def grad(y_pred, y_true,
metric=SO3.bi_invariant_metric,
representation='vector'):
"""Closed-form for the gradient of pose_loss.
Parameters
----------
y_pred : array-like
Prediction on SO(3).
y_true : array-like
Ground-truth on SO(3).
metric : RiemannianMetric
Metric used to compute the loss and gradient.
representation : str, {'vector', 'matrix'}
Representation chosen for points in SE(3).
Returns
-------
lie_grad : array-like
Tangent vector at point y_pred.
"""
y_pred = gs.expand_dims(y_pred, axis=0)
y_true = gs.expand_dims(y_true, axis=0)
if representation == 'vector':
lie_grad = lie_group.grad(y_pred, y_true, SO3, metric)
if representation == 'quaternion':
quat_scalar = y_pred[:, :1]
quat_vec = y_pred[:, 1:]
quat_vec_norm = gs.linalg.norm(quat_vec, axis=1)
quat_sq_norm = quat_vec_norm ** 2 + quat_scalar ** 2
quat_arctan2 = gs.arctan2(quat_vec_norm, quat_scalar)
differential_scalar = - 2 * quat_vec / (quat_sq_norm)
differential_scalar = gs.to_ndarray(differential_scalar, to_ndim=2)
differential_scalar = gs.transpose(differential_scalar)
differential_vec = (2 * (quat_scalar / quat_sq_norm
- 2 * quat_arctan2 / quat_vec_norm)
* (gs.einsum('ni,nj->nij', quat_vec, quat_vec)
/ quat_vec_norm ** 2)
+ 2 * quat_arctan2 / quat_vec_norm * gs.eye(3))
differential_vec = gs.squeeze(differential_vec)
differential = gs.concatenate(
[differential_scalar, differential_vec],
axis=1)
y_pred = SO3.rotation_vector_from_quaternion(y_pred)
y_true = SO3.rotation_vector_from_quaternion(y_true)
lie_grad = lie_group.grad(y_pred, y_true, SO3, metric)
lie_grad = gs.matmul(lie_grad, differential)
lie_grad = gs.squeeze(lie_grad, axis=0)
return lie_grad
def grad(y_pred,
y_true,
metric=SE3.left_canonical_metric,
representation='vector'):
"""
Closed-form for the gradient of pose_loss.
:return: tangent vector at point y_pred.
"""
if gs.ndim(y_pred) == 1:
y_pred = gs.expand_dims(y_pred, axis=0)
if gs.ndim(y_true) == 1:
y_true = gs.expand_dims(y_true, axis=0)
if representation == 'vector':
grad = lie_group.grad(y_pred, y_true, SE3, metric)
if representation == 'quaternion':
y_pred_rot_vec = SO3.rotation_vector_from_quaternion(y_pred[:, :4])
y_pred_pose = gs.hstack([y_pred_rot_vec, y_pred[:, 4:]])
y_true_rot_vec = SO3.rotation_vector_from_quaternion(y_true[:, :4])
y_true_pose = gs.hstack([y_true_rot_vec, y_true[:, 4:]])
grad = lie_group.grad(y_pred_pose, y_true_pose, SE3, metric)
quat_scalar = y_pred[:, :1]
quat_vec = y_pred[:, 1:4]
quat_vec_norm = gs.linalg.norm(quat_vec, axis=1)
quat_sq_norm = quat_vec_norm**2 + quat_scalar**2
quat_arctan2 = gs.arctan2(quat_vec_norm, quat_scalar)
differential_scalar = -2 * quat_vec / (quat_sq_norm)
differential_vec = (
2 *
(quat_scalar / quat_sq_norm - 2 * quat_arctan2 / quat_vec_norm) *
(gs.einsum('ni,nj->nij', quat_vec, quat_vec) / quat_vec_norm *
quat_vec_norm) + 2 * quat_arctan2 / quat_vec_norm * gs.eye(3))
differential_scalar_t = gs.transpose(differential_scalar, axes=(1, 0))
upper_left_block = gs.hstack(
(differential_scalar_t, differential_vec[0]))
upper_right_block = gs.zeros((3, 3))
lower_right_block = gs.eye(3)
lower_left_block = gs.zeros((3, 4))
top = gs.hstack((upper_left_block, upper_right_block))
bottom = gs.hstack((lower_left_block, lower_right_block))
differential = gs.vstack((top, bottom))
differential = gs.expand_dims(differential, axis=0)
grad = gs.einsum('ni,nij->ni', grad, differential)
