internal/ceres/local_parameterization_test.cc - ceres-solver - Git at Google

 // Ceres Solver - A fast non-linear least squares minimizer
 // Copyright 2010, 2011, 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: sameeragarwal@google.com (Sameer Agarwal)

 #include <cmath>
 #include "ceres/fpclassify.h"
 #include "ceres/internal/autodiff.h"
 #include "ceres/internal/eigen.h"
 #include "ceres/local_parameterization.h"
 #include "ceres/rotation.h"
 #include "gtest/gtest.h"

 namespace ceres {
 namespace internal {

 TEST(IdentityParameterization, EverythingTest) {
   IdentityParameterization parameterization(3);
   EXPECT_EQ(parameterization.GlobalSize(), 3);
   EXPECT_EQ(parameterization.LocalSize(), 3);

   double x[3] = {1.0, 2.0, 3.0};
   double delta[3] = {0.0, 1.0, 2.0};
   double x_plus_delta[3] = {0.0, 0.0, 0.0};
   parameterization.Plus(x, delta, x_plus_delta);
   EXPECT_EQ(x_plus_delta[0], 1.0);
   EXPECT_EQ(x_plus_delta[1], 3.0);
   EXPECT_EQ(x_plus_delta[2], 5.0);

   double jacobian[9];
   parameterization.ComputeJacobian(x, jacobian);
   int k = 0;
   for (int i = 0; i < 3; ++i) {
     for (int j = 0; j < 3; ++j, ++k) {
       EXPECT_EQ(jacobian[k], (i == j) ? 1.0 : 0.0);
     }
   }

   Matrix global_matrix = Matrix::Ones(10, 3);
   Matrix local_matrix = Matrix::Zero(10, 3);
   parameterization.MultiplyByJacobian(x,
                                       10,
                                       global_matrix.data(),
                                       local_matrix.data());
   EXPECT_EQ((local_matrix - global_matrix).norm(), 0.0);
 }

 TEST(SubsetParameterization, DeathTests) {
   vector<int> constant_parameters;
   EXPECT_DEATH_IF_SUPPORTED(
       SubsetParameterization parameterization(1, constant_parameters),
       "at least");

   constant_parameters.push_back(0);
   EXPECT_DEATH_IF_SUPPORTED(
       SubsetParameterization parameterization(1, constant_parameters),
       "Number of parameters");

   constant_parameters.push_back(1);
   EXPECT_DEATH_IF_SUPPORTED(
       SubsetParameterization parameterization(2, constant_parameters),
       "Number of parameters");

   constant_parameters.push_back(1);
   EXPECT_DEATH_IF_SUPPORTED(
       SubsetParameterization parameterization(2, constant_parameters),
       "duplicates");
 }

 TEST(SubsetParameterization, NormalFunctionTest) {
   const int kGlobalSize = 4;
   const int kLocalSize = 3;

   double x[kGlobalSize] = {1.0, 2.0, 3.0, 4.0};
   for (int i = 0; i < kGlobalSize; ++i) {
     vector<int> constant_parameters;
     constant_parameters.push_back(i);
     SubsetParameterization parameterization(kGlobalSize, constant_parameters);
     double delta[kLocalSize] = {1.0, 2.0, 3.0};
     double x_plus_delta[kGlobalSize] = {0.0, 0.0, 0.0};

     parameterization.Plus(x, delta, x_plus_delta);
     int k = 0;
     for (int j = 0; j < kGlobalSize; ++j) {
       if (j == i)  {
         EXPECT_EQ(x_plus_delta[j], x[j]);
       } else {
         EXPECT_EQ(x_plus_delta[j], x[j] + delta[k++]);
       }
     }

     double jacobian[kGlobalSize * kLocalSize];
     parameterization.ComputeJacobian(x, jacobian);
     int delta_cursor = 0;
     int jacobian_cursor = 0;
     for (int j = 0; j < kGlobalSize; ++j) {
       if (j != i) {
         for (int k = 0; k < kLocalSize; ++k, jacobian_cursor++) {
           EXPECT_EQ(jacobian[jacobian_cursor], delta_cursor == k ? 1.0 : 0.0);
         }
         ++delta_cursor;
       } else {
         for (int k = 0; k < kLocalSize; ++k, jacobian_cursor++) {
           EXPECT_EQ(jacobian[jacobian_cursor], 0.0);
         }
       }
     }

     Matrix global_matrix = Matrix::Ones(10, kGlobalSize);
     for (int row = 0; row < kGlobalSize; ++row) {
       for (int col = 0; col < kGlobalSize; ++col) {
         global_matrix(row, col) = col;
       }
     }

