pnp_solver.cpp
15.6 KB
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
// This file is part of OpenCV project.
// It is subject to the license terms in the LICENSE file found in the top-level directory
// of this distribution and at http://opencv.org/license.html.
#include "../precomp.hpp"
#include "../usac.hpp"
#include "../polynom_solver.h"
#if defined(HAVE_EIGEN)
#include <Eigen/Eigen>
#include <Eigen/QR>
#elif defined(HAVE_LAPACK)
#include "opencv_lapack.h"
#endif
namespace cv { namespace usac {
class PnPMinimalSolver6PtsImpl : public PnPMinimalSolver6Pts {
private:
const Mat * points_mat;
const float * const points;
public:
// linear 6 points required (11 equations)
int getSampleSize() const override { return 6; }
int getMaxNumberOfSolutions () const override { return 1; }
explicit PnPMinimalSolver6PtsImpl (const Mat &points_) :
points_mat(&points_), points ((float*)points_.data) {}
/*
DLT:
d x = P X, x = (u, v, 1), X = (X, Y, Z, 1), P = K[R t]
is 3x4 projection matrix with rows p1, p2, p3. d is depth
u = p1^T X / p3^T X
v = p2^T X / p3^T X
(p1^T - u p3^T) X = 0
(p2^T - v p3^T) X = 0
(p11 - u p31) X + (p12 - u p32) Y + (p13 - u p33) Z + (p14 - u p34) = 0
(p12 - v p31) X + (p22 - v p32) Y + (p23 - v p33) Z + (p24 - v p34) = 0
[X, Y, Z, 1, 0, 0, 0, 0, -u X, -u Y, -u Z, -u] [p11] [0]
[0, 0, 0, 0, X, Y, Z, 1, -v X, -v Y, -v Z, -v] [p12] [0]
. = [0]
.
. [p34] [0]
minimum 11 equations, each point gives 2 equation, so at least 6 points are required.
@points is array Nx5
u1 v1 X1 Y1 Z1
...
uN vN XN YN ZN
@P is output projection matrix
A1 =
[X1, Y1, Z1, 1, 0, 0, 0, 0, -u1 X1, -u1 Y1, -u1 Z1, -u1] [p11] [0]
[X2, Y2, Z2, 1, 0, 0, 0, 0, -u2 X2, -u2 Y2, -u2 Z2, -u2] [p12] [0]
[X3, Y3, Z3, 1, 0, 0, 0, 0, -u3 X3, -u3 Y3, -u3 Z3, -u3] [p13] [0]
[X4, Y4, Z4, 1, 0, 0, 0, 0, -u4 X4, -u4 Y4, -u4 Z4, -u4] [p14] [0]
[X5, Y5, Z5, 1, 0, 0, 0, 0, -u5 X5, -u5 Y5, -u5 Z5, -u5] [p21] [0]
[p22]
A2 = (without first 4 columns)
[0, 0, 0, 0, X1, Y1, Z1, 1, -v1 X1, -v1 Y1, -v1 Z1, -v1] [p23] = [0]
[0, 0, 0, 0, X2, Y2, Z2, 1, -v2 X2, -v2 Y2, -v2 Z2, -v2] [p24] [0]
[0, 0, 0, 0, X3, Y3, Z3, 1, -v3 X3, -v3 Y3, -v3 Z3, -v3] [p31] [0]
[0, 0, 0, 0, X4, Y4, Z4, 1, -v4 X4, -v4 Y4, -v4 Z4, -v4] [p32] [0]
[0, 0, 0, 0, X5, Y5, Z5, 1, -v5 X5, -v5 Y5, -v5 Z5, -v5] [p33] [0]
[0, 0, 0, 0, X6, Y6, Z6, 1, -v6 X6, -v6 Y6, -v6 Z6, -v6] [p34=1] [0]
P = null A; dim null A = n - rank(A) = 12 - 11 = 1
*/
int estimate (const std::vector<int> &sample, std::vector<Mat> &models) const override {
std::vector<double> A1 (60, 0), A2(56, 0); // 5x12, 7x8
