| // 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 "ceres/rotation.h" |
| |
| #include <cmath> |
| #include <limits> |
| #include <string> |
| |
| #include "ceres/internal/eigen.h" |
| #include "ceres/internal/port.h" |
| #include "ceres/is_close.h" |
| #include "ceres/jet.h" |
| #include "ceres/stringprintf.h" |
| #include "ceres/test_util.h" |
| #include "glog/logging.h" |
| #include "gmock/gmock.h" |
| #include "gtest/gtest.h" |
| |
| namespace ceres { |
| namespace internal { |
| |
| using std::max; |
| using std::min; |
| using std::numeric_limits; |
| using std::string; |
| using std::swap; |
| |
| const double kPi = 3.14159265358979323846; |
| const double kHalfSqrt2 = 0.707106781186547524401; |
| |
| static double RandDouble() { |
| double r = rand(); |
| return r / RAND_MAX; |
| } |
| |
| // A tolerance value for floating-point comparisons. |
| static double const kTolerance = numeric_limits<double>::epsilon() * 10; |
| |
| // Looser tolerance used for numerically unstable conversions. |
| static double const kLooseTolerance = 1e-9; |
| |
| // Use as: |
| // double quaternion[4]; |
| // EXPECT_THAT(quaternion, IsNormalizedQuaternion()); |
| MATCHER(IsNormalizedQuaternion, "") { |
| if (arg == NULL) { |
| *result_listener << "Null quaternion"; |
| return false; |
| } |
| |
| double norm2 = |
| arg[0] * arg[0] + arg[1] * arg[1] + arg[2] * arg[2] + arg[3] * arg[3]; |
| if (fabs(norm2 - 1.0) > kTolerance) { |
| *result_listener << "squared norm is " << norm2; |
| return false; |
| } |
| |
| return true; |
| } |
| |
| // Use as: |
| // double expected_quaternion[4]; |
| // double actual_quaternion[4]; |
| // EXPECT_THAT(actual_quaternion, IsNearQuaternion(expected_quaternion)); |
| MATCHER_P(IsNearQuaternion, expected, "") { |
| if (arg == NULL) { |
| *result_listener << "Null quaternion"; |
| return false; |
| } |
| |
| // Quaternions are equivalent upto a sign change. So we will compare |
| // both signs before declaring failure. |
| bool near = true; |
| for (int i = 0; i < 4; i++) { |
| if (fabs(arg[i] - expected[i]) > kTolerance) { |
| near = false; |
| break; |
| } |
| } |
| |
| if (near) { |
| return true; |
| } |
| |
| near = true; |
| for (int i = 0; i < 4; i++) { |
| if (fabs(arg[i] + expected[i]) > kTolerance) { |
| near = false; |
| break; |
| } |
| } |
| |
| if (near) { |
| return true; |
| } |
| |
| // clang-format off |
| *result_listener << "expected : " |
| << expected[0] << " " |
| << expected[1] << " " |
| << expected[2] << " " |
| << expected[3] << " " |
| << "actual : " |
| << arg[0] << " " |
| << arg[1] << " " |
| << arg[2] << " " |
| << arg[3]; |
| // clang-format on |
| return false; |
| } |
| |
| // Use as: |
| // double expected_axis_angle[3]; |
| // double actual_axis_angle[3]; |
| // EXPECT_THAT(actual_axis_angle, IsNearAngleAxis(expected_axis_angle)); |
| MATCHER_P(IsNearAngleAxis, expected, "") { |
| if (arg == NULL) { |
| *result_listener << "Null axis/angle"; |
| return false; |
| } |
| |
| Eigen::Vector3d a(arg[0], arg[1], arg[2]); |
| Eigen::Vector3d e(expected[0], expected[1], expected[2]); |
| const double e_norm = e.norm(); |
| |
| double delta_norm = numeric_limits<double>::max(); |
| if (e_norm > 0) { |
| // Deal with the sign ambiguity near PI. Since the sign can flip, |
| // we take the smaller of the two differences. |
| if (fabs(e_norm - kPi) < kLooseTolerance) { |
| delta_norm = min((a - e).norm(), (a + e).norm()) / e_norm; |
| } else { |
| delta_norm = (a - e).norm() / e_norm; |
| } |
| } else { |
| delta_norm = a.norm(); |
| } |
| |
| if (delta_norm <= kLooseTolerance) { |
| return true; |
| } |
| |
| // clang-format off |
