File size: 12,866 Bytes
15fa80a
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
# Copyright (c) 2018-present, Facebook, Inc.
# All rights reserved.
#
# This source code is licensed under the license found in the
# LICENSE file in the root directory of this source tree.
#

import torch
import numpy as np

_EPS4 = np.finfo(float).eps * 4.0

_FLOAT_EPS = np.finfo(np.float64).eps

# PyTorch-backed implementations
def qinv(q):
    assert q.shape[-1] == 4, 'q must be a tensor of shape (*, 4)'
    mask = torch.ones_like(q)
    mask[..., 1:] = -mask[..., 1:]
    return q * mask


def qinv_np(q):
    assert q.shape[-1] == 4, 'q must be a tensor of shape (*, 4)'
    return qinv(torch.from_numpy(q).float()).numpy()


def qnormalize(q):
    assert q.shape[-1] == 4, 'q must be a tensor of shape (*, 4)'
    return q / torch.norm(q, dim=-1, keepdim=True)


def qmul(q, r):
    """
    Multiply quaternion(s) q with quaternion(s) r.
    Expects two equally-sized tensors of shape (*, 4), where * denotes any number of dimensions.
    Returns q*r as a tensor of shape (*, 4).
    """
    assert q.shape[-1] == 4
    assert r.shape[-1] == 4

    original_shape = q.shape

    # Compute outer product
    terms = torch.bmm(r.view(-1, 4, 1), q.view(-1, 1, 4))

    w = terms[:, 0, 0] - terms[:, 1, 1] - terms[:, 2, 2] - terms[:, 3, 3]
    x = terms[:, 0, 1] + terms[:, 1, 0] - terms[:, 2, 3] + terms[:, 3, 2]
    y = terms[:, 0, 2] + terms[:, 1, 3] + terms[:, 2, 0] - terms[:, 3, 1]
    z = terms[:, 0, 3] - terms[:, 1, 2] + terms[:, 2, 1] + terms[:, 3, 0]
    return torch.stack((w, x, y, z), dim=1).view(original_shape)


def qrot(q, v):
    """
    Rotate vector(s) v about the rotation described by quaternion(s) q.
    Expects a tensor of shape (*, 4) for q and a tensor of shape (*, 3) for v,
    where * denotes any number of dimensions.
    Returns a tensor of shape (*, 3).
    """
    assert q.shape[-1] == 4
    assert v.shape[-1] == 3
    assert q.shape[:-1] == v.shape[:-1]

    original_shape = list(v.shape)
    # print(q.shape)
    q = q.contiguous().view(-1, 4)
    v = v.contiguous().view(-1, 3)

    qvec = q[:, 1:]
    uv = torch.cross(qvec, v, dim=1)
    uuv = torch.cross(qvec, uv, dim=1)
    return (v + 2 * (q[:, :1] * uv + uuv)).view(original_shape)


def qeuler(q, order, epsilon=0, deg=True):
    """
    Convert quaternion(s) q to Euler angles.
    Expects a tensor of shape (*, 4), where * denotes any number of dimensions.
    Returns a tensor of shape (*, 3).
    """
    assert q.shape[-1] == 4

    original_shape = list(q.shape)
    original_shape[-1] = 3
    q = q.view(-1, 4)

    q0 = q[:, 0]
    q1 = q[:, 1]
    q2 = q[:, 2]
    q3 = q[:, 3]