int cnt1 = 0, cnt2 = 0;
for (int i = 0; i < 6; i++) {
const int smpl = 5 * sample[i];
const double u = points[smpl ], v = points[smpl + 1];
const double X = points[smpl + 2], Y = points[smpl + 3], Z = points[smpl + 4];
if (i != 5) {
A1[cnt1++] = X;
A1[cnt1++] = Y;
A1[cnt1++] = Z;
A1[cnt1++] = 1;
cnt1 += 4; // skip zeros
A1[cnt1++] = -u * X;
A1[cnt1++] = -u * Y;
A1[cnt1++] = -u * Z;
A1[cnt1++] = -u;
}
A2[cnt2++] = X;
A2[cnt2++] = Y;
A2[cnt2++] = Z;
A2[cnt2++] = 1;
A2[cnt2++] = -v * X;
A2[cnt2++] = -v * Y;
A2[cnt2++] = -v * Z;
A2[cnt2++] = -v;
}
// matrix is sparse -> do not test for singularity
Math::eliminateUpperTriangular(A1, 5, 12);
int offset = 4*12;
// add last eliminated row of A1
for (int i = 0; i < 8; i++)
A2[cnt2++] = A1[offset + i + 4/* skip 4 first cols*/];
// must be full-rank
if (!Math::eliminateUpperTriangular(A2, 7, 8))
return 0;
// fixed scale to 1. In general the projection matrix is up-to-scale.
// P = alpha * P^, alpha = 1 / P^_[3,4]
Mat P = Mat_<double>(3,4);
auto * p = (double *) P.data;
p[11] = 1;
// start from the last row
for (int i = 6; i >= 0; i--) {
double acc = 0;
for (int j = i+1; j < 8; j++)
acc -= A2[i*8+j]*p[j+4];
p[i+4] = acc / A2[i*8+i];
// due to numerical errors return 0 solutions
if (std::isnan(p[i+4]))
return 0;
}
for (int i = 3; i >= 0; i--) {
double acc = 0;
for (int j = i+1; j < 12; j++)
acc -= A1[i*12+j]*p[j];
p[i] = acc / A1[i*12+i];
if (std::isnan(p[i]))
return 0;
}
models = std::vector<Mat>{P};
return 1;
}
Ptr<MinimalSolver> clone () const override {
return makePtr<PnPMinimalSolver6PtsImpl>(*points_mat);
}
};
Ptr<PnPMinimalSolver6Pts> PnPMinimalSolver6Pts::create(const Mat &points_) {
return makePtr<PnPMinimalSolver6PtsImpl>(points_);
}
class PnPNonMinimalSolverImpl : public PnPNonMinimalSolver {
private:
const Mat * points_mat;
const float * const points;
public:
explicit PnPNonMinimalSolverImpl (const Mat &points_) :
points_mat(&points_), points ((float*)points_.data){}
int estimate (const std::vector<int> &sample, int sample_size,
std::vector<Mat> &models, const std::vector<double> &weights) const override {
if (sample_size < 6)
return 0;
double AtA [144] = {0}; // 12x12
double a1[12] = {0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0},
a2[12] = {0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0};
if (weights.empty())
for (int i = 0; i < sample_size; i++) {
const int smpl = 5 * sample[i];
const double u = points[smpl], v = points[smpl + 1];
const double X = points[smpl + 2], Y = points[smpl + 3], Z = points[smpl + 4];
a1[0 ] = -X;
a1[1 ] = -Y;
a1[2 ] = -Z;