| *result_listener << " arg:" |
| << " " << arg[0] |
| << " " << arg[1] |
| << " " << arg[2] |
| << " was expected to be:" |
| << " " << expected[0] |
| << " " << expected[1] |
| << " " << expected[2]; |
| // clang-format on |
| return false; |
| } |
| |
| // Use as: |
| // double matrix[9]; |
| // EXPECT_THAT(matrix, IsOrthonormal()); |
| MATCHER(IsOrthonormal, "") { |
| if (arg == NULL) { |
| *result_listener << "Null matrix"; |
| return false; |
| } |
| |
| for (int c1 = 0; c1 < 3; c1++) { |
| for (int c2 = 0; c2 < 3; c2++) { |
| double v = 0; |
| for (int i = 0; i < 3; i++) { |
| v += arg[i + 3 * c1] * arg[i + 3 * c2]; |
| } |
| double expected = (c1 == c2) ? 1 : 0; |
| if (fabs(expected - v) > kTolerance) { |
| *result_listener << "Columns " << c1 << " and " << c2 |
| << " should have dot product " << expected |
| << " but have " << v; |
| return false; |
| } |
| } |
| } |
| |
| return true; |
| } |
| |
| // Use as: |
| // double matrix1[9]; |
| // double matrix2[9]; |
| // EXPECT_THAT(matrix1, IsNear3x3Matrix(matrix2)); |
| MATCHER_P(IsNear3x3Matrix, expected, "") { |
| if (arg == NULL) { |
| *result_listener << "Null matrix"; |
| return false; |
| } |
| |
| for (int i = 0; i < 9; i++) { |
| if (fabs(arg[i] - expected[i]) > kTolerance) { |
| *result_listener << "component " << i << " should be " << expected[i]; |
| return false; |
| } |
| } |
| |
| return true; |
| } |
| |
| // Transforms a zero axis/angle to a quaternion. |
| TEST(Rotation, ZeroAngleAxisToQuaternion) { |
| double axis_angle[3] = {0, 0, 0}; |
| double quaternion[4]; |
| double expected[4] = {1, 0, 0, 0}; |
| AngleAxisToQuaternion(axis_angle, quaternion); |
| EXPECT_THAT(quaternion, IsNormalizedQuaternion()); |
| EXPECT_THAT(quaternion, IsNearQuaternion(expected)); |
| } |
| |
| // Test that exact conversion works for small angles. |
| TEST(Rotation, SmallAngleAxisToQuaternion) { |
| // Small, finite value to test. |
| double theta = 1.0e-2; |
| double axis_angle[3] = {theta, 0, 0}; |
| double quaternion[4]; |
| double expected[4] = {cos(theta / 2), sin(theta / 2.0), 0, 0}; |
| AngleAxisToQuaternion(axis_angle, quaternion); |
| EXPECT_THAT(quaternion, IsNormalizedQuaternion()); |
| EXPECT_THAT(quaternion, IsNearQuaternion(expected)); |
| } |
| |
| // Test that approximate conversion works for very small angles. |
| TEST(Rotation, TinyAngleAxisToQuaternion) { |
| // Very small value that could potentially cause underflow. |
| double theta = pow(numeric_limits<double>::min(), 0.75); |
| double axis_angle[3] = {theta, 0, 0}; |
| double quaternion[4]; |
| double expected[4] = {cos(theta / 2), sin(theta / 2.0), 0, 0}; |
| AngleAxisToQuaternion(axis_angle, quaternion); |
| EXPECT_THAT(quaternion, IsNormalizedQuaternion()); |
| EXPECT_THAT(quaternion, IsNearQuaternion(expected)); |
| } |
| |
| // Transforms a rotation by pi/2 around X to a quaternion. |
| TEST(Rotation, XRotationToQuaternion) { |
| double axis_angle[3] = {kPi / 2, 0, 0}; |
| double quaternion[4]; |
| double expected[4] = {kHalfSqrt2, kHalfSqrt2, 0, 0}; |
| AngleAxisToQuaternion(axis_angle, quaternion); |
| EXPECT_THAT(quaternion, IsNormalizedQuaternion()); |
| EXPECT_THAT(quaternion, IsNearQuaternion(expected)); |
| } |
| |
| // Transforms a unit quaternion to an axis angle. |
| TEST(Rotation, UnitQuaternionToAngleAxis) { |
| double quaternion[4] = {1, 0, 0, 0}; |
| double axis_angle[3]; |
| double expected[3] = {0, 0, 0}; |
| QuaternionToAngleAxis(quaternion, axis_angle); |
| EXPECT_THAT(axis_angle, IsNearAngleAxis(expected)); |
| } |
| |
| // Transforms a quaternion that rotates by pi about the Y axis to an axis angle. |
| TEST(Rotation, YRotationQuaternionToAngleAxis) { |
