Increase performance for conversions including axis angles (#1948)

Summary:
This is an extension of https://github.com/facebookresearch/pytorch3d/issues/1544 with various speed, stability, and readability improvements. (I could not find a way to make a commit to the existing PR). This PR is still based on the [Rodrigues' rotation formula](https://en.wikipedia.org/wiki/Rotation_formalisms_in_three_dimensions#Rotation_matrix_%E2%86%94_Euler_axis/angle).

The motivation is the same; this change speeds up the conversions up to 10x, depending on the device, batch size, etc.

### Notes
- As the angles get very close to `π`, the existing implementation and the proposed one start to differ. However, (my understanding is that) this is not a problem as the axis can not be stably inferred from the rotation matrix in this case in general.
- bottler , I tried to follow similar conventions as existing functions to deal with weird angles, let me know if something needs to be changed to merge this.

Pull Request resolved: https://github.com/facebookresearch/pytorch3d/pull/1948

Reviewed By: MichaelRamamonjisoa

Differential Revision: D69193009

Pulled By: bottler

fbshipit-source-id: e5ed34b45b625114ec4419bb89e22a6aefad4eeb

pytorch3d/transforms/rotation_conversions.py

@@ -463,7 +463,7 @@ def quaternion_apply(quaternion: torch.Tensor, point: torch.Tensor) -> torch.Ten
     return out[..., 1:]
 
 
-def axis_angle_to_matrix(axis_angle: torch.Tensor) -> torch.Tensor:
+def axis_angle_to_matrix(axis_angle: torch.Tensor, fast: bool = False) -> torch.Tensor:
     """
     Convert rotations given as axis/angle to rotation matrices.
 
@@ -472,27 +472,93 @@ def axis_angle_to_matrix(axis_angle: torch.Tensor) -> torch.Tensor:
             as a tensor of shape (..., 3), where the magnitude is
             the angle turned anticlockwise in radians around the
             vector's direction.
+        fast: Whether to use the new faster implementation (based on the
+            Rodrigues formula) instead of the original implementation (which
+            first converted to a quaternion and then back to a rotation matrix).
 
     Returns:
         Rotation matrices as tensor of shape (..., 3, 3).
     """
-    return quaternion_to_matrix(axis_angle_to_quaternion(axis_angle))
+    if not fast:
+        return quaternion_to_matrix(axis_angle_to_quaternion(axis_angle))
+
+    shape = axis_angle.shape
+    device, dtype = axis_angle.device, axis_angle.dtype
+
+    angles = torch.norm(axis_angle, p=2, dim=-1, keepdim=True).unsqueeze(-1)
+
+    rx, ry, rz = axis_angle[..., 0], axis_angle[..., 1], axis_angle[..., 2]
+    zeros = torch.zeros(shape[:-1], dtype=dtype, device=device)
+    cross_product_matrix = torch.stack(
+        [zeros, -rz, ry, rz, zeros, -rx, -ry, rx, zeros], dim=-1
+    ).view(shape + (3,))
+    cross_product_matrix_sqrd = cross_product_matrix @ cross_product_matrix
+
+    identity = torch.eye(3, dtype=dtype, device=device)
+    angles_sqrd = angles * angles
+    angles_sqrd = torch.where(angles_sqrd == 0, 1, angles_sqrd)
+    return (
+        identity.expand(cross_product_matrix.shape)
+        + torch.sinc(angles / torch.pi) * cross_product_matrix
+        + ((1 - torch.cos(angles)) / angles_sqrd) * cross_product_matrix_sqrd
+    )
 
 
-def matrix_to_axis_angle(matrix: torch.Tensor) -> torch.Tensor:
+def matrix_to_axis_angle(matrix: torch.Tensor, fast: bool = False) -> torch.Tensor:
     """
     Convert rotations given as rotation matrices to axis/angle.
 
     Args:
         matrix: Rotation matrices as tensor of shape (..., 3, 3).
+        fast: Whether to use the new faster implementation (based on the
+            Rodrigues formula) instead of the original implementation (which
+            first converted to a quaternion and then back to a rotation matrix).
 
