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
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
|
// Copyright (c) 2009 libmv authors.
//
// Permission is hereby granted, free of charge, to any person obtaining a copy
// of this software and associated documentation files (the "Software"), to
// deal in the Software without restriction, including without limitation the
// rights to use, copy, modify, merge, publish, distribute, sublicense, and/or
// sell copies of the Software, and to permit persons to whom the Software is
// furnished to do so, subject to the following conditions:
//
// The above copyright notice and this permission notice shall be included in
// all copies or substantial portions of the Software.
//
// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
// IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
// FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
// AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
// LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
// FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS
// IN THE SOFTWARE.
#include <cmath>
#include <limits>
#include <Eigen/SVD>
#include <Eigen/Geometry>
#include "libmv/base/vector.h"
#include "libmv/logging/logging.h"
#include "libmv/multiview/euclidean_resection.h"
#include "libmv/multiview/projection.h"
namespace libmv {
namespace euclidean_resection {
typedef unsigned int uint;
bool EuclideanResection(const Mat2X &x_camera,
const Mat3X &X_world,
Mat3 *R, Vec3 *t,
ResectionMethod method) {
switch (method) {
case RESECTION_ANSAR_DANIILIDIS:
EuclideanResectionAnsarDaniilidis(x_camera, X_world, R, t);
break;
case RESECTION_EPNP:
return EuclideanResectionEPnP(x_camera, X_world, R, t);
break;
default:
LOG(FATAL) << "Unknown resection method.";
}
return false;
}
bool EuclideanResection(const Mat &x_image,
const Mat3X &X_world,
const Mat3 &K,
Mat3 *R, Vec3 *t,
ResectionMethod method) {
CHECK(x_image.rows() == 2 || x_image.rows() == 3)
<< "Invalid size for x_image: "
<< x_image.rows() << "x" << x_image.cols();
Mat2X x_camera;
if (x_image.rows() == 2) {
EuclideanToNormalizedCamera(x_image, K, &x_camera);
} else if (x_image.rows() == 3) {
HomogeneousToNormalizedCamera(x_image, K, &x_camera);
}
return EuclideanResection(x_camera, X_world, R, t, method);
}
void AbsoluteOrientation(const Mat3X &X,
const Mat3X &Xp,
Mat3 *R,
Vec3 *t) {
int num_points = X.cols();
Vec3 C = X.rowwise().sum() / num_points; // Centroid of X.
Vec3 Cp = Xp.rowwise().sum() / num_points; // Centroid of Xp.
// Normalize the two point sets.
Mat3X Xn(3, num_points), Xpn(3, num_points);
for( int i = 0; i < num_points; ++i ){
Xn.col(i) = X.col(i) - C;
Xpn.col(i) = Xp.col(i) - Cp;
}
// Construct the N matrix (pg. 635).
double Sxx = Xn.row(0).dot(Xpn.row(0));
double Syy = Xn.row(1).dot(Xpn.row(1));
double Szz = Xn.row(2).dot(Xpn.row(2));
double Sxy = Xn.row(0).dot(Xpn.row(1));
double Syx = Xn.row(1).dot(Xpn.row(0));
double Sxz = Xn.row(0).dot(Xpn.row(2));
double Szx = Xn.row(2).dot(Xpn.row(0));
double Syz = Xn.row(1).dot(Xpn.row(2));
double Szy = Xn.row(2).dot(Xpn.row(1));
Mat4 N;
N << Sxx + Syy + Szz, Syz - Szy, Szx - Sxz, Sxy - Syx,
Syz - Szy, Sxx - Syy - Szz, Sxy + Syx, Szx + Sxz,
Szx - Sxz, Sxy + Syx, -Sxx + Syy - Szz, Syz + Szy,
Sxy - Syx, Szx + Sxz, Syz + Szy, -Sxx - Syy + Szz;
// Find the unit quaternion q that maximizes qNq. It is the eigenvector
// corresponding to the lagest eigenvalue.
Vec4 q = N.jacobiSvd(Eigen::ComputeFullU).matrixU().col(0);
// Retrieve the 3x3 rotation matrix.
