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 //
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
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20 // ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
21 // LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
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24 // INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
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26 // ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
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28 //
29 // Author: moll.markus@arcor.de (Markus Moll)
30 // sameeragarwal@google.com (Sameer Agarwal)
31
32 #include "ceres/polynomial.h"
33
34 #include <cmath>
35 #include <cstddef>
36 #include <vector>
37
38 #include "Eigen/Dense"
39 #include "ceres/internal/port.h"
40 #include "ceres/stringprintf.h"
41 #include "glog/logging.h"
42
43 namespace ceres {
44 namespace internal {
45 namespace {
46
47 // Balancing function as described by B. N. Parlett and C. Reinsch,
48 // "Balancing a Matrix for Calculation of Eigenvalues and Eigenvectors".
49 // In: Numerische Mathematik, Volume 13, Number 4 (1969), 293-304,
50 // Springer Berlin / Heidelberg. DOI: 10.1007/BF02165404
BalanceCompanionMatrix(Matrix * companion_matrix_ptr)51 void BalanceCompanionMatrix(Matrix* companion_matrix_ptr) {
52 CHECK_NOTNULL(companion_matrix_ptr);
53 Matrix& companion_matrix = *companion_matrix_ptr;
54 Matrix companion_matrix_offdiagonal = companion_matrix;
55 companion_matrix_offdiagonal.diagonal().setZero();
56
57 const int degree = companion_matrix.rows();
58
59 // gamma <= 1 controls how much a change in the scaling has to
60 // lower the 1-norm of the companion matrix to be accepted.
61 //
62 // gamma = 1 seems to lead to cycles (numerical issues?), so
63 // we set it slightly lower.
64 const double gamma = 0.9;
65
66 // Greedily scale row/column pairs until there is no change.
67 bool scaling_has_changed;
68 do {
69 scaling_has_changed = false;
70
71 for (int i = 0; i < degree; ++i) {
72 const double row_norm = companion_matrix_offdiagonal.row(i).lpNorm<1>();
73 const double col_norm = companion_matrix_offdiagonal.col(i).lpNorm<1>();
74
75 // Decompose row_norm/col_norm into mantissa * 2^exponent,
76 // where 0.5 <= mantissa < 1. Discard mantissa (return value
77 // of frexp), as only the exponent is needed.
78 int exponent = 0;
79 std::frexp(row_norm / col_norm, &exponent);
80 exponent /= 2;
81
82 if (exponent != 0) {
83 const double scaled_col_norm = std::ldexp(col_norm, exponent);
84 const double scaled_row_norm = std::ldexp(row_norm, -exponent);
85 if (scaled_col_norm + scaled_row_norm < gamma * (col_norm + row_norm)) {
86 // Accept the new scaling. (Multiplication by powers of 2 should not
87 // introduce rounding errors (ignoring non-normalized numbers and
88 // over- or underflow))
89 scaling_has_changed = true;
90 companion_matrix_offdiagonal.row(i) *= std::ldexp(1.0, -exponent);
91 companion_matrix_offdiagonal.col(i) *= std::ldexp(1.0, exponent);
92 }
93 }
94 }
95 } while (scaling_has_changed);
96
97 companion_matrix_offdiagonal.diagonal() = companion_matrix.diagonal();
98 companion_matrix = companion_matrix_offdiagonal;
99 VLOG(3) << "Balanced companion matrix is\n" << companion_matrix;
100 }
101
BuildCompanionMatrix(const Vector & polynomial,Matrix * companion_matrix_ptr)102 void BuildCompanionMatrix(const Vector& polynomial,
103 Matrix* companion_matrix_ptr) {
104 CHECK_NOTNULL(companion_matrix_ptr);
105 Matrix& companion_matrix = *companion_matrix_ptr;
106
107 const int degree = polynomial.size() - 1;
108
109 companion_matrix.resize(degree, degree);
110 companion_matrix.setZero();
111 companion_matrix.diagonal(-1).setOnes();
112 companion_matrix.col(degree - 1) = -polynomial.reverse().head(degree);
113 }
114
115 // Remove leading terms with zero coefficients.
RemoveLeadingZeros(const Vector & polynomial_in)116 Vector RemoveLeadingZeros(const Vector& polynomial_in) {
117 int i = 0;
118 while (i < (polynomial_in.size() - 1) && polynomial_in(i) == 0.0) {
119 ++i;
120 }
121 return polynomial_in.tail(polynomial_in.size() - i);
122 }
123
FindLinearPolynomialRoots(const Vector & polynomial,Vector * real,Vector * imaginary)124 void FindLinearPolynomialRoots(const Vector& polynomial,
125 Vector* real,
126 Vector* imaginary) {
127 CHECK_EQ(polynomial.size(), 2);
128 if (real != NULL) {
129 real->resize(1);
130 (*real)(0) = -polynomial(1) / polynomial(0);
131 }
132
133 if (imaginary != NULL) {
134 imaginary->setZero(1);
135 }
136 }
137
FindQuadraticPolynomialRoots(const Vector & polynomial,Vector * real,Vector * imaginary)138 void FindQuadraticPolynomialRoots(const Vector& polynomial,
139 Vector* real,
140 Vector* imaginary) {
141 CHECK_EQ(polynomial.size(), 3);
142 const double a = polynomial(0);
143 const double b = polynomial(1);
144 const double c = polynomial(2);
145 const double D = b * b - 4 * a * c;
146 const double sqrt_D = sqrt(fabs(D));
147 if (real != NULL) {
148 real->setZero(2);
149 }
150 if (imaginary != NULL) {
151 imaginary->setZero(2);
152 }
153
154 // Real roots.