| double quaternion[4] = {0, 0, 1, 0}; |
| double axis_angle[3]; |
| double expected[3] = {0, kPi, 0}; |
| QuaternionToAngleAxis(quaternion, axis_angle); |
| EXPECT_THAT(axis_angle, IsNearAngleAxis(expected)); |
| } |
| |
| // Transforms a quaternion that rotates by pi/3 about the Z axis to an axis |
| // angle. |
| TEST(Rotation, ZRotationQuaternionToAngleAxis) { |
| double quaternion[4] = {sqrt(3) / 2, 0, 0, 0.5}; |
| double axis_angle[3]; |
| double expected[3] = {0, 0, kPi / 3}; |
| QuaternionToAngleAxis(quaternion, axis_angle); |
| EXPECT_THAT(axis_angle, IsNearAngleAxis(expected)); |
| } |
| |
| // Test that exact conversion works for small angles. |
| TEST(Rotation, SmallQuaternionToAngleAxis) { |
| // Small, finite value to test. |
| double theta = 1.0e-2; |
| double quaternion[4] = {cos(theta / 2), sin(theta / 2.0), 0, 0}; |
| double axis_angle[3]; |
| double expected[3] = {theta, 0, 0}; |
| QuaternionToAngleAxis(quaternion, axis_angle); |
| EXPECT_THAT(axis_angle, IsNearAngleAxis(expected)); |
| } |
| |
| // Test that approximate conversion works for very small angles. |
| TEST(Rotation, TinyQuaternionToAngleAxis) { |
| // Very small value that could potentially cause underflow. |
| double theta = pow(numeric_limits<double>::min(), 0.75); |
| double quaternion[4] = {cos(theta / 2), sin(theta / 2.0), 0, 0}; |
| double axis_angle[3]; |
| double expected[3] = {theta, 0, 0}; |
| QuaternionToAngleAxis(quaternion, axis_angle); |
| EXPECT_THAT(axis_angle, IsNearAngleAxis(expected)); |
| } |
| |
| TEST(Rotation, QuaternionToAngleAxisAngleIsLessThanPi) { |
| double quaternion[4]; |
| double angle_axis[3]; |
| |
| const double half_theta = 0.75 * kPi; |
| |
| quaternion[0] = cos(half_theta); |
| quaternion[1] = 1.0 * sin(half_theta); |
| quaternion[2] = 0.0; |
| quaternion[3] = 0.0; |
| QuaternionToAngleAxis(quaternion, angle_axis); |
| const double angle = |
| sqrt(angle_axis[0] * angle_axis[0] + angle_axis[1] * angle_axis[1] + |
| angle_axis[2] * angle_axis[2]); |
| EXPECT_LE(angle, kPi); |
| } |
| |
| static constexpr int kNumTrials = 10000; |
| |
| // Takes a bunch of random axis/angle values, converts them to quaternions, |
| // and back again. |
| TEST(Rotation, AngleAxisToQuaterionAndBack) { |
| srand(5); |
| for (int i = 0; i < kNumTrials; i++) { |
| double axis_angle[3]; |
| // Make an axis by choosing three random numbers in [-1, 1) and |
| // normalizing. |
| double norm = 0; |
| for (int i = 0; i < 3; i++) { |
| axis_angle[i] = RandDouble() * 2 - 1; |
| norm += axis_angle[i] * axis_angle[i]; |
| } |
| norm = sqrt(norm); |
| |
| // Angle in [-pi, pi). |
| double theta = kPi * 2 * RandDouble() - kPi; |
| for (int i = 0; i < 3; i++) { |
| axis_angle[i] = axis_angle[i] * theta / norm; |
| } |
| |
| double quaternion[4]; |
| double round_trip[3]; |
| // We use ASSERTs here because if there's one failure, there are |
| // probably many and spewing a million failures doesn't make anyone's |
| // day. |
| AngleAxisToQuaternion(axis_angle, quaternion); |
| ASSERT_THAT(quaternion, IsNormalizedQuaternion()); |
| QuaternionToAngleAxis(quaternion, round_trip); |
| ASSERT_THAT(round_trip, IsNearAngleAxis(axis_angle)); |
| } |
| } |
| |
| // Takes a bunch of random quaternions, converts them to axis/angle, |
| // and back again. |
| TEST(Rotation, QuaterionToAngleAxisAndBack) { |
| srand(5); |
| for (int i = 0; i < kNumTrials; i++) { |
| double quaternion[4]; |
| // Choose four random numbers in [-1, 1) and normalize. |
| double norm = 0; |
| for (int i = 0; i < 4; i++) { |
| quaternion[i] = RandDouble() * 2 - 1; |
| norm += quaternion[i] * quaternion[i]; |
| } |
| norm = sqrt(norm); |
