Squashed 'third_party/ceres/' content from commit e51e9b4

Change-Id: I763587619d57e594d3fa158dc3a7fe0b89a1743b
git-subtree-dir: third_party/ceres
git-subtree-split: e51e9b46f6ca88ab8b2266d0e362771db6d98067
diff --git a/internal/ceres/local_parameterization_test.cc b/internal/ceres/local_parameterization_test.cc
new file mode 100644
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--- /dev/null
+++ b/internal/ceres/local_parameterization_test.cc
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+// Ceres Solver - A fast non-linear least squares minimizer
+// Copyright 2015 Google Inc. All rights reserved.
+// http://ceres-solver.org/
+//
+// 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 <limits>
+#include <memory>
+
+#include "Eigen/Geometry"
+#include "ceres/autodiff_local_parameterization.h"
+#include "ceres/householder_vector.h"
+#include "ceres/internal/autodiff.h"
+#include "ceres/internal/eigen.h"
+#include "ceres/local_parameterization.h"
+#include "ceres/random.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, NegativeParameterIndexDeathTest) {
+  std::vector<int> constant_parameters;
+  constant_parameters.push_back(-1);
+  EXPECT_DEATH_IF_SUPPORTED(
+      SubsetParameterization parameterization(2, constant_parameters),
+      "greater than equal to zero");
+}
+
+TEST(SubsetParameterization, GreaterThanSizeParameterIndexDeathTest) {
+  std::vector<int> constant_parameters;
+  constant_parameters.push_back(2);
+  EXPECT_DEATH_IF_SUPPORTED(
+      SubsetParameterization parameterization(2, constant_parameters),
+      "less than the size");
+}
+
+TEST(SubsetParameterization, DuplicateParametersDeathTest) {
+  std::vector<int> constant_parameters;
+  constant_parameters.push_back(1);
+  constant_parameters.push_back(1);
+  EXPECT_DEATH_IF_SUPPORTED(
+      SubsetParameterization parameterization(2, constant_parameters),
+      "duplicates");
+}
+
+TEST(SubsetParameterization,
+     ProductParameterizationWithZeroLocalSizeSubsetParameterization1) {
+  std::vector<int> constant_parameters;
+  constant_parameters.push_back(0);
+  LocalParameterization* subset_param =
+      new SubsetParameterization(1, constant_parameters);
+  LocalParameterization* identity_param = new IdentityParameterization(2);
+  ProductParameterization product_param(subset_param, identity_param);
+  EXPECT_EQ(product_param.GlobalSize(), 3);
+  EXPECT_EQ(product_param.LocalSize(), 2);
+  double x[] = {1.0, 1.0, 1.0};
+  double delta[] = {2.0, 3.0};
+  double x_plus_delta[] = {0.0, 0.0, 0.0};
+  EXPECT_TRUE(product_param.Plus(x, delta, x_plus_delta));
+  EXPECT_EQ(x_plus_delta[0], x[0]);
+  EXPECT_EQ(x_plus_delta[1], x[1] + delta[0]);
+  EXPECT_EQ(x_plus_delta[2], x[2] + delta[1]);
+
+  Matrix actual_jacobian(3, 2);
+  EXPECT_TRUE(product_param.ComputeJacobian(x, actual_jacobian.data()));
+}
+
+TEST(SubsetParameterization,
+     ProductParameterizationWithZeroLocalSizeSubsetParameterization2) {
+  std::vector<int> constant_parameters;
+  constant_parameters.push_back(0);
+  LocalParameterization* subset_param =
+      new SubsetParameterization(1, constant_parameters);
+  LocalParameterization* identity_param = new IdentityParameterization(2);
+  ProductParameterization product_param(identity_param, subset_param);
+  EXPECT_EQ(product_param.GlobalSize(), 3);
+  EXPECT_EQ(product_param.LocalSize(), 2);
+  double x[] = {1.0, 1.0, 1.0};
+  double delta[] = {2.0, 3.0};
+  double x_plus_delta[] = {0.0, 0.0, 0.0};
+  EXPECT_TRUE(product_param.Plus(x, delta, x_plus_delta));
+  EXPECT_EQ(x_plus_delta[0], x[0] + delta[0]);
+  EXPECT_EQ(x_plus_delta[1], x[1] + delta[1]);
+  EXPECT_EQ(x_plus_delta[2], x[2]);
+
+  Matrix actual_jacobian(3, 2);
+  EXPECT_TRUE(product_param.ComputeJacobian(x, actual_jacobian.data()));
+}
+
+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) {
