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04afe87 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 | """Rotation conversion utilities for MEI representation.
All functions support arbitrary batch dimensions (..., ).
Coordinate convention: Y-up, right-handed (X-right, Y-up, Z-forward).
Core rotation functions ported from HY-Motion (hymotion/utils/geometry.py),
torch -> numpy.
"""
import numpy as np
# ============================================================
# Helpers (from HY-Motion)
# ============================================================
def _sqrt_positive_part(x: np.ndarray) -> np.ndarray:
"""Returns np.sqrt(np.maximum(0, x))."""
ret = np.zeros_like(x)
positive_mask = x > 0
ret[positive_mask] = np.sqrt(x[positive_mask])
return ret
def standardize_quaternion(quaternions: np.ndarray) -> np.ndarray:
"""
Convert a unit quaternion to a standard form: one in which the real
part is non negative.
Args:
quaternions: Quaternions with real part first,
as array of shape (..., 4).
Returns:
Standardized quaternions as array of shape (..., 4).
"""
return np.where(quaternions[..., 0:1] < 0, -quaternions, quaternions)
# ============================================================
# Axis-angle <-> Quaternion <-> Rotation matrix
# ============================================================
def axis_angle_to_quaternion(axis_angle: np.ndarray) -> np.ndarray:
"""Convert rotations given as axis/angle to quaternions.
Args:
axis_angle: Rotations given as a vector in axis angle form,
as an array 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 array of shape (..., 4).
"""
angles = np.linalg.norm(axis_angle, axis=-1, keepdims=True)
half_angles = angles * 0.5
# sin(angle/2) / angle, exact; limit -> 0.5 as angle -> 0
nonzero = angles != 0
safe_angles = np.where(nonzero, angles, np.ones_like(angles))
sin_half_angles_over_angles = np.where(
nonzero, np.sin(half_angles) / safe_angles, 0.5
)
quaternions = np.concatenate(
[np.cos(half_angles), axis_angle * sin_half_angles_over_angles], axis=-1
)
return quaternions
def quaternion_to_matrix(quaternions: np.ndarray) -> np.ndarray:
"""Convert rotations given as quaternions to rotation matrices.
Args:
quaternions: quaternions with real part first,
as array of shape (..., 4).
Returns:
Rotation matrices as array of shape (..., 3, 3).
"""
r, i, j, k = (
quaternions[..., 0],
quaternions[..., 1],
quaternions[..., 2],
quaternions[..., 3],
)
two_s = 2.0 / (quaternions * quaternions).sum(-1)
o = np.stack(
(
1 - two_s * (j * j + k * k),
two_s * (i * j - k * r),
two_s * (i * k + j * r),
two_s * (i * j + k * r),
1 - two_s * (i * i + k * k),
two_s * (j * k - i * r),
two_s * (i * k - j * r),
two_s * (j * k + i * r),
1 - two_s * (i * i + j * j),
),
axis=-1,
)
return o.reshape(quaternions.shape[:-1] + (3, 3))
def axis_angle_to_matrix(axis_angle: np.ndarray) -> np.ndarray:
"""Convert rotations given as axis/angle to rotation matrices.
Args:
axis_angle: Rotations given as a vector in axis angle form,
as an array of shape (..., 3), where the magnitude is
the angle turned anticlockwise in radians around the
vector's direction.
Returns:
Rotation matrices as array of shape (..., 3, 3).
"""
return quaternion_to_matrix(axis_angle_to_quaternion(axis_angle))
def matrix_to_quaternion(matrix: np.ndarray) -> np.ndarray:
"""Convert rotations given as rotation matrices to quaternions.
Args:
matrix: Rotation matrices as array of shape (..., 3, 3).
Returns:
quaternions with real part first, as array of shape (..., 4).
"""
if matrix.shape[-1] != 3 or matrix.shape[-2] != 3:
raise ValueError(f"Invalid rotation matrix shape {matrix.shape}.")
batch_dim = matrix.shape[:-2]
m00, m01, m02, m10, m11, m12, m20, m21, m22 = np.split(
matrix.reshape(batch_dim + (9,)), 9, axis=-1
)
m00 = m00[..., 0]
m01 = m01[..., 0]
m02 = m02[..., 0]
m10 = m10[..., 0]
m11 = m11[..., 0]
m12 = m12[..., 0]
m20 = m20[..., 0]
m21 = m21[..., 0]
m22 = m22[..., 0]
q_abs = _sqrt_positive_part(
np.stack(
[
1.0 + m00 + m11 + m22,
1.0 + m00 - m11 - m22,
1.0 - m00 + m11 - m22,
1.0 - m00 - m11 + m22,
],
axis=-1,
)
)
# we produce the desired quaternion multiplied by each of r, i, j, k
quat_by_rijk = np.stack(
[
np.stack(
[q_abs[..., 0] ** 2, m21 - m12, m02 - m20, m10 - m01], axis=-1
),
np.stack(
[m21 - m12, q_abs[..., 1] ** 2, m10 + m01, m02 + m20], axis=-1
),
np.stack(
[m02 - m20, m10 + m01, q_abs[..., 2] ** 2, m12 + m21], axis=-1
),
np.stack(
[m10 - m01, m20 + m02, m21 + m12, q_abs[..., 3] ** 2], axis=-1
),
],
axis=-2,
)
