:
inv_skew_mat = -self.rotations.skew_matrix_from_vector(rot_vec)
eye = gs.to_ndarray(gs.eye(self.n), to_ndim=3)
eye = gs.tile(eye, [n_points, 1, 1])
jacobian_block_line_2 = gs.concatenate([inv_skew_mat, eye],
axis=2)
jacobian = gs.concatenate(
[jacobian_block_line_1, jacobian_block_line_2], axis=1)
assert gs.ndim(jacobian) == 3
elif point_type == 'matrix':
raise NotImplementedError()
return jacobian
def exp_from_identity(self, tangent_vec, point_type=None):
"""Compute group exponential of the tangent vector at the identity.
Parameters
----------
tangent_vec: array-like, shape=[n_samples, {dimension, [n + 1, n + 1]}]
point_type: str, {'vector', 'matrix'}, optional
default: self.default_point_type
Returns
-------
group_exp: array-like, shape=[n_samples, {dimension, [n + 1, n + 1]}]
the group exponential of the tangent vectors calculated
at the identity
"""
if point_type is None:
point_type = self.default_point_type
if point_type == 'vector':
tangent_vec = gs.to_ndarray(tangent_vec, to_ndim=2)
rotations = self.rotations
dim_rotations = rotations.dimension
rot_vec = tangent_vec[:, :dim_rotations]
rot_vec = self.rotations.regularize(rot_vec, point_type=point_type)
translation = tangent_vec[:, dim_rotations:]
angle = gs.linalg.norm(rot_vec, axis=1)
angle = gs.to_ndarray(angle, to_ndim=2, axis=1)
skew_mat = self.rotations.skew_matrix_from_vector(rot_vec)
sq_skew_mat = gs.matmul(skew_mat, skew_mat)
mask_0 = gs.equal(angle, 0.)
mask_close_0 = gs.isclose(angle, 0.) & ~mask_0
mask_else = ~mask_0 & ~mask_close_0
mask_0_float = gs.cast(mask_0, gs.float32)
mask_close_0_float = gs.cast(mask_close_0, gs.float32)
mask_else_float = gs.cast(mask_else, gs.float32)
angle += mask_0_float * gs.ones_like(angle)
coef_1 = gs.zeros_like(angle)
coef_2 = gs.zeros_like(angle)
coef_1 += mask_0_float * 1. / 2. * gs.ones_like(angle)
coef_2 += mask_0_float * 1. / 6. * gs.ones_like(angle)
coef_1 += mask_close_0_float * (
TAYLOR_COEFFS_1_AT_0[0] + TAYLOR_COEFFS_1_AT_0[2] * angle**2 +
TAYLOR_COEFFS_1_AT_0[4] * angle**4 +
TAYLOR_COEFFS_1_AT_0[6] * angle**6)
coef_2 += mask_close_0_float * (
TAYLOR_COEFFS_2_AT_0[0] + TAYLOR_COEFFS_2_AT_0[2] * angle**2 +
TAYLOR_COEFFS_2_AT_0[4] * angle**4 +
TAYLOR_COEFFS_2_AT_0[6] * angle**6)
coef_1 += mask_else_float * ((1. - gs.cos(angle)) / angle**2)
coef_2 += mask_else_float * ((angle - gs.sin(angle)) / angle**3)
n_tangent_vecs, _ = tangent_vec.shape
exp_translation = gs.zeros((n_tangent_vecs, self.n))
for i in range(n_tangent_vecs):
translation_i = translation[i]
term_1_i = coef_1[i] * gs.dot(translation_i,
gs.transpose(skew_mat[i]))
term_2_i = coef_2[i] * gs.dot(translation_i,
gs.transpose(sq_skew_mat[i]))
mask_i_float = gs.get_mask_i_float(i, n_tangent_vecs)
exp_translation += mask_i_float * (translation_i + term_1_i +
term_2_i)
group_exp = gs.concatenate([rot_vec, exp_translation], axis=1)
group_exp = self.regularize(group_exp, point_type=point_type)
return group_exp
elif point_type == 'matrix':
raise NotImplementedError()
def log_from_identity(self, point, point_type=None):
"""Compute the group logarithm of the point at the identity.
