essential_solver.cpp
15.7 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
// 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"
#if defined(HAVE_EIGEN)
#include <Eigen/Eigen>
#elif defined(HAVE_LAPACK)
#include "opencv_lapack.h"
#endif
namespace cv { namespace usac {
// Essential matrix solver:
/*
* H. Stewenius, C. Engels, and D. Nister. Recent developments on direct relative orientation.
* ISPRS J. of Photogrammetry and Remote Sensing, 60:284,294, 2006
* http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.61.9329&rep=rep1&type=pdf
*/
class EssentialMinimalSolverStewenius5ptsImpl : public EssentialMinimalSolverStewenius5pts {
private:
// Points must be calibrated K^-1 x
const Mat * points_mat;
#if defined(HAVE_EIGEN) || defined(HAVE_LAPACK)
const float * const pts;
#endif
public:
explicit EssentialMinimalSolverStewenius5ptsImpl (const Mat &points_) :
points_mat(&points_)
#if defined(HAVE_EIGEN) || defined(HAVE_LAPACK)
, pts((float*)points_.data)
#endif
{}
#if defined(HAVE_LAPACK) || defined(HAVE_EIGEN)
int estimate (const std::vector<int> &sample, std::vector<Mat> &models) const override {
// (1) Extract 4 null vectors from linear equations of epipolar constraint
std::vector<double> coefficients(45); // 5 pts=rows, 9 columns
auto *coefficients_ = &coefficients[0];
for (int i = 0; i < 5; i++) {
const int smpl = 4 * sample[i];
const auto x1 = pts[smpl], y1 = pts[smpl+1], x2 = pts[smpl+2], y2 = pts[smpl+3];
(*coefficients_++) = x2 * x1;
(*coefficients_++) = x2 * y1;
(*coefficients_++) = x2;
(*coefficients_++) = y2 * x1;
(*coefficients_++) = y2 * y1;
(*coefficients_++) = y2;
(*coefficients_++) = x1;
(*coefficients_++) = y1;
(*coefficients_++) = 1;
}
const int num_cols = 9, num_e_mat = 4;
double ee[36]; // 9*4
// eliminate linear equations
if (!Math::eliminateUpperTriangular(coefficients, 5, num_cols))
return 0;
for (int i = 0; i < num_e_mat; i++)
for (int j = 5; j < num_cols; j++)
ee[num_cols * i + j] = (i + 5 == j) ? 1 : 0;
// use back-substitution
for (int e = 0; e < num_e_mat; e++) {
const int curr_e = num_cols * e;
// start from the last row
for (int i = 4; i >= 0; i--) {
const int row_i = i * num_cols;
double acc = 0;
for (int j = i + 1; j < num_cols; j++)
acc -= coefficients[row_i + j] * ee[curr_e + j];
ee[curr_e + i] = acc / coefficients[row_i + i];
// due to numerical errors return 0 solutions
if (std::isnan(ee[curr_e + i]))
return 0;
}
}
const Matx<double, 4, 9> null_space(ee);
const Matx<double, 4, 1> null_space_mat[3][3] = {
{null_space.col(0), null_space.col(3), null_space.col(6)},
{null_space.col(1), null_space.col(4), null_space.col(7)},
{null_space.col(2), null_space.col(5), null_space.col(8)}};
// (2) Use the rank constraint and the trace constraint to build ten third-order polynomial
// equations in the three unknowns. The monomials are ordered in GrLex order and
// represented in a 10×20 matrix, where each row corresponds to an equation and each column
// corresponds to a monomial
Matx<double, 1, 10> eet[3][3];
for (int i = 0; i < 3; i++)
for (int j = 0; j < 3; j++)
// compute EE Transpose
// Shorthand for multiplying the Essential matrix with its transpose.