    if order == 'xyz':
        x = torch.atan2(2 * (q0 * q1 - q2 * q3), 1 - 2 * (q1 * q1 + q2 * q2))
        y = torch.asin(torch.clamp(2 * (q1 * q3 + q0 * q2), -1 + epsilon, 1 - epsilon))
        z = torch.atan2(2 * (q0 * q3 - q1 * q2), 1 - 2 * (q2 * q2 + q3 * q3))
    elif order == 'yzx':
        x = torch.atan2(2 * (q0 * q1 - q2 * q3), 1 - 2 * (q1 * q1 + q3 * q3))
        y = torch.atan2(2 * (q0 * q2 - q1 * q3), 1 - 2 * (q2 * q2 + q3 * q3))
        z = torch.asin(torch.clamp(2 * (q1 * q2 + q0 * q3), -1 + epsilon, 1 - epsilon))
    elif order == 'zxy':
        x = torch.asin(torch.clamp(2 * (q0 * q1 + q2 * q3), -1 + epsilon, 1 - epsilon))
        y = torch.atan2(2 * (q0 * q2 - q1 * q3), 1 - 2 * (q1 * q1 + q2 * q2))
        z = torch.atan2(2 * (q0 * q3 - q1 * q2), 1 - 2 * (q1 * q1 + q3 * q3))
    elif order == 'xzy':
        x = torch.atan2(2 * (q0 * q1 + q2 * q3), 1 - 2 * (q1 * q1 + q3 * q3))
        y = torch.atan2(2 * (q0 * q2 + q1 * q3), 1 - 2 * (q2 * q2 + q3 * q3))
        z = torch.asin(torch.clamp(2 * (q0 * q3 - q1 * q2), -1 + epsilon, 1 - epsilon))
    elif order == 'yxz':
        x = torch.asin(torch.clamp(2 * (q0 * q1 - q2 * q3), -1 + epsilon, 1 - epsilon))
        y = torch.atan2(2 * (q1 * q3 + q0 * q2), 1 - 2 * (q1 * q1 + q2 * q2))
        z = torch.atan2(2 * (q1 * q2 + q0 * q3), 1 - 2 * (q1 * q1 + q3 * q3))
    elif order == 'zyx':
        x = torch.atan2(2 * (q0 * q1 + q2 * q3), 1 - 2 * (q1 * q1 + q2 * q2))
        y = torch.asin(torch.clamp(2 * (q0 * q2 - q1 * q3), -1 + epsilon, 1 - epsilon))
        z = torch.atan2(2 * (q0 * q3 + q1 * q2), 1 - 2 * (q2 * q2 + q3 * q3))
    else:
        raise

    if deg:
        return torch.stack((x, y, z), dim=1).view(original_shape) * 180 / np.pi
    else:
        return torch.stack((x, y, z), dim=1).view(original_shape)


# Numpy-backed implementations

def qmul_np(q, r):
    q = torch.from_numpy(q).contiguous().float()
    r = torch.from_numpy(r).contiguous().float()
    return qmul(q, r).numpy()


def qrot_np(q, v):
    q = torch.from_numpy(q).contiguous().float()
    v = torch.from_numpy(v).contiguous().float()
    return qrot(q, v).numpy()


def qeuler_np(q, order, epsilon=0, use_gpu=False):
    if use_gpu:
        q = torch.from_numpy(q).cuda().float()
        return qeuler(q, order, epsilon).cpu().numpy()
    else:
        q = torch.from_numpy(q).contiguous().float()
        return qeuler(q, order, epsilon).numpy()


def qfix(q):
    """
    Enforce quaternion continuity across the time dimension by selecting
    the representation (q or -q) with minimal distance (or, equivalently, maximal dot product)
    between two consecutive frames.

    Expects a tensor of shape (L, J, 4), where L is the sequence length and J is the number of joints.
    Returns a tensor of the same shape.
    """
    assert len(q.shape) == 3
    assert q.shape[-1] == 4

    result = q.copy()
    dot_products = np.sum(q[1:] * q[:-1], axis=2)
    mask = dot_products < 0
    mask = (np.cumsum(mask, axis=0) % 2).astype(bool)
    result[1:][mask] *= -1
    return result


def euler2quat(e, order, deg=True):
    """
    Convert Euler angles to quaternions.
    """
    assert e.shape[-1] == 3

    original_shape = list(e.shape)
    original_shape[-1] = 4

    e = e.view(-1, 3)

