1 // Ceres Solver - A fast non-linear least squares minimizer
2 // Copyright 2012 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 //
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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: keir@google.com (Keir Mierle)
30 // sameeragarwal@google.com (Sameer Agarwal)
31 //
32 // This tests the TrustRegionMinimizer loop using a direct Evaluator
33 // implementation, rather than having a test that goes through all the
34 // Program and Problem machinery.
35
36 #include <cmath>
37 #include "ceres/cost_function.h"
38 #include "ceres/dense_qr_solver.h"
39 #include "ceres/dense_sparse_matrix.h"
40 #include "ceres/evaluator.h"
41 #include "ceres/internal/port.h"
42 #include "ceres/linear_solver.h"
43 #include "ceres/minimizer.h"
44 #include "ceres/problem.h"
45 #include "ceres/trust_region_minimizer.h"
46 #include "ceres/trust_region_strategy.h"
47 #include "gtest/gtest.h"
48
49 namespace ceres {
50 namespace internal {
51
52 // Templated Evaluator for Powell's function. The template parameters
53 // indicate which of the four variables/columns of the jacobian are
54 // active. This is equivalent to constructing a problem and using the
55 // SubsetLocalParameterization. This allows us to test the support for
56 // the Evaluator::Plus operation besides checking for the basic
57 // performance of the trust region algorithm.
58 template <bool col1, bool col2, bool col3, bool col4>
59 class PowellEvaluator2 : public Evaluator {
60 public:
PowellEvaluator2()61 PowellEvaluator2()
62 : num_active_cols_(
63 (col1 ? 1 : 0) +
64 (col2 ? 1 : 0) +
65 (col3 ? 1 : 0) +
66 (col4 ? 1 : 0)) {
67 VLOG(1) << "Columns: "
68 << col1 << " "
69 << col2 << " "
70 << col3 << " "
71 << col4;
72 }
73
~PowellEvaluator2()74 virtual ~PowellEvaluator2() {}
75
76 // Implementation of Evaluator interface.
CreateJacobian() const77 virtual SparseMatrix* CreateJacobian() const {
78 CHECK(col1 || col2 || col3 || col4);
79 DenseSparseMatrix* dense_jacobian =
80 new DenseSparseMatrix(NumResiduals(), NumEffectiveParameters());
81 dense_jacobian->SetZero();
82 return dense_jacobian;
83 }
84
Evaluate(const Evaluator::EvaluateOptions & evaluate_options,const double * state,double * cost,double * residuals,double * gradient,SparseMatrix * jacobian)85 virtual bool Evaluate(const Evaluator::EvaluateOptions& evaluate_options,
86 const double* state,
87 double* cost,
88 double* residuals,
89 double* gradient,
90 SparseMatrix* jacobian) {
91 const double x1 = state[0];
92 const double x2 = state[1];
93 const double x3 = state[2];
94 const double x4 = state[3];
95
96 VLOG(1) << "State: "
97 << "x1=" << x1 << ", "
98 << "x2=" << x2 << ", "
99 << "x3=" << x3 << ", "
100 << "x4=" << x4 << ".";
101
102 const double f1 = x1 + 10.0 * x2;
103 const double f2 = sqrt(5.0) * (x3 - x4);
104 const double f3 = pow(x2 - 2.0 * x3, 2.0);
105 const double f4 = sqrt(10.0) * pow(x1 - x4, 2.0);
106
107 VLOG(1) << "Function: "
108 << "f1=" << f1 << ", "
109 << "f2=" << f2 << ", "
110 << "f3=" << f3 << ", "
111 << "f4=" << f4 << ".";
112
113 *cost = (f1*f1 + f2*f2 + f3*f3 + f4*f4) / 2.0;