grad = gs.squeeze(grad, axis=0)
return grad
def grad(y_pred, y_true,
metric=SO3.bi_invariant_metric,
representation='vector'):
y_pred = gs.expand_dims(y_pred, axis=0)
y_true = gs.expand_dims(y_true, axis=0)
if representation == 'vector':
grad = lie_group.grad(y_pred, y_true, SO3, metric)
if representation == 'quaternion':
quat_scalar = y_pred[:, :1]
quat_vec = y_pred[:, 1:]
quat_vec_norm = gs.linalg.norm(quat_vec, axis=1)
quat_sq_norm = quat_vec_norm ** 2 + quat_scalar ** 2
quat_arctan2 = gs.arctan2(quat_vec_norm, quat_scalar)
differential_scalar = - 2 * quat_vec / (quat_sq_norm)
differential_scalar = gs.to_ndarray(differential_scalar, to_ndim=2)
differential_scalar = gs.transpose(differential_scalar)
differential_vec = (2 * (quat_scalar / quat_sq_norm
- 2 * quat_arctan2 / quat_vec_norm)
* (gs.einsum('ni,nj->nij', quat_vec, quat_vec)
/ quat_vec_norm ** 2)
+ 2 * quat_arctan2 / quat_vec_norm * gs.eye(3))
differential_vec = gs.squeeze(differential_vec)
differential = gs.concatenate(
[differential_scalar, differential_vec],
axis=1)
y_pred = SO3.rotation_vector_from_quaternion(y_pred)
y_true = SO3.rotation_vector_from_quaternion(y_true)
grad = lie_group.grad(y_pred, y_true, SO3, metric)
grad = gs.matmul(grad, differential)
grad = gs.squeeze(grad, axis=0)
return grad
def grad(y_pred,
y_true,
metric=SE3.left_canonical_metric,
representation='vector'):
"""Closed-form for the gradient of pose_loss.
Parameters
----------
y_pred : array-like
Prediction on SE(3).
y_true : array-like
Ground-truth on SE(3).
metric : RiemannianMetric
Metric used to compute the loss and gradient.
representation : str, {'vector', 'matrix'}
Representation chosen for points in SE(3).
Returns
-------
lie_grad : array-like
Tangent vector at point y_pred.
"""
if gs.ndim(y_pred) == 1:
y_pred = gs.expand_dims(y_pred, axis=0)
if gs.ndim(y_true) == 1:
y_true = gs.expand_dims(y_true, axis=0)
if representation == 'vector':
lie_grad = lie_group.grad(y_pred, y_true, SE3, metric)
if representation == 'quaternion':
y_pred_rot_vec = SO3.rotation_vector_from_quaternion(y_pred[:, :4])
y_pred_pose = gs.hstack([y_pred_rot_vec, y_pred[:, 4:]])
y_true_rot_vec = SO3.rotation_vector_from_quaternion(y_true[:, :4])
y_true_pose = gs.hstack([y_true_rot_vec, y_true[:, 4:]])
lie_grad = lie_group.grad(y_pred_pose, y_true_pose, SE3, metric)
quat_scalar = y_pred[:, :1]
quat_vec = y_pred[:, 1:4]
quat_vec_norm = gs.linalg.norm(quat_vec, axis=1)
quat_sq_norm = quat_vec_norm**2 + quat_scalar**2
quat_arctan2 = gs.arctan2(quat_vec_norm, quat_scalar)
differential_scalar = -2 * quat_vec / (quat_sq_norm)
differential_vec = (
2 *
(quat_scalar / quat_sq_norm - 2 * quat_arctan2 / quat_vec_norm) *
(gs.einsum('ni,nj->nij', quat_vec, quat_vec) / quat_vec_norm *
quat_vec_norm) + 2 * quat_arctan2 / quat_vec_norm * gs.eye(3))
differential_scalar_t = gs.transpose(differential_scalar, axes=(1, 0))
upper_left_block = gs.hstack(
(differential_scalar_t, differential_vec[0]))
upper_right_block = gs.zeros((3, 3))
lower_right_block = gs.eye(3)
lower_left_block = gs.zeros((3, 4))
top = gs.hstack((upper_left_block, upper_right_block))
bottom = gs.hstack((lower_left_block, lower_right_block))
differential = gs.vstack((top, bottom))
differential = gs.expand_dims(differential, axis=0)
lie_grad = gs.einsum('ni,nij->ni', lie_grad, differential)
lie_grad = gs.squeeze(lie_grad, axis=0)
return lie_grad