     Matrix local_matrix = Matrix::Zero(10, kLocalSize);
     parameterization.MultiplyByJacobian(x,
                                         10,
                                         global_matrix.data(),
                                         local_matrix.data());
     Matrix expected_local_matrix =
         global_matrix * MatrixRef(jacobian, kGlobalSize, kLocalSize);
     EXPECT_EQ((local_matrix - expected_local_matrix).norm(), 0.0);
   };
 }

 // Functor needed to implement automatically differentiated Plus for
 // quaternions.
 struct QuaternionPlus {
   template<typename T>
   bool operator()(const T* x, const T* delta, T* x_plus_delta) const {
     const T squared_norm_delta =
         delta[0] * delta[0] + delta[1] * delta[1] + delta[2] * delta[2];

     T q_delta[4];
     if (squared_norm_delta > T(0.0)) {
       T norm_delta = sqrt(squared_norm_delta);
       const T sin_delta_by_delta = sin(norm_delta) / norm_delta;
       q_delta[0] = cos(norm_delta);
       q_delta[1] = sin_delta_by_delta * delta[0];
       q_delta[2] = sin_delta_by_delta * delta[1];
       q_delta[3] = sin_delta_by_delta * delta[2];
     } else {
       // We do not just use q_delta = [1,0,0,0] here because that is a
       // constant and when used for automatic differentiation will
       // lead to a zero derivative. Instead we take a first order
       // approximation and evaluate it at zero.
       q_delta[0] = T(1.0);
       q_delta[1] = delta[0];
       q_delta[2] = delta[1];
       q_delta[3] = delta[2];
     }

     QuaternionProduct(q_delta, x, x_plus_delta);
     return true;
   }
 };

 void QuaternionParameterizationTestHelper(const double* x,
                                           const double* delta,
                                           const double* q_delta) {
   const int kGlobalSize = 4;
   const int kLocalSize = 3;

   const double kTolerance = 1e-14;
   double x_plus_delta_ref[kGlobalSize] = {0.0, 0.0, 0.0, 0.0};
   QuaternionProduct(q_delta, x, x_plus_delta_ref);

   double x_plus_delta[kGlobalSize] = {0.0, 0.0, 0.0, 0.0};
   QuaternionParameterization parameterization;
   parameterization.Plus(x, delta, x_plus_delta);
   for (int i = 0; i < kGlobalSize; ++i) {
     EXPECT_NEAR(x_plus_delta[i], x_plus_delta_ref[i], kTolerance);
   }

   const double x_plus_delta_norm =
       sqrt(x_plus_delta[0] * x_plus_delta[0] +
            x_plus_delta[1] * x_plus_delta[1] +
            x_plus_delta[2] * x_plus_delta[2] +
            x_plus_delta[3] * x_plus_delta[3]);

   EXPECT_NEAR(x_plus_delta_norm, 1.0, kTolerance);

   double jacobian_ref[12];
   double zero_delta[kLocalSize] = {0.0, 0.0, 0.0};
   const double* parameters[2] = {x, zero_delta};
   double* jacobian_array[2] = { NULL, jacobian_ref };

   // Autodiff jacobian at delta_x = 0.
   internal::AutoDiff<QuaternionPlus,
                      double,
                      kGlobalSize,
                      kLocalSize>::Differentiate(QuaternionPlus(),
                                                 parameters,
                                                 kGlobalSize,
                                                 x_plus_delta,
                                                 jacobian_array);

   double jacobian[12];
   parameterization.ComputeJacobian(x, jacobian);
   for (int i = 0; i < 12; ++i) {
     EXPECT_TRUE(IsFinite(jacobian[i]));
     EXPECT_NEAR(jacobian[i], jacobian_ref[i], kTolerance)
         << "Jacobian mismatch: i = " << i
         << "\n Expected \n"
         << ConstMatrixRef(jacobian_ref, kGlobalSize, kLocalSize)
         << "\n Actual \n"
         << ConstMatrixRef(jacobian, kGlobalSize, kLocalSize);
   }

   Matrix global_matrix = Matrix::Random(10, kGlobalSize);
   Matrix local_matrix = Matrix::Zero(10, kLocalSize);
   parameterization.MultiplyByJacobian(x,
                                       10,
                                       global_matrix.data(),
                                       local_matrix.data());
   Matrix expected_local_matrix =
       global_matrix * MatrixRef(jacobian, kGlobalSize, kLocalSize);
   EXPECT_EQ((local_matrix - expected_local_matrix).norm(), 0.0);
 }