Parameters
----------
point: array-like, shape=[n_samples, {dimension, [n + 1, n + 1]}]
point_type: str, {'vector', 'matrix'}, optional
default: self.default_point_type
Returns
-------
group_log: array-like, shape=[n_samples, {dimension, [n + 1, n + 1]}]
the group logarithm in the Lie algbra
"""
if point_type is None:
point_type = self.default_point_type
point = self.regularize(point, point_type=point_type)
rotations = self.rotations
dim_rotations = rotations.dimension
if point_type == 'vector':
rot_vec = point[:, :dim_rotations]
angle = gs.linalg.norm(rot_vec, axis=1)
angle = gs.to_ndarray(angle, to_ndim=2, axis=1)
translation = point[:, dim_rotations:]
skew_rot_vec = rotations.skew_matrix_from_vector(rot_vec)
sq_skew_rot_vec = gs.matmul(skew_rot_vec, skew_rot_vec)
mask_close_0 = gs.isclose(angle, 0.)
mask_close_pi = gs.isclose(angle, gs.pi)
mask_else = ~mask_close_0 & ~mask_close_pi
mask_close_0_float = gs.cast(mask_close_0, gs.float32)
mask_close_pi_float = gs.cast(mask_close_pi, gs.float32)
mask_else_float = gs.cast(mask_else, gs.float32)
mask_0 = gs.isclose(angle, 0., atol=1e-6)
mask_0_float = gs.cast(mask_0, gs.float32)
angle += mask_0_float * gs.ones_like(angle)
coef_1 = -0.5 * gs.ones_like(angle)
coef_2 = gs.zeros_like(angle)
coef_2 += mask_close_0_float * (1. / 12. + angle**2 / 720. +
angle**4 / 30240. +
angle**6 / 1209600.)
delta_angle = angle - gs.pi
coef_2 += mask_close_pi_float * (
1. / PI2 + (PI2 - 8.) * delta_angle / (4. * PI3) -
((PI2 - 12.) * delta_angle**2 / (4. * PI4)) +
((-192. + 12. * PI2 + PI4) * delta_angle**3 / (48. * PI5)) -
((-240. + 12. * PI2 + PI4) * delta_angle**4 / (48. * PI6)) +
((-2880. + 120. * PI2 + 10. * PI4 + PI6) * delta_angle**5 /
(480. * PI7)) -
((-3360 + 120. * PI2 + 10. * PI4 + PI6) * delta_angle**6 /
(480. * PI8)))
psi = 0.5 * angle * gs.sin(angle) / (1 - gs.cos(angle))
coef_2 += mask_else_float * (1 - psi) / (angle**2)
n_points, _ = point.shape
log_translation = gs.zeros((n_points, self.n))
for i in range(n_points):
translation_i = translation[i]
term_1_i = coef_1[i] * gs.dot(translation_i,
gs.transpose(skew_rot_vec[i]))
term_2_i = coef_2[i] * gs.dot(translation_i,
gs.transpose(sq_skew_rot_vec[i]))
mask_i_float = gs.get_mask_i_float(i, n_points)
log_translation += mask_i_float * (translation_i + term_1_i +
term_2_i)
group_log = gs.concatenate([rot_vec, log_translation], axis=1)
assert gs.ndim(group_log) == 2
elif point_type == 'matrix':
raise NotImplementedError()
return group_log
def random_uniform(self, n_samples=1, point_type=None):
"""Sample in SE(n) with the uniform distribution.
Parameters
----------
n_samples: int, optional
default: 1
point_typ: str, {'vector', 'matrix'}, optional
default: self.default_point_type
Returns
-------
random_transfo: array-like,
shape=[n_samples, {dimension, [n + 1, n + 1]}]
an array of random elements in SE(n) having the given point_type
"""
if point_type is None:
point_type = self.default_point_type
random_rot_vec = self.rotations.random_uniform(n_samples,
point_type=point_type)
random_translation = self.translations.random_uniform(n_samples)
if point_type == 'vector':
random_transfo = gs.concatenate(
[random_rot_vec, random_translation], axis=1)
elif point_type == 'matrix':
raise NotImplementedError()
random_transfo = self.regularize(random_transfo, point_type=point_type)
return random_transfo
def exponential_matrix(self, rot_vec):
"""Compute exponential of rotation matrix represented by rot_vec.
Parameters
----------
rot_vec : array-like, shape=[n_samples, dimension]
Returns
-------
exponential_mat: The matrix exponential of rot_vec
"""
rot_vec = self.rotations.regularize(rot_vec)
n_rot_vecs, _ = rot_vec.shape
angle = gs.linalg.norm(rot_vec, axis=1)
angle = gs.to_ndarray(angle, to_ndim=2, axis=1)
skew_rot_vec = self.rotations.skew_matrix_from_vector(rot_vec)
coef_1 = gs.empty_like(angle)
coef_2 = gs.empty_like(coef_1)
mask_0 = gs.equal(angle, 0)
mask_0 = gs.squeeze(mask_0, axis=1)
mask_close_to_0 = gs.isclose(angle, 0)
mask_close_to_0 = gs.squeeze(mask_close_to_0, axis=1)
mask_else = ~mask_0 & ~mask_close_to_0
coef_1[mask_close_to_0] = (1. / 2. - angle[mask_close_to_0]**2 / 24.)
coef_2[mask_close_to_0] = (1. / 6. - angle[mask_close_to_0]**3 / 120.)