| |
| for (int i = 0; i < 4; i++) { |
| quaternion[i] = quaternion[i] / norm; |
| } |
| |
| double axis_angle[3]; |
| double round_trip[4]; |
| QuaternionToAngleAxis(quaternion, axis_angle); |
| AngleAxisToQuaternion(axis_angle, round_trip); |
| ASSERT_THAT(round_trip, IsNormalizedQuaternion()); |
| ASSERT_THAT(round_trip, IsNearQuaternion(quaternion)); |
| } |
| } |
| |
| // Transforms a zero axis/angle to a rotation matrix. |
| TEST(Rotation, ZeroAngleAxisToRotationMatrix) { |
| double axis_angle[3] = {0, 0, 0}; |
| double matrix[9]; |
| double expected[9] = {1, 0, 0, 0, 1, 0, 0, 0, 1}; |
| AngleAxisToRotationMatrix(axis_angle, matrix); |
| EXPECT_THAT(matrix, IsOrthonormal()); |
| EXPECT_THAT(matrix, IsNear3x3Matrix(expected)); |
| } |
| |
| TEST(Rotation, NearZeroAngleAxisToRotationMatrix) { |
| double axis_angle[3] = {1e-24, 2e-24, 3e-24}; |
| double matrix[9]; |
| double expected[9] = {1, 0, 0, 0, 1, 0, 0, 0, 1}; |
| AngleAxisToRotationMatrix(axis_angle, matrix); |
| EXPECT_THAT(matrix, IsOrthonormal()); |
| EXPECT_THAT(matrix, IsNear3x3Matrix(expected)); |
| } |
| |
| // Transforms a rotation by pi/2 around X to a rotation matrix and back. |
| TEST(Rotation, XRotationToRotationMatrix) { |
| double axis_angle[3] = {kPi / 2, 0, 0}; |
| double matrix[9]; |
| // The rotation matrices are stored column-major. |
| double expected[9] = {1, 0, 0, 0, 0, 1, 0, -1, 0}; |
| AngleAxisToRotationMatrix(axis_angle, matrix); |
| EXPECT_THAT(matrix, IsOrthonormal()); |
| EXPECT_THAT(matrix, IsNear3x3Matrix(expected)); |
| double round_trip[3]; |
| RotationMatrixToAngleAxis(matrix, round_trip); |
| EXPECT_THAT(round_trip, IsNearAngleAxis(axis_angle)); |
| } |
| |
| // Transforms an axis angle that rotates by pi about the Y axis to a |
| // rotation matrix and back. |
| TEST(Rotation, YRotationToRotationMatrix) { |
| double axis_angle[3] = {0, kPi, 0}; |
| double matrix[9]; |
| double expected[9] = {-1, 0, 0, 0, 1, 0, 0, 0, -1}; |
| AngleAxisToRotationMatrix(axis_angle, matrix); |
| EXPECT_THAT(matrix, IsOrthonormal()); |
| EXPECT_THAT(matrix, IsNear3x3Matrix(expected)); |
| |
| double round_trip[3]; |
| RotationMatrixToAngleAxis(matrix, round_trip); |
| EXPECT_THAT(round_trip, IsNearAngleAxis(axis_angle)); |
| } |
| |
| TEST(Rotation, NearPiAngleAxisRoundTrip) { |
| double in_axis_angle[3]; |
| double matrix[9]; |
| double out_axis_angle[3]; |
| |
| srand(5); |
| for (int i = 0; i < kNumTrials; i++) { |
| // Make an axis by choosing three random numbers in [-1, 1) and |
| // normalizing. |
| double norm = 0; |
| for (int i = 0; i < 3; i++) { |
| in_axis_angle[i] = RandDouble() * 2 - 1; |
| norm += in_axis_angle[i] * in_axis_angle[i]; |
| } |
| norm = sqrt(norm); |
| |
| // Angle in [pi - kMaxSmallAngle, pi). |
| const double kMaxSmallAngle = 1e-8; |
| double theta = kPi - kMaxSmallAngle * RandDouble(); |
| |
| for (int i = 0; i < 3; i++) { |
| in_axis_angle[i] *= (theta / norm); |
| } |
| AngleAxisToRotationMatrix(in_axis_angle, matrix); |
| RotationMatrixToAngleAxis(matrix, out_axis_angle); |
| EXPECT_THAT(in_axis_angle, IsNearAngleAxis(out_axis_angle)); |
| } |
| } |
| |
| TEST(Rotation, AtPiAngleAxisRoundTrip) { |
| // A rotation of kPi about the X axis; |
| // clang-format off |
| static constexpr double kMatrix[3][3] = { |
| {1.0, 0.0, 0.0}, |
| {0.0, -1.0, 0.0}, |
| {0.0, 0.0, -1.0} |
| }; |
| // clang-format on |
| |
| double in_matrix[9]; |
| // Fill it from kMatrix in col-major order. |
| for (int j = 0, k = 0; j < 3; ++j) { |
| for (int i = 0; i < 3; ++i, ++k) { |
| in_matrix[k] = kMatrix[i][j]; |
| } |
| } |
| |
| const double expected_axis_angle[3] = {kPi, 0, 0}; |
| |
| double out_matrix[9]; |
