def setup_method(self):
s2 = Hypersphere(dim=2)
r2 = Euclidean(dim=2)
r3 = s2.embedding_space
initial_point = [0.0, 0.0, 1.0]
initial_tangent_vec_a = [1.0, 0.0, 0.0]
initial_tangent_vec_b = [0.0, 1.0, 0.0]
initial_tangent_vec_c = [-1.0, 0.0, 0.0]
curve_fun_a = s2.metric.geodesic(
initial_point=initial_point, initial_tangent_vec=initial_tangent_vec_a
)
curve_fun_b = s2.metric.geodesic(
initial_point=initial_point, initial_tangent_vec=initial_tangent_vec_b
)
curve_fun_c = s2.metric.geodesic(
initial_point=initial_point, initial_tangent_vec=initial_tangent_vec_c
)
self.curve_fun_a = curve_fun_a
self.n_sampling_points = 10
self.sampling_times = gs.linspace(0.0, 1.0, self.n_sampling_points)
self.curve_a = curve_fun_a(self.sampling_times)
self.curve_b = curve_fun_b(self.sampling_times)
self.curve_c = curve_fun_c(self.sampling_times)
self.space_curves_in_euclidean_3d = DiscreteCurves(ambient_manifold=r3)
self.space_curves_in_sphere_2d = DiscreteCurves(ambient_manifold=s2)
self.space_closed_curves_in_euclidean_2d = ClosedDiscreteCurves(
ambient_manifold=r2
)
self.l2_metric_s2 = L2CurvesMetric(ambient_manifold=s2)
self.l2_metric_r3 = L2CurvesMetric(ambient_manifold=r3)
self.srv_metric_r3 = (
self.space_curves_in_euclidean_3d.square_root_velocity_metric
)
self.quotient_srv_metric_r3 = (
self.space_curves_in_euclidean_3d.quotient_square_root_velocity_metric
)
self.a = 1
self.b = 1
self.elastic_metric = ElasticMetric(self.a, self.b)
self.n_discretized_curves = 5
self.times = gs.linspace(0.0, 1.0, self.n_discretized_curves)
gs.random.seed(1234)
def test_srv_metric_pointwise_inner_products(self, times, curve_a, curve_b,
curve_c, n_discretized_curves,
n_sampling_points):
l2_metric_s2 = L2CurvesMetric(ambient_manifold=s2)
srv_metric_r3 = SRVMetric(ambient_manifold=r3)
curves_ab = l2_metric_s2.geodesic(curve_a, curve_b)
curves_bc = l2_metric_s2.geodesic(curve_b, curve_c)
curves_ab = curves_ab(times)
curves_bc = curves_bc(times)
tangent_vecs = l2_metric_s2.log(point=curves_bc, base_point=curves_ab)
result = srv_metric_r3.l2_metric.pointwise_inner_products(
tangent_vec_a=tangent_vecs,
tangent_vec_b=tangent_vecs,
base_curve=curves_ab)
expected_shape = (n_discretized_curves, n_sampling_points)
self.assertAllClose(gs.shape(result), expected_shape)
result = srv_metric_r3.l2_metric.pointwise_inner_products(
tangent_vec_a=tangent_vecs[0],
tangent_vec_b=tangent_vecs[0],
base_curve=curves_ab[0],
)
expected_shape = (n_sampling_points, )
self.assertAllClose(gs.shape(result), expected_shape)
def test_l2_metric_geodesic(self, ambient_manifold, curve_a, curve_b,
times, n_sampling_points):
"""Test the geodesic method of L2Metric."""
l2_metric_s2 = L2CurvesMetric(ambient_manifold=s2)
curves_ab = l2_metric_s2.geodesic(curve_a, curve_b)
curves_ab = curves_ab(times)
result = curves_ab
expected = []
for k in range(n_sampling_points):
geod = l2_metric_s2.ambient_metric.geodesic(
initial_point=curve_a[k, :], end_point=curve_b[k, :])
expected.append(geod(times))
expected = gs.stack(expected, axis=1)
self.assertAllClose(result, expected)
def test_srv_transform_and_inverse(self, times, curve_a, curve_b):
"""Test of SRVT and its inverse.
N.B: Here curves_ab are seen as curves in R3 and not S2.
"""
l2_metric_s2 = L2CurvesMetric(ambient_manifold=s2)
srv_metric_r3 = SRVMetric(ambient_manifold=r3)
curves_ab = l2_metric_s2.geodesic(curve_a, curve_b)
curves_ab = curves_ab(times)
curves = curves_ab
srv_curves = srv_metric_r3.srv_transform(curves)
starting_points = curves[:, 0, :]
result = srv_metric_r3.srv_transform_inverse(srv_curves, starting_points)
expected = curves
self.assertAllClose(result, expected)
def test_srv_norm(self, curve_a, curve_b, times):
l2_metric_s2 = L2CurvesMetric(ambient_manifold=s2)
srv_metric_r3 = SRVMetric(ambient_manifold=r3)
curves_ab = l2_metric_s2.geodesic(curve_a, curve_b)
curves_ab = curves_ab(times)
srvs_ab = srv_metric_r3.srv_transform(curves_ab)
result = srv_metric_r3.l2_curves_metric.norm(srvs_ab)
products = srvs_ab * srvs_ab
sums = [gs.sum(product) for product in products]
squared_norm = gs.array(sums) / (srvs_ab.shape[-2] + 1)
expected = gs.sqrt(squared_norm)
self.assertAllClose(result, expected)
result = result.shape
expected = [srvs_ab.shape[0]]
self.assertAllClose(result, expected)
def test_srv_inner_product(self, curve_a, curve_b, curve_c, times):
l2_metric_s2 = L2CurvesMetric(ambient_manifold=s2)
srv_metric_r3 = SRVMetric(ambient_manifold=r3)
curves_ab = l2_metric_s2.geodesic(curve_a, curve_b)
curves_bc = l2_metric_s2.geodesic(curve_b, curve_c)
curves_ab = curves_ab(times)
curves_bc = curves_bc(times)
srvs_ab = srv_metric_r3.srv_transform(curves_ab)
srvs_bc = srv_metric_r3.srv_transform(curves_bc)
result = srv_metric_r3.l2_curves_metric.inner_product(srvs_ab, srvs_bc)
products = srvs_ab * srvs_bc
expected = [gs.sum(product) for product in products]
expected = gs.array(expected) / (srvs_ab.shape[-2] + 1)
self.assertAllClose(result, expected)
result = result.shape
expected = [srvs_ab.shape[0]]
self.assertAllClose(result, expected)