155 if (D >= 0) {
156 if (real != NULL) {
157 // Stable quadratic roots according to BKP Horn.
158 // http://people.csail.mit.edu/bkph/articles/Quadratics.pdf
159 if (b >= 0) {
160 (*real)(0) = (-b - sqrt_D) / (2.0 * a);
161 (*real)(1) = (2.0 * c) / (-b - sqrt_D);
162 } else {
163 (*real)(0) = (2.0 * c) / (-b + sqrt_D);
164 (*real)(1) = (-b + sqrt_D) / (2.0 * a);
165 }
166 }
167 return;
168 }
169
170 // Use the normal quadratic formula for the complex case.
171 if (real != NULL) {
172 (*real)(0) = -b / (2.0 * a);
173 (*real)(1) = -b / (2.0 * a);
174 }
175 if (imaginary != NULL) {
176 (*imaginary)(0) = sqrt_D / (2.0 * a);
177 (*imaginary)(1) = -sqrt_D / (2.0 * a);
178 }
179 }
180 } // namespace
181
FindPolynomialRoots(const Vector & polynomial_in,Vector * real,Vector * imaginary)182 bool FindPolynomialRoots(const Vector& polynomial_in,
183 Vector* real,
184 Vector* imaginary) {
185 if (polynomial_in.size() == 0) {
186 LOG(ERROR) << "Invalid polynomial of size 0 passed to FindPolynomialRoots";
187 return false;
188 }
189
190 Vector polynomial = RemoveLeadingZeros(polynomial_in);
191 const int degree = polynomial.size() - 1;
192
193 VLOG(3) << "Input polynomial: " << polynomial_in.transpose();
194 if (polynomial.size() != polynomial_in.size()) {
195 VLOG(3) << "Trimmed polynomial: " << polynomial.transpose();
196 }
197
198 // Is the polynomial constant?
199 if (degree == 0) {
200 LOG(WARNING) << "Trying to extract roots from a constant "
201 << "polynomial in FindPolynomialRoots";
202 // We return true with no roots, not false, as if the polynomial is constant
203 // it is correct that there are no roots. It is not the case that they were
204 // there, but that we have failed to extract them.
205 return true;
206 }
207
208 // Linear
209 if (degree == 1) {
210 FindLinearPolynomialRoots(polynomial, real, imaginary);
211 return true;
212 }
213
214 // Quadratic
215 if (degree == 2) {
216 FindQuadraticPolynomialRoots(polynomial, real, imaginary);
217 return true;
218 }
219
220 // The degree is now known to be at least 3. For cubic or higher
221 // roots we use the method of companion matrices.
222
223 // Divide by leading term
224 const double leading_term = polynomial(0);
225 polynomial /= leading_term;
226
227 // Build and balance the companion matrix to the polynomial.
228 Matrix companion_matrix(degree, degree);
229 BuildCompanionMatrix(polynomial, &companion_matrix);
230 BalanceCompanionMatrix(&companion_matrix);
231
232 // Find its (complex) eigenvalues.
233 Eigen::EigenSolver<Matrix> solver(companion_matrix, false);
234 if (solver.info() != Eigen::Success) {
235 LOG(ERROR) << "Failed to extract eigenvalues from companion matrix.";
236 return false;
237 }
238
239 // Output roots
240 if (real != NULL) {
241 *real = solver.eigenvalues().real();
242 } else {
243 LOG(WARNING) << "NULL pointer passed as real argument to "
244 << "FindPolynomialRoots. Real parts of the roots will not "
245 << "be returned.";
246 }
247 if (imaginary != NULL) {
248 *imaginary = solver.eigenvalues().imag();
249 }
250 return true;
251 }
252
DifferentiatePolynomial(const Vector & polynomial)253 Vector DifferentiatePolynomial(const Vector& polynomial) {
254 const int degree = polynomial.rows() - 1;
255 CHECK_GE(degree, 0);
256
257 // Degree zero polynomials are constants, and their derivative does
258 // not result in a smaller degree polynomial, just a degree zero
259 // polynomial with value zero.