| double axis_angle[3]; |
| RotationMatrixToAngleAxis(in_matrix, axis_angle); |
| AngleAxisToRotationMatrix(axis_angle, out_matrix); |
| |
| LOG(INFO) << "AngleAxis = " << axis_angle[0] << " " << axis_angle[1] << " " |
| << axis_angle[2]; |
| LOG(INFO) << "Expected AngleAxis = " << kPi << " 0 0"; |
| double out_rowmajor[3][3]; |
| for (int j = 0, k = 0; j < 3; ++j) { |
| for (int i = 0; i < 3; ++i, ++k) { |
| out_rowmajor[i][j] = out_matrix[k]; |
| } |
| } |
| LOG(INFO) << "Rotation:"; |
| LOG(INFO) << "EXPECTED | ACTUAL"; |
| for (int i = 0; i < 3; ++i) { |
| string line; |
| for (int j = 0; j < 3; ++j) { |
| StringAppendF(&line, "%g ", kMatrix[i][j]); |
| } |
| line += " | "; |
| for (int j = 0; j < 3; ++j) { |
| StringAppendF(&line, "%g ", out_rowmajor[i][j]); |
| } |
| LOG(INFO) << line; |
| } |
| |
| EXPECT_THAT(axis_angle, IsNearAngleAxis(expected_axis_angle)); |
| EXPECT_THAT(out_matrix, IsNear3x3Matrix(in_matrix)); |
| } |
| |
| // Transforms an axis angle that rotates by pi/3 about the Z axis to a |
| // rotation matrix. |
| TEST(Rotation, ZRotationToRotationMatrix) { |
| double axis_angle[3] = {0, 0, kPi / 3}; |
| double matrix[9]; |
| // This is laid-out row-major on the screen but is actually stored |
| // column-major. |
| // clang-format off |
| double expected[9] = { 0.5, sqrt(3) / 2, 0, // Column 1 |
| -sqrt(3) / 2, 0.5, 0, // Column 2 |
| 0, 0, 1 }; // Column 3 |
| // clang-format on |
| AngleAxisToRotationMatrix(axis_angle, matrix); |
| EXPECT_THAT(matrix, IsOrthonormal()); |
| EXPECT_THAT(matrix, IsNear3x3Matrix(expected)); |
| double round_trip[3]; |
| RotationMatrixToAngleAxis(matrix, round_trip); |
| EXPECT_THAT(round_trip, IsNearAngleAxis(axis_angle)); |
| } |
| |
| // Takes a bunch of random axis/angle values, converts them to rotation |
| // matrices, and back again. |
| TEST(Rotation, AngleAxisToRotationMatrixAndBack) { |
| srand(5); |
| for (int i = 0; i < kNumTrials; i++) { |
| double axis_angle[3]; |
| // Make an axis by choosing three random numbers in [-1, 1) and |
| // normalizing. |
| double norm = 0; |
| for (int i = 0; i < 3; i++) { |
| axis_angle[i] = RandDouble() * 2 - 1; |
| norm += axis_angle[i] * axis_angle[i]; |
| } |
| norm = sqrt(norm); |
| |
| // Angle in [-pi, pi). |
| double theta = kPi * 2 * RandDouble() - kPi; |
| for (int i = 0; i < 3; i++) { |
| axis_angle[i] = axis_angle[i] * theta / norm; |
| } |
| |
| double matrix[9]; |
| double round_trip[3]; |
| AngleAxisToRotationMatrix(axis_angle, matrix); |
| ASSERT_THAT(matrix, IsOrthonormal()); |
| RotationMatrixToAngleAxis(matrix, round_trip); |
| |
| for (int i = 0; i < 3; ++i) { |
| EXPECT_NEAR(round_trip[i], axis_angle[i], kLooseTolerance); |
| } |
| } |
| } |
| |
| // Takes a bunch of random axis/angle values near zero, converts them |
| // to rotation matrices, and back again. |
| TEST(Rotation, AngleAxisToRotationMatrixAndBackNearZero) { |
| srand(5); |
| for (int i = 0; i < kNumTrials; i++) { |
| double axis_angle[3]; |
| // Make an axis by choosing three random numbers in [-1, 1) and |
| // normalizing. |
| double norm = 0; |
| for (int i = 0; i < 3; i++) { |
| axis_angle[i] = RandDouble() * 2 - 1; |
| norm += axis_angle[i] * axis_angle[i]; |
| } |
| norm = sqrt(norm); |
| |
| // Tiny theta. |
| double theta = 1e-16 * (kPi * 2 * RandDouble() - kPi); |
| for (int i = 0; i < 3; i++) { |
| axis_angle[i] = axis_angle[i] * theta / norm; |
| } |
| |
| double matrix[9]; |
| double round_trip[3]; |
| AngleAxisToRotationMatrix(axis_angle, matrix); |
| ASSERT_THAT(matrix, IsOrthonormal()); |
| RotationMatrixToAngleAxis(matrix, round_trip); |
| |
| for (int i = 0; i < 3; ++i) { |
| EXPECT_NEAR( |