eet[i][j] = 2 * (multPolysDegOne(null_space_mat[i][0].val, null_space_mat[j][0].val) +
multPolysDegOne(null_space_mat[i][1].val, null_space_mat[j][1].val) +
multPolysDegOne(null_space_mat[i][2].val, null_space_mat[j][2].val));
const Matx<double, 1, 10> trace = eet[0][0] + eet[1][1] + eet[2][2];
Mat_<double> constraint_mat(10, 20);
// Trace constraint
for (int i = 0; i < 3; i++)
for (int j = 0; j < 3; j++)
Mat(multPolysDegOneAndTwo(eet[i][0].val, null_space_mat[0][j].val) +
multPolysDegOneAndTwo(eet[i][1].val, null_space_mat[1][j].val) +
multPolysDegOneAndTwo(eet[i][2].val, null_space_mat[2][j].val) -
0.5 * multPolysDegOneAndTwo(trace.val, null_space_mat[i][j].val))
.copyTo(constraint_mat.row(3 * i + j));
// Rank = zero determinant constraint
Mat(multPolysDegOneAndTwo(
(multPolysDegOne(null_space_mat[0][1].val, null_space_mat[1][2].val) -
multPolysDegOne(null_space_mat[0][2].val, null_space_mat[1][1].val)).val,
null_space_mat[2][0].val) +
multPolysDegOneAndTwo(
(multPolysDegOne(null_space_mat[0][2].val, null_space_mat[1][0].val) -
multPolysDegOne(null_space_mat[0][0].val, null_space_mat[1][2].val)).val,
null_space_mat[2][1].val) +
multPolysDegOneAndTwo(
(multPolysDegOne(null_space_mat[0][0].val, null_space_mat[1][1].val) -
multPolysDegOne(null_space_mat[0][1].val, null_space_mat[1][0].val)).val,
null_space_mat[2][2].val)).copyTo(constraint_mat.row(9));
#ifdef HAVE_EIGEN
const Eigen::Matrix<double, 10, 20, Eigen::RowMajor> constraint_mat_eig((double *) constraint_mat.data);
// (3) Compute the Gröbner basis. This turns out to be as simple as performing a
// Gauss-Jordan elimination on the 10×20 matrix
const Eigen::Matrix<double, 10, 10> eliminated_mat_eig = constraint_mat_eig.block<10, 10>(0, 0)
.fullPivLu().solve(constraint_mat_eig.block<10, 10>(0, 10));
// (4) Compute the 10×10 action matrix for multiplication by one of the un-knowns.
// This is a simple matter of extracting the correct elements fromthe eliminated
// 10×20 matrix and organising them to form the action matrix.
Eigen::Matrix<double, 10, 10> action_mat_eig = Eigen::Matrix<double, 10, 10>::Zero();
action_mat_eig.block<3, 10>(0, 0) = eliminated_mat_eig.block<3, 10>(0, 0);
action_mat_eig.block<2, 10>(3, 0) = eliminated_mat_eig.block<2, 10>(4, 0);
action_mat_eig.row(5) = eliminated_mat_eig.row(7);
action_mat_eig(6, 0) = -1.0;
action_mat_eig(7, 1) = -1.0;
action_mat_eig(8, 3) = -1.0;
action_mat_eig(9, 6) = -1.0;
// (5) Compute the left eigenvectors of the action matrix
Eigen::EigenSolver<Eigen::Matrix<double, 10, 10>> eigensolver(action_mat_eig);
const Eigen::VectorXcd &eigenvalues = eigensolver.eigenvalues();
const auto * const eig_vecs_ = (double *) eigensolver.eigenvectors().real().data();
#else
Matx<double, 10, 10> A = constraint_mat.colRange(0, 10),
B = constraint_mat.colRange(10, 20), eliminated_mat;
if (!solve(A, B, eliminated_mat, DECOMP_LU)) return 0;
Mat eliminated_mat_dyn = Mat(eliminated_mat);
Mat action_mat = Mat_<double>::zeros(10, 10);
eliminated_mat_dyn.rowRange(0,3).copyTo(action_mat.rowRange(0,3));
eliminated_mat_dyn.rowRange(4,6).copyTo(action_mat.rowRange(3,5));
eliminated_mat_dyn.row(7).copyTo(action_mat.row(5));
auto * action_mat_data = (double *) action_mat.data;
action_mat_data[60] = -1.0; // 6 row, 0 col
action_mat_data[71] = -1.0; // 7 row, 1 col
action_mat_data[83] = -1.0; // 8 row, 3 col
action_mat_data[96] = -1.0; // 9 row, 6 col
int mat_order = 10, info, lda = 10, ldvl = 10, ldvr = 1, lwork = 100;
double wr[10], wi[10] = {0}, eig_vecs[100], work[100]; // 10 = mat_order, 100 = lwork
char jobvl = 'V', jobvr = 'N'; // only left eigen vectors are computed
dgeev_(&jobvl, &jobvr, &mat_order, action_mat_data, &lda, wr, wi, eig_vecs, &ldvl,
nullptr, &ldvr, work, &lwork, &info);
if (info != 0) return 0;
#endif
models = std::vector<Mat>(); models.reserve(10);
// Read off the values for the three unknowns at all the solution points and
// back-substitute to obtain the solutions for the essential matrix.