260 if (degree == 0) {
261 return Eigen::VectorXd::Zero(1);
262 }
263
264 Vector derivative(degree);
265 for (int i = 0; i < degree; ++i) {
266 derivative(i) = (degree - i) * polynomial(i);
267 }
268
269 return derivative;
270 }
271
MinimizePolynomial(const Vector & polynomial,const double x_min,const double x_max,double * optimal_x,double * optimal_value)272 void MinimizePolynomial(const Vector& polynomial,
273 const double x_min,
274 const double x_max,
275 double* optimal_x,
276 double* optimal_value) {
277 // Find the minimum of the polynomial at the two ends.
278 //
279 // We start by inspecting the middle of the interval. Technically
280 // this is not needed, but we do this to make this code as close to
281 // the minFunc package as possible.
282 *optimal_x = (x_min + x_max) / 2.0;
283 *optimal_value = EvaluatePolynomial(polynomial, *optimal_x);
284
285 const double x_min_value = EvaluatePolynomial(polynomial, x_min);
286 if (x_min_value < *optimal_value) {
287 *optimal_value = x_min_value;
288 *optimal_x = x_min;
289 }
290
291 const double x_max_value = EvaluatePolynomial(polynomial, x_max);
292 if (x_max_value < *optimal_value) {
293 *optimal_value = x_max_value;
294 *optimal_x = x_max;
295 }
296
297 // If the polynomial is linear or constant, we are done.
298 if (polynomial.rows() <= 2) {
299 return;
300 }
301
302 const Vector derivative = DifferentiatePolynomial(polynomial);
303 Vector roots_real;
304 if (!FindPolynomialRoots(derivative, &roots_real, NULL)) {
305 LOG(WARNING) << "Unable to find the critical points of "
306 << "the interpolating polynomial.";
307 return;
308 }
309
310 // This is a bit of an overkill, as some of the roots may actually
311 // have a complex part, but its simpler to just check these values.
312 for (int i = 0; i < roots_real.rows(); ++i) {
313 const double root = roots_real(i);
314 if ((root < x_min) || (root > x_max)) {
315 continue;
316 }
317
318 const double value = EvaluatePolynomial(polynomial, root);
319 if (value < *optimal_value) {
320 *optimal_value = value;
321 *optimal_x = root;
322 }
323 }
324 }
325
ToDebugString() const326 string FunctionSample::ToDebugString() const {
327 return StringPrintf("[x: %.8e, value: %.8e, gradient: %.8e, "
328 "value_is_valid: %d, gradient_is_valid: %d]",
329 x, value, gradient, value_is_valid, gradient_is_valid);
330 }
331
FindInterpolatingPolynomial(const vector<FunctionSample> & samples)332 Vector FindInterpolatingPolynomial(const vector<FunctionSample>& samples) {
333 const int num_samples = samples.size();
334 int num_constraints = 0;
335 for (int i = 0; i < num_samples; ++i) {
336 if (samples[i].value_is_valid) {
337 ++num_constraints;
338 }
339 if (samples[i].gradient_is_valid) {
340 ++num_constraints;
341 }
342 }
343
344 const int degree = num_constraints - 1;
345
346 Matrix lhs = Matrix::Zero(num_constraints, num_constraints);
347 Vector rhs = Vector::Zero(num_constraints);
348
349 int row = 0;
350 for (int i = 0; i < num_samples; ++i) {
351 const FunctionSample& sample = samples[i];
352 if (sample.value_is_valid) {
353 for (int j = 0; j <= degree; ++j) {
354 lhs(row, j) = pow(sample.x, degree - j);
355 }
356 rhs(row) = sample.value;
357 ++row;
358 }
359
360 if (sample.gradient_is_valid) {
361 for (int j = 0; j < degree; ++j) {
362 lhs(row, j) = (degree - j) * pow(sample.x, degree - j - 1);
363 }
364 rhs(row) = sample.gradient;
365 ++row;
366 }
367 }
368
369 return lhs.fullPivLu().solve(rhs);
370 }
371
MinimizeInterpolatingPolynomial(const vector<FunctionSample> & samples,double x_min,double x_max,double * optimal_x,double * optimal_value)372 void MinimizeInterpolatingPolynomial(const vector<FunctionSample>& samples,
373 double x_min,
374 double x_max,
375 double* optimal_x,
376 double* optimal_value) {
377 const Vector polynomial = FindInterpolatingPolynomial(samples);
378 MinimizePolynomial(polynomial, x_min, x_max, optimal_x, optimal_value);
379 for (int i = 0; i < samples.size(); ++i) {
380 const FunctionSample& sample = samples[i];
381 if ((sample.x < x_min) || (sample.x > x_max)) {
382 continue;
383 }
384
385 const double value = EvaluatePolynomial(polynomial, sample.x);
386 if (value < *optimal_value) {
387 *optimal_x = sample.x;
388 *optimal_value = value;
389 }
390 }
391 }
392
393 } // namespace internal
394 } // namespace ceres
395