| round_trip[i], axis_angle[i], numeric_limits<double>::epsilon()); |
| } |
| } |
| } |
| |
| // Transposes a 3x3 matrix. |
| static void Transpose3x3(double m[9]) { |
| swap(m[1], m[3]); |
| swap(m[2], m[6]); |
| swap(m[5], m[7]); |
| } |
| |
| // Convert Euler angles from radians to degrees. |
| static void ToDegrees(double euler_angles[3]) { |
| for (int i = 0; i < 3; ++i) { |
| euler_angles[i] *= 180.0 / kPi; |
| } |
| } |
| |
| // Compare the 3x3 rotation matrices produced by the axis-angle |
| // rotation 'aa' and the Euler angle rotation 'ea' (in radians). |
| static void CompareEulerToAngleAxis(double aa[3], double ea[3]) { |
| double aa_matrix[9]; |
| AngleAxisToRotationMatrix(aa, aa_matrix); |
| Transpose3x3(aa_matrix); // Column to row major order. |
| |
| double ea_matrix[9]; |
| ToDegrees(ea); // Radians to degrees. |
| const int kRowStride = 3; |
| EulerAnglesToRotationMatrix(ea, kRowStride, ea_matrix); |
| |
| EXPECT_THAT(aa_matrix, IsOrthonormal()); |
| EXPECT_THAT(ea_matrix, IsOrthonormal()); |
| EXPECT_THAT(ea_matrix, IsNear3x3Matrix(aa_matrix)); |
| } |
| |
| // Test with rotation axis along the x/y/z axes. |
| // Also test zero rotation. |
| TEST(EulerAnglesToRotationMatrix, OnAxis) { |
| int n_tests = 0; |
| for (double x = -1.0; x <= 1.0; x += 1.0) { |
| for (double y = -1.0; y <= 1.0; y += 1.0) { |
| for (double z = -1.0; z <= 1.0; z += 1.0) { |
| if ((x != 0) + (y != 0) + (z != 0) > 1) continue; |
| double axis_angle[3] = {x, y, z}; |
| double euler_angles[3] = {x, y, z}; |
| CompareEulerToAngleAxis(axis_angle, euler_angles); |
| ++n_tests; |
| } |
| } |
| } |
| CHECK_EQ(7, n_tests); |
| } |
| |
| // Test that a random rotation produces an orthonormal rotation |
| // matrix. |
| TEST(EulerAnglesToRotationMatrix, IsOrthonormal) { |
| srand(5); |
| for (int trial = 0; trial < kNumTrials; ++trial) { |
| double euler_angles_degrees[3]; |
| for (int i = 0; i < 3; ++i) { |
| euler_angles_degrees[i] = RandDouble() * 360.0 - 180.0; |
| } |
| double rotation_matrix[9]; |
| EulerAnglesToRotationMatrix(euler_angles_degrees, 3, rotation_matrix); |
| EXPECT_THAT(rotation_matrix, IsOrthonormal()); |
| } |
| } |
| |
| // Tests using Jets for specific behavior involving auto differentiation |
| // near singularity points. |
| |
| typedef Jet<double, 3> J3; |
| typedef Jet<double, 4> J4; |
| |
| namespace { |
| |
| J3 MakeJ3(double a, double v0, double v1, double v2) { |
| J3 j; |
| j.a = a; |
| j.v[0] = v0; |
| j.v[1] = v1; |
| j.v[2] = v2; |
| return j; |
| } |
| |
| J4 MakeJ4(double a, double v0, double v1, double v2, double v3) { |
| J4 j; |
| j.a = a; |
| j.v[0] = v0; |
| j.v[1] = v1; |
| j.v[2] = v2; |
| j.v[3] = v3; |
| return j; |
| } |
| |
| bool IsClose(double x, double y) { |
| EXPECT_FALSE(IsNaN(x)); |
| EXPECT_FALSE(IsNaN(y)); |
| return internal::IsClose(x, y, kTolerance, NULL, NULL); |
| } |
| |
| } // namespace |
| |
| template <int N> |
| bool IsClose(const Jet<double, N>& x, const Jet<double, N>& y) { |
| if (!IsClose(x.a, y.a)) { |
| return false; |
| } |
| for (int i = 0; i < N; i++) { |
| if (!IsClose(x.v[i], y.v[i])) { |
| return false; |
| } |
| } |
| return true; |
| } |
| |
| template <int M, int N> |
| void ExpectJetArraysClose(const Jet<double, N>* x, const Jet<double, N>* y) { |
| for (int i = 0; i < M; i++) { |
| if (!IsClose(x[i], y[i])) { |
| LOG(ERROR) << "Jet " << i << "/" << M << " not equal"; |
| LOG(ERROR) << "x[" << i << "]: " << x[i]; |
| LOG(ERROR) << "y[" << i << "]: " << y[i]; |
| Jet<double, N> d, zero; |
| d.a = y[i].a - x[i].a; |
| for (int j = 0; j < N; j++) { |
| d.v[j] = y[i].v[j] - x[i].v[j]; |
| } |
| LOG(ERROR) << "diff: " << d; |