     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))
+    if not fast:
+        return quaternion_to_axis_angle(matrix_to_quaternion(matrix))
+
+    if matrix.size(-1) != 3 or matrix.size(-2) != 3:
+        raise ValueError(f"Invalid rotation matrix shape {matrix.shape}.")
+
+    omegas = torch.stack(
+        [
+            matrix[..., 2, 1] - matrix[..., 1, 2],
+            matrix[..., 0, 2] - matrix[..., 2, 0],
+            matrix[..., 1, 0] - matrix[..., 0, 1],
+        ],
+        dim=-1,
+    )
+    norms = torch.norm(omegas, p=2, dim=-1, keepdim=True)
+    traces = torch.diagonal(matrix, dim1=-2, dim2=-1).sum(-1).unsqueeze(-1)
+    angles = torch.atan2(norms, traces - 1)
+
+    zeros = torch.zeros(3, dtype=matrix.dtype, device=matrix.device)
+    omegas = torch.where(torch.isclose(angles, torch.zeros_like(angles)), zeros, omegas)
+
+    near_pi = torch.isclose(((traces - 1) / 2).abs(), torch.ones_like(traces)).squeeze(
+        -1
+    )
+
+    axis_angles = torch.empty_like(omegas)
+    axis_angles[~near_pi] = (
+        0.5 * omegas[~near_pi] / torch.sinc(angles[~near_pi] / torch.pi)
+    )
+
+    # this derives from: nnT = (R + 1) / 2
+    n = 0.5 * (
+        matrix[near_pi][..., 0, :]
+        + torch.eye(1, 3, dtype=matrix.dtype, device=matrix.device)
+    )
+    axis_angles[near_pi] = angles[near_pi] * n / torch.norm(n)
+
+    return axis_angles
 
 
 def axis_angle_to_quaternion(axis_angle: torch.Tensor) -> torch.Tensor:
@@ -509,22 +575,10 @@ def axis_angle_to_quaternion(axis_angle: torch.Tensor) -> torch.Tensor:
         quaternions with real part first, as tensor of shape (..., 4).
     """
     angles = torch.norm(axis_angle, p=2, dim=-1, keepdim=True)
-    half_angles = angles * 0.5
+    sin_half_angles_over_angles = 0.5 * torch.sinc(angles * 0.5 / torch.pi)
-    eps = 1e-6
+    return torch.cat(
-    small_angles = angles.abs() < eps
+        [torch.cos(angles * 0.5), axis_angle * sin_half_angles_over_angles], dim=-1
-    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 - (angles[small_angles] * 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: torch.Tensor) -> torch.Tensor:
@@ -543,18 +597,9 @@ def quaternion_to_axis_angle(quaternions: torch.Tensor) -> torch.Tensor:
     """
     norms = torch.norm(quaternions[..., 1:], p=2, dim=-1, keepdim=True)
     half_angles = torch.atan2(norms, quaternions[..., :1])
-    angles = 2 * half_angles
+    sin_half_angles_over_angles = 0.5 * torch.sinc(half_angles / torch.pi)
-    eps = 1e-6
+    # angles/2 are between [-pi/2, pi/2], thus sin_half_angles_over_angles
-    small_angles = angles.abs() < eps
+    # can't be zero
-    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 - (angles[small_angles] * angles[small_angles]) / 48
-    )
     return quaternions[..., 1:] / sin_half_angles_over_angles
 
 

tests/test_rotation_conversions.py

@@ -204,6 +204,9 @@ class TestRotationConversion(TestCaseMixin, unittest.TestCase):
         n_repetitions = 20
         data = torch.rand(n_repetitions, 3)
         matrices = axis_angle_to_matrix(data)
+        self.assertClose(data, matrix_to_axis_angle(matrices), atol=2e-6)
+        self.assertClose(data, matrix_to_axis_angle(matrices, fast=True), atol=2e-6)
+        matrices = axis_angle_to_matrix(data, fast=True)
         mdata = matrix_to_axis_angle(matrices)
         self.assertClose(data, mdata, atol=2e-6)
 
@@ -221,8 +224,10 @@ class TestRotationConversion(TestCaseMixin, unittest.TestCase):
         """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)
+        euler_angles_fast = matrix_to_axis_angle(data)
-        self.assertClose(data, mdata)
+        self.assertClose(data, axis_angle_to_matrix(euler_angles))
+        self.assertClose(data, axis_angle_to_matrix(euler_angles_fast))
+        self.assertClose(data, axis_angle_to_matrix(euler_angles, fast=True))
 
     def test_quaternion_application(self):
         """Applying a quaternion is the same as applying the matrix."""