    ## if euler angles in degrees
    if deg:
        e = e * np.pi / 180.

    x = e[:, 0]
    y = e[:, 1]
    z = e[:, 2]

    rx = torch.stack((torch.cos(x / 2), torch.sin(x / 2), torch.zeros_like(x), torch.zeros_like(x)), dim=1)
    ry = torch.stack((torch.cos(y / 2), torch.zeros_like(y), torch.sin(y / 2), torch.zeros_like(y)), dim=1)
    rz = torch.stack((torch.cos(z / 2), torch.zeros_like(z), torch.zeros_like(z), torch.sin(z / 2)), dim=1)

    result = None
    for coord in order:
        if coord == 'x':
            r = rx
        elif coord == 'y':
            r = ry
        elif coord == 'z':
            r = rz
        else:
            raise
        if result is None:
            result = r
        else:
            result = qmul(result, r)

    # Reverse antipodal representation to have a non-negative "w"
    if order in ['xyz', 'yzx', 'zxy']:
        result *= -1

    return result.view(original_shape)


def expmap_to_quaternion(e):
    """
    Convert axis-angle rotations (aka exponential maps) to quaternions.
    Stable formula from "Practical Parameterization of Rotations Using the Exponential Map".
    Expects a tensor of shape (*, 3), where * denotes any number of dimensions.
    Returns a tensor of shape (*, 4).
    """
    assert e.shape[-1] == 3

    original_shape = list(e.shape)
    original_shape[-1] = 4
    e = e.reshape(-1, 3)

    theta = np.linalg.norm(e, axis=1).reshape(-1, 1)
    w = np.cos(0.5 * theta).reshape(-1, 1)
    xyz = 0.5 * np.sinc(0.5 * theta / np.pi) * e
    return np.concatenate((w, xyz), axis=1).reshape(original_shape)


def euler_to_quaternion(e, order):
    """
    Convert Euler angles to quaternions.
    """
    assert e.shape[-1] == 3

    original_shape = list(e.shape)
    original_shape[-1] = 4

    e = e.reshape(-1, 3)

    x = e[:, 0]
    y = e[:, 1]
    z = e[:, 2]

    rx = np.stack((np.cos(x / 2), np.sin(x / 2), np.zeros_like(x), np.zeros_like(x)), axis=1)
    ry = np.stack((np.cos(y / 2), np.zeros_like(y), np.sin(y / 2), np.zeros_like(y)), axis=1)
    rz = np.stack((np.cos(z / 2), np.zeros_like(z), np.zeros_like(z), np.sin(z / 2)), axis=1)

    result = None
    for coord in order:
        if coord == 'x':
            r = rx
        elif coord == 'y':
            r = ry
        elif coord == 'z':
            r = rz
        else:
            raise
        if result is None:
            result = r
        else:
            result = qmul_np(result, r)

    # Reverse antipodal representation to have a non-negative "w"
    if order in ['xyz', 'yzx', 'zxy']:
        result *= -1

    return result.reshape(original_shape)


def quaternion_to_matrix(quaternions):
    """
    Convert rotations given as quaternions to rotation matrices.
    Args:
        quaternions: quaternions with real part first,
            as tensor of shape (..., 4).
    Returns:
        Rotation matrices as tensor of shape (..., 3, 3).
    """
    r, i, j, k = torch.unbind(quaternions, -1)
    two_s = 2.0 / (quaternions * quaternions).sum(-1)

    o = torch.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),
        ),
        -1,
    )
    return o.reshape(quaternions.shape[:-1] + (3, 3))


def quaternion_to_matrix_np(quaternions):
    q = torch.from_numpy(quaternions).contiguous().float()
    return quaternion_to_matrix(q).numpy()


def quaternion_to_cont6d_np(quaternions):
    rotation_mat = quaternion_to_matrix_np(quaternions)
    cont_6d = np.concatenate([rotation_mat[..., 0], rotation_mat[..., 1]], axis=-1)
    return cont_6d