# TODO(nina): Check if the discontinuity at 0 is expected.
coef_1[mask_0] = 0
coef_2[mask_0] = 0
coef_1[mask_else] = (angle[mask_else]**(-2) *
(1. - gs.cos(angle[mask_else])))
coef_2[mask_else] = (angle[mask_else]**(-2) *
(1. -
(gs.sin(angle[mask_else]) / angle[mask_else])))
term_1 = gs.zeros((n_rot_vecs, self.n, self.n))
term_2 = gs.zeros_like(term_1)
for i in range(n_rot_vecs):
term_1[i] = gs.eye(self.n) + skew_rot_vec[i] * coef_1[i]
term_2[i] = gs.matmul(skew_rot_vec[i], skew_rot_vec[i]) * coef_2[i]
exponential_mat = term_1 + term_2
assert exponential_mat.ndim == 3
return exponential_mat
def exponential_barycenter(self, points, weights=None, point_type=None):
"""Compute the group exponential barycenter in SE(n).
Parameters
----------
points: array-like, shape=[n_samples, {dimension, [n + 1, n + 1]}]
weights: array-like, shape=[n_samples], optional
default: weight 1 / n_samples for each point
point_type: str, {'vector', 'matrix'}, optional
default: self.default_point_type
Returns
-------
exp_bar: array-like, shape=[{dimension, [n + 1, n + 1]}]
the exponential barycenter
"""
if point_type is None:
point_type = self.default_point_type
n_points = points.shape[0]
assert n_points > 0
if weights is None:
weights = gs.ones((n_points, 1))
weights = gs.to_ndarray(weights, to_ndim=2, axis=1)
n_weights, _ = weights.shape
assert n_points == n_weights
dim = self.dimension
rotations = self.rotations
dim_rotations = rotations.dimension
if point_type == 'vector':
rotation_vectors = points[:, :dim_rotations]
translations = points[:, dim_rotations:dim]
assert rotation_vectors.shape == (n_points, dim_rotations)
assert translations.shape == (n_points, self.n)
mean_rotation = rotations.exponential_barycenter(
points=rotation_vectors, weights=weights)
mean_rotation_mat = rotations.matrix_from_rotation_vector(
mean_rotation)
matrix = gs.zeros((1, ) + (self.n, ) * 2)
translation_aux = gs.zeros((1, self.n))
inv_rot_mats = rotations.matrix_from_rotation_vector(
-rotation_vectors)
matrix_aux = gs.matmul(mean_rotation_mat, inv_rot_mats)
assert matrix_aux.shape == (n_points, ) + (dim_rotations, ) * 2
vec_aux = rotations.rotation_vector_from_matrix(matrix_aux)
matrix_aux = self.exponential_matrix(vec_aux)
matrix_aux = gs.linalg.inv(matrix_aux)
for i in range(n_points):
matrix += weights[i] * matrix_aux[i]
translation_aux += weights[i] * gs.dot(
gs.matmul(matrix_aux[i], inv_rot_mats[i]), translations[i])
mean_translation = gs.dot(
translation_aux,
gs.transpose(gs.linalg.inv(matrix), axes=(0, 2, 1)))
exp_bar = gs.zeros((1, dim))
exp_bar[0, :dim_rotations] = mean_rotation
exp_bar[0, dim_rotations:dim] = mean_translation
elif point_type == 'matrix':
vector_points = self.rotation_vector_from_matrix(points)
vector_exp_bar = self.exponential_barycenter(vector_points,
weights,
point_type='vector')
exp_bar = self.matrix_from_rotation_vector(vector_exp_bar)
return exp_bar
class SpecialEuclidean(LieGroup):
"""Class for the special euclidean group SE(n).
i.e. the Lie group of rigid transformations. Elements of SE(n) can either
be represented as vectors (in 3d) or as matrices in general. The matrix
representation corresponds to homogeneous coordinates.
"""
def __init__(self, n, default_point_type=None, epsilon=0.):
"""Initiate an object of class SpecialEuclidean.
Parameter
---------
n : int
the dimension of the euclidean space that SE(n) acts upon
point_type : str, {'vector', 'matrix'}, optional
whether to represent elmenents of SE(n) by vectors or matrices
if None is given, point_type is set to 'vector' for dimension 3
and 'matrix' otherwise
epsilon : float, optional
precision to use for calculations involving potential division by
rotations
default: 0
"""
if not (isinstance(n, int) and n > 1):
raise ValueError('n must be an integer > 1.')
self.n = n
self.dimension = int((n * (n - 1)) / 2 + n)
self.epsilon = epsilon
super(SpecialEuclidean,
self).__init__(dim=self.dimension,
default_point_type=default_point_type)
if default_point_type is None:
self.default_point_type = 'vector' if n == 3 else 'matrix'
self.rotations = SpecialOrthogonal(
n=n, epsilon=epsilon, default_point_type=default_point_type)
self.translations = Euclidean(dim=n)
def get_identity(self, point_type=None):
"""Get the identity of the group.