Vec4 qq = q.array() * q.array();
double q0q1 = q(0) * q(1);
double q0q2 = q(0) * q(2);
double q0q3 = q(0) * q(3);
double q1q2 = q(1) * q(2);
double q1q3 = q(1) * q(3);
double q2q3 = q(2) * q(3);
(*R) << qq(0) + qq(1) - qq(2) - qq(3),
2 * (q1q2 - q0q3),
2 * (q1q3 + q0q2),
2 * (q1q2+ q0q3),
qq(0) - qq(1) + qq(2) - qq(3),
2 * (q2q3 - q0q1),
2 * (q1q3 - q0q2),
2 * (q2q3 + q0q1),
qq(0) - qq(1) - qq(2) + qq(3);
// Fix the handedness of the R matrix.
if (R->determinant() < 0) {
R->row(2) = -R->row(2);
}
// Compute the final translation.
*t = Cp - *R * C;
}
// Convert i and j indices of the original variables into their quadratic
// permutation single index. It follows that t_ij = t_ji.
static int IJToPointIndex(int i, int j, int num_points) {
// Always make sure that j is bigger than i. This handles t_ij = t_ji.
if (j < i) {
std::swap(i, j);
}
int idx;
int num_permutation_rows = num_points * (num_points - 1) / 2;
// All t_ii's are located at the end of the t vector after all t_ij's.
if (j == i) {
idx = num_permutation_rows + i;
} else {
int offset = (num_points - i - 1) * (num_points - i) / 2;
idx = (num_permutation_rows - offset + j - i - 1);
}
return idx;
};
// Convert i and j indexes of the solution for lambda to their linear indexes.
static int IJToIndex(int i, int j, int num_lambda) {
if (j < i) {
std::swap(i, j);
}
int A = num_lambda * (num_lambda + 1) / 2;
int B = num_lambda - i;
int C = B * (B + 1) / 2;
int idx = A - C + j - i;
return idx;
};
static int Sign(double value) {
return (value < 0) ? -1 : 1;
};
// Organizes a square matrix into a single row constraint on the elements of
// Lambda to create the constraints in equation (5) in "Linear Pose Estimation
// from Points or Lines", by Ansar, A. and Daniilidis, PAMI 2003. vol. 25, no.
// 5.
static Vec MatrixToConstraint(const Mat &A,
int num_k_columns,
int num_lambda) {
Vec C(num_k_columns);
C.setZero();
int idx = 0;
for (int i = 0; i < num_lambda; ++i) {
for( int j = i; j < num_lambda; ++j) {
C(idx) = A(i, j);
if (i != j){
C(idx) += A(j, i);
}
++ idx;
}
}
return C;
}
// Normalizes the columns of vectors.
static void NormalizeColumnVectors(Mat3X *vectors) {
int num_columns = vectors->cols();
for (int i = 0; i < num_columns; ++i){
vectors->col(i).normalize();
}
}
void EuclideanResectionAnsarDaniilidis(const Mat2X &x_camera,
const Mat3X &X_world,
Mat3 *R,
Vec3 *t) {
CHECK(x_camera.cols() == X_world.cols());
CHECK(x_camera.cols() > 3);
int num_points = x_camera.cols();
// Copy the normalized camera coords into 3 vectors and normalize them so
// that they are unit vectors from the camera center.
Mat3X x_camera_unit(3, num_points);
x_camera_unit.block(0, 0, 2, num_points) = x_camera;
x_camera_unit.row(2).setOnes();
NormalizeColumnVectors(&x_camera_unit);
int num_m_rows = num_points * (num_points - 1) / 2;
int num_tt_variables = num_points * (num_points + 1) / 2;
int num_m_columns = num_tt_variables + 1;
Mat M(num_m_columns, num_m_columns);
M.setZero();
Matu ij_index(num_tt_variables, 2);
// Create the constraint equations for the t_ij variables (7) and arrange
// them into the M matrix (8). Also store the initial (i, j) indices.