114
115 VLOG(1) << "Cost: " << *cost;
116
117 if (residuals != NULL) {
118 residuals[0] = f1;
119 residuals[1] = f2;
120 residuals[2] = f3;
121 residuals[3] = f4;
122 }
123
124 if (jacobian != NULL) {
125 DenseSparseMatrix* dense_jacobian;
126 dense_jacobian = down_cast<DenseSparseMatrix*>(jacobian);
127 dense_jacobian->SetZero();
128
129 ColMajorMatrixRef jacobian_matrix = dense_jacobian->mutable_matrix();
130 CHECK_EQ(jacobian_matrix.cols(), num_active_cols_);
131
132 int column_index = 0;
133 if (col1) {
134 jacobian_matrix.col(column_index++) <<
135 1.0,
136 0.0,
137 0.0,
138 sqrt(10.0) * 2.0 * (x1 - x4) * (1.0 - x4);
139 }
140 if (col2) {
141 jacobian_matrix.col(column_index++) <<
142 10.0,
143 0.0,
144 2.0*(x2 - 2.0*x3)*(1.0 - 2.0*x3),
145 0.0;
146 }
147
148 if (col3) {
149 jacobian_matrix.col(column_index++) <<
150 0.0,
151 sqrt(5.0),
152 2.0*(x2 - 2.0*x3)*(x2 - 2.0),
153 0.0;
154 }
155
156 if (col4) {
157 jacobian_matrix.col(column_index++) <<
158 0.0,
159 -sqrt(5.0),
160 0.0,
161 sqrt(10.0) * 2.0 * (x1 - x4) * (x1 - 1.0);
162 }
163 VLOG(1) << "\n" << jacobian_matrix;
164 }
165
166 if (gradient != NULL) {
167 int column_index = 0;
168 if (col1) {
169 gradient[column_index++] = f1 + f4 * sqrt(10.0) * 2.0 * (x1 - x4);
170 }
171
172 if (col2) {
173 gradient[column_index++] = f1 * 10.0 + f3 * 2.0 * (x2 - 2.0 * x3);
174 }
175
176 if (col3) {
177 gradient[column_index++] =
178 f2 * sqrt(5.0) + f3 * (2.0 * 2.0 * (2.0 * x3 - x2));
179 }
180
181 if (col4) {
182 gradient[column_index++] =
183 -f2 * sqrt(5.0) + f4 * sqrt(10.0) * 2.0 * (x4 - x1);
184 }
185 }
186
187 return true;
188 }
189
Plus(const double * state,const double * delta,double * state_plus_delta) const190 virtual bool Plus(const double* state,
191 const double* delta,
192 double* state_plus_delta) const {
193 int delta_index = 0;
194 state_plus_delta[0] = (col1 ? state[0] + delta[delta_index++] : state[0]);
195 state_plus_delta[1] = (col2 ? state[1] + delta[delta_index++] : state[1]);
196 state_plus_delta[2] = (col3 ? state[2] + delta[delta_index++] : state[2]);
197 state_plus_delta[3] = (col4 ? state[3] + delta[delta_index++] : state[3]);
198 return true;
199 }
200
NumEffectiveParameters() const201 virtual int NumEffectiveParameters() const { return num_active_cols_; }
NumParameters() const202 virtual int NumParameters() const { return 4; }
NumResiduals() const203 virtual int NumResiduals() const { return 4; }
204
205 private:
206 const int num_active_cols_;
207 };
208
209 // Templated function to hold a subset of the columns fixed and check
210 // if the solver converges to the optimal values or not.
211 template<bool col1, bool col2, bool col3, bool col4>
IsTrustRegionSolveSuccessful(TrustRegionStrategyType strategy_type)212 void IsTrustRegionSolveSuccessful(TrustRegionStrategyType strategy_type) {
213 Solver::Options solver_options;
214 LinearSolver::Options linear_solver_options;
215 DenseQRSolver linear_solver(linear_solver_options);
216
217 double parameters[4] = { 3, -1, 0, 1.0 };
218
219 // If the column is inactive, then set its value to the optimal
220 // value.
