axis_angle representation of rotations

Summary: We can represent a rotation as a vector in the axis direction, whose length is the rotation anticlockwise in radians around that axis.

Reviewed By: gkioxari

Differential Revision: D24306293

fbshipit-source-id: 2e0f138eda8329f6cceff600a6e5f17a00e4deb7

Author: bottler

pytorch3d/transforms/rotation_conversions.py

@@ -413,6 +413,101 @@ def quaternion_apply(quaternion, point):
     return out[..., 1:]
 
 
+def axis_angle_to_matrix(axis_angle):
+    """
+    Convert rotations given as axis/angle to rotation matrices.
+
+    Args:
+        axis_angle: Rotations given as a vector in axis angle form,
+            as a tensor of shape (..., 3), where the magnitude is
+            the angle turned anticlockwise in radians around the
+            vector's direction.
+
+    Returns:
+        Rotation matrices as tensor of shape (..., 3, 3).
+    """
+    return quaternion_to_matrix(axis_angle_to_quaternion(axis_angle))
+
+
+def matrix_to_axis_angle(matrix):
+    """
+    Convert rotations given as rotation matrices to axis/angle.
+
+    Args:
+        matrix: Rotation matrices as tensor of shape (..., 3, 3).
+
+    Returns:
+        Rotations given as a vector in axis angle form, as a tensor
+            of shape (..., 3), where the magnitude is the angle
+            turned anticlockwise in radians around the vector's
+            direction.
+    """
+    return quaternion_to_axis_angle(matrix_to_quaternion(matrix))
+
+
+def axis_angle_to_quaternion(axis_angle):
+    """
+    Convert rotations given as axis/angle to quaternions.
+
+    Args:
+        axis_angle: Rotations given as a vector in axis angle form,
+            as a tensor of shape (..., 3), where the magnitude is
+            the angle turned anticlockwise in radians around the
+            vector's direction.
+
+    Returns:
+        quaternions with real part first, as tensor of shape (..., 4).
+    """
+    angles = torch.norm(axis_angle, p=2, dim=-1, keepdim=True)
+    half_angles = 0.5 * angles
+    eps = 1e-6
+    small_angles = angles.abs() < eps
+    sin_half_angles_over_angles = torch.empty_like(angles)
+    sin_half_angles_over_angles[~small_angles] = (
+        torch.sin(half_angles[~small_angles]) / angles[~small_angles]
+    )
+    # for x small, sin(x/2) is about x/2 - (x/2)^3/6
+    # so sin(x/2)/x is about 1/2 - (x*x)/48
+    sin_half_angles_over_angles[small_angles] = (
+        0.5 - torch.square(angles[small_angles]) / 48
+    )
+    quaternions = torch.cat(
+        [torch.cos(half_angles), axis_angle * sin_half_angles_over_angles], dim=-1
+    )
+    return quaternions
+
+
+def quaternion_to_axis_angle(quaternions):
+    """
+    Convert rotations given as quaternions to axis/angle.
+
+    Args:
+        quaternions: quaternions with real part first,
+            as tensor of shape (..., 4).
+
+    Returns:
+        Rotations given as a vector in axis angle form, as a tensor
+            of shape (..., 3), where the magnitude is the angle
+            turned anticlockwise in radians around the vector's
+            direction.
+    """
+    norms = torch.norm(quaternions[..., 1:], p=2, dim=-1, keepdim=True)
+    half_angles = torch.atan2(norms, quaternions[..., :1])
+    angles = 2 * half_angles
+    eps = 1e-6
+    small_angles = angles.abs() < eps
+    sin_half_angles_over_angles = torch.empty_like(angles)
+    sin_half_angles_over_angles[~small_angles] = (
+        torch.sin(half_angles[~small_angles]) / angles[~small_angles]
+    )
+    # for x small, sin(x/2) is about x/2 - (x/2)^3/6
+    # so sin(x/2)/x is about 1/2 - (x*x)/48
+    sin_half_angles_over_angles[small_angles] = (
+        0.5 - torch.square(angles[small_angles]) / 48
+    )
+    return quaternions[..., 1:] / sin_half_angles_over_angles
+
+
 def rotation_6d_to_matrix(d6: torch.Tensor) -> torch.Tensor:
     """
     Converts 6D rotation representation by Zhou et al. [1] to rotation matrix