+    std::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;
+  }
+};
+
+template<typename Parameterization, typename Plus>
+void QuaternionParameterizationTestHelper(
+    const double* x, const double* delta,
+    const double* x_plus_delta_ref) {
+  const int kGlobalSize = 4;
+  const int kLocalSize = 3;
+
+  const double kTolerance = 1e-14;
+
+  double x_plus_delta[kGlobalSize] = {0.0, 0.0, 0.0, 0.0};
+  Parameterization parameterization;
+  parameterization.Plus(x, delta, x_plus_delta);
+  for (int i = 0; i < kGlobalSize; ++i) {
+    EXPECT_NEAR(x_plus_delta[i], x_plus_delta[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::AutoDifferentiate<StaticParameterDims<kGlobalSize, kLocalSize>>(
+      Plus(),
+      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_NEAR((local_matrix - expected_local_matrix).norm(),
+              0.0,
+              10.0 * std::numeric_limits<double>::epsilon());
+}
+
+template <int N>
+void Normalize(double* x) {
+  VectorRef(x, N).normalize();
+}
+
+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};
+  double x_plus_delta[4] = {0.0, 0.0, 0.0, 0.0};
+  QuaternionProduct(q_delta, x, x_plus_delta);
+  QuaternionParameterizationTestHelper<QuaternionParameterization,
+                                       QuaternionPlus>(x, delta, x_plus_delta);
+}
+
+TEST(QuaternionParameterization, NearZeroTest) {
+  double x[4] = {0.52, 0.25, 0.15, 0.45};
+  Normalize<4>(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];
+
+  double x_plus_delta[4] = {0.0, 0.0, 0.0, 0.0};
+  QuaternionProduct(q_delta, x, x_plus_delta);
+  QuaternionParameterizationTestHelper<QuaternionParameterization,
+                                       QuaternionPlus>(x, delta, x_plus_delta);
+}
+
+TEST(QuaternionParameterization, AwayFromZeroTest) {
+  double x[4] = {0.52, 0.25, 0.15, 0.45};
+  Normalize<4>(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];
+
+  double x_plus_delta[4] = {0.0, 0.0, 0.0, 0.0};
+  QuaternionProduct(q_delta, x, x_plus_delta);
+  QuaternionParameterizationTestHelper<QuaternionParameterization,
+                                       QuaternionPlus>(x, delta, x_plus_delta);
+}
+
+// Functor needed to implement automatically differentiated Plus for
+// Eigen's quaternion.
+struct EigenQuaternionPlus {
+  template<typename T>
+  bool operator()(const T* x, const T* delta, T* x_plus_delta) const {
+    const T norm_delta =
+        sqrt(delta[0] * delta[0] + delta[1] * delta[1] + delta[2] * delta[2]);
+
+    Eigen::Quaternion<T> q_delta;
+    if (norm_delta > T(0.0)) {
+      const T sin_delta_by_delta = sin(norm_delta) / norm_delta;
+      q_delta.coeffs() << sin_delta_by_delta * delta[0],
+          sin_delta_by_delta * delta[1], sin_delta_by_delta * delta[2],
+          cos(norm_delta);
+    } else {
+      // We do not just use q_delta = [0,0,0,1] 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.coeffs() <<  delta[0], delta[1], delta[2], T(1.0);
+    }
+
+    Eigen::Map<Eigen::Quaternion<T>> x_plus_delta_ref(x_plus_delta);
+    Eigen::Map<const Eigen::Quaternion<T>> x_ref(x);
+    x_plus_delta_ref = q_delta * x_ref;
+    return true;
+  }
+};
+
+TEST(EigenQuaternionParameterization, ZeroTest) {
+  Eigen::Quaterniond x(0.5, 0.5, 0.5, 0.5);
+  double delta[3] = {0.0, 0.0, 0.0};
+  Eigen::Quaterniond q_delta(1.0, 0.0, 0.0, 0.0);
+  Eigen::Quaterniond x_plus_delta = q_delta * x;
+  QuaternionParameterizationTestHelper<EigenQuaternionParameterization,
+                                       EigenQuaternionPlus>(
+      x.coeffs().data(), delta, x_plus_delta.coeffs().data());
+}
+
+TEST(EigenQuaternionParameterization, NearZeroTest) {
+  Eigen::Quaterniond x(0.52, 0.25, 0.15, 0.45);
+  x.normalize();
+
+  double delta[3] = {0.24, 0.15, 0.10};
+  for (int i = 0; i < 3; ++i) {
+    delta[i] = delta[i] * 1e-14;
+  }
+
+  // Note: w is first in the constructor.