a1[8 ] = u * X;
a1[9 ] = u * Y;
a1[10] = u * Z;
a1[11] = u;
a2[4 ] = -X;
a2[5 ] = -Y;
a2[6 ] = -Z;
a2[8 ] = v * X;
a2[9 ] = v * Y;
a2[10] = v * Z;
a2[11] = v;
// fill covarinace matrix
for (int j = 0; j < 12; j++)
for (int z = j; z < 12; z++)
AtA[j * 12 + z] += a1[j] * a1[z] + a2[j] * a2[z];
}
else
for (int i = 0; i < sample_size; i++) {
const int smpl = 5 * sample[i];
const double weight = weights[i], u = points[smpl], v = points[smpl + 1];
const double weight_X = weight * points[smpl + 2],
weight_Y = weight * points[smpl + 3],
weight_Z = weight * points[smpl + 4];
a1[0 ] = -weight_X;
a1[1 ] = -weight_Y;
a1[2 ] = -weight_Z;
a1[3 ] = -weight;
a1[8 ] = u * weight_X;
a1[9 ] = u * weight_Y;
a1[10] = u * weight_Z;
a1[11] = u * weight;
a2[4 ] = -weight_X;
a2[5 ] = -weight_Y;
a2[6 ] = -weight_Z;
a2[7 ] = -weight;
a2[8 ] = v * weight_X;
a2[9 ] = v * weight_Y;
a2[10] = v * weight_Z;
a2[11] = v * weight;
// fill covarinace matrix
for (int j = 0; j < 12; j++)
for (int z = j; z < 12; z++)
AtA[j * 12 + z] += a1[j] * a1[z] + a2[j] * a2[z];
}
// copy symmetric part of covariance matrix
for (int j = 1; j < 12; j++)
for (int z = 0; z < j; z++)
AtA[j*12+z] = AtA[z*12+j];
#ifdef HAVE_EIGEN
models = std::vector<Mat>{ Mat_<double>(3,4) };
Eigen::HouseholderQR<Eigen::Matrix<double, 12, 12>> qr((Eigen::Matrix<double, 12, 12>(AtA)));
const Eigen::Matrix<double, 12, 12> &Q = qr.householderQ();
// extract the last nullspace
Eigen::Map<Eigen::Matrix<double, 12, 1>>((double *)models[0].data) = Q.col(11);
#else
Matx<double, 12, 12> Vt;
Vec<double, 12> D;
if (! eigen(Matx<double, 12, 12>(AtA), D, Vt)) return 0;
models = std::vector<Mat>{ Mat(Vt.row(11).reshape<3,4>()) };
#endif
return 1;
}
int getMinimumRequiredSampleSize() const override { return 6; }
int getMaxNumberOfSolutions () const override { return 1; }
Ptr<NonMinimalSolver> clone () const override {
return makePtr<PnPNonMinimalSolverImpl>(*points_mat);
}
};
Ptr<PnPNonMinimalSolver> PnPNonMinimalSolver::create(const Mat &points) {
return makePtr<PnPNonMinimalSolverImpl>(points);
}
class P3PSolverImpl : public P3PSolver {
private:
/*
* calibrated normalized points
* K^-1 [u v 1]^T / ||K^-1 [u v 1]^T||
*/
const Mat * points_mat, * calib_norm_points_mat, * K_mat, &K;
const float * const calib_norm_points, * const points;
const double VAL_THR = 1e-4;
public:
/*
* @points_ is matrix N x 5
* u v x y z. (u,v) is image point, (x y z) is world point
*/
P3PSolverImpl (const Mat &points_, const Mat &calib_norm_points_, const Mat &K_) :
points_mat(&points_), calib_norm_points_mat(&calib_norm_points_), K_mat (&K_),