 TEST(QuaternionParameterization, ZeroTest) {
   double x[4] = {0.5, 0.5, 0.5, 0.5};
   double delta[3] = {0.0, 0.0, 0.0};
   double q_delta[4] = {1.0, 0.0, 0.0, 0.0};
   QuaternionParameterizationTestHelper(x, delta, q_delta);
 }


 TEST(QuaternionParameterization, NearZeroTest) {
   double x[4] = {0.52, 0.25, 0.15, 0.45};
   double norm_x = sqrt(x[0] * x[0] +
                        x[1] * x[1] +
                        x[2] * x[2] +
                        x[3] * x[3]);
   for (int i = 0; i < 4; ++i) {
     x[i] = x[i] / norm_x;
   }

   double delta[3] = {0.24, 0.15, 0.10};
   for (int i = 0; i < 3; ++i) {
     delta[i] = delta[i] * 1e-14;
   }

   double q_delta[4];
   q_delta[0] = 1.0;
   q_delta[1] = delta[0];
   q_delta[2] = delta[1];
   q_delta[3] = delta[2];

   QuaternionParameterizationTestHelper(x, delta, q_delta);
 }

 TEST(QuaternionParameterization, AwayFromZeroTest) {
   double x[4] = {0.52, 0.25, 0.15, 0.45};
   double norm_x = sqrt(x[0] * x[0] +
                        x[1] * x[1] +
                        x[2] * x[2] +
                        x[3] * x[3]);

   for (int i = 0; i < 4; ++i) {
     x[i] = x[i] / norm_x;
   }

   double delta[3] = {0.24, 0.15, 0.10};
   const double delta_norm = sqrt(delta[0] * delta[0] +
                                  delta[1] * delta[1] +
                                  delta[2] * delta[2]);
   double q_delta[4];
   q_delta[0] = cos(delta_norm);
   q_delta[1] = sin(delta_norm) / delta_norm * delta[0];
   q_delta[2] = sin(delta_norm) / delta_norm * delta[1];
   q_delta[3] = sin(delta_norm) / delta_norm * delta[2];

   QuaternionParameterizationTestHelper(x, delta, q_delta);
 }


 }  // namespace internal
 }  // namespace ceres
	// Ceres Solver - A fast non-linear least squares minimizer
	// Copyright 2010, 2011, 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: sameeragarwal@google.com (Sameer Agarwal)

	#include <cmath>
	#include "ceres/fpclassify.h"
	#include "ceres/internal/autodiff.h"
	#include "ceres/internal/eigen.h"
	#include "ceres/local_parameterization.h"
	#include "ceres/rotation.h"
	#include "gtest/gtest.h"

	namespace ceres {
	namespace internal {

	TEST(IdentityParameterization, EverythingTest) {
	IdentityParameterization parameterization(3);
	EXPECT_EQ(parameterization.GlobalSize(), 3);
	EXPECT_EQ(parameterization.LocalSize(), 3);

	double x[3] = {1.0, 2.0, 3.0};
	double delta[3] = {0.0, 1.0, 2.0};
	double x_plus_delta[3] = {0.0, 0.0, 0.0};
	parameterization.Plus(x, delta, x_plus_delta);
	EXPECT_EQ(x_plus_delta[0], 1.0);
	EXPECT_EQ(x_plus_delta[1], 3.0);
	EXPECT_EQ(x_plus_delta[2], 5.0);

	double jacobian[9];
	parameterization.ComputeJacobian(x, jacobian);
	int k = 0;
	for (int i = 0; i < 3; ++i) {
	for (int j = 0; j < 3; ++j, ++k) {
	EXPECT_EQ(jacobian[k], (i == j) ? 1.0 : 0.0);
	}
	}

	Matrix global_matrix = Matrix::Ones(10, 3);
	Matrix local_matrix = Matrix::Zero(10, 3);
	parameterization.MultiplyByJacobian(x,
	10,
	global_matrix.data(),
	local_matrix.data());
	EXPECT_EQ((local_matrix - global_matrix).norm(), 0.0);
	}