Parameters
----------
point_type : str, {'vector', 'matrix'}, optional
the point_type of the returned value
default: self.default_point_type
Returns
-------
identity : array-like, shape={[dim], [n + 1, n + 1]}
"""
if point_type is None:
point_type = self.default_point_type
identity = gs.zeros(self.dimension)
if point_type == 'matrix':
identity = gs.eye(self.n + 1)
return identity
identity = property(get_identity)
def get_point_type_shape(self, point_type=None):
"""Get the shape of the instance given the default_point_style."""
return self.get_identity(point_type).shape
@geomstats.vectorization.decorator(['else', 'point', 'point_type'])
def belongs(self, point, point_type=None):
"""Evaluate if a point belongs to SE(n).
Parameters
----------
point : array-like, shape=[n_samples, {dim, [n + 1, n + 1]}]
the point of which to check whether it belongs to SE(n)
point_type : str, {'vector', 'matrix'}, optional
default: self.default_point_type
Returns
-------
belongs : array-like, shape=[n_samples, 1]
array of booleans indicating whether point belongs to SE(n)
"""
if point_type == 'vector':
n_points, vec_dim = gs.shape(point)
belongs = vec_dim == self.dimension
belongs = gs.tile([belongs], (point.shape[0], ))
belongs = gs.logical_and(belongs,
self.rotations.belongs(point[:, :self.n]))
return gs.flatten(belongs)
if point_type == 'matrix':
n_points, point_dim1, point_dim2 = point.shape
belongs = (point_dim1 == point_dim2 == self.n + 1)
belongs = [belongs] * n_points
rotation = point[:, :self.n, :self.n]
rot_belongs = self.rotations.belongs(rotation,
point_type=point_type)
belongs = gs.logical_and(belongs, rot_belongs)
last_line_except_last_term = point[:, self.n:, :-1]
all_but_last_zeros = ~gs.any(last_line_except_last_term,
axis=(1, 2))
belongs = gs.logical_and(belongs, all_but_last_zeros)
last_term = point[:, self.n:, self.n:]
belongs = gs.logical_and(belongs,
gs.all(last_term == 1, axis=(1, 2)))
return gs.flatten(belongs)
raise ValueError('Invalid point_type, expected \'vector\' or '
'\'matrix\'.')
@geomstats.vectorization.decorator(['else', 'point', 'point_type'])
def regularize(self, point, point_type=None):
"""Regularize a point to the default representation for SE(n).
Parameters
----------
point : array-like, shape=[n_samples, {dim, [n + 1, n + 1]}]
the point which should be regularized
point_type : str, {'vector', 'matrix'}, optional
default: self.default_point_type
Returns
-------
point : array-like, shape=[n_samples, {dim, [n + 1, n + 1]}]
"""
if point_type == 'vector':
rotations = self.rotations
dim_rotations = rotations.dim
rot_vec = point[:, :dim_rotations]
regularized_rot_vec = rotations.regularize(rot_vec,
point_type=point_type)
translation = point[:, dim_rotations:]
return gs.concatenate([regularized_rot_vec, translation], axis=1)
if point_type == 'matrix':
return gs.to_ndarray(point, to_ndim=3)
raise ValueError('Invalid point_type, expected \'vector\' or '
'\'matrix\'.')
@geomstats.vectorization.decorator(['else', 'point', 'else', 'point_type'])
def regularize_tangent_vec_at_identity(self,
tangent_vec,
metric=None,
point_type=None):
"""Regularize a tangent vector at the identity.
Parameters
----------
tangent_vec: array-like, shape=[n_samples, {dim, [n + 1, n + 1]}]
metric : RiemannianMetric, optional
point_type : str, {'vector', 'matrix'}, optional
default: self.default_point_type
Returns
-------
regularized_vec : the regularized tangent vector
"""
if point_type == 'vector':
return self.regularize_tangent_vec(tangent_vec,
self.identity,
metric,
point_type=point_type)
if point_type == 'matrix':
translation_mask = gs.hstack(
[gs.ones((self.n, ) * 2), 2 * gs.ones((self.n, 1))])
translation_mask = gs.concatenate(
[translation_mask, gs.zeros((1, self.n + 1))], axis=0)
tangent_vec = tangent_vec * gs.where(translation_mask != 0.,
gs.array(1.), gs.array(0.))
tangent_vec = (tangent_vec -
GeneralLinear.transpose(tangent_vec)) / 2.
return tangent_vec * translation_mask
raise ValueError('Invalid point_type, expected \'vector\' or '
'\'matrix\'.')
@geomstats.vectorization.decorator(
['else', 'point', 'point', 'else', 'point_type'])
def regularize_tangent_vec(self,
tangent_vec,
base_point,
metric=None,
point_type=None):
"""Regularize a tangent vector at a base point.