tests/test_rotation_conversions.py

@@ -8,12 +8,16 @@
 import torch
 from common_testing import TestCaseMixin
 from pytorch3d.transforms.rotation_conversions import (
+    axis_angle_to_matrix,
+    axis_angle_to_quaternion,
     euler_angles_to_matrix,
+    matrix_to_axis_angle,
     matrix_to_euler_angles,
     matrix_to_quaternion,
     matrix_to_rotation_6d,
     quaternion_apply,
     quaternion_multiply,
+    quaternion_to_axis_angle,
     quaternion_to_matrix,
     random_quaternions,
     random_rotation,
@@ -60,13 +64,13 @@ def test_from_quat(self):
         """quat -> mtx -> quat"""
         data = random_quaternions(13, dtype=torch.float64)
         mdata = matrix_to_quaternion(quaternion_to_matrix(data))
-        self.assertTrue(torch.allclose(data, mdata))
+        self.assertClose(data, mdata)
 
     def test_to_quat(self):
         """mtx -> quat -> mtx"""
         data = random_rotations(13, dtype=torch.float64)
         mdata = quaternion_to_matrix(matrix_to_quaternion(data))
-        self.assertTrue(torch.allclose(data, mdata))
+        self.assertClose(data, mdata)
 
     def test_quat_grad_exists(self):
         """Quaternion calculations are differentiable."""
@@ -107,21 +111,21 @@ def test_from_euler(self):
         for convention in self._tait_bryan_conventions():
             matrices = euler_angles_to_matrix(data, convention)
             mdata = matrix_to_euler_angles(matrices, convention)
-            self.assertTrue(torch.allclose(data, mdata))
+            self.assertClose(data, mdata)
 
         data[:, 1] += half_pi
         for convention in self._proper_euler_conventions():
             matrices = euler_angles_to_matrix(data, convention)
             mdata = matrix_to_euler_angles(matrices, convention)
-            self.assertTrue(torch.allclose(data, mdata))
+            self.assertClose(data, mdata)
 
     def test_to_euler(self):
         """mtx -> euler -> mtx"""
         data = random_rotations(13, dtype=torch.float64)
         for convention in self._all_euler_angle_conventions():
             euler_angles = matrix_to_euler_angles(data, convention)
             mdata = euler_angles_to_matrix(euler_angles, convention)
-            self.assertTrue(torch.allclose(data, mdata))
+            self.assertClose(data, mdata)
 
     def test_euler_grad_exists(self):
         """Euler angle calculations are differentiable."""
@@ -143,7 +147,7 @@ def test_quaternion_multiplication(self):
         ab_matrix = torch.matmul(a_matrix, b_matrix)
         ab_from_matrix = matrix_to_quaternion(ab_matrix)
         self.assertEqual(ab.shape, ab_from_matrix.shape)
-        self.assertTrue(torch.allclose(ab, ab_from_matrix))
+        self.assertClose(ab, ab_from_matrix)
 
     def test_matrix_to_quaternion_corner_case(self):
         """Check no bad gradients from sqrt(0)."""
@@ -159,14 +163,39 @@ def test_matrix_to_quaternion_corner_case(self):
 
         self.assertClose(matrix, 0.95 * torch.eye(3))
 
+    def test_from_axis_angle(self):
+        """axis_angle -> mtx -> axis_angle"""
+        n_repetitions = 20
+        data = torch.rand(n_repetitions, 3)
+        matrices = axis_angle_to_matrix(data)
+        mdata = matrix_to_axis_angle(matrices)
+        self.assertClose(data, mdata, atol=2e-6)
+
+    def test_from_axis_angle_has_grad(self):
+        n_repetitions = 20
+        data = torch.rand(n_repetitions, 3, requires_grad=True)
+        matrices = axis_angle_to_matrix(data)
+        mdata = matrix_to_axis_angle(matrices)
+        quats = axis_angle_to_quaternion(data)
+        mdata2 = quaternion_to_axis_angle(quats)
+        (grad,) = torch.autograd.grad(mdata.sum() + mdata2.sum(), data)
+        self.assertTrue(torch.isfinite(grad).all())
+
+    def test_to_axis_angle(self):
+        """mtx -> axis_angle -> mtx"""
+        data = random_rotations(13, dtype=torch.float64)
+        euler_angles = matrix_to_axis_angle(data)
+        mdata = axis_angle_to_matrix(euler_angles)
+        self.assertClose(data, mdata)
+
     def test_quaternion_application(self):
         """Applying a quaternion is the same as applying the matrix."""
         quaternions = random_quaternions(3, torch.float64, requires_grad=True)
         matrices = quaternion_to_matrix(quaternions)
         points = torch.randn(3, 3, dtype=torch.float64, requires_grad=True)
         transform1 = quaternion_apply(quaternions, points)
         transform2 = torch.matmul(matrices, points[..., None])[..., 0]
-        self.assertTrue(torch.allclose(transform1, transform2))
+        self.assertClose(transform1, transform2)
 
         [p, q] = torch.autograd.grad(transform1.sum(), [points, quaternions])
         self.assertTrue(torch.isfinite(p).all())