+  Eigen::Quaterniond q_delta(1.0, delta[0], delta[1], delta[2]);
+
+  Eigen::Quaterniond x_plus_delta = q_delta * x;
+  QuaternionParameterizationTestHelper<EigenQuaternionParameterization,
+                                       EigenQuaternionPlus>(
+      x.coeffs().data(), delta, x_plus_delta.coeffs().data());
+}
+
+TEST(EigenQuaternionParameterization, AwayFromZeroTest) {
+  Eigen::Quaterniond x(0.52, 0.25, 0.15, 0.45);
+  x.normalize();
+
+  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]);
+
+  // Note: w is first in the constructor.
+  Eigen::Quaterniond q_delta(cos(delta_norm),
+                             sin(delta_norm) / delta_norm * delta[0],
+                             sin(delta_norm) / delta_norm * delta[1],
+                             sin(delta_norm) / delta_norm * delta[2]);
+
+  Eigen::Quaterniond x_plus_delta = q_delta * x;
+  QuaternionParameterizationTestHelper<EigenQuaternionParameterization,
+                                       EigenQuaternionPlus>(
+      x.coeffs().data(), delta, x_plus_delta.coeffs().data());
+}
+
+// Functor needed to implement automatically differentiated Plus for
+// homogeneous vectors. Note this explicitly defined for vectors of size 4.
+struct HomogeneousVectorParameterizationPlus {
+  template<typename Scalar>
+  bool operator()(const Scalar* p_x, const Scalar* p_delta,
+                  Scalar* p_x_plus_delta) const {
+    Eigen::Map<const Eigen::Matrix<Scalar, 4, 1>> x(p_x);
+    Eigen::Map<const Eigen::Matrix<Scalar, 3, 1>> delta(p_delta);
+    Eigen::Map<Eigen::Matrix<Scalar, 4, 1>> x_plus_delta(p_x_plus_delta);
+
+    const Scalar squared_norm_delta =
+        delta[0] * delta[0] + delta[1] * delta[1] + delta[2] * delta[2];
+
+    Eigen::Matrix<Scalar, 4, 1> y;
+    Scalar one_half(0.5);
+    if (squared_norm_delta > Scalar(0.0)) {
+      Scalar norm_delta = sqrt(squared_norm_delta);
+      Scalar norm_delta_div_2 = 0.5 * norm_delta;
+      const Scalar sin_delta_by_delta = sin(norm_delta_div_2) /
+          norm_delta_div_2;
+      y[0] = sin_delta_by_delta * delta[0] * one_half;
+      y[1] = sin_delta_by_delta * delta[1] * one_half;
+      y[2] = sin_delta_by_delta * delta[2] * one_half;
+      y[3] = cos(norm_delta_div_2);
+
+    } else {
+      // We do not just use y = [0,0,0,1] 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.
+      y[0] = delta[0] * one_half;
+      y[1] = delta[1] * one_half;
+      y[2] = delta[2] * one_half;
+      y[3] = Scalar(1.0);
+    }
+
+    Eigen::Matrix<Scalar, Eigen::Dynamic, 1> v(4);
+    Scalar beta;
+    internal::ComputeHouseholderVector<Scalar>(x, &v, &beta);
+
+    x_plus_delta = x.norm() * (y - v * (beta * v.dot(y)));
+
+    return true;
+  }
+};
+
+void HomogeneousVectorParameterizationHelper(const double* x,
+                                             const double* delta) {
+  const double kTolerance = 1e-14;
+
+  HomogeneousVectorParameterization homogeneous_vector_parameterization(4);
+
+  // Ensure the update maintains the norm.