for (int i = 0; i < 10; i++)
// process only real solutions
#ifdef HAVE_EIGEN
if (eigenvalues(i).imag() == 0) {
Mat_<double> model(3, 3);
auto * model_data = (double *) model.data;
const int eig_i = 20 * i + 12; // eigen stores imaginary values too
for (int j = 0; j < 9; j++)
model_data[j] = ee[j ] * eig_vecs_[eig_i ] + ee[j+9 ] * eig_vecs_[eig_i+2] +
ee[j+18] * eig_vecs_[eig_i+4] + ee[j+27] * eig_vecs_[eig_i+6];
#else
if (wi[i] == 0) {
Mat_<double> model (3,3);
auto * model_data = (double *) model.data;
const int eig_i = 10 * i + 6;
for (int j = 0; j < 9; j++)
model_data[j] = ee[j ]*eig_vecs[eig_i ] + ee[j+9 ]*eig_vecs[eig_i+1] +
ee[j+18]*eig_vecs[eig_i+2] + ee[j+27]*eig_vecs[eig_i+3];
#endif
models.emplace_back(model);
}
return static_cast<int>(models.size());
#else
int estimate (const std::vector<int> &/*sample*/, std::vector<Mat> &/*models*/) const override {
CV_Error(cv::Error::StsNotImplemented, "To use essential matrix solver LAPACK or Eigen has to be installed!");
#endif
}
// number of possible solutions is 0,2,4,6,8,10
int getMaxNumberOfSolutions () const override { return 10; }
int getSampleSize() const override { return 5; }
Ptr<MinimalSolver> clone () const override {
return makePtr<EssentialMinimalSolverStewenius5ptsImpl>(*points_mat);
}
private:
/*
* Multiply two polynomials of degree one with unknowns x y z
* @p = (p1 x + p2 y + p3 z + p4) [p1 p2 p3 p4]
* @q = (q1 x + q2 y + q3 z + q4) [q1 q2 q3 a4]
* @result is a new polynomial in x^2 xy y^2 xz yz z^2 x y z 1 of size 10
*/
static inline Matx<double,1,10> multPolysDegOne(const double * const p,
const double * const q) {
return
{p[0]*q[0], p[0]*q[1]+p[1]*q[0], p[1]*q[1], p[0]*q[2]+p[2]*q[0], p[1]*q[2]+p[2]*q[1],
p[2]*q[2], p[0]*q[3]+p[3]*q[0], p[1]*q[3]+p[3]*q[1], p[2]*q[3]+p[3]*q[2], p[3]*q[3]};
}
/*
* Multiply two polynomials with unknowns x y z
* @p is of size 10 and @q is of size 4
* @p = (p1 x^2 + p2 xy + p3 y^2 + p4 xz + p5 yz + p6 z^2 + p7 x + p8 y + p9 z + p10)
* @q = (q1 x + q2 y + q3 z + a4) [q1 q2 q3 q4]
* @result is a new polynomial of size 20
* x^3 x^2y xy^2 y^3 x^2z xyz y^2z xz^2 yz^2 z^3 x^2 xy y^2 xz yz z^2 x y z 1
*/
static inline Matx<double, 1, 20> multPolysDegOneAndTwo(const double * const p,
const double * const q) {
return Matx<double, 1, 20>
({p[0]*q[0], p[0]*q[1]+p[1]*q[0], p[1]*q[1]+p[2]*q[0], p[2]*q[1], p[0]*q[2]+p[3]*q[0],
p[1]*q[2]+p[3]*q[1]+p[4]*q[0], p[2]*q[2]+p[4]*q[1], p[3]*q[2]+p[5]*q[0],
p[4]*q[2]+p[5]*q[1], p[5]*q[2], p[0]*q[3]+p[6]*q[0], p[1]*q[3]+p[6]*q[1]+p[7]*q[0],
p[2]*q[3]+p[7]*q[1], p[3]*q[3]+p[6]*q[2]+p[8]*q[0], p[4]*q[3]+p[7]*q[2]+p[8]*q[1],
p[5]*q[3]+p[8]*q[2], p[6]*q[3]+p[9]*q[0], p[7]*q[3]+p[9]*q[1], p[8]*q[3]+p[9]*q[2],
p[9]*q[3]});
}
};
Ptr<EssentialMinimalSolverStewenius5pts> EssentialMinimalSolverStewenius5pts::create
(const Mat &points_) {
return makePtr<EssentialMinimalSolverStewenius5ptsImpl>(points_);
}
class EssentialNonMinimalSolverImpl : public EssentialNonMinimalSolver {
private:
const Mat * points_mat;
const float * const points;
public:
/*
* Input calibrated points K^-1 x.
