ellipse_approximation.cc revision 79397c21138f54fcff6ec067b44b847f1f7e0e98
1// Ceres Solver - A fast non-linear least squares minimizer 2// Copyright 2014 Google Inc. All rights reserved. 3// http://code.google.com/p/ceres-solver/ 4// 5// Redistribution and use in source and binary forms, with or without 6// modification, are permitted provided that the following conditions are met: 7// 8// * Redistributions of source code must retain the above copyright notice, 9// this list of conditions and the following disclaimer. 10// * Redistributions in binary form must reproduce the above copyright notice, 11// this list of conditions and the following disclaimer in the documentation 12// and/or other materials provided with the distribution. 13// * Neither the name of Google Inc. nor the names of its contributors may be 14// used to endorse or promote products derived from this software without 15// specific prior written permission. 16// 17// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" 18// AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 19// IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE 20// ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE 21// LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR 22// CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF 23// SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS 24// INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN 25// CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) 26// ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE 27// POSSIBILITY OF SUCH DAMAGE. 28// 29// Author: richie.stebbing@gmail.com (Richard Stebbing) 30// 31// This fits points randomly distributed on an ellipse with an approximate 32// line segment contour. This is done by jointly optimizing the control points 33// of the line segment contour along with the preimage positions for the data 34// points. The purpose of this example is to show an example use case for 35// dynamic_sparsity, and how it can benefit problems which are numerically 36// dense but dynamically sparse. 37 38#include <cmath> 39#include <vector> 40#include "ceres/ceres.h" 41#include "glog/logging.h" 42 43// Data generated with the following Python code. 44// import numpy as np 45// np.random.seed(1337) 46// t = np.linspace(0.0, 2.0 * np.pi, 212, endpoint=False) 47// t += 2.0 * np.pi * 0.01 * np.random.randn(t.size) 48// theta = np.deg2rad(15) 49// a, b = np.cos(theta), np.sin(theta) 50// R = np.array([[a, -b], 51// [b, a]]) 52// Y = np.dot(np.c_[4.0 * np.cos(t), np.sin(t)], R.T) 53 54const int kYRows = 212; 55const int kYCols = 2; 56const double kYData[kYRows * kYCols] = { 57 +3.871364e+00, +9.916027e-01, 58 +3.864003e+00, +1.034148e+00, 59 +3.850651e+00, +1.072202e+00, 60 +3.868350e+00, +1.014408e+00, 61 +3.796381e+00, +1.153021e+00, 62 +3.857138e+00, +1.056102e+00, 63 +3.787532e+00, +1.162215e+00, 64 +3.704477e+00, +1.227272e+00, 65 +3.564711e+00, +1.294959e+00, 66 +3.754363e+00, +1.191948e+00, 67 +3.482098e+00, +1.322725e+00, 68 +3.602777e+00, +1.279658e+00, 69 +3.585433e+00, +1.286858e+00, 70 +3.347505e+00, +1.356415e+00, 71 +3.220855e+00, +1.378914e+00, 72 +3.558808e+00, +1.297174e+00, 73 +3.403618e+00, +1.343809e+00, 74 +3.179828e+00, +1.384721e+00, 75 +3.054789e+00, +1.398759e+00, 76 +3.294153e+00, +1.366808e+00, 77 +3.247312e+00, +1.374813e+00, 78 +2.988547e+00, +1.404247e+00, 79 +3.114508e+00, +1.392698e+00, 80 +2.899226e+00, +1.409802e+00, 81 +2.533256e+00, +1.414778e+00, 82 +2.654773e+00, +1.415909e+00, 83 +2.565100e+00, +1.415313e+00, 84 +2.976456e+00, +1.405118e+00, 85 +2.484200e+00, +1.413640e+00, 86 +2.324751e+00, +1.407476e+00, 87 +1.930468e+00, +1.378221e+00, 88 +2.329017e+00, +1.407688e+00, 89 +1.760640e+00, +1.360319e+00, 90 +2.147375e+00, +1.396603e+00, 91 +1.741989e+00, +1.358178e+00, 92 +1.743859e+00, +1.358394e+00, 93 +1.557372e+00, +1.335208e+00, 94 +1.280551e+00, +1.295087e+00, 95 +1.429880e+00, +1.317546e+00, 96 +1.213485e+00, +1.284400e+00, 97 +9.168172e-01, +1.232870e+00, 98 +1.311141e+00, +1.299839e+00, 99 +1.231969e+00, +1.287382e+00, 100 +7.453773e-01, +1.200049e+00, 101 +6.151587e-01, +1.173683e+00, 102 +5.935666e-01, +1.169193e+00, 103 +2.538707e-01, +1.094227e+00, 104 +6.806136e-01, +1.187089e+00, 105 +2.805447e-01, +1.100405e+00, 106 +6.184807e-01, +1.174371e+00, 107 +1.170550e-01, +1.061762e+00, 108 +2.890507e-01, +1.102365e+00, 109 +3.834234e-01, +1.123772e+00, 110 +3.980161e-04, +1.033061e+00, 111 -3.651680e-01, +9.370367e-01, 112 -8.386351e-01, +7.987201e-01, 113 -8.105704e-01, +8.073702e-01, 114 -8.735139e-01, +7.878886e-01, 115 -9.913836e-01, +7.506100e-01, 116 -8.784011e-01, +7.863636e-01, 117 -1.181440e+00, +6.882566e-01, 118 -1.229556e+00, +6.720191e-01, 119 -1.035839e+00, +7.362765e-01, 120 -8.031520e-01, +8.096470e-01, 121 -1.539136e+00, +5.629549e-01, 122 -1.755423e+00, +4.817306e-01, 123 -1.337589e+00, +6.348763e-01, 124 -1.836966e+00, +4.499485e-01, 125 -1.913367e+00, +4.195617e-01, 126 -2.126467e+00, +3.314900e-01, 127 -1.927625e+00, +4.138238e-01, 128 -2.339862e+00, +2.379074e-01, 129 -1.881736e+00, +4.322152e-01, 130 -2.116753e+00, +3.356163e-01, 131 -2.255733e+00, +2.754930e-01, 132 -2.555834e+00, +1.368473e-01, 133 -2.770277e+00, +2.895711e-02, 134 -2.563376e+00, +1.331890e-01, 135 -2.826715e+00, -9.000818e-04, 136 -2.978191e+00, -8.457804e-02, 137 -3.115855e+00, -1.658786e-01, 138 -2.982049e+00, -8.678322e-02, 139 -3.307892e+00, -2.902083e-01, 140 -3.038346e+00, -1.194222e-01, 141 -3.190057e+00, -2.122060e-01, 142 -3.279086e+00, -2.705777e-01, 143 -3.322028e+00, -2.999889e-01, 144 -3.122576e+00, -1.699965e-01, 145 -3.551973e+00, -4.768674e-01, 146 -3.581866e+00, -5.032175e-01, 147 -3.497799e+00, -4.315203e-01, 148 -3.565384e+00, -4.885602e-01, 149 -3.699493e+00, -6.199815e-01, 150 -3.585166e+00, -5.061925e-01, 151 -3.758914e+00, -6.918275e-01, 152 -3.741104e+00, -6.689131e-01, 153 -3.688331e+00, -6.077239e-01, 154 -3.810425e+00, -7.689015e-01, 155 -3.791829e+00, -7.386911e-01, 156 -3.789951e+00, -7.358189e-01, 157 -3.823100e+00, -7.918398e-01, 158 -3.857021e+00, -8.727074e-01, 159 -3.858250e+00, -8.767645e-01, 160 -3.872100e+00, -9.563174e-01, 161 -3.864397e+00, -1.032630e+00, 162 -3.846230e+00, -1.081669e+00, 163 -3.834799e+00, -1.102536e+00, 164 -3.866684e+00, -1.022901e+00, 165 -3.808643e+00, -1.139084e+00, 166 -3.868840e+00, -1.011569e+00, 167 -3.791071e+00, -1.158615e+00, 168 -3.797999e+00, -1.151267e+00, 169 -3.696278e+00, -1.232314e+00, 170 -3.779007e+00, -1.170504e+00, 171 -3.622855e+00, -1.270793e+00, 172 -3.647249e+00, -1.259166e+00, 173 -3.655412e+00, -1.255042e+00, 174 -3.573218e+00, -1.291696e+00, 175 -3.638019e+00, -1.263684e+00, 176 -3.498409e+00, -1.317750e+00, 177 -3.304143e+00, -1.364970e+00, 178 -3.183001e+00, -1.384295e+00, 179 -3.202456e+00, -1.381599e+00, 180 -3.244063e+00, -1.375332e+00, 181 -3.233308e+00, -1.377019e+00, 182 -3.060112e+00, -1.398264e+00, 183 -3.078187e+00, -1.396517e+00, 184 -2.689594e+00, -1.415761e+00, 185 -2.947662e+00, -1.407039e+00, 186 -2.854490e+00, -1.411860e+00, 187 -2.660499e+00, -1.415900e+00, 188 -2.875955e+00, -1.410930e+00, 189 -2.675385e+00, -1.415848e+00, 190 -2.813155e+00, -1.413363e+00, 191 -2.417673e+00, -1.411512e+00, 192 -2.725461e+00, -1.415373e+00, 193 -2.148334e+00, -1.396672e+00, 194 -2.108972e+00, -1.393738e+00, 195 -2.029905e+00, -1.387302e+00, 196 -2.046214e+00, -1.388687e+00, 197 -2.057402e+00, -1.389621e+00, 198 -1.650250e+00, -1.347160e+00, 199 -1.806764e+00, -1.365469e+00, 200 -1.206973e+00, -1.283343e+00, 201 -8.029259e-01, -1.211308e+00, 202 -1.229551e+00, -1.286993e+00, 203 -1.101507e+00, -1.265754e+00, 204 -9.110645e-01, -1.231804e+00, 205 -1.110046e+00, -1.267211e+00, 206 -8.465274e-01, -1.219677e+00, 207 -7.594163e-01, -1.202818e+00, 208 -8.023823e-01, -1.211203e+00, 209 -3.732519e-01, -1.121494e+00, 210 -1.918373e-01, -1.079668e+00, 211 -4.671988e-01, -1.142253e+00, 212 -4.033645e-01, -1.128215e+00, 213 -1.920740e-01, -1.079724e+00, 214 -3.022157e-01, -1.105389e+00, 215 -1.652831e-01, -1.073354e+00, 216 +4.671625e-01, -9.085886e-01, 217 +5.940178e-01, -8.721832e-01, 218 +3.147557e-01, -9.508290e-01, 219 +6.383631e-01, -8.591867e-01, 220 +9.888923e-01, -7.514088e-01, 221 +7.076339e-01, -8.386023e-01, 222 +1.326682e+00, -6.386698e-01, 223 +1.149834e+00, -6.988221e-01, 224 +1.257742e+00, -6.624207e-01, 225 +1.492352e+00, -5.799632e-01, 226 +1.595574e+00, -5.421766e-01, 227 +1.240173e+00, -6.684113e-01, 228 +1.706612e+00, -5.004442e-01, 229 +1.873984e+00, -4.353002e-01, 230 +1.985633e+00, -3.902561e-01, 231 +1.722880e+00, -4.942329e-01, 232 +2.095182e+00, -3.447402e-01, 233 +2.018118e+00, -3.768991e-01, 234 +2.422702e+00, -1.999563e-01, 235 +2.370611e+00, -2.239326e-01, 236 +2.152154e+00, -3.205250e-01, 237 +2.525121e+00, -1.516499e-01, 238 +2.422116e+00, -2.002280e-01, 239 +2.842806e+00, +9.536372e-03, 240 +3.030128e+00, +1.146027e-01, 241 +2.888424e+00, +3.433444e-02, 242 +2.991609e+00, +9.226409e-02, 243 +2.924807e+00, +5.445844e-02, 244 +3.007772e+00, +1.015875e-01, 245 +2.781973e+00, -2.282382e-02, 246 +3.164737e+00, +1.961781e-01, 247 +3.237671e+00, +2.430139e-01, 248 +3.046123e+00, +1.240014e-01, 249 +3.414834e+00, +3.669060e-01, 250 +3.436591e+00, +3.833600e-01, 251 +3.626207e+00, +5.444311e-01, 252 +3.223325e+00, +2.336361e-01, 253 +3.511963e+00, +4.431060e-01, 254 +3.698380e+00, +6.187442e-01, 255 +3.670244e+00, +5.884943e-01, 256 +3.558833e+00, +4.828230e-01, 257 +3.661807e+00, +5.797689e-01, 258 +3.767261e+00, +7.030893e-01, 259 +3.801065e+00, +7.532650e-01, 260 +3.828523e+00, +8.024454e-01, 261 +3.840719e+00, +8.287032e-01, 262 +3.848748e+00, +8.485921e-01, 263 +3.865801e+00, +9.066551e-01, 264 +3.870983e+00, +9.404873e-01, 265 +3.870263e+00, +1.001884e+00, 266 +3.864462e+00, +1.032374e+00, 267 +3.870542e+00, +9.996121e-01, 268 +3.865424e+00, +1.028474e+00 269}; 270ceres::ConstMatrixRef kY(kYData, kYRows, kYCols); 271 272class PointToLineSegmentContourCostFunction : public ceres::CostFunction { 273 public: 274 PointToLineSegmentContourCostFunction(const int num_segments, 275 const Eigen::Vector2d y) 276 : num_segments_(num_segments), y_(y) { 277 // The first parameter is the preimage position. 278 mutable_parameter_block_sizes()->push_back(1); 279 // The next parameters are the control points for the line segment contour. 280 for (int i = 0; i < num_segments_; ++i) { 281 mutable_parameter_block_sizes()->push_back(2); 282 } 283 set_num_residuals(2); 284 } 285 286 virtual bool Evaluate(const double* const* x, 287 double* residuals, 288 double** jacobians) const { 289 // Convert the preimage position `t` into a segment index `i0` and the 290 // line segment interpolation parameter `u`. `i1` is the index of the next 291 // control point. 292 const double t = ModuloNumSegments(*x[0]); 293 CHECK_GE(t, 0.0); 294 CHECK_LT(t, num_segments_); 295 const int i0 = floor(t), i1 = (i0 + 1) % num_segments_; 296 const double u = t - i0; 297 298 // Linearly interpolate between control points `i0` and `i1`. 299 residuals[0] = y_[0] - ((1.0 - u) * x[1 + i0][0] + u * x[1 + i1][0]); 300 residuals[1] = y_[1] - ((1.0 - u) * x[1 + i0][1] + u * x[1 + i1][1]); 301 302 if (jacobians == NULL) { 303 return true; 304 } 305 306 if (jacobians[0] != NULL) { 307 jacobians[0][0] = x[1 + i0][0] - x[1 + i1][0]; 308 jacobians[0][1] = x[1 + i0][1] - x[1 + i1][1]; 309 } 310 for (int i = 0; i < num_segments_; ++i) { 311 if (jacobians[i + 1] != NULL) { 312 ceres::MatrixRef(jacobians[i + 1], 2, 2).setZero(); 313 if (i == i0) { 314 jacobians[i + 1][0] = -(1.0 - u); 315 jacobians[i + 1][3] = -(1.0 - u); 316 } else if (i == i1) { 317 jacobians[i + 1][0] = -u; 318 jacobians[i + 1][3] = -u; 319 } 320 } 321 } 322 return true; 323 } 324 325 static ceres::CostFunction* Create(const int num_segments, 326 const Eigen::Vector2d y) { 327 return new PointToLineSegmentContourCostFunction(num_segments, y); 328 } 329 330 private: 331 inline double ModuloNumSegments(const double& t) const { 332 return t - num_segments_ * floor(t / num_segments_); 333 } 334 335 const int num_segments_; 336 const Eigen::Vector2d y_; 337}; 338 339struct EuclideanDistanceFunctor { 340 EuclideanDistanceFunctor(const double& sqrt_weight) 341 : sqrt_weight_(sqrt_weight) {} 342 343 template <typename T> 344 bool operator()(const T* x0, const T* x1, T* residuals) const { 345 residuals[0] = T(sqrt_weight_) * (x0[0] - x1[0]); 346 residuals[1] = T(sqrt_weight_) * (x0[1] - x1[1]); 347 return true; 348 } 349 350 static ceres::CostFunction* Create(const double& sqrt_weight) { 351 return new ceres::AutoDiffCostFunction<EuclideanDistanceFunctor, 2, 2, 2>( 352 new EuclideanDistanceFunctor(sqrt_weight)); 353 } 354 355 private: 356 const double sqrt_weight_; 357}; 358 359bool SolveWithFullReport(ceres::Solver::Options options, 360 ceres::Problem* problem, 361 bool