| EXPECT_TRUE(IsClose(x[i], y[i])); |
| } |
| } |
| } |
| |
| // Log-10 of a value well below machine precision. |
| static const int kSmallTinyCutoff = |
| static_cast<int>(2 * log(numeric_limits<double>::epsilon()) / log(10.0)); |
| |
| // Log-10 of a value just below values representable by double. |
| static const int kTinyZeroLimit = |
| static_cast<int>(1 + log(numeric_limits<double>::min()) / log(10.0)); |
| |
| // Test that exact conversion works for small angles when jets are used. |
| TEST(Rotation, SmallAngleAxisToQuaternionForJets) { |
| // Examine small x rotations that are still large enough |
| // to be well within the range represented by doubles. |
| for (int i = -2; i >= kSmallTinyCutoff; i--) { |
| double theta = pow(10.0, i); |
| J3 axis_angle[3] = {J3(theta, 0), J3(0, 1), J3(0, 2)}; |
| J3 quaternion[4]; |
| J3 expected[4] = { |
| MakeJ3(cos(theta / 2), -sin(theta / 2) / 2, 0, 0), |
| MakeJ3(sin(theta / 2), cos(theta / 2) / 2, 0, 0), |
| MakeJ3(0, 0, sin(theta / 2) / theta, 0), |
| MakeJ3(0, 0, 0, sin(theta / 2) / theta), |
| }; |
| AngleAxisToQuaternion(axis_angle, quaternion); |
| ExpectJetArraysClose<4, 3>(quaternion, expected); |
| } |
| } |
| |
| // Test that conversion works for very small angles when jets are used. |
| TEST(Rotation, TinyAngleAxisToQuaternionForJets) { |
| // Examine tiny x rotations that extend all the way to where |
| // underflow occurs. |
| for (int i = kSmallTinyCutoff; i >= kTinyZeroLimit; i--) { |
| double theta = pow(10.0, i); |
| J3 axis_angle[3] = {J3(theta, 0), J3(0, 1), J3(0, 2)}; |
| J3 quaternion[4]; |
| // To avoid loss of precision in the test itself, |
| // a finite expansion is used here, which will |
| // be exact up to machine precision for the test values used. |
| J3 expected[4] = { |
| MakeJ3(1.0, 0, 0, 0), |
| MakeJ3(0, 0.5, 0, 0), |
| MakeJ3(0, 0, 0.5, 0), |
| MakeJ3(0, 0, 0, 0.5), |
| }; |
| AngleAxisToQuaternion(axis_angle, quaternion); |
| ExpectJetArraysClose<4, 3>(quaternion, expected); |
| } |
| } |
| |
| // Test that derivatives are correct for zero rotation. |
| TEST(Rotation, ZeroAngleAxisToQuaternionForJets) { |
| J3 axis_angle[3] = {J3(0, 0), J3(0, 1), J3(0, 2)}; |
| J3 quaternion[4]; |
| J3 expected[4] = { |
| MakeJ3(1.0, 0, 0, 0), |
| MakeJ3(0, 0.5, 0, 0), |
| MakeJ3(0, 0, 0.5, 0), |
| MakeJ3(0, 0, 0, 0.5), |
| }; |
| AngleAxisToQuaternion(axis_angle, quaternion); |
| ExpectJetArraysClose<4, 3>(quaternion, expected); |
| } |
| |
| // Test that exact conversion works for small angles. |
| TEST(Rotation, SmallQuaternionToAngleAxisForJets) { |
| // Examine small x rotations that are still large enough |
| // to be well within the range represented by doubles. |
| for (int i = -2; i >= kSmallTinyCutoff; i--) { |
| double theta = pow(10.0, i); |
| double s = sin(theta); |
| double c = cos(theta); |
| J4 quaternion[4] = {J4(c, 0), J4(s, 1), J4(0, 2), J4(0, 3)}; |
| J4 axis_angle[3]; |
| // clang-format off |
| J4 expected[3] = { |
| MakeJ4(2*theta, -2*s, 2*c, 0, 0), |
| MakeJ4(0, 0, 0, 2*theta/s, 0), |
| MakeJ4(0, 0, 0, 0, 2*theta/s), |
| }; |
| // clang-format on |
| QuaternionToAngleAxis(quaternion, axis_angle); |
| ExpectJetArraysClose<3, 4>(axis_angle, expected); |
| } |
| } |
| |
| // Test that conversion works for very small angles. |
| TEST(Rotation, TinyQuaternionToAngleAxisForJets) { |
| // Examine tiny x rotations that extend all the way to where |
| // underflow occurs. |
| for (int i = kSmallTinyCutoff; i >= kTinyZeroLimit; i--) { |
| double theta = pow(10.0, i); |
| double s = sin(theta); |
| double c = cos(theta); |
| J4 quaternion[4] = {J4(c, 0), J4(s, 1), J4(0, 2), J4(0, 3)}; |
| J4 axis_angle[3]; |