221 parameters[0] = (col1 ? parameters[0] : 0.0);
222 parameters[1] = (col2 ? parameters[1] : 0.0);
223 parameters[2] = (col3 ? parameters[2] : 0.0);
224 parameters[3] = (col4 ? parameters[3] : 0.0);
225
226 PowellEvaluator2<col1, col2, col3, col4> powell_evaluator;
227 scoped_ptr<SparseMatrix> jacobian(powell_evaluator.CreateJacobian());
228
229 Minimizer::Options minimizer_options(solver_options);
230 minimizer_options.gradient_tolerance = 1e-26;
231 minimizer_options.function_tolerance = 1e-26;
232 minimizer_options.parameter_tolerance = 1e-26;
233 minimizer_options.evaluator = &powell_evaluator;
234 minimizer_options.jacobian = jacobian.get();
235
236 TrustRegionStrategy::Options trust_region_strategy_options;
237 trust_region_strategy_options.trust_region_strategy_type = strategy_type;
238 trust_region_strategy_options.linear_solver = &linear_solver;
239 trust_region_strategy_options.initial_radius = 1e4;
240 trust_region_strategy_options.max_radius = 1e20;
241 trust_region_strategy_options.min_lm_diagonal = 1e-6;
242 trust_region_strategy_options.max_lm_diagonal = 1e32;
243 scoped_ptr<TrustRegionStrategy> strategy(
244 TrustRegionStrategy::Create(trust_region_strategy_options));
245 minimizer_options.trust_region_strategy = strategy.get();
246
247 TrustRegionMinimizer minimizer;
248 Solver::Summary summary;
249 minimizer.Minimize(minimizer_options, parameters, &summary);
250
251 // The minimum is at x1 = x2 = x3 = x4 = 0.
252 EXPECT_NEAR(0.0, parameters[0], 0.001);
253 EXPECT_NEAR(0.0, parameters[1], 0.001);
254 EXPECT_NEAR(0.0, parameters[2], 0.001);
255 EXPECT_NEAR(0.0, parameters[3], 0.001);
256 };
257
TEST(TrustRegionMinimizer,PowellsSingularFunctionUsingLevenbergMarquardt)258 TEST(TrustRegionMinimizer, PowellsSingularFunctionUsingLevenbergMarquardt) {
259 // This case is excluded because this has a local minimum and does
260 // not find the optimum. This should not affect the correctness of
261 // this test since we are testing all the other 14 combinations of
262 // column activations.
263 //
264 // IsSolveSuccessful<true, true, false, true>();
265
266 const TrustRegionStrategyType kStrategy = LEVENBERG_MARQUARDT;
267 IsTrustRegionSolveSuccessful<true, true, true, true >(kStrategy);
268 IsTrustRegionSolveSuccessful<true, true, true, false>(kStrategy);
269 IsTrustRegionSolveSuccessful<true, false, true, true >(kStrategy);
270 IsTrustRegionSolveSuccessful<false, true, true, true >(kStrategy);
271 IsTrustRegionSolveSuccessful<true, true, false, false>(kStrategy);
272 IsTrustRegionSolveSuccessful<true, false, true, false>(kStrategy);
273 IsTrustRegionSolveSuccessful<false, true, true, false>(kStrategy);
274 IsTrustRegionSolveSuccessful<true, false, false, true >(kStrategy);
275 IsTrustRegionSolveSuccessful<false, true, false, true >(kStrategy);
276 IsTrustRegionSolveSuccessful<false, false, true, true >(kStrategy);
277 IsTrustRegionSolveSuccessful<true, false, false, false>(kStrategy);
278 IsTrustRegionSolveSuccessful<false, true, false, false>(kStrategy);
279 IsTrustRegionSolveSuccessful<false, false, true, false>(kStrategy);
280 IsTrustRegionSolveSuccessful<false, false, false, true >(kStrategy);
281 }
282
TEST(TrustRegionMinimizer,PowellsSingularFunctionUsingDogleg)283 TEST(TrustRegionMinimizer, PowellsSingularFunctionUsingDogleg) {
284 // The following two cases are excluded because they encounter a
285 // local minimum.