dynamic_sparsity) { 362 options.dynamic_sparsity = dynamic_sparsity; 363 364 ceres::Solver::Summary summary; 365 ceres::Solve(options, problem, &summary); 366 367 std::cout << "####################" << std::endl; 368 std::cout << "dynamic_sparsity = " << dynamic_sparsity << std::endl; 369 std::cout << "####################" << std::endl; 370 std::cout << summary.FullReport() << std::endl; 371 372 return summary.termination_type == ceres::CONVERGENCE; 373} 374 375int main(int argc, char** argv) { 376 google::InitGoogleLogging(argv[0]); 377 378 // Problem configuration. 379 const int num_segments = 151; 380 const double regularization_weight = 1e-2; 381 382 // Eigen::MatrixXd is column major so we define our own MatrixXd which is 383 // row major. Eigen::VectorXd can be used directly. 384 typedef Eigen::Matrix<double, 385 Eigen::Dynamic, Eigen::Dynamic, 386 Eigen::RowMajor> MatrixXd; 387 using Eigen::VectorXd; 388 389 // `X` is the matrix of control points which make up the contour of line 390 // segments. The number of control points is equal to the number of line 391 // segments because the contour is closed. 392 // 393 // Initialize `X` to points on the unit circle. 394 VectorXd w(num_segments + 1); 395 w.setLinSpaced(num_segments + 1, 0.0, 2.0 * M_PI); 396 w.conservativeResize(num_segments); 397 MatrixXd X(num_segments, 2); 398 X.col(0) = w.array().cos(); 399 X.col(1) = w.array().sin(); 400 401 // Each data point has an associated preimage position on the line segment 402 // contour. For each data point we initialize the preimage positions to 403 // the index of the closest control point. 404 const int num_observations = kY.rows(); 405 VectorXd t(num_observations); 406 for (int i = 0; i < num_observations; ++i) { 407 (X.rowwise() - kY.row(i)).rowwise().squaredNorm().minCoeff(&t[i]); 408 } 409 410 ceres::Problem problem; 411 412 // For each data point add a residual which measures its distance to its 413 // corresponding position on the line segment contour. 414 std::vector<double*> parameter_blocks(1 + num_segments); 415 parameter_blocks[0] = NULL; 416 for (int i = 0; i < num_segments; ++i) { 417 parameter_blocks[i + 1] = X.data() + 2 * i; 418 } 419 for (int i = 0; i < num_observations; ++i) { 420 parameter_blocks[0] = &t[i]; 421 problem.AddResidualBlock( 422 PointToLineSegmentContourCostFunction::Create(num_segments, kY.row(i)), 423 NULL, 424 parameter_blocks); 425 } 426 427 // Add regularization to minimize the length of the line segment contour. 428 for (int i = 0; i < num_segments; ++i) { 429 problem.AddResidualBlock( 430 EuclideanDistanceFunctor::Create(sqrt(regularization_weight)), 431 NULL, 432 X.data() + 2 * i, 433 X.data() + 2 * ((i + 1) % num_segments)); 434 } 435 436 ceres::Solver::Options options; 437 options.max_num_iterations = 100; 438 options.linear_solver_type = ceres::SPARSE_NORMAL_CHOLESKY; 439 440 // First, solve `X` and `t` jointly with dynamic_sparsity = true. 441 MatrixXd X0 = X; 442 VectorXd t0 = t; 443 CHECK(SolveWithFullReport(options, &problem, true)); 444 445 // Second, solve with dynamic_sparsity = false. 446 X = X0; 447 t = t0; 448 CHECK(SolveWithFullReport(options, &problem, false)); 449 450 return 0; 451} 452