int row=0;
for (int i = 0; i < num_points; ++i) {
for (int j = i+1; j < num_points; ++j) {
M(row, row) = -2 * x_camera_unit.col(i).dot(x_camera_unit.col(j));
M(row, num_m_rows + i) = x_camera_unit.col(i).dot(x_camera_unit.col(i));
M(row, num_m_rows + j) = x_camera_unit.col(j).dot(x_camera_unit.col(j));
Vec3 Xdiff = X_world.col(i) - X_world.col(j);
double center_to_point_distance = Xdiff.norm();
M(row, num_m_columns - 1) =
- center_to_point_distance * center_to_point_distance;
ij_index(row, 0) = i;
ij_index(row, 1) = j;
++row;
}
ij_index(i + num_m_rows, 0) = i;
ij_index(i + num_m_rows, 1) = i;
}
int num_lambda = num_points + 1; // Dimension of the null space of M.
Mat V = M.jacobiSvd(Eigen::ComputeFullV).matrixV().block(0,
num_m_rows,
num_m_columns,
num_lambda);
// TODO(vess): The number of constraint equations in K (num_k_rows) must be
// (num_points + 1) * (num_points + 2)/2. This creates a performance issue
// for more than 4 points. It is fine for 4 points at the moment with 18
// instead of 15 equations.
int num_k_rows = num_m_rows + num_points *
(num_points*(num_points-1)/2 - num_points+1);
int num_k_columns = num_lambda * (num_lambda + 1) / 2;
Mat K(num_k_rows, num_k_columns);
K.setZero();
// Construct the first part of the K matrix corresponding to (t_ii, t_jk) for
// i != j.
int counter_k_row = 0;
for (int idx1 = num_m_rows; idx1 < num_tt_variables; ++idx1) {
for (int idx2 = 0; idx2 < num_m_rows; ++idx2) {
unsigned int i = ij_index(idx1, 0);
unsigned int j = ij_index(idx2, 0);
unsigned int k = ij_index(idx2, 1);
if( i != j && i != k ){
int idx3 = IJToPointIndex(i, j, num_points);
int idx4 = IJToPointIndex(i, k, num_points);
K.row(counter_k_row) =
MatrixToConstraint(V.row(idx1).transpose() * V.row(idx2)-
V.row(idx3).transpose() * V.row(idx4),
num_k_columns,
num_lambda);
++counter_k_row;
}
}
}
// Construct the second part of the K matrix corresponding to (t_ii,t_jk) for
// j==k.
for (int idx1 = num_m_rows; idx1 < num_tt_variables; ++idx1) {
for (int idx2 = idx1 + 1; idx2 < num_tt_variables; ++idx2) {
unsigned int i = ij_index(idx1, 0);
unsigned int j = ij_index(idx2, 0);
unsigned int k = ij_index(idx2, 1);
int idx3 = IJToPointIndex(i, j, num_points);
int idx4 = IJToPointIndex(i, k, num_points);
K.row(counter_k_row) =
MatrixToConstraint(V.row(idx1).transpose() * V.row(idx2)-
V.row(idx3).transpose() * V.row(idx4),
num_k_columns,
num_lambda);
++counter_k_row;
}
}
Vec L_sq = K.jacobiSvd(Eigen::ComputeFullV).matrixV().col(num_k_columns - 1);
// Pivot on the largest element for numerical stability. Afterwards recover
// the sign of the lambda solution.
double max_L_sq_value = fabs(L_sq(IJToIndex(0, 0, num_lambda)));
int max_L_sq_index = 1;
for (int i = 1; i < num_lambda; ++i) {
double abs_sq_value = fabs(L_sq(IJToIndex(i, i, num_lambda)));
if (max_L_sq_value < abs_sq_value) {
max_L_sq_value = abs_sq_value;
max_L_sq_index = i;
}
}
// Ensure positiveness of the largest value corresponding to lambda_ii.