+  double x_plus_delta[4] = {0.0, 0.0, 0.0, 0.0};
+  homogeneous_vector_parameterization.Plus(x, delta, x_plus_delta);
+
+  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]);
+
+  const double x_norm = sqrt(x[0] * x[0] + x[1] * x[1] +
+                             x[2] * x[2] + x[3] * x[3]);
+
+  EXPECT_NEAR(x_plus_delta_norm, x_norm, kTolerance);
+
+  // Autodiff jacobian at delta_x = 0.
+  AutoDiffLocalParameterization<HomogeneousVectorParameterizationPlus, 4, 3>
+      autodiff_jacobian;
+
+  double jacobian_autodiff[12];
+  double jacobian_analytic[12];
+
+  homogeneous_vector_parameterization.ComputeJacobian(x, jacobian_analytic);
+  autodiff_jacobian.ComputeJacobian(x, jacobian_autodiff);
+
+  for (int i = 0; i < 12; ++i) {
+    EXPECT_TRUE(ceres::IsFinite(jacobian_analytic[i]));
+    EXPECT_NEAR(jacobian_analytic[i], jacobian_autodiff[i], kTolerance)
+        << "Jacobian mismatch: i = " << i << ", " << jacobian_analytic[i] << " "
+        << jacobian_autodiff[i];
+  }
+}
+
+TEST(HomogeneousVectorParameterization, ZeroTest) {
+  double x[4] = {0.0, 0.0, 0.0, 1.0};
+  Normalize<4>(x);
+  double delta[3] = {0.0, 0.0, 0.0};
+
+  HomogeneousVectorParameterizationHelper(x, delta);
+}
+
+TEST(HomogeneousVectorParameterization, NearZeroTest1) {
+  double x[4] = {1e-5, 1e-5, 1e-5, 1.0};
+  Normalize<4>(x);
+  double delta[3] = {0.0, 1.0, 0.0};
+
+  HomogeneousVectorParameterizationHelper(x, delta);
+}
+
+TEST(HomogeneousVectorParameterization, NearZeroTest2) {
+  double x[4] = {0.001, 0.0, 0.0, 0.0};
+  double delta[3] = {0.0, 1.0, 0.0};
+
+  HomogeneousVectorParameterizationHelper(x, delta);
+}
+
+TEST(HomogeneousVectorParameterization, AwayFromZeroTest1) {
+  double x[4] = {0.52, 0.25, 0.15, 0.45};
+  Normalize<4>(x);
+  double delta[3] = {0.0, 1.0, -0.5};
+
+  HomogeneousVectorParameterizationHelper(x, delta);
+}
+
+TEST(HomogeneousVectorParameterization, AwayFromZeroTest2) {
+  double x[4] = {0.87, -0.25, -0.34, 0.45};
+  Normalize<4>(x);
+  double delta[3] = {0.0, 0.0, -0.5};
+
+  HomogeneousVectorParameterizationHelper(x, delta);
+}
+
+TEST(HomogeneousVectorParameterization, AwayFromZeroTest3) {
+  double x[4] = {0.0, 0.0, 0.0, 2.0};
+  double delta[3] = {0.0, 0.0, 0};
+
+  HomogeneousVectorParameterizationHelper(x, delta);
+}
+
+TEST(HomogeneousVectorParameterization, AwayFromZeroTest4) {
+  double x[4] = {0.2, -1.0, 0.0, 2.0};
+  double delta[3] = {1.4, 0.0, -0.5};
+
+  HomogeneousVectorParameterizationHelper(x, delta);
+}
+
+TEST(HomogeneousVectorParameterization, AwayFromZeroTest5) {
+  double x[4] = {2.0, 0.0, 0.0, 0.0};
+  double delta[3] = {1.4, 0.0, -0.5};
+
+  HomogeneousVectorParameterizationHelper(x, delta);
+}
+
+TEST(HomogeneousVectorParameterization, DeathTests) {
+  EXPECT_DEATH_IF_SUPPORTED(HomogeneousVectorParameterization x(1), "size");
+}
+
+
+class ProductParameterizationTest : public ::testing::Test {
+ protected :
+  virtual void SetUp() {
+    const int global_size1 = 5;
+    std::vector<int> constant_parameters1;
+    constant_parameters1.push_back(2);
+    param1_.reset(new SubsetParameterization(global_size1,
+                                             constant_parameters1));
+
+    const int global_size2 = 3;
+    std::vector<int> constant_parameters2;
+    constant_parameters2.push_back(0);
+    constant_parameters2.push_back(1);
+    param2_.reset(new SubsetParameterization(global_size2,