Parameters
----------
tangent_vec: array-like, shape=[n_samples, {dim, [n + 1, n + 1]}]
base_point : array-like, shape=[n_samples, {dim, [n + 1, n + 1]}]
metric : RiemannianMetric, optional
default: self.left_canonical_metric
point_type: str, {'vector', 'matrix'}, optional
default: self.default_point_type
Returns
-------
regularized_vec : the regularized tangent vector
"""
if metric is None:
metric = self.left_canonical_metric
if point_type == 'vector':
rotations = self.rotations
dim_rotations = rotations.dim
rot_tangent_vec = tangent_vec[:, :dim_rotations]
rot_base_point = base_point[:, :dim_rotations]
metric_mat = metric.inner_product_mat_at_identity
rot_metric_mat = metric_mat[:dim_rotations, :dim_rotations]
rot_metric = InvariantMetric(
group=rotations,
inner_product_mat_at_identity=rot_metric_mat,
left_or_right=metric.left_or_right)
rotations_vec = rotations.regularize_tangent_vec(
tangent_vec=rot_tangent_vec,
base_point=rot_base_point,
metric=rot_metric,
point_type=point_type)
return gs.concatenate(
[rotations_vec, tangent_vec[:, dim_rotations:]], axis=1)
if point_type == 'matrix':
tangent_vec_at_id = self.compose(self.inverse(base_point),
tangent_vec)
regularized = self.regularize_tangent_vec_at_identity(
tangent_vec_at_id, point_type=point_type)
return self.compose(base_point, regularized)
raise ValueError('Invalid point_type, expected \'vector\' or '
'\'matrix\'.')
@geomstats.vectorization.decorator(['else', 'vector'])
def matrix_from_vector(self, vec):
"""Convert point in vector point-type to matrix.
Parameters
----------
vec: array-like, shape=[n_samples, dim]
Returns
-------
mat: array-like, shape=[n_samples, {dim, [n+1, n+1]}]
"""
vec = self.regularize(vec, point_type='vector')
n_vecs, _ = vec.shape
rot_vec = vec[:, :self.rotations.dim]
trans_vec = vec[:, self.rotations.dim:]
rot_mat = self.rotations.matrix_from_rotation_vector(rot_vec)
trans_vec = gs.reshape(trans_vec, (n_vecs, self.n, 1))
mat = gs.concatenate((rot_mat, trans_vec), axis=2)
last_lines = gs.array(gs.get_mask_i_float(self.n, self.n + 1))
last_lines = gs.to_ndarray(last_lines, to_ndim=2)
last_lines = gs.to_ndarray(last_lines, to_ndim=3)
mat = gs.concatenate((mat, last_lines), axis=1)
return mat
@geomstats.vectorization.decorator(
['else', 'point', 'point', 'point_type'])
def compose(self, point_a, point_b, point_type=None):
r"""Compose two elements of SE(n).
Parameters
----------
point_1 : array-like, shape=[n_samples, {dim, [n + 1, n + 1]}]
point_2 : array-like, shape=[n_samples, {dim, [n + 1, n + 1]}]
point_type: str, {'vector', 'matrix'}, optional
default: self.default_point_type
Equation
---------
(:math: `(R_1, t_1) \\cdot (R_2, t_2) = (R_1 R_2, R_1 t_2 + t_1)`)
Returns
-------
composition : the composition of point_1 and point_2
"""
rotations = self.rotations
dim_rotations = rotations.dim
point_a = self.regularize(point_a, point_type=point_type)
point_b = self.regularize(point_b, point_type=point_type)
if point_type == 'vector':
n_points_a, _ = point_a.shape
n_points_b, _ = point_b.shape
if not (point_a.shape == point_b.shape or n_points_a == 1
or n_points_b == 1):
raise ValueError()
rot_vec_a = point_a[:, :dim_rotations]
rot_mat_a = rotations.matrix_from_rotation_vector(rot_vec_a)
rot_vec_b = point_b[:, :dim_rotations]
rot_mat_b = rotations.matrix_from_rotation_vector(rot_vec_b)
translation_a = point_a[:, dim_rotations:]
translation_b = point_b[:, dim_rotations:]
composition_rot_mat = gs.matmul(rot_mat_a, rot_mat_b)
composition_rot_vec = rotations.rotation_vector_from_matrix(
composition_rot_mat)
composition_translation = gs.einsum(
'...j,...kj->...k', translation_b, rot_mat_a) + translation_a
composition = gs.concatenate(
(composition_rot_vec, composition_translation), axis=-1)
return self.regularize(composition, point_type=point_type)
if point_type == 'matrix':
return GeneralLinear.compose(point_a, point_b)
raise ValueError('Invalid point_type, expected \'vector\' or '
'\'matrix\'.')
@geomstats.vectorization.decorator(['else', 'point', 'point_type'])
def inverse(self, point, point_type=None):
r"""Compute the group inverse in SE(n).
