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