+                                             constant_parameters2));
+
+    const int global_size3 = 4;
+    std::vector<int> constant_parameters3;
+    constant_parameters3.push_back(1);
+    param3_.reset(new SubsetParameterization(global_size3,
+                                             constant_parameters3));
+
+    const int global_size4 = 2;
+    std::vector<int> constant_parameters4;
+    constant_parameters4.push_back(1);
+    param4_.reset(new SubsetParameterization(global_size4,
+                                             constant_parameters4));
+  }
+
+  std::unique_ptr<LocalParameterization> param1_;
+  std::unique_ptr<LocalParameterization> param2_;
+  std::unique_ptr<LocalParameterization> param3_;
+  std::unique_ptr<LocalParameterization> param4_;
+};
+
+TEST_F(ProductParameterizationTest, LocalAndGlobalSize2) {
+  LocalParameterization* param1 = param1_.release();
+  LocalParameterization* param2 = param2_.release();
+
+  ProductParameterization product_param(param1, param2);
+  EXPECT_EQ(product_param.LocalSize(),
+            param1->LocalSize() + param2->LocalSize());
+  EXPECT_EQ(product_param.GlobalSize(),
+            param1->GlobalSize() + param2->GlobalSize());
+}
+
+
+TEST_F(ProductParameterizationTest, LocalAndGlobalSize3) {
+  LocalParameterization* param1 = param1_.release();
+  LocalParameterization* param2 = param2_.release();
+  LocalParameterization* param3 = param3_.release();
+
+  ProductParameterization product_param(param1, param2, param3);
+  EXPECT_EQ(product_param.LocalSize(),
+            param1->LocalSize() + param2->LocalSize() + param3->LocalSize());
+  EXPECT_EQ(product_param.GlobalSize(),
+            param1->GlobalSize() + param2->GlobalSize() + param3->GlobalSize());
+}
+
+TEST_F(ProductParameterizationTest, LocalAndGlobalSize4) {
+  LocalParameterization* param1 = param1_.release();
+  LocalParameterization* param2 = param2_.release();
+  LocalParameterization* param3 = param3_.release();
+  LocalParameterization* param4 = param4_.release();
+
+  ProductParameterization product_param(param1, param2, param3, param4);
+  EXPECT_EQ(product_param.LocalSize(),
+            param1->LocalSize() +
+            param2->LocalSize() +
+            param3->LocalSize() +
+            param4->LocalSize());
+  EXPECT_EQ(product_param.GlobalSize(),
+            param1->GlobalSize() +
+            param2->GlobalSize() +
+            param3->GlobalSize() +
+            param4->GlobalSize());
+}
+
+TEST_F(ProductParameterizationTest, Plus) {
+  LocalParameterization* param1 = param1_.release();
+  LocalParameterization* param2 = param2_.release();
+  LocalParameterization* param3 = param3_.release();
+  LocalParameterization* param4 = param4_.release();
+
+  ProductParameterization product_param(param1, param2, param3, param4);
+  std::vector<double> x(product_param.GlobalSize(), 0.0);
+  std::vector<double> delta(product_param.LocalSize(), 0.0);
+  std::vector<double> x_plus_delta_expected(product_param.GlobalSize(), 0.0);
+  std::vector<double> x_plus_delta(product_param.GlobalSize(), 0.0);
+
+  for (int i = 0; i < product_param.GlobalSize(); ++i) {
+    x[i] = RandNormal();
+  }
+
+  for (int i = 0; i < product_param.LocalSize(); ++i) {
+    delta[i] = RandNormal();
+  }
+
+  EXPECT_TRUE(product_param.Plus(&x[0], &delta[0], &x_plus_delta_expected[0]));