Parameters
----------
point: array-like, shape=[n_samples, {dim, [n + 1, n + 1]}]
point_type: str, {'vector', 'matrix'}, optional
default: self.default_point_type
Returns
-------
inverse_point : array-like,
shape=[n_samples, {dim, [n + 1, n + 1]}]
the inverted point
Notes
-----
:math:`(R, t)^{-1} = (R^{-1}, R^{-1}.(-t))`
"""
rotations = self.rotations
dim_rotations = rotations.dim
point = self.regularize(point)
if point_type == 'vector':
rot_vec = point[:, :dim_rotations]
translation = point[:, dim_rotations:]
inverse_rotation = -rot_vec
inv_rot_mat = rotations.matrix_from_rotation_vector(
inverse_rotation)
inverse_translation = gs.einsum(
'ni,nij->nj', -translation,
gs.transpose(inv_rot_mat, axes=(0, 2, 1)))
inverse_point = gs.concatenate(
[inverse_rotation, inverse_translation], axis=-1)
return self.regularize(inverse_point, point_type=point_type)
if point_type == 'matrix':
inv_rot = gs.transpose(point[:, :self.n, :self.n], axes=(0, 2, 1))
inv_trans = gs.matmul(inv_rot, -point[:, :self.n, self.n:])
last_line = point[:, self.n:, :]
inverse_point = gs.concatenate((inv_rot, inv_trans), axis=2)
return gs.concatenate((inverse_point, last_line), axis=1)
raise ValueError('Invalid point_type, expected \'vector\' or '
'\'matrix\'.')
@geomstats.vectorization.decorator(['else', 'point', 'else', 'point_type'])
def jacobian_translation(self,
point,
left_or_right='left',
point_type=None):
"""Compute the Jacobian matrix resulting from translation.
Compute the matrix of the differential
of the left/right translations from the identity to point in SE(n).
Parameters
----------
point: array-like, shape=[n_samples, {dim, [n + 1, n + 1]}]
left_or_right: str, {'left', 'right'}, optional
default: 'left'
whether to compute the jacobian of the left or right translation
point_type : str, {'vector', 'matrix'}, optional
default: self.default_point_type
Returns
-------
jacobian : array-like, shape=[n_samples, dim]
The jacobian of the left / right translation
"""
if point_type is None:
point_type = self.default_point_type
if left_or_right not in ('left', 'right'):
raise ValueError('`left_or_right` must be `left` or `right`.')
rotations = self.rotations
translations = self.translations
dim_rotations = rotations.dim
dim_translations = translations.dim
point = self.regularize(point, point_type=point_type)
if point_type == 'vector':
n_points, _ = point.shape
rot_vec = point[:, :dim_rotations]
jacobian_rot = self.rotations.jacobian_translation(
point=rot_vec,
left_or_right=left_or_right,
point_type=point_type)
block_zeros_1 = gs.zeros(
(n_points, dim_rotations, dim_translations))
jacobian_block_line_1 = gs.concatenate(
[jacobian_rot, block_zeros_1], axis=2)
if left_or_right == 'left':
rot_mat = self.rotations.matrix_from_rotation_vector(rot_vec)
jacobian_trans = rot_mat
block_zeros_2 = gs.zeros(
(n_points, dim_translations, dim_rotations))
jacobian_block_line_2 = gs.concatenate(
[block_zeros_2, jacobian_trans], axis=2)
else:
inv_skew_mat = -self.rotations.skew_matrix_from_vector(rot_vec)
eye = gs.to_ndarray(gs.eye(self.n), to_ndim=3)
eye = gs.tile(eye, [n_points, 1, 1])
jacobian_block_line_2 = gs.concatenate([inv_skew_mat, eye],
axis=2)
return gs.concatenate(
[jacobian_block_line_1, jacobian_block_line_2], axis=1)
if point_type == 'matrix':
return point
raise ValueError('Invalid point_type, expected \'vector\' or '
'\'matrix\'.')
@geomstats.vectorization.decorator(['else', 'point', 'point_type'])
def exp_from_identity(self, tangent_vec, point_type=None):
"""Compute group exponential of the tangent vector at the identity.
Parameters
----------
tangent_vec: array-like, shape=[n_samples, {dim, [n + 1, n + 1]}]
point_type: str, {'vector', 'matrix'}, optional
default: self.default_point_type
Returns
-------
group_exp: array-like, shape=[n_samples, {dim, [n + 1, n + 1]}]
the group exponential of the tangent vectors calculated
at the identity
"""
if point_type == 'vector':
rotations = self.rotations
dim_rotations = rotations.dim
rot_vec = tangent_vec[:, :dim_rotations]
rot_vec = self.rotations.regularize(rot_vec, point_type=point_type)
translation = tangent_vec[:, dim_rotations:]
angle = gs.linalg.norm(rot_vec, axis=1)
angle = gs.to_ndarray(angle, to_ndim=2, axis=1)
skew_mat = self.rotations.skew_matrix_from_vector(rot_vec)
sq_skew_mat = gs.matmul(skew_mat, skew_mat)
mask_0 = gs.equal(angle, 0.)