K(K_), calib_norm_points((float*)calib_norm_points_.data), points((float*)points_.data) {}
int estimate (const std::vector<int> &sample, std::vector<Mat> &models) const override {
/*
* The description of this solution can be found here:
* http://cmp.felk.cvut.cz/~pajdla/gvg/GVG-2016-Lecture.pdf
* pages: 51-59
*/
const int idx1 = 5*sample[0], idx2 = 5*sample[1], idx3 = 5*sample[2];
const Vec3d X1 (points[idx1+2], points[idx1+3], points[idx1+4]);
const Vec3d X2 (points[idx2+2], points[idx2+3], points[idx2+4]);
const Vec3d X3 (points[idx3+2], points[idx3+3], points[idx3+4]);
// find distance between world points d_ij = ||Xi - Xj||
const double d12 = norm(X1 - X2);
const double d23 = norm(X2 - X3);
const double d31 = norm(X3 - X1);
if (d12 < VAL_THR || d23 < VAL_THR || d31 < VAL_THR)
return 0;
const int c_idx1 = 3*sample[0], c_idx2 = 3*sample[1], c_idx3 = 3*sample[2];
const Vec3d cx1 (calib_norm_points[c_idx1], calib_norm_points[c_idx1+1], calib_norm_points[c_idx1+2]);
const Vec3d cx2 (calib_norm_points[c_idx2], calib_norm_points[c_idx2+1], calib_norm_points[c_idx2+2]);
const Vec3d cx3 (calib_norm_points[c_idx3], calib_norm_points[c_idx3+1], calib_norm_points[c_idx3+2]);
// find cosine angles, cos(x1,x2) = K^-1 x1.dot(K^-1 x2) / (||K^-1 x1|| * ||K^-1 x2||)
// calib_norm_points are already K^-1 x / ||K^-1 x||, so we perform only dot product
const double c12 = cx1(0)*cx2(0) + cx1(1)*cx2(1) + cx1(2)*cx2(2);
const double c23 = cx2(0)*cx3(0) + cx2(1)*cx3(1) + cx2(2)*cx3(2);
const double c31 = cx3(0)*cx1(0) + cx3(1)*cx1(1) + cx3(2)*cx1(2);
Matx33d Z, Zw;
auto * z = Z.val, * zw = Zw.val;
// find coefficients of polynomial a4 x^4 + ... + a0 = 0
const double c12_p2 = c12*c12, c23_p2 = c23*c23, c31_p2 = c31*c31;
const double d12_p2 = d12*d12, d12_p4 = d12_p2*d12_p2;
const double d23_p2 = d23*d23, d23_p4 = d23_p2*d23_p2, d23_p6 = d23_p2*d23_p4, d23_p8 = d23_p4*d23_p4;
const double d31_p2 = d31*d31, d31_p4 = d31_p2*d31_p2;
const double a4 = -4*d23_p4*d12_p2*d31_p2*c23_p2+d23_p8-2*d23_p6*d12_p2-2*d23_p6*d31_p2+d23_p4*d12_p4+2*d23_p4*d12_p2*d31_p2+d23_p4*d31_p4;
const double a3 = 8*d23_p4*d12_p2*d31_p2*c12*c23_p2+4*d23_p6*d12_p2*c31*c23-4*d23_p4*d12_p4*c31*c23+4*d23_p4*d12_p2*d31_p2*c31*c23-4*d23_p8*c12+4*d23_p6*d12_p2*c12+8*d23_p6*d31_p2*c12-4*d23_p4*d12_p2*d31_p2*c12-4*d23_p4*d31_p4*c12;
const double a2 = -8*d23_p6*d12_p2*c31*c12*c23-8*d23_p4*d12_p2*d31_p2*c31*c12*c23+4*d23_p8*c12_p2-4*d23_p6*d12_p2*c31_p2-8*d23_p6*d31_p2*c12_p2+4*d23_p4*d12_p4*c31_p2+4*d23_p4*d12_p4*c23_p2-4*d23_p4*d12_p2*d31_p2*c23_p2+4*d23_p4*d31_p4*c12_p2+2*d23_p8-4*d23_p6*d31_p2-2*d23_p4*d12_p4+2*d23_p4*d31_p4;