286 //
287 // IsTrustRegionSolveSuccessful<true, true, false, true >(kStrategy);
288 // IsTrustRegionSolveSuccessful<true, true, true, true >(kStrategy);
289
290 const TrustRegionStrategyType kStrategy = DOGLEG;
291 IsTrustRegionSolveSuccessful<true, true, true, false>(kStrategy);
292 IsTrustRegionSolveSuccessful<true, false, true, true >(kStrategy);
293 IsTrustRegionSolveSuccessful<false, true, true, true >(kStrategy);
294 IsTrustRegionSolveSuccessful<true, true, false, false>(kStrategy);
295 IsTrustRegionSolveSuccessful<true, false, true, false>(kStrategy);
296 IsTrustRegionSolveSuccessful<false, true, true, false>(kStrategy);
297 IsTrustRegionSolveSuccessful<true, false, false, true >(kStrategy);
298 IsTrustRegionSolveSuccessful<false, true, false, true >(kStrategy);
299 IsTrustRegionSolveSuccessful<false, false, true, true >(kStrategy);
300 IsTrustRegionSolveSuccessful<true, false, false, false>(kStrategy);
301 IsTrustRegionSolveSuccessful<false, true, false, false>(kStrategy);
302 IsTrustRegionSolveSuccessful<false, false, true, false>(kStrategy);
303 IsTrustRegionSolveSuccessful<false, false, false, true >(kStrategy);
304 }
305
306
307 class CurveCostFunction : public CostFunction {
308 public:
CurveCostFunction(int num_vertices,double target_length)309 CurveCostFunction(int num_vertices, double target_length)
310 : num_vertices_(num_vertices), target_length_(target_length) {
311 set_num_residuals(1);
312 for (int i = 0; i < num_vertices_; ++i) {
313 mutable_parameter_block_sizes()->push_back(2);
314 }
315 }
316
Evaluate(double const * const * parameters,double * residuals,double ** jacobians) const317 bool Evaluate(double const* const* parameters,
318 double* residuals,
319 double** jacobians) const {
320 residuals[0] = target_length_;
321
322 for (int i = 0; i < num_vertices_; ++i) {
323 int prev = (num_vertices_ + i - 1) % num_vertices_;
324 double length = 0.0;
325 for (int dim = 0; dim < 2; dim++) {
326 const double diff = parameters[prev][dim] - parameters[i][dim];
327 length += diff * diff;
328 }
329 residuals[0] -= sqrt(length);
330 }
331
332 if (jacobians == NULL) {
333 return true;
334 }
335
336 for (int i = 0; i < num_vertices_; ++i) {
337 if (jacobians[i] != NULL) {
338 int prev = (num_vertices_ + i - 1) % num_vertices_;
339 int next = (i + 1) % num_vertices_;
340
341 double u[2], v[2];
342 double norm_u = 0., norm_v = 0.;
343 for (int dim = 0; dim < 2; dim++) {
344 u[dim] = parameters[i][dim] - parameters[prev][dim];
345 norm_u += u[dim] * u[dim];
346 v[dim] = parameters[next][dim] - parameters[i][dim];
347 norm_v += v[dim] * v[dim];
348 }
349
350 norm_u = sqrt(norm_u);
351 norm_v = sqrt(norm_v);
352
353 for (int dim = 0; dim < 2; dim++) {
354 jacobians[i][dim] = 0.;
355
356 if (norm_u > std::numeric_limits< double >::min()) {
357 jacobians[i][dim] -= u[dim] / norm_u;
358 }
359
360 if (norm_v > std::numeric_limits< double >::min()) {
361 jacobians[i][dim] += v[dim] / norm_v;
362 }
363 }
364 }
365 }
366
367 return true;
368 }
369
370 private:
371 int num_vertices_;
372 double target_length_;
373 };
374
TEST(TrustRegionMinimizer,JacobiScalingTest)375 TEST(TrustRegionMinimizer, JacobiScalingTest) {
376 int N = 6;
377 std::vector< double* > y(N);
378 const double pi = 3.1415926535897932384626433;
379 for (int i = 0; i < N; i++) {
380 double theta = i * 2. * pi/ static_cast< double >(N);
381 y[i] = new double[2];
382 y[i][0] = cos(theta);
383 y[i][1] = sin(theta);
384 }
385
386 Problem problem;
387 problem.AddResidualBlock(new CurveCostFunction(N, 10.), NULL, y);
388 Solver::Options options;
389 options.linear_solver_type = ceres::DENSE_QR;
390 Solver::Summary summary;
391 Solve(options, &problem, &summary);
392 EXPECT_LE(summary.final_cost, 1e-10);
393
394 for (int i = 0; i < N; i++) {
395 delete []y[i];
396 }
397 }
398
399 } // namespace internal
400 } // namespace ceres
401