mask_close_0 = gs.isclose(angle, 0.) & ~mask_0
mask_else = ~mask_0 & ~mask_close_0
mask_0_float = gs.cast(mask_0, gs.float32)
mask_close_0_float = gs.cast(mask_close_0, gs.float32)
mask_else_float = gs.cast(mask_else, gs.float32)
angle += mask_0_float * gs.ones_like(angle)
coef_1 = gs.zeros_like(angle)
coef_2 = gs.zeros_like(angle)
coef_1 += mask_0_float * 1. / 2. * gs.ones_like(angle)
coef_2 += mask_0_float * 1. / 6. * gs.ones_like(angle)
coef_1 += mask_close_0_float * (
TAYLOR_COEFFS_1_AT_0[0] + TAYLOR_COEFFS_1_AT_0[2] * angle**2 +
TAYLOR_COEFFS_1_AT_0[4] * angle**4 +
TAYLOR_COEFFS_1_AT_0[6] * angle**6)
coef_2 += mask_close_0_float * (
TAYLOR_COEFFS_2_AT_0[0] + TAYLOR_COEFFS_2_AT_0[2] * angle**2 +
TAYLOR_COEFFS_2_AT_0[4] * angle**4 +
TAYLOR_COEFFS_2_AT_0[6] * angle**6)
coef_1 += mask_else_float * ((1. - gs.cos(angle)) / angle**2)
coef_2 += mask_else_float * ((angle - gs.sin(angle)) / angle**3)
n_tangent_vecs, _ = tangent_vec.shape
exp_translation = gs.zeros((n_tangent_vecs, self.n))
for i in range(n_tangent_vecs):
translation_i = translation[i]
term_1_i = coef_1[i] * gs.dot(translation_i,
gs.transpose(skew_mat[i]))
term_2_i = coef_2[i] * gs.dot(translation_i,
gs.transpose(sq_skew_mat[i]))
mask_i_float = gs.get_mask_i_float(i, n_tangent_vecs)
exp_translation += gs.outer(
mask_i_float, translation_i + term_1_i + term_2_i)
group_exp = gs.concatenate([rot_vec, exp_translation], axis=1)
group_exp = self.regularize(group_exp, point_type=point_type)
return group_exp
if point_type == 'matrix':
return GeneralLinear.exp(tangent_vec)
raise ValueError('Invalid point_type, expected \'vector\' or '
'\'matrix\'.')
@geomstats.vectorization.decorator(['else', 'point', 'point_type'])
def log_from_identity(self, point, point_type=None):
"""Compute the group logarithm of the point at the identity.
Parameters
----------
point: array-like, shape=[n_samples, {dim, [n + 1, n + 1]}]
point_type: str, {'vector', 'matrix'}, optional
default: self.default_point_type
Returns
-------
group_log: array-like, shape=[n_samples, {dim, [n + 1, n + 1]}]
the group logarithm in the Lie algbra
"""
point = self.regularize(point, point_type=point_type)
rotations = self.rotations
dim_rotations = rotations.dim
if point_type == 'vector':
rot_vec = point[:, :dim_rotations]
angle = gs.linalg.norm(rot_vec, axis=1)
angle = gs.to_ndarray(angle, to_ndim=2, axis=1)
translation = point[:, dim_rotations:]
skew_rot_vec = rotations.skew_matrix_from_vector(rot_vec)
sq_skew_rot_vec = gs.matmul(skew_rot_vec, skew_rot_vec)
mask_close_0 = gs.isclose(angle, 0.)
mask_close_pi = gs.isclose(angle, gs.pi)
mask_else = ~mask_close_0 & ~mask_close_pi
mask_close_0_float = gs.cast(mask_close_0, gs.float32)
mask_close_pi_float = gs.cast(mask_close_pi, gs.float32)
mask_else_float = gs.cast(mask_else, gs.float32)
mask_0 = gs.isclose(angle, 0., atol=1e-6)
mask_0_float = gs.cast(mask_0, gs.float32)
angle += mask_0_float * gs.ones_like(angle)
coef_1 = -0.5 * gs.ones_like(angle)
coef_2 = gs.zeros_like(angle)
coef_2 += mask_close_0_float * (1. / 12. + angle**2 / 720. +
angle**4 / 30240. +
angle**6 / 1209600.)
