def pointwise_inner_product(self, tangent_vec_a, tangent_vec_b,
base_curve):
"""Compute the pointwise inner product of pair of tangent vectors.
Compute the point-wise inner-product between two tangent vectors
at a base curve.
Parameters
----------
tangent_vec_a : array-like, shape=[..., n_sampling_points, ambient_dim]
Tangent vector to discrete curve.
tangent_vec_b : array-like, shape=[..., n_sampling_points, ambient_dim]
Tangent vector to discrete curve.
base_curve : array-like, shape=[..., n_sampling_points, ambient_dim]
Point representing a discrete curve.
Returns
-------
inner_prod : array-like, shape=[..., n_sampling_points]
Point-wise inner-product.
"""
def inner_prod_aux(vec_a, vec_b, curve):
inner_prod = self.ambient_metric.inner_product(vec_a, vec_b, curve)
return gs.squeeze(inner_prod)
inner_prod = gs.vectorize(
(tangent_vec_a, tangent_vec_b, base_curve),
inner_prod_aux,
dtype=gs.float32,
multiple_args=True,
signature='(i,j),(i,j),(i,j)->(i)')
return inner_prod
def pointwise_inner_product(self, tangent_vec_a, tangent_vec_b,
base_curve):
"""
Compute the inner products of the components of a (series of)
pair(s) of tangent vectors at (a) base curve(s).
"""
base_curve = gs.to_ndarray(base_curve, to_ndim=3)
tangent_vec_a = gs.to_ndarray(tangent_vec_a, to_ndim=3)
tangent_vec_b = gs.to_ndarray(tangent_vec_b, to_ndim=3)
n_tangent_vecs = tangent_vec_a.shape[0]
n_sampling_points = tangent_vec_a.shape[1]
inner_prod = gs.zeros([n_tangent_vecs, n_sampling_points])
def inner_prod_aux(vec_a, vec_b, curve):
inner_prod = self.embedding_metric.inner_product(
vec_a, vec_b, curve)
return gs.squeeze(inner_prod)
inner_prod = gs.vectorize((tangent_vec_a, tangent_vec_b, base_curve),
lambda x, y, z: inner_prod_aux(x, y, z),
dtype=gs.float32,
multiple_args=True,
signature='(i,j),(i,j),(i,j)->(i)')
return inner_prod
def landmarks_on_geodesic(t):
t = gs.cast(t, gs.float32)
t = gs.to_ndarray(t, to_ndim=1)
tangent_vecs = gs.einsum("...,...ij->...ij", t,
initial_tangent_vec)
def point_on_landmarks(tangent_vec):
if gs.ndim(tangent_vec) < 2:
raise RuntimeError
exp = self.exp(tangent_vec=tangent_vec,
base_point=initial_point)
return exp
landmarks_at_time_t = gs.vectorize(tangent_vecs,
point_on_landmarks,
signature="(i,j)->(i,j)")
return landmarks_at_time_t
def landmarks_on_geodesic(t):
t = gs.cast(t, gs.float32)
t = gs.to_ndarray(t, to_ndim=1)
t = gs.to_ndarray(t, to_ndim=2, axis=1)
new_initial_landmarks = gs.to_ndarray(initial_landmarks,
to_ndim=landmarks_ndim + 1)
new_initial_tangent_vec = gs.to_ndarray(initial_tangent_vec,
to_ndim=landmarks_ndim + 1)
tangent_vecs = gs.einsum('il,nkm->ikm', t, new_initial_tangent_vec)
def point_on_landmarks(tangent_vec):
assert gs.ndim(tangent_vec) >= 2
exp = self.exp(tangent_vec=tangent_vec,
base_landmarks=new_initial_landmarks)
return exp
landmarks_at_time_t = gs.vectorize(tangent_vecs,
point_on_landmarks,
signature='(i,j)->(i,j)')
return landmarks_at_time_t
def curve_on_geodesic(t):
t = gs.cast(t, gs.float32)
t = gs.to_ndarray(t, to_ndim=1)
t = gs.to_ndarray(t, to_ndim=2, axis=1)
new_initial_curve = gs.to_ndarray(initial_curve,
to_ndim=curve_ndim + 1)
new_initial_tangent_vec = gs.to_ndarray(initial_tangent_vec,
to_ndim=curve_ndim + 1)
tangent_vecs = gs.einsum('il,nkm->ikm', t, new_initial_tangent_vec)
def point_on_curve(tangent_vec):
assert gs.ndim(tangent_vec) >= 2
exp = self.exp(tangent_vec=tangent_vec,
base_curve=new_initial_curve)
return exp
curve_at_time_t = gs.vectorize(tangent_vecs,
point_on_curve,
signature='(i,j)->(i,j)')
return curve_at_time_t
def landmarks_on_geodesic(t):
t = gs.cast(t, gs.float32)
t = gs.to_ndarray(t, to_ndim=1)
t = gs.to_ndarray(t, to_ndim=2, axis=1)
new_initial_landmarks = gs.to_ndarray(initial_landmarks,
to_ndim=landmarks_ndim + 1)
new_initial_tangent_vec = gs.to_ndarray(initial_tangent_vec,
to_ndim=landmarks_ndim + 1)
tangent_vecs = gs.einsum("il,nkm->ikm", t, new_initial_tangent_vec)
def point_on_landmarks(tangent_vec):
if gs.ndim(tangent_vec) < 2:
raise RuntimeError
exp = self.exp(tangent_vec=tangent_vec,
base_point=new_initial_landmarks)
return exp
landmarks_at_time_t = gs.vectorize(tangent_vecs,
point_on_landmarks,
signature="(i,j)->(i,j)")
return landmarks_at_time_t