L_sq = L_sq * Sign(L_sq(IJToIndex(max_L_sq_index,
max_L_sq_index,
num_lambda)));
Vec L(num_lambda);
L(max_L_sq_index) = sqrt(L_sq(IJToIndex(max_L_sq_index,
max_L_sq_index,
num_lambda)));
for (int i = 0; i < num_lambda; ++i) {
if (i != max_L_sq_index) {
L(i) = L_sq(IJToIndex(max_L_sq_index, i, num_lambda)) / L(max_L_sq_index);
}
}
// Correct the scale using the fact that the last constraint is equal to 1.
L = L / (V.row(num_m_columns - 1).dot(L));
Vec X = V * L;
// Recover the distances from the camera center to the 3D points Q.
Vec d(num_points);
d.setZero();
for (int c_point = num_m_rows; c_point < num_tt_variables; ++c_point) {
d(c_point - num_m_rows) = sqrt(X(c_point));
}
// Create the 3D points in the camera system.
Mat X_cam(3, num_points);
for (int c_point = 0; c_point < num_points; ++c_point ) {
X_cam.col(c_point) = d(c_point) * x_camera_unit.col(c_point);
}
// Recover the camera translation and rotation.
AbsoluteOrientation(X_world, X_cam, R, t);
}
// Selects 4 virtual control points using mean and PCA.
static void SelectControlPoints(const Mat3X &X_world,
Mat *X_centered,
Mat34 *X_control_points) {
size_t num_points = X_world.cols();
// The first virtual control point, C0, is the centroid.
Vec mean, variance;
MeanAndVarianceAlongRows(X_world, &mean, &variance);
X_control_points->col(0) = mean;
// Computes PCA
X_centered->resize (3, num_points);
for (size_t c = 0; c < num_points; c++) {
X_centered->col(c) = X_world.col (c) - mean;
}
Mat3 X_centered_sq = (*X_centered) * X_centered->transpose();
Eigen::JacobiSVD<Mat3> X_centered_sq_svd(X_centered_sq, Eigen::ComputeFullU);
Vec3 w = X_centered_sq_svd.singularValues();
Mat3 u = X_centered_sq_svd.matrixU();
for (size_t c = 0; c < 3; c++) {
double k = sqrt (w (c) / num_points);
X_control_points->col (c + 1) = mean + k * u.col (c);
}
}
// Computes the barycentric coordinates for all real points
static void ComputeBarycentricCoordinates(const Mat3X &X_world_centered,
const Mat34 &X_control_points,
Mat4X *alphas) {
size_t num_points = X_world_centered.cols();
Mat3 C2 ;
for (size_t c = 1; c < 4; c++) {
C2.col(c-1) = X_control_points.col(c) - X_control_points.col(0);
}
Mat3 C2inv = C2.inverse();
Mat3X a = C2inv * X_world_centered;
alphas->resize(4, num_points);
alphas->setZero();
alphas->block(1, 0, 3, num_points) = a;
for (size_t c = 0; c < num_points; c++) {
(*alphas)(0, c) = 1.0 - alphas->col(c).sum();
}
}
// Estimates the coordinates of all real points in the camera coordinate frame
static void ComputePointsCoordinatesInCameraFrame(
const Mat4X &alphas,
const Vec4 &betas,
const Eigen::Matrix<double, 12, 12> &U,
Mat3X *X_camera) {
size_t num_points = alphas.cols();
// Estimates the control points in the camera reference frame.
Mat34 C2b; C2b.setZero();
for (size_t cu = 0; cu < 4; cu++) {
for (size_t c = 0; c < 4; c++) {
C2b.col(c) += betas(cu) * U.block(11 - cu, c * 3, 1, 3).transpose();
}
}
// Estimates the 3D points in the camera reference frame
X_camera->resize(3, num_points);
for (size_t c = 0; c < num_points; c++) {
X_camera->col(c) = C2b * alphas.col(c);
}
// Check the sign of the z coordinate of the points (should be positive)
uint num_z_neg = 0;
for (size_t i = 0; i < X_camera->cols(); ++i) {
if ((*X_camera)(2,i) < 0) {
num_z_neg++;
}
}
// If more than 50% of z are negative, we change the signs
if (num_z_neg > 0.5 * X_camera->cols()) {
C2b = -C2b;
*X_camera = -(*X_camera);
}
}
bool EuclideanResectionEPnP(const Mat2X &x_camera,
const Mat3X &X_world,
Mat3 *R, Vec3 *t) {
CHECK(x_camera.cols() == X_world.cols());
CHECK(x_camera.cols() > 3);
size_t num_points = X_world.cols();
// Select the control points.