delta_angle = angle - gs.pi
coef_2 += mask_close_pi_float * (
1. / PI2 + (PI2 - 8.) * delta_angle / (4. * PI3) -
((PI2 - 12.) * delta_angle**2 / (4. * PI4)) +
((-192. + 12. * PI2 + PI4) * delta_angle**3 / (48. * PI5)) -
((-240. + 12. * PI2 + PI4) * delta_angle**4 / (48. * PI6)) +
((-2880. + 120. * PI2 + 10. * PI4 + PI6) * delta_angle**5 /
(480. * PI7)) -
((-3360 + 120. * PI2 + 10. * PI4 + PI6) * delta_angle**6 /
(480. * PI8)))
psi = 0.5 * angle * gs.sin(angle) / (1 - gs.cos(angle))
coef_2 += mask_else_float * (1 - psi) / (angle**2)
n_points, _ = point.shape
log_translation = gs.zeros((n_points, self.n))
for i in range(n_points):
translation_i = translation[i]
term_1_i = coef_1[i] * gs.dot(translation_i,
gs.transpose(skew_rot_vec[i]))
term_2_i = coef_2[i] * gs.dot(translation_i,
gs.transpose(sq_skew_rot_vec[i]))
mask_i_float = gs.get_mask_i_float(i, n_points)
log_translation += gs.outer(
mask_i_float, translation_i + term_1_i + term_2_i)
return gs.concatenate([rot_vec, log_translation], axis=1)
if point_type == 'matrix':
return GeneralLinear.log(point)
raise ValueError('Invalid point_type, expected \'vector\' or '
'\'matrix\'.')
def random_uniform(self, n_samples=1, point_type=None):
"""Sample in SE(n) with the uniform distribution.
Parameters
----------
n_samples: int, optional
default: 1
point_type: str, {'vector', 'matrix'}, optional
default: self.default_point_type
Returns
-------
random_point: array-like,
shape=[n_samples, {dim, [n + 1, n + 1]}]
An array of random elements in SE(n) having the given point_type.
"""
if point_type is None:
point_type = self.default_point_type
random_translation = self.translations.random_uniform(n_samples)
if point_type == 'vector':
random_rot_vec = self.rotations.random_uniform(
n_samples, point_type=point_type)
return gs.concatenate([random_rot_vec, random_translation],
axis=-1)
if point_type == 'matrix':
random_rotation = self.rotations.random_uniform(
n_samples, point_type=point_type)
random_rotation = gs.to_ndarray(random_rotation, to_ndim=3)
random_translation = gs.to_ndarray(random_translation, to_ndim=2)
random_translation = gs.transpose(
gs.to_ndarray(random_translation, to_ndim=3, axis=1),
(0, 2, 1))
random_point = gs.concatenate(
(random_rotation, random_translation), axis=2)
last_line = gs.zeros((n_samples, 1, self.n + 1))
random_point = gs.concatenate((random_point, last_line), axis=1)
random_point = gs.assignment(random_point, 1, (-1, -1), axis=0)
if gs.shape(random_point)[0] == 1:
random_point = gs.squeeze(random_point, axis=0)
return random_point
raise ValueError('Invalid point_type, expected \'vector\' or '
'\'matrix\'.')
def _exponential_matrix(self, rot_vec):
"""Compute exponential of rotation matrix represented by rot_vec.
Parameters
----------
rot_vec : array-like, shape=[n_samples, dim]
Returns
-------
exponential_mat : The matrix exponential of rot_vec
"""
# TODO(nguigs): find usecase for this method
rot_vec = self.rotations.regularize(rot_vec)
n_rot_vecs, _ = rot_vec.shape
angle = gs.linalg.norm(rot_vec, axis=1)
angle = gs.to_ndarray(angle, to_ndim=2, axis=1)
skew_rot_vec = self.rotations.skew_matrix_from_vector(rot_vec)
coef_1 = gs.empty_like(angle)
coef_2 = gs.empty_like(coef_1)
mask_0 = gs.equal(angle, 0)
mask_0 = gs.squeeze(mask_0, axis=1)
mask_close_to_0 = gs.isclose(angle, 0)
mask_close_to_0 = gs.squeeze(mask_close_to_0, axis=1)
mask_else = ~mask_0 & ~mask_close_to_0
coef_1[mask_close_to_0] = (1. / 2. - angle[mask_close_to_0]**2 / 24.)
coef_2[mask_close_to_0] = (1. / 6. - angle[mask_close_to_0]**3 / 120.)
# TODO(nina): Check if the discontinuity at 0 is expected.
coef_1[mask_0] = 0
coef_2[mask_0] = 0
coef_1[mask_else] = (angle[mask_else]**(-2) *
(1. - gs.cos(angle[mask_else])))
coef_2[mask_else] = (angle[mask_else]**(-2) *
(1. -
(gs.sin(angle[mask_else]) / angle[mask_else])))
term_1 = gs.zeros((n_rot_vecs, self.n, self.n))
term_2 = gs.zeros_like(term_1)
for i in range(n_rot_vecs):
term_1[i] = gs.eye(self.n) + skew_rot_vec[i] * coef_1[i]
term_2[i] = gs.matmul(skew_rot_vec[i], skew_rot_vec[i]) * coef_2[i]
exponential_mat = term_1 + term_2
return exponential_mat
