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+// Ceres Solver - A fast non-linear least squares minimizer
+// Copyright 2012 Google Inc. All rights reserved.
+// http://code.google.com/p/ceres-solver/
+//
+// Redistribution and use in source and binary forms, with or without
+// modification, are permitted provided that the following conditions are met:
+//
+// * Redistributions of source code must retain the above copyright notice,
+// this list of conditions and the following disclaimer.
+// * Redistributions in binary form must reproduce the above copyright notice,
+// this list of conditions and the following disclaimer in the documentation
+// and/or other materials provided with the distribution.
+// * Neither the name of Google Inc. nor the names of its contributors may be
+// used to endorse or promote products derived from this software without
+// specific prior written permission.
+//
+// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
+// AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
+// IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
+// ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
+// LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
+// CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
+// SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
+// INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
+// CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
+// ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
+// POSSIBILITY OF SUCH DAMAGE.
+//
+// Author: moll.markus@arcor.de (Markus Moll)
+
+#include <limits>
+#include "ceres/internal/eigen.h"
+#include "ceres/internal/scoped_ptr.h"
+#include "ceres/dense_qr_solver.h"
+#include "ceres/dogleg_strategy.h"
+#include "ceres/linear_solver.h"
+#include "ceres/trust_region_strategy.h"
+#include "glog/logging.h"
+#include "gtest/gtest.h"
+
+namespace ceres {
+namespace internal {
+namespace {
+
+class Fixture : public testing::Test {
+ protected:
+ scoped_ptr<DenseSparseMatrix> jacobian_;
+ Vector residual_;
+ Vector x_;
+ TrustRegionStrategy::Options options_;
+};
+
+// A test problem where
+//
+// J^T J = Q diag([1 2 4 8 16 32]) Q^T
+//
+// where Q is a randomly chosen orthonormal basis of R^6.
+// The residual is chosen so that the minimum of the quadratic function is
+// at (1, 1, 1, 1, 1, 1). It is therefore at a distance of sqrt(6) ~ 2.45
+// from the origin.
+class DoglegStrategyFixtureEllipse : public Fixture {
+ protected:
+ virtual void SetUp() {
+ Matrix basis(6, 6);
+ // The following lines exceed 80 characters for better readability.
+ basis << -0.1046920933796121, -0.7449367449921986, -0.4190744502875876, -0.4480450716142566, 0.2375351607929440, -0.0363053418882862,
+ 0.4064975684355914, 0.2681113508511354, -0.7463625494601520, -0.0803264850508117, -0.4463149623021321, 0.0130224954867195,
+ -0.5514387729089798, 0.1026621026168657, -0.5008316122125011, 0.5738122212666414, 0.2974664724007106, 0.1296020877535158,
+ 0.5037835370947156, 0.2668479925183712, -0.1051754618492798, -0.0272739396578799, 0.7947481647088278, -0.1776623363955670,
+ -0.4005458426625444, 0.2939330589634109, -0.0682629380550051, -0.2895448882503687, -0.0457239396341685, -0.8139899477847840,
+ -0.3247764582762654, 0.4528151365941945, -0.0276683863102816, -0.6155994592510784, 0.1489240599972848, 0.5362574892189350;
+
+ Vector Ddiag(6);
+ Ddiag << 1.0, 2.0, 4.0, 8.0, 16.0, 32.0;
+
+ Matrix sqrtD = Ddiag.array().sqrt().matrix().asDiagonal();
+ Matrix jacobian = sqrtD * basis;
+ jacobian_.reset(new DenseSparseMatrix(jacobian));
+
+ Vector minimum(6);
+ minimum << 1.0, 1.0, 1.0, 1.0, 1.0, 1.0;
+ residual_ = -jacobian * minimum;
+
+ x_.resize(6);
+ x_.setZero();
+
+ options_.lm_min_diagonal = 1.0;
+ options_.lm_max_diagonal = 1.0;
+ }
+};
+
+// A test problem where
+//
+// J^T J = diag([1 2 4 8 16 32]) .
+//
+// The residual is chosen so that the minimum of the quadratic function is
+// at (0, 0, 1, 0, 0, 0). It is therefore at a distance of 1 from the origin.
+// The gradient at the origin points towards the global minimum.