# We floor here at 0.1 but the exact level is not important; if q_abs is small,
# the candidate won't be picked.
flr = 0.1
quat_candidates = quat_by_rijk / (2.0 * np.maximum(q_abs[..., None], flr))
# 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)
best = q_abs.argmax(axis=-1) # (*batch_dim,)
# Advanced indexing to select the best candidate per element
flat_candidates = quat_candidates.reshape(-1, 4, 4)
flat_best = best.reshape(-1)
out = flat_candidates[np.arange(flat_candidates.shape[0]), flat_best, :]
out = out.reshape(batch_dim + (4,))
return standardize_quaternion(out)
def quaternion_to_axis_angle(quaternions: np.ndarray) -> np.ndarray:
"""Convert rotations given as quaternions to axis/angle.
Args:
quaternions: quaternions with real part first,
as array of shape (..., 4).
Returns:
Rotations given as a vector in axis angle form, as an array
of shape (..., 3), where the magnitude is the angle
turned anticlockwise in radians around the vector's
direction.
"""
norms = np.linalg.norm(quaternions[..., 1:], axis=-1, keepdims=True)
half_angles = np.arctan2(norms, quaternions[..., :1])
angles = 2 * half_angles
# sin(half_angle) / angle, exact; limit -> 0.5 as angle -> 0
nonzero = angles != 0
safe_angles = np.where(nonzero, angles, np.ones_like(angles))
sin_half_angles_over_angles = np.where(
nonzero, np.sin(half_angles) / safe_angles, 0.5
)
return quaternions[..., 1:] / sin_half_angles_over_angles
def matrix_to_axis_angle(matrix: np.ndarray) -> np.ndarray:
"""Convert rotations given as rotation matrices to axis/angle.
Args:
matrix: Rotation matrices as array of shape (..., 3, 3).
Returns:
Rotations given as a vector in axis angle form, as an array
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))
# ============================================================
# 6D continuous rotation representation (Zhou et al., CVPR 2019)
# ============================================================
def rotation_6d_to_matrix(rot6d: np.ndarray) -> np.ndarray:
"""Convert 6D rotation representation to 3x3 rotation matrix.
Based on Zhou et al., "On the Continuity of Rotation Representations
in Neural Networks", CVPR 2019.
Args:
rot6d: array of shape (*, 6) of 6d rotation representations.
Returns:
rotation matrices of size (*, 3, 3).
"""
x = rot6d.reshape(*rot6d.shape[:-1], 3, 2)
a1 = x[..., 0]
a2 = x[..., 1]
b1 = a1 / np.maximum(np.linalg.norm(a1, axis=-1, keepdims=True), 1e-12)
b2 = a2 - np.sum(b1 * a2, axis=-1, keepdims=True) * b1
b2 = b2 / np.maximum(np.linalg.norm(b2, axis=-1, keepdims=True), 1e-12)
b3 = np.cross(b1, b2, axis=-1)
return np.stack((b1, b2, b3), axis=-1)
def matrix_to_rotation_6d(matrix: np.ndarray) -> np.ndarray:
"""Convert 3x3 rotation matrix to 6D rotation representation.
Args:
matrix: rotation matrices of shape (*, 3, 3).
Returns:
6D rotation representation of shape (*, 6).
"""
v1 = matrix[..., 0:1]
v2 = matrix[..., 1:2]
rot6d = np.concatenate([v1, v2], axis=-1).reshape(*matrix.shape[:-2], 6)
return rot6d
# ============================================================
# Yaw (Y-axis) rotation helpers (MEI-specific)
# ============================================================
def yaw_rotation_matrix(angle: np.ndarray) -> np.ndarray:
"""Create rotation matrices for yaw (Y-axis rotation).
R_y(theta) maps local Z-forward to the heading direction in world XZ plane.
Args:
angle: (...) yaw angles in radians.
Returns:
R: (..., 3, 3) rotation matrices.
"""
c = np.cos(angle)
s = np.sin(angle)
z = np.zeros_like(angle)
o = np.ones_like(angle)
return np.stack([
c, z, s,
z, o, z,
-s, z, c,
], axis=-1).reshape(*angle.shape, 3, 3)
def wrap_angle(angle: np.ndarray) -> np.ndarray:
"""Wrap angle to [-pi, pi]."""
return (angle + np.pi) % (2 * np.pi) - np.pi
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