	TEST(SubsetParameterization, DeathTests) {
	vector<int> constant_parameters;
	EXPECT_DEATH_IF_SUPPORTED(
	SubsetParameterization parameterization(1, constant_parameters),
	"at least");

	constant_parameters.push_back(0);
	EXPECT_DEATH_IF_SUPPORTED(
	SubsetParameterization parameterization(1, constant_parameters),
	"Number of parameters");

	constant_parameters.push_back(1);
	EXPECT_DEATH_IF_SUPPORTED(
	SubsetParameterization parameterization(2, constant_parameters),
	"Number of parameters");

	constant_parameters.push_back(1);
	EXPECT_DEATH_IF_SUPPORTED(
	SubsetParameterization parameterization(2, constant_parameters),
	"duplicates");
	}

	TEST(SubsetParameterization, NormalFunctionTest) {
	const int kGlobalSize = 4;
	const int kLocalSize = 3;

	double x[kGlobalSize] = {1.0, 2.0, 3.0, 4.0};
	for (int i = 0; i < kGlobalSize; ++i) {
	vector<int> constant_parameters;
	constant_parameters.push_back(i);
	SubsetParameterization parameterization(kGlobalSize, constant_parameters);
	double delta[kLocalSize] = {1.0, 2.0, 3.0};
	double x_plus_delta[kGlobalSize] = {0.0, 0.0, 0.0};

	parameterization.Plus(x, delta, x_plus_delta);
	int k = 0;
	for (int j = 0; j < kGlobalSize; ++j) {
	if (j == i) {
	EXPECT_EQ(x_plus_delta[j], x[j]);
	} else {
	EXPECT_EQ(x_plus_delta[j], x[j] + delta[k++]);
	}
	}

	double jacobian[kGlobalSize * kLocalSize];
	parameterization.ComputeJacobian(x, jacobian);
	int delta_cursor = 0;
	int jacobian_cursor = 0;
	for (int j = 0; j < kGlobalSize; ++j) {
	if (j != i) {
	for (int k = 0; k < kLocalSize; ++k, jacobian_cursor++) {
	EXPECT_EQ(jacobian[jacobian_cursor], delta_cursor == k ? 1.0 : 0.0);
	}
	++delta_cursor;
	} else {
	for (int k = 0; k < kLocalSize; ++k, jacobian_cursor++) {
	EXPECT_EQ(jacobian[jacobian_cursor], 0.0);
	}
	}
	}

	Matrix global_matrix = Matrix::Ones(10, kGlobalSize);
	for (int row = 0; row < kGlobalSize; ++row) {
	for (int col = 0; col < kGlobalSize; ++col) {
	global_matrix(row, col) = col;
	}
	}

	Matrix local_matrix = Matrix::Zero(10, kLocalSize);
	parameterization.MultiplyByJacobian(x,
	10,
	global_matrix.data(),
	local_matrix.data());
	Matrix expected_local_matrix =
	global_matrix * MatrixRef(jacobian, kGlobalSize, kLocalSize);
	EXPECT_EQ((local_matrix - expected_local_matrix).norm(), 0.0);
	};
	}

	// Functor needed to implement automatically differentiated Plus for
	// quaternions.
	struct QuaternionPlus {
	template<typename T>
	bool operator()(const T* x, const T* delta, T* x_plus_delta) const {
	const T squared_norm_delta =
	delta[0] * delta[0] + delta[1] * delta[1] + delta[2] * delta[2];

	T q_delta[4];
	if (squared_norm_delta > T(0.0)) {
	T norm_delta = sqrt(squared_norm_delta);
	const T sin_delta_by_delta = sin(norm_delta) / norm_delta;
	q_delta[0] = cos(norm_delta);
	q_delta[1] = sin_delta_by_delta * delta[0];
	q_delta[2] = sin_delta_by_delta * delta[1];
	q_delta[3] = sin_delta_by_delta * delta[2];
	} else {
	// We do not just use q_delta = [1,0,0,0] here because that is a
	// constant and when used for automatic differentiation will
	// lead to a zero derivative. Instead we take a first order
	// approximation and evaluate it at zero.
	q_delta[0] = T(1.0);
	q_delta[1] = delta[0];
	q_delta[2] = delta[1];
	q_delta[3] = delta[2];
	}