+class DoglegStrategyFixtureValley : public Fixture {
+ protected:
+ virtual void SetUp() {
+ Vector Ddiag(6);
+ Ddiag << 1.0, 2.0, 4.0, 8.0, 16.0, 32.0;
+
+ Matrix jacobian = Ddiag.asDiagonal();
+ jacobian_.reset(new DenseSparseMatrix(jacobian));
+
+ Vector minimum(6);
+ minimum << 0.0, 0.0, 1.0, 0.0, 0.0, 0.0;
+ residual_ = -jacobian * minimum;
+
+ x_.resize(6);
+ x_.setZero();
+
+ options_.lm_min_diagonal = 1.0;
+ options_.lm_max_diagonal = 1.0;
+ }
+};
+
+const double kTolerance = 1e-14;
+const double kToleranceLoose = 1e-5;
+const double kEpsilon = std::numeric_limits<double>::epsilon();
+
+} // namespace
+
+// The DoglegStrategy must never return a step that is longer than the current
+// trust region radius.
+TEST_F(DoglegStrategyFixtureEllipse, TrustRegionObeyedTraditional) {
+ scoped_ptr<LinearSolver> linear_solver(
+ new DenseQRSolver(LinearSolver::Options()));
+ options_.linear_solver = linear_solver.get();
+ // The global minimum is at (1, 1, ..., 1), so the distance to it is sqrt(6.0).
+ // By restricting the trust region to a radius of 2.0, we test if the trust
+ // region is actually obeyed.
+ options_.dogleg_type = TRADITIONAL_DOGLEG;
+ options_.initial_radius = 2.0;
+ options_.max_radius = 2.0;
+
+ DoglegStrategy strategy(options_);
+ TrustRegionStrategy::PerSolveOptions pso;
+
+ TrustRegionStrategy::Summary summary = strategy.ComputeStep(pso,
+ jacobian_.get(),
+ residual_.data(),
+ x_.data());
+
+ EXPECT_NE(summary.termination_type, FAILURE);
+ EXPECT_LE(x_.norm(), options_.initial_radius * (1.0 + 4.0 * kEpsilon));
+}
+
+TEST_F(DoglegStrategyFixtureEllipse, TrustRegionObeyedSubspace) {
+ scoped_ptr<LinearSolver> linear_solver(
+ new DenseQRSolver(LinearSolver::Options()));
+ options_.linear_solver = linear_solver.get();
+ options_.dogleg_type = SUBSPACE_DOGLEG;
+ options_.initial_radius = 2.0;
+ options_.max_radius = 2.0;
+
+ DoglegStrategy strategy(options_);
+ TrustRegionStrategy::PerSolveOptions pso;
+
+ TrustRegionStrategy::Summary summary = strategy.ComputeStep(pso,
+ jacobian_.get(),
+ residual_.data(),
+ x_.data());
+
+ EXPECT_NE(summary.termination_type, FAILURE);
+ EXPECT_LE(x_.norm(), options_.initial_radius * (1.0 + 4.0 * kEpsilon));
+}
+
+TEST_F(DoglegStrategyFixtureEllipse, CorrectGaussNewtonStep) {
+ scoped_ptr<LinearSolver> linear_solver(
+ new DenseQRSolver(LinearSolver::Options()));
+ options_.linear_solver = linear_solver.get();
+ options_.dogleg_type = SUBSPACE_DOGLEG;
+ options_.initial_radius = 10.0;
+ options_.max_radius = 10.0;
+
+ DoglegStrategy strategy(options_);
+ TrustRegionStrategy::PerSolveOptions pso;
+
+ TrustRegionStrategy::Summary summary = strategy.ComputeStep(pso,
+ jacobian_.get(),
+ residual_.data(),
+ x_.data());
+
+ EXPECT_NE(summary.termination_type, FAILURE);
+ EXPECT_NEAR(x_(0), 1.0, kToleranceLoose);
+ EXPECT_NEAR(x_(1), 1.0, kToleranceLoose);
+ EXPECT_NEAR(x_(2), 1.0, kToleranceLoose);
+ EXPECT_NEAR(x_(3), 1.0, kToleranceLoose);
+ EXPECT_NEAR(x_(4), 1.0, kToleranceLoose);
+ EXPECT_NEAR(x_(5), 1.0, kToleranceLoose);
+}
+
+// Test if the subspace basis is a valid orthonormal basis of the space spanned
+// by the gradient and the Gauss-Newton point.