Mat34 X_control_points;
Mat X_centered;
SelectControlPoints(X_world, &X_centered, &X_control_points);
// Compute the barycentric coordinates.
Mat4X alphas(4, num_points);
ComputeBarycentricCoordinates(X_centered, X_control_points, &alphas);
// Estimates the M matrix with the barycentric coordinates
Mat M(2 * num_points, 12);
Eigen::Matrix<double, 2, 12> sub_M;
for (size_t c = 0; c < num_points; c++) {
double a0 = alphas(0, c);
double a1 = alphas(1, c);
double a2 = alphas(2, c);
double a3 = alphas(3, c);
double ui = x_camera(0, c);
double vi = x_camera(1, c);
M.block(2*c, 0, 2, 12) << a0, 0,
a0*(-ui), a1, 0,
a1*(-ui), a2, 0,
a2*(-ui), a3, 0,
a3*(-ui), 0,
a0, a0*(-vi), 0,
a1, a1*(-vi), 0,
a2, a2*(-vi), 0,
a3, a3*(-vi);
}
// TODO(julien): Avoid the transpose by rewriting the u2.block() calls.
Eigen::JacobiSVD<Mat> MtMsvd(M.transpose()*M, Eigen::ComputeFullU);
Eigen::Matrix<double, 12, 12> u2 = MtMsvd.matrixU().transpose();
// Estimate the L matrix.
Eigen::Matrix<double, 6, 3> dv1;
Eigen::Matrix<double, 6, 3> dv2;
Eigen::Matrix<double, 6, 3> dv3;
Eigen::Matrix<double, 6, 3> dv4;
dv1.row(0) = u2.block(11, 0, 1, 3) - u2.block(11, 3, 1, 3);
dv1.row(1) = u2.block(11, 0, 1, 3) - u2.block(11, 6, 1, 3);
dv1.row(2) = u2.block(11, 0, 1, 3) - u2.block(11, 9, 1, 3);
dv1.row(3) = u2.block(11, 3, 1, 3) - u2.block(11, 6, 1, 3);
dv1.row(4) = u2.block(11, 3, 1, 3) - u2.block(11, 9, 1, 3);
dv1.row(5) = u2.block(11, 6, 1, 3) - u2.block(11, 9, 1, 3);
dv2.row(0) = u2.block(10, 0, 1, 3) - u2.block(10, 3, 1, 3);
dv2.row(1) = u2.block(10, 0, 1, 3) - u2.block(10, 6, 1, 3);
dv2.row(2) = u2.block(10, 0, 1, 3) - u2.block(10, 9, 1, 3);
dv2.row(3) = u2.block(10, 3, 1, 3) - u2.block(10, 6, 1, 3);
dv2.row(4) = u2.block(10, 3, 1, 3) - u2.block(10, 9, 1, 3);
dv2.row(5) = u2.block(10, 6, 1, 3) - u2.block(10, 9, 1, 3);
dv3.row(0) = u2.block( 9, 0, 1, 3) - u2.block( 9, 3, 1, 3);
dv3.row(1) = u2.block( 9, 0, 1, 3) - u2.block( 9, 6, 1, 3);
dv3.row(2) = u2.block( 9, 0, 1, 3) - u2.block( 9, 9, 1, 3);
dv3.row(3) = u2.block( 9, 3, 1, 3) - u2.block( 9, 6, 1, 3);
dv3.row(4) = u2.block( 9, 3, 1, 3) - u2.block( 9, 9, 1, 3);
dv3.row(5) = u2.block( 9, 6, 1, 3) - u2.block( 9, 9, 1, 3);
dv4.row(0) = u2.block( 8, 0, 1, 3) - u2.block( 8, 3, 1, 3);
dv4.row(1) = u2.block( 8, 0, 1, 3) - u2.block( 8, 6, 1, 3);
dv4.row(2) = u2.block( 8, 0, 1, 3) - u2.block( 8, 9, 1, 3);
dv4.row(3) = u2.block( 8, 3, 1, 3) - u2.block( 8, 6, 1, 3);
dv4.row(4) = u2.block( 8, 3, 1, 3) - u2.block( 8, 9, 1, 3);
dv4.row(5) = u2.block( 8, 6, 1, 3) - u2.block( 8, 9, 1, 3);
Eigen::Matrix<double, 6, 10> L;
for (size_t r = 0; r < 6; r++) {
L.row(r) << dv1.row(r).dot(dv1.row(r)),