+  int x_cursor = 0;
+  int delta_cursor = 0;
+
+  EXPECT_TRUE(param1->Plus(&x[x_cursor],
+                           &delta[delta_cursor],
+                           &x_plus_delta[x_cursor]));
+  x_cursor += param1->GlobalSize();
+  delta_cursor += param1->LocalSize();
+
+  EXPECT_TRUE(param2->Plus(&x[x_cursor],
+                           &delta[delta_cursor],
+                           &x_plus_delta[x_cursor]));
+  x_cursor += param2->GlobalSize();
+  delta_cursor += param2->LocalSize();
+
+  EXPECT_TRUE(param3->Plus(&x[x_cursor],
+                           &delta[delta_cursor],
+                           &x_plus_delta[x_cursor]));
+  x_cursor += param3->GlobalSize();
+  delta_cursor += param3->LocalSize();
+
+  EXPECT_TRUE(param4->Plus(&x[x_cursor],
+                           &delta[delta_cursor],
+                           &x_plus_delta[x_cursor]));
+  x_cursor += param4->GlobalSize();
+  delta_cursor += param4->LocalSize();
+
+  for (int i = 0; i < x.size(); ++i) {
+    EXPECT_EQ(x_plus_delta[i], x_plus_delta_expected[i]);
+  }
+}
+
+TEST_F(ProductParameterizationTest, ComputeJacobian) {
+  LocalParameterization* param1 = param1_.release();
+  LocalParameterization* param2 = param2_.release();
+  LocalParameterization* param3 = param3_.release();
+  LocalParameterization* param4 = param4_.release();
+
+  ProductParameterization product_param(param1, param2, param3, param4);
+  std::vector<double> x(product_param.GlobalSize(), 0.0);
+
+  for (int i = 0; i < product_param.GlobalSize(); ++i) {
+    x[i] = RandNormal();
+  }
+
+  Matrix jacobian = Matrix::Random(product_param.GlobalSize(),
+                                   product_param.LocalSize());
+  EXPECT_TRUE(product_param.ComputeJacobian(&x[0], jacobian.data()));
+  int x_cursor = 0;
+  int delta_cursor = 0;
+
+  Matrix jacobian1(param1->GlobalSize(), param1->LocalSize());
+  EXPECT_TRUE(param1->ComputeJacobian(&x[x_cursor], jacobian1.data()));
+  jacobian.block(x_cursor, delta_cursor,
+                 param1->GlobalSize(),
+                 param1->LocalSize())
+      -= jacobian1;
+  x_cursor += param1->GlobalSize();
+  delta_cursor += param1->LocalSize();
+
+  Matrix jacobian2(param2->GlobalSize(), param2->LocalSize());
+  EXPECT_TRUE(param2->ComputeJacobian(&x[x_cursor], jacobian2.data()));
+  jacobian.block(x_cursor, delta_cursor,
+                 param2->GlobalSize(),
+                 param2->LocalSize())
+      -= jacobian2;
+  x_cursor += param2->GlobalSize();
+  delta_cursor += param2->LocalSize();
+
+  Matrix jacobian3(param3->GlobalSize(), param3->LocalSize());
+  EXPECT_TRUE(param3->ComputeJacobian(&x[x_cursor], jacobian3.data()));
+  jacobian.block(x_cursor, delta_cursor,
+                 param3->GlobalSize(),
+                 param3->LocalSize())
+      -= jacobian3;
+  x_cursor += param3->GlobalSize();
+  delta_cursor += param3->LocalSize();
+
+  Matrix jacobian4(param4->GlobalSize(), param4->LocalSize());
+  EXPECT_TRUE(param4->ComputeJacobian(&x[x_cursor], jacobian4.data()));
+  jacobian.block(x_cursor, delta_cursor,
+                 param4->GlobalSize(),
+                 param4->LocalSize())
+      -= jacobian4;
+  x_cursor += param4->GlobalSize();
+  delta_cursor += param4->LocalSize();
+
+  EXPECT_NEAR(jacobian.norm(), 0.0, std::numeric_limits<double>::epsilon());
+}
+
+}  // namespace internal
+}  // namespace ceres