	QuaternionProduct(q_delta, x, x_plus_delta);
	return true;
	}
	};

	void QuaternionParameterizationTestHelper(const double* x,
	const double* delta,
	const double* q_delta) {
	const int kGlobalSize = 4;
	const int kLocalSize = 3;

	const double kTolerance = 1e-14;
	double x_plus_delta_ref[kGlobalSize] = {0.0, 0.0, 0.0, 0.0};
	QuaternionProduct(q_delta, x, x_plus_delta_ref);

	double x_plus_delta[kGlobalSize] = {0.0, 0.0, 0.0, 0.0};
	QuaternionParameterization parameterization;
	parameterization.Plus(x, delta, x_plus_delta);
	for (int i = 0; i < kGlobalSize; ++i) {
	EXPECT_NEAR(x_plus_delta[i], x_plus_delta_ref[i], kTolerance);
	}

	const double x_plus_delta_norm =
	sqrt(x_plus_delta[0] * x_plus_delta[0] +
	x_plus_delta[1] * x_plus_delta[1] +
	x_plus_delta[2] * x_plus_delta[2] +
	x_plus_delta[3] * x_plus_delta[3]);

	EXPECT_NEAR(x_plus_delta_norm, 1.0, kTolerance);

	double jacobian_ref[12];
	double zero_delta[kLocalSize] = {0.0, 0.0, 0.0};
	const double* parameters[2] = {x, zero_delta};
	double* jacobian_array[2] = { NULL, jacobian_ref };

	// Autodiff jacobian at delta_x = 0.
	internal::AutoDiff<QuaternionPlus,
	double,
	kGlobalSize,
	kLocalSize>::Differentiate(QuaternionPlus(),
	parameters,
	kGlobalSize,
	x_plus_delta,
	jacobian_array);

	double jacobian[12];
	parameterization.ComputeJacobian(x, jacobian);
	for (int i = 0; i < 12; ++i) {
	EXPECT_TRUE(IsFinite(jacobian[i]));
	EXPECT_NEAR(jacobian[i], jacobian_ref[i], kTolerance)
	<< "Jacobian mismatch: i = " << i
	<< "\n Expected \n"
	<< ConstMatrixRef(jacobian_ref, kGlobalSize, kLocalSize)
	<< "\n Actual \n"
	<< ConstMatrixRef(jacobian, kGlobalSize, kLocalSize);
	}

	Matrix global_matrix = Matrix::Random(10, kGlobalSize);
	Matrix local_matrix = Matrix::Zero(10, kLocalSize);
	parameterization.MultiplyByJacobian(x,
	10,
	global_matrix.data(),
	local_matrix.data());
	Matrix expected_local_matrix =
	global_matrix * MatrixRef(jacobian, kGlobalSize, kLocalSize);
	EXPECT_EQ((local_matrix - expected_local_matrix).norm(), 0.0);
	}

	TEST(QuaternionParameterization, ZeroTest) {
	double x[4] = {0.5, 0.5, 0.5, 0.5};
	double delta[3] = {0.0, 0.0, 0.0};
	double q_delta[4] = {1.0, 0.0, 0.0, 0.0};
	QuaternionParameterizationTestHelper(x, delta, q_delta);
	}


	TEST(QuaternionParameterization, NearZeroTest) {
	double x[4] = {0.52, 0.25, 0.15, 0.45};
	double norm_x = sqrt(x[0] * x[0] +
	x[1] * x[1] +
	x[2] * x[2] +
	x[3] * x[3]);
	for (int i = 0; i < 4; ++i) {
	x[i] = x[i] / norm_x;
	}

	double delta[3] = {0.24, 0.15, 0.10};
	for (int i = 0; i < 3; ++i) {
	delta[i] = delta[i] * 1e-14;
	}

	double q_delta[4];
	q_delta[0] = 1.0;
	q_delta[1] = delta[0];
	q_delta[2] = delta[1];
	q_delta[3] = delta[2];

	QuaternionParameterizationTestHelper(x, delta, q_delta);
	}

	TEST(QuaternionParameterization, AwayFromZeroTest) {
	double x[4] = {0.52, 0.25, 0.15, 0.45};
	double norm_x = sqrt(x[0] * x[0] +
	x[1] * x[1] +
	x[2] * x[2] +
	x[3] * x[3]);

	for (int i = 0; i < 4; ++i) {
	x[i] = x[i] / norm_x;
	}

	double delta[3] = {0.24, 0.15, 0.10};
	const double delta_norm = sqrt(delta[0] * delta[0] +
	delta[1] * delta[1] +
	delta[2] * delta[2]);
	double q_delta[4];
	q_delta[0] = cos(delta_norm);
	q_delta[1] = sin(delta_norm) / delta_norm * delta[0];
	q_delta[2] = sin(delta_norm) / delta_norm * delta[1];
	q_delta[3] = sin(delta_norm) / delta_norm * delta[2];

	QuaternionParameterizationTestHelper(x, delta, q_delta);
	}


	} // namespace internal
	} // namespace ceres