2.0 * dv1.row(r).dot(dv2.row(r)),
dv2.row(r).dot(dv2.row(r)),
2.0 * dv1.row(r).dot(dv3.row(r)),
2.0 * dv2.row(r).dot(dv3.row(r)),
dv3.row(r).dot(dv3.row(r)),
2.0 * dv1.row(r).dot(dv4.row(r)),
2.0 * dv2.row(r).dot(dv4.row(r)),
2.0 * dv3.row(r).dot(dv4.row(r)),
dv4.row(r).dot(dv4.row(r));
}
Vec6 rho;
rho << (X_control_points.col(0) - X_control_points.col(1)).squaredNorm(),
(X_control_points.col(0) - X_control_points.col(2)).squaredNorm(),
(X_control_points.col(0) - X_control_points.col(3)).squaredNorm(),
(X_control_points.col(1) - X_control_points.col(2)).squaredNorm(),
(X_control_points.col(1) - X_control_points.col(3)).squaredNorm(),
(X_control_points.col(2) - X_control_points.col(3)).squaredNorm();
// There are three possible solutions based on the three approximations of L
// (betas). Below, each one is solved for then the best one is chosen.
Mat3X X_camera;
Mat3 K; K.setIdentity();
vector<Mat3> Rs(3);
vector<Vec3> ts(3);
Vec rmse(3);
// At one point this threshold was 1e-3, and caused no end of problems in
// Blender by causing frames to not resect when they would have worked fine.
// When the resect failed, the projective followup is run leading to worse
// results, and often the dreaded "flipping" where objects get flipped
// between frames. Instead, disable the check for now, always succeeding. The
// ultimate check is always reprojection error anyway.
//
// TODO(keir): Decide if setting this to infinity, effectively disabling the
// check, is the right approach. So far this seems the case.
double kSuccessThreshold = std::numeric_limits<double>::max();
// Find the first possible solution for R, t corresponding to:
// Betas = [b00 b01 b11 b02 b12 b22 b03 b13 b23 b33]
// Betas_approx_1 = [b00 b01 b02 b03]
Vec4 betas = Vec4::Zero();
Eigen::Matrix<double, 6, 4> l_6x4;
for (size_t r = 0; r < 6; r++) {
l_6x4.row(r) << L(r, 0), L(r, 1), L(r, 3), L(r, 6);
}
Eigen::JacobiSVD<Mat> svd_of_l4(l_6x4,
Eigen::ComputeFullU | Eigen::ComputeFullV);
Vec4 b4 = svd_of_l4.solve(rho);
if ((l_6x4 * b4).isApprox(rho, kSuccessThreshold)) {
if (b4(0) < 0) {
b4 = -b4;
}
b4(0) = std::sqrt(b4(0));
betas << b4(0), b4(1) / b4(0), b4(2) / b4(0), b4(3) / b4(0);
ComputePointsCoordinatesInCameraFrame(alphas, betas, u2, &X_camera);
AbsoluteOrientation(X_world, X_camera, &Rs[0], &ts[0]);
rmse(0) = RootMeanSquareError(x_camera, X_world, K, Rs[0], ts[0]);
} else {
LOG(ERROR) << "First approximation of beta not good enough.";
ts[0].setZero();
rmse(0) = std::numeric_limits<double>::max();
}
// Find the second possible solution for R, t corresponding to:
// Betas = [b00 b01 b11 b02 b12 b22 b03 b13 b23 b33]
