Sign issue about quaternion_to_matrix and matrix_to_quaternion

Summary:
As reported on github, `matrix_to_quaternion` was incorrect for rotations by 180˚. We resolved the sign of the component `i` based on the sign of `i*r`, assuming `r > 0`, which is untrue if `r == 0`.

This diff handles special cases and ensures we use the non-zero elements to copy the sign from.

Reviewed By: bottler

Differential Revision: D29149465

fbshipit-source-id: cd508cc31567fc37ea3463dd7e8c8e8d5d64a235

pytorch3d/transforms/rotation_conversions.py

@@ -82,7 +82,7 @@ def _copysign(a, b):
     return torch.where(signs_differ, -a, a)
 
 
-def _sqrt_positive_part(x):
+def _sqrt_positive_part(x: torch.Tensor) -> torch.Tensor:
     """
     Returns torch.sqrt(torch.max(0, x))
     but with a zero subgradient where x is 0.
@@ -93,7 +93,7 @@ def _sqrt_positive_part(x):
     return ret
 
 
-def matrix_to_quaternion(matrix):
+def matrix_to_quaternion(matrix: torch.Tensor) -> torch.Tensor:
     """
     Convert rotations given as rotation matrices to quaternions.
 
@@ -105,17 +105,44 @@ def matrix_to_quaternion(matrix):
     """
     if matrix.size(-1) != 3 or matrix.size(-2) != 3:
         raise ValueError(f"Invalid rotation matrix  shape f{matrix.shape}.")
-    m00 = matrix[..., 0, 0]
-    m11 = matrix[..., 1, 1]
-    m22 = matrix[..., 2, 2]
-    o0 = 0.5 * _sqrt_positive_part(1 + m00 + m11 + m22)
-    x = 0.5 * _sqrt_positive_part(1 + m00 - m11 - m22)
-    y = 0.5 * _sqrt_positive_part(1 - m00 + m11 - m22)
-    z = 0.5 * _sqrt_positive_part(1 - m00 - m11 + m22)
-    o1 = _copysign(x, matrix[..., 2, 1] - matrix[..., 1, 2])
-    o2 = _copysign(y, matrix[..., 0, 2] - matrix[..., 2, 0])
-    o3 = _copysign(z, matrix[..., 1, 0] - matrix[..., 0, 1])
-    return torch.stack((o0, o1, o2, o3), -1)
+
+    batch_dim = matrix.shape[:-2]
+    m00, m01, m02, m10, m11, m12, m20, m21, m22 = torch.unbind(
+        matrix.reshape(*batch_dim, 9), dim=-1
+    )
+
+    q_abs = _sqrt_positive_part(
+        torch.stack(
+            [
+                1.0 + m00 + m11 + m22,
+                1.0 + m00 - m11 - m22,
+                1.0 - m00 + m11 - m22,
+                1.0 - m00 - m11 + m22,
+            ],
+            dim=-1,
+        )
+    )
+
+    # we produce the desired quaternion multiplied by each of r, i, j, k
+    quat_by_rijk = torch.stack(
+        [
+            torch.stack([q_abs[..., 0] ** 2, m21 - m12, m02 - m20, m10 - m01], dim=-1),
+            torch.stack([m21 - m12, q_abs[..., 1] ** 2, m10 + m01, m02 + m20], dim=-1),
+            torch.stack([m02 - m20, m10 + m01, q_abs[..., 2] ** 2, m12 + m21], dim=-1),
+            torch.stack([m10 - m01, m20 + m02, m21 + m12, q_abs[..., 3] ** 2], dim=-1),
+        ],
+        dim=-2,
+    )
+
+    # clipping is not important here; if q_abs is small, the candidate won't be picked
+    quat_candidates = quat_by_rijk / (2.0 * q_abs[..., None].clip(0.1))
+
+    # if not for numerical problems, quat_candidates[i] should be same (up to a sign),
+    # forall i; we pick the best-conditioned one (with the largest denominator)
+
+    return quat_candidates[
+        F.one_hot(q_abs.argmax(dim=-1), num_classes=4) > 0.5, :  # pyre-ignore[16]
+    ].reshape(*batch_dim, 4)
 
 
 def _axis_angle_rotation(axis: str, angle):

tests/test_rotation_conversions.py

@@ -4,7 +4,9 @@
 import itertools
 import math
 import unittest
+from typing import Optional, Union
 