def quaternion_to_cont6d(quaternions):
    rotation_mat = quaternion_to_matrix(quaternions)
    cont_6d = torch.cat([rotation_mat[..., 0], rotation_mat[..., 1]], dim=-1)
    return cont_6d


def cont6d_to_matrix(cont6d):
    assert cont6d.shape[-1] == 6, "The last dimension must be 6"
    x_raw = cont6d[..., 0:3]
    y_raw = cont6d[..., 3:6]

    x = x_raw / torch.norm(x_raw, dim=-1, keepdim=True)
    z = torch.cross(x, y_raw, dim=-1)
    z = z / torch.norm(z, dim=-1, keepdim=True)

    y = torch.cross(z, x, dim=-1)

    x = x[..., None]
    y = y[..., None]
    z = z[..., None]

    mat = torch.cat([x, y, z], dim=-1)
    return mat


def cont6d_to_matrix_np(cont6d):
    q = torch.from_numpy(cont6d).contiguous().float()
    return cont6d_to_matrix(q).numpy()


def qpow(q0, t, dtype=torch.float):
    ''' q0 : tensor of quaternions
    t: tensor of powers
    '''
    q0 = qnormalize(q0)
    theta0 = torch.acos(q0[..., 0])

    ## if theta0 is close to zero, add epsilon to avoid NaNs
    mask = (theta0 <= 10e-10) * (theta0 >= -10e-10)
    theta0 = (1 - mask) * theta0 + mask * 10e-10
    v0 = q0[..., 1:] / torch.sin(theta0).view(-1, 1)

    if isinstance(t, torch.Tensor):
        q = torch.zeros(t.shape + q0.shape)
        theta = t.view(-1, 1) * theta0.view(1, -1)
    else:  ## if t is a number
        q = torch.zeros(q0.shape)
        theta = t * theta0

    q[..., 0] = torch.cos(theta)
    q[..., 1:] = v0 * torch.sin(theta).unsqueeze(-1)

    return q.to(dtype)


def qslerp(q0, q1, t):
    '''
    q0: starting quaternion
    q1: ending quaternion
    t: array of points along the way

    Returns:
    Tensor of Slerps: t.shape + q0.shape
    '''

    q0 = qnormalize(q0)
    q1 = qnormalize(q1)
    q_ = qpow(qmul(q1, qinv(q0)), t)

    return qmul(q_,
                q0.contiguous().view(torch.Size([1] * len(t.shape)) + q0.shape).expand(t.shape + q0.shape).contiguous())


def qbetween(v0, v1):
    '''
    find the quaternion used to rotate v0 to v1
    '''
    assert v0.shape[-1] == 3, 'v0 must be of the shape (*, 3)'
    assert v1.shape[-1] == 3, 'v1 must be of the shape (*, 3)'

    v = torch.cross(v0, v1)
    w = torch.sqrt((v0 ** 2).sum(dim=-1, keepdim=True) * (v1 ** 2).sum(dim=-1, keepdim=True)) + (v0 * v1).sum(dim=-1,
                                                                                                              keepdim=True)
    return qnormalize(torch.cat([w, v], dim=-1))


def qbetween_np(v0, v1):
    '''
    find the quaternion used to rotate v0 to v1
    '''
    assert v0.shape[-1] == 3, 'v0 must be of the shape (*, 3)'
    assert v1.shape[-1] == 3, 'v1 must be of the shape (*, 3)'

    v0 = torch.from_numpy(v0).float()
    v1 = torch.from_numpy(v1).float()
    return qbetween(v0, v1).numpy()


def lerp(p0, p1, t):
    if not isinstance(t, torch.Tensor):
        t = torch.Tensor([t])

    new_shape = t.shape + p0.shape
    new_view_t = t.shape + torch.Size([1] * len(p0.shape))
    new_view_p = torch.Size([1] * len(t.shape)) + p0.shape
    p0 = p0.view(new_view_p).expand(new_shape)
    p1 = p1.view(new_view_p).expand(new_shape)
    t = t.view(new_view_t).expand(new_shape)

    return p0 + t * (p1 - p0)