// Betas_approx_2 = [b00 b01 b11]
betas.setZero();
Eigen::Matrix<double, 6, 3> l_6x3;
l_6x3 = L.block(0, 0, 6, 3);
Eigen::JacobiSVD<Mat> svdOfL3(l_6x3,
Eigen::ComputeFullU | Eigen::ComputeFullV);
Vec3 b3 = svdOfL3.solve(rho);
VLOG(2) << " rho = " << rho;
VLOG(2) << " l_6x3 * b3 = " << l_6x3 * b3;
if ((l_6x3 * b3).isApprox(rho, kSuccessThreshold)) {
if (b3(0) < 0) {
betas(0) = std::sqrt(-b3(0));
betas(1) = (b3(2) < 0) ? std::sqrt(-b3(2)) : 0;
} else {
betas(0) = std::sqrt(b3(0));
betas(1) = (b3(2) > 0) ? std::sqrt(b3(2)) : 0;
}
if (b3(1) < 0) {
betas(0) = -betas(0);
}
betas(2) = 0;
betas(3) = 0;
ComputePointsCoordinatesInCameraFrame(alphas, betas, u2, &X_camera);
AbsoluteOrientation(X_world, X_camera, &Rs[1], &ts[1]);
rmse(1) = RootMeanSquareError(x_camera, X_world, K, Rs[1], ts[1]);
} else {
LOG(ERROR) << "Second approximation of beta not good enough.";
ts[1].setZero();
rmse(1) = std::numeric_limits<double>::max();
}
// Find the third possible solution for R, t corresponding to:
// Betas = [b00 b01 b11 b02 b12 b22 b03 b13 b23 b33]
// Betas_approx_3 = [b00 b01 b11 b02 b12]
betas.setZero();
Eigen::Matrix<double, 6, 5> l_6x5;
l_6x5 = L.block(0, 0, 6, 5);
Eigen::JacobiSVD<Mat> svdOfL5(l_6x5,
Eigen::ComputeFullU | Eigen::ComputeFullV);
Vec5 b5 = svdOfL5.solve(rho);
if ((l_6x5 * b5).isApprox(rho, kSuccessThreshold)) {
if (b5(0) < 0) {
betas(0) = std::sqrt(-b5(0));
if (b5(2) < 0) {
betas(1) = std::sqrt(-b5(2));
} else {
b5(2) = 0;
}
} else {
betas(0) = std::sqrt(b5(0));
if (b5(2) > 0) {
betas(1) = std::sqrt(b5(2));
} else {
b5(2) = 0;
}
}
if (b5(1) < 0) {
betas(0) = -betas(0);
}
betas(2) = b5(3) / betas(0);
betas(3) = 0;
ComputePointsCoordinatesInCameraFrame(alphas, betas, u2, &X_camera);
AbsoluteOrientation(X_world, X_camera, &Rs[2], &ts[2]);
rmse(2) = RootMeanSquareError(x_camera, X_world, K, Rs[2], ts[2]);
} else {
LOG(ERROR) << "Third approximation of beta not good enough.";
ts[2].setZero();
rmse(2) = std::numeric_limits<double>::max();
}
// Finally, with all three solutions, select the (R, t) with the best RMSE.
VLOG(2) << "RMSE for solution 0: " << rmse(0);
VLOG(2) << "RMSE for solution 1: " << rmse(0);
VLOG(2) << "RMSE for solution 2: " << rmse(0);
size_t n = 0;
if (rmse(1) < rmse(0)) {
n = 1;
}
if (rmse(2) < rmse(n)) {
n = 2;
}
if (rmse(n) == std::numeric_limits<double>::max()) {
LOG(ERROR) << "All three possibilities failed. Reporting failure.";
return false;
}
VLOG(1) << "RMSE for best solution #" << n << ": " << rmse(n);
*R = Rs[n];
*t = ts[n];
// TODO(julien): Improve the solutions with non-linear refinement.
return true;
}
} // namespace resection
} // namespace libmv
|