+import numpy as np
 import torch
 from common_testing import TestCaseMixin
 from pytorch3d.transforms.rotation_conversions import (
@@ -64,7 +66,7 @@ class TestRotationConversion(TestCaseMixin, unittest.TestCase):
         """quat -> mtx -> quat"""
         data = random_quaternions(13, dtype=torch.float64)
         mdata = matrix_to_quaternion(quaternion_to_matrix(data))
-        self.assertClose(data, mdata)
+        self._assert_quaternions_close(data, mdata)
 
     def test_to_quat(self):
         """mtx -> quat -> mtx"""
@@ -146,8 +148,7 @@ class TestRotationConversion(TestCaseMixin, unittest.TestCase):
         b_matrix = quaternion_to_matrix(b)
         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.assertClose(ab, ab_from_matrix)
+        self._assert_quaternions_close(ab, ab_from_matrix)
 
     def test_matrix_to_quaternion_corner_case(self):
         """Check no bad gradients from sqrt(0)."""
@@ -161,7 +162,34 @@ class TestRotationConversion(TestCaseMixin, unittest.TestCase):
         loss.backward()
         optimizer.step()
 
-        self.assertClose(matrix, 0.95 * torch.eye(3))
+        self.assertClose(matrix, matrix, msg="Result has non-finite values")
+        delta = 1e-2
+        self.assertLess(
+            matrix.trace(),
+            3.0 - delta,
+            msg="Identity initialisation unchanged by a gradient step",
+        )
+
+    def test_matrix_to_quaternion_by_pi(self):
+        # We check that rotations by pi around each of the 26
+        # nonzero vectors containing nothing but 0, 1 and -1
+        # are mapped to the right quaternions.
+        # This is representative across the directions.
+        options = [0.0, -1.0, 1.0]
+        axes = [
+            torch.tensor(vec)
+            for vec in itertools.islice(  # exclude [0, 0, 0]
+                itertools.product(options, options, options), 1, None
+            )
+        ]
+
+        axes = torch.nn.functional.normalize(torch.stack(axes), dim=-1)
+        # Rotation by pi around unit vector x is given by
+        # the matrix 2 x x^T - Id.
+        R = 2 * torch.matmul(axes[..., None], axes[..., None, :]) - torch.eye(3)
+        quats_hat = matrix_to_quaternion(R)
+        R_hat = quaternion_to_matrix(quats_hat)
+        self.assertClose(R, R_hat, atol=1e-3)
 
     def test_from_axis_angle(self):
         """axis_angle -> mtx -> axis_angle"""
@@ -228,3 +256,20 @@ class TestRotationConversion(TestCaseMixin, unittest.TestCase):
         self.assertClose(
             torch.matmul(r, r.permute(0, 2, 1)), torch.eye(3).expand_as(r), atol=1e-6
         )
+
+    def _assert_quaternions_close(
+        self,
+        input: Union[torch.Tensor, np.ndarray],
+        other: Union[torch.Tensor, np.ndarray],
+        *,
+        rtol: float = 1e-05,
+        atol: float = 1e-08,
+        equal_nan: bool = False,
+        msg: Optional[str] = None,
+    ):
+        self.assertEqual(np.shape(input), np.shape(other))
+        dot = (input * other).sum(-1)
+        ones = torch.ones_like(dot)
+        self.assertClose(
+            dot.abs(), ones, rtol=rtol, atol=atol, equal_nan=equal_nan, msg=msg
+        )