* Linear 8 points algorithm is used for estimation.
*/
explicit EssentialNonMinimalSolverImpl (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 < getMinimumRequiredSampleSize())
return 0;
// ------- 8 points algorithm with Eigen and covariance matrix --------------
double a[9] = {0, 0, 0, 0, 0, 0, 0, 0, 1};
double AtA[81] = {0}; // 9x9
if (weights.empty()) {
for (int i = 0; i < sample_size; i++) {
const int pidx = 4*sample[i];
const double x1 = points[pidx ], y1 = points[pidx+1],
x2 = points[pidx+2], y2 = points[pidx+3];
a[0] = x2*x1;
a[1] = x2*y1;
a[2] = x2;
a[3] = y2*x1;
a[4] = y2*y1;
a[5] = y2;
a[6] = x1;
a[7] = y1;
// calculate covariance for eigen
for (int row = 0; row < 9; row++)
for (int col = row; col < 9; col++)
AtA[row*9+col] += a[row]*a[col];
}
} else {
for (int i = 0; i < sample_size; i++) {
const int smpl = 4*sample[i];
const double weight = weights[i];
const double x1 = points[smpl ], y1 = points[smpl+1],
x2 = points[smpl+2], y2 = points[smpl+3];
const double weight_times_x2 = weight * x2,
weight_times_y2 = weight * y2;
a[0] = weight_times_x2 * x1;
a[1] = weight_times_x2 * y1;
a[2] = weight_times_x2;
a[3] = weight_times_y2 * x1;
a[4] = weight_times_y2 * y1;
a[5] = weight_times_y2;
a[6] = weight * x1;
a[7] = weight * y1;
a[8] = weight;
// calculate covariance for eigen
for (int row = 0; row < 9; row++)
for (int col = row; col < 9; col++)
AtA[row*9+col] += a[row]*a[col];
}
}
// copy symmetric part of covariance matrix
for (int j = 1; j < 9; j++)
for (int z = 0; z < j; z++)
AtA[j*9+z] = AtA[z*9+j];
#ifdef HAVE_EIGEN
models = std::vector<Mat>{ Mat_<double>(3,3) };
const Eigen::JacobiSVD<Eigen::Matrix<double, 9, 9>> svd((Eigen::Matrix<double, 9, 9>(AtA)),
Eigen::ComputeFullV);
// extract the last nullspace
Eigen::Map<Eigen::Matrix<double, 9, 1>>((double *)models[0].data) = svd.matrixV().col(8);
#else
Matx<double, 9, 9> AtA_(AtA), U, Vt;
Vec<double, 9> W;
SVD::compute(AtA_, W, U, Vt, SVD::FULL_UV + SVD::MODIFY_A);
models = std::vector<Mat> { Mat_<double>(3, 3, Vt.val + 72 /*=8*9*/) };
#endif
FundamentalDegeneracy::recoverRank(models[0], false /*E*/);
return 1;
}
int getMinimumRequiredSampleSize() const override { return 8; }
int getMaxNumberOfSolutions () const override { return 1; }
Ptr<NonMinimalSolver> clone () const override {
return makePtr<EssentialNonMinimalSolverImpl>(*points_mat);
}
};
Ptr<EssentialNonMinimalSolver> EssentialNonMinimalSolver::create (const Mat &points_) {
return makePtr<EssentialNonMinimalSolverImpl>(points_);
}
}}