+TEST_F(DoglegStrategyFixtureEllipse, ValidSubspaceBasis) {
+ scoped_ptr<LinearSolver> linear_solver(
+ new DenseQRSolver(LinearSolver::Options()));
+ options_.linear_solver = linear_solver.get();
+ options_.dogleg_type = SUBSPACE_DOGLEG;
+ options_.initial_radius = 2.0;
+ options_.max_radius = 2.0;
+
+ DoglegStrategy strategy(options_);
+ TrustRegionStrategy::PerSolveOptions pso;
+
+ TrustRegionStrategy::Summary summary = strategy.ComputeStep(pso,
+ jacobian_.get(),
+ residual_.data(),
+ x_.data());
+
+ // Check if the basis is orthonormal.
+ const Matrix basis = strategy.subspace_basis();
+ EXPECT_NEAR(basis.col(0).norm(), 1.0, kTolerance);
+ EXPECT_NEAR(basis.col(1).norm(), 1.0, kTolerance);
+ EXPECT_NEAR(basis.col(0).dot(basis.col(1)), 0.0, kTolerance);
+
+ // Check if the gradient projects onto itself.
+ const Vector gradient = strategy.gradient();
+ EXPECT_NEAR((gradient - basis*(basis.transpose()*gradient)).norm(),
+ 0.0,
+ kTolerance);
+
+ // Check if the Gauss-Newton point projects onto itself.
+ const Vector gn = strategy.gauss_newton_step();
+ EXPECT_NEAR((gn - basis*(basis.transpose()*gn)).norm(),
+ 0.0,
+ kTolerance);
+}
+
+// Test if the step is correct if the gradient and the Gauss-Newton step point
+// in the same direction and the Gauss-Newton step is outside the trust region,
+// i.e. the trust region is active.
+TEST_F(DoglegStrategyFixtureValley, CorrectStepLocalOptimumAlongGradient) {
+ scoped_ptr<LinearSolver> linear_solver(
+ new DenseQRSolver(LinearSolver::Options()));
+ options_.linear_solver = linear_solver.get();
+ options_.dogleg_type = SUBSPACE_DOGLEG;
+ options_.initial_radius = 0.25;
+ options_.max_radius = 0.25;
+
+ DoglegStrategy strategy(options_);
+ TrustRegionStrategy::PerSolveOptions pso;
+
+ TrustRegionStrategy::Summary summary = strategy.ComputeStep(pso,
+ jacobian_.get(),
+ residual_.data(),
+ x_.data());
+
+ EXPECT_NE(summary.termination_type, FAILURE);
+ EXPECT_NEAR(x_(0), 0.0, kToleranceLoose);
+ EXPECT_NEAR(x_(1), 0.0, kToleranceLoose);
+ EXPECT_NEAR(x_(2), options_.initial_radius, kToleranceLoose);
+ EXPECT_NEAR(x_(3), 0.0, kToleranceLoose);
+ EXPECT_NEAR(x_(4), 0.0, kToleranceLoose);
+ EXPECT_NEAR(x_(5), 0.0, kToleranceLoose);
+}
+
+// Test if the step is correct if the gradient and the Gauss-Newton step point
+// in the same direction and the Gauss-Newton step is inside the trust region,
+// i.e. the trust region is inactive.
+TEST_F(DoglegStrategyFixtureValley, CorrectStepGlobalOptimumAlongGradient) {
+ scoped_ptr<LinearSolver> linear_solver(
+ new DenseQRSolver(LinearSolver::Options()));
+ options_.linear_solver = linear_solver.get();
+ options_.dogleg_type = SUBSPACE_DOGLEG;
+ options_.initial_radius = 2.0;
+ options_.max_radius = 2.0;
+
+ DoglegStrategy strategy(options_);
+ TrustRegionStrategy::PerSolveOptions pso;
+
+ TrustRegionStrategy::Summary summary = strategy.ComputeStep(pso,
+ jacobian_.get(),
+ residual_.data(),
+ x_.data());
+
+ EXPECT_NE(summary.termination_type, FAILURE);
+ EXPECT_NEAR(x_(0), 0.0, kToleranceLoose);
+ EXPECT_NEAR(x_(1), 0.0, kToleranceLoose);
+ EXPECT_NEAR(x_(2), 1.0, kToleranceLoose);
+ EXPECT_NEAR(x_(3), 0.0, kToleranceLoose);
+ EXPECT_NEAR(x_(4), 0.0, kToleranceLoose);
+ EXPECT_NEAR(x_(5), 0.0, kToleranceLoose);
+}
+
+} // namespace internal
+} // namespace ceres
+
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