const double a1 = 8*d23_p6*d12_p2*c31_p2*c12+4*d23_p6*d12_p2*c31*c23-4*d23_p4*d12_p4*c31*c23+4*d23_p4*d12_p2*d31_p2*c31*c23-4*d23_p8*c12-4*d23_p6*d12_p2*c12+8*d23_p6*d31_p2*c12+4*d23_p4*d12_p2*d31_p2*c12-4*d23_p4*d31_p4*c12;
const double a0 = -4*d23_p6*d12_p2*c31_p2+d23_p8-2*d23_p4*d12_p2*d31_p2+2*d23_p6*d12_p2+d23_p4*d31_p4+d23_p4*d12_p4-2*d23_p6*d31_p2;
double roots[4] = {0};
int num_roots = solve_deg4(a4, a3, a2, a1, a0, roots[0], roots[1], roots[2], roots[3]);
models = std::vector<Mat>(); models.reserve(num_roots);
for (double root : roots) {
if (root <= 0) continue;
const double n12 = root, n12_p2 = n12 * n12;
const double n13 = (d12_p2*(d23_p2-d31_p2*n12_p2)+(d23_p2-d31_p2)*(d23_p2*(1+n12_p2-2*n12*c12)-d12_p2*n12_p2))
/ (2*d12_p2*(d23_p2*c31 - d31_p2*c23*n12) + 2*(d31_p2-d23_p2)*d12_p2*c23*n12);
const double n1 = d12 / sqrt(1 + n12_p2 - 2*n12*c12); // 1+n12^2-2n12c12 is always > 0
const double n2 = n1 * n12;
const double n3 = n1 * n13;
if (n1 <= 0 || n2 <= 0 || n3 <= 0)
continue;
// compute and check errors
if (fabs((sqrt(n1*n1 + n2*n2 - 2*n1*n2*c12) - d12) / d12) > VAL_THR ||
fabs((sqrt(n2*n2 + n3*n3 - 2*n2*n3*c23) - d23) / d23) > VAL_THR ||
fabs((sqrt(n3*n3 + n1*n1 - 2*n3*n1*c31) - d31) / d31) > VAL_THR)
continue;
const Vec3d nX1 = n1 * cx1;
Vec3d Z2 = n2 * cx2 - nX1; Z2 /= norm(Z2);
Vec3d Z3 = n3 * cx3 - nX1; Z3 /= norm(Z3);
Vec3d Z1 = Z2.cross(Z3); Z1 /= norm(Z1);
const Vec3d Z3crZ1 = Z3.cross(Z1);
z[0] = Z1(0); z[3] = Z1(1); z[6] = Z1(2);
z[1] = Z2(0); z[4] = Z2(1); z[7] = Z2(2);
z[2] = Z3crZ1(0); z[5] = Z3crZ1(1); z[8] = Z3crZ1(2);
Vec3d Zw2 = (X2 - X1) / d12;
Vec3d Zw3 = (X3 - X1) / d31;
Vec3d Zw1 = Zw2.cross(Zw3); Zw1 /= norm(Zw1);
const Vec3d Z3crZ1w = Zw3.cross(Zw1);
zw[0] = Zw1(0); zw[3] = Zw1(1); zw[6] = Zw1(2);
zw[1] = Zw2(0); zw[4] = Zw2(1); zw[7] = Zw2(2);
zw[2] = Z3crZ1w(0); zw[5] = Z3crZ1w(1); zw[8] = Z3crZ1w(2);
const Matx33d R = Math::rotVec2RotMat(Math::rotMat2RotVec(Z * Zw.inv()));
Mat P, KR = K * R;
hconcat(KR, -KR * (X1 - R.t() * nX1), P);
models.emplace_back(P);
}
return static_cast<int>(models.size());
}
int getSampleSize() const override { return 3; }
int getMaxNumberOfSolutions () const override { return 4; }
Ptr<MinimalSolver> clone () const override {
return makePtr<P3PSolverImpl>(*points_mat, *calib_norm_points_mat, *K_mat);
}
};
Ptr<P3PSolver> P3PSolver::create(const Mat &points_, const Mat &calib_norm_pts, const Mat &K) {
return makePtr<P3PSolverImpl>(points_, calib_norm_pts, K);
}
}}