| // To avoid loss of precision in the test itself, |
| // a finite expansion is used here, which will |
| // be exact up to machine precision for the test values used. |
| // clang-format off |
| J4 expected[3] = { |
| MakeJ4(2*theta, -2*s, 2.0, 0, 0), |
| MakeJ4(0, 0, 0, 2.0, 0), |
| MakeJ4(0, 0, 0, 0, 2.0), |
| }; |
| // clang-format on |
| QuaternionToAngleAxis(quaternion, axis_angle); |
| ExpectJetArraysClose<3, 4>(axis_angle, expected); |
| } |
| } |
| |
| // Test that conversion works for no rotation. |
| TEST(Rotation, ZeroQuaternionToAngleAxisForJets) { |
| J4 quaternion[4] = {J4(1, 0), J4(0, 1), J4(0, 2), J4(0, 3)}; |
| J4 axis_angle[3]; |
| J4 expected[3] = { |
| MakeJ4(0, 0, 2.0, 0, 0), |
| MakeJ4(0, 0, 0, 2.0, 0), |
| MakeJ4(0, 0, 0, 0, 2.0), |
| }; |
| QuaternionToAngleAxis(quaternion, axis_angle); |
| ExpectJetArraysClose<3, 4>(axis_angle, expected); |
| } |
| |
| TEST(Quaternion, RotatePointGivesSameAnswerAsRotationByMatrixCanned) { |
| // Canned data generated in octave. |
| double const q[4] = { |
| +0.1956830471754074, |
| -0.0150618562474847, |
| +0.7634572982788086, |
| -0.3019454777240753, |
| }; |
| double const Q[3][3] = { |
| // Scaled rotation matrix. |
| {-0.6355194033477252, +0.0951730541682254, +0.3078870197911186}, |
| {-0.1411693904792992, +0.5297609702153905, -0.4551502574482019}, |
| {-0.2896955822708862, -0.4669396571547050, -0.4536309793389248}, |
| }; |
| double const R[3][3] = { |
| // With unit rows and columns. |
| {-0.8918859164053080, +0.1335655625725649, +0.4320876677394745}, |
| {-0.1981166751680096, +0.7434648665444399, -0.6387564287225856}, |
| {-0.4065578619806013, -0.6553016349046693, -0.6366242786393164}, |
| }; |
| |
| // Compute R from q and compare to known answer. |
| double Rq[3][3]; |
| QuaternionToScaledRotation<double>(q, Rq[0]); |
| ExpectArraysClose(9, Q[0], Rq[0], kTolerance); |
| |
| // Now do the same but compute R with normalization. |
| QuaternionToRotation<double>(q, Rq[0]); |
| ExpectArraysClose(9, R[0], Rq[0], kTolerance); |
| } |
| |
| TEST(Quaternion, RotatePointGivesSameAnswerAsRotationByMatrix) { |
| // Rotation defined by a unit quaternion. |
| double const q[4] = { |
| +0.2318160216097109, |
| -0.0178430356832060, |
| +0.9044300776717159, |
| -0.3576998641394597, |
| }; |
| double const p[3] = { |
| +0.11, |
| -13.15, |
| 1.17, |
| }; |
| |
| double R[3 * 3]; |
| QuaternionToRotation(q, R); |
| |
| double result1[3]; |
| UnitQuaternionRotatePoint(q, p, result1); |
| |
| double result2[3]; |
| VectorRef(result2, 3) = ConstMatrixRef(R, 3, 3) * ConstVectorRef(p, 3); |
| ExpectArraysClose(3, result1, result2, kTolerance); |
| } |
| |
| // Verify that (a * b) * c == a * (b * c). |
| TEST(Quaternion, MultiplicationIsAssociative) { |
| double a[4]; |
| double b[4]; |
| double c[4]; |
| for (int i = 0; i < 4; ++i) { |
| a[i] = 2 * RandDouble() - 1; |
| b[i] = 2 * RandDouble() - 1; |
| c[i] = 2 * RandDouble() - 1; |
| } |
| |
| double ab[4]; |
| double ab_c[4]; |
| QuaternionProduct(a, b, ab); |
| QuaternionProduct(ab, c, ab_c); |
| |
| double bc[4]; |
| double a_bc[4]; |
| QuaternionProduct(b, c, bc); |
| QuaternionProduct(a, bc, a_bc); |
| |
| ASSERT_NEAR(ab_c[0], a_bc[0], kTolerance); |
| ASSERT_NEAR(ab_c[1], a_bc[1], kTolerance); |
| ASSERT_NEAR(ab_c[2], a_bc[2], kTolerance); |
| ASSERT_NEAR(ab_c[3], a_bc[3], kTolerance); |
| } |
| |
| TEST(AngleAxis, RotatePointGivesSameAnswerAsRotationMatrix) { |
| double angle_axis[3]; |
| double R[9]; |
| double p[3]; |
| double angle_axis_rotated_p[3]; |
| double rotation_matrix_rotated_p[3]; |
| |
| for (int i = 0; i < 10000; ++i) { |
| double theta = (2.0 * i * 0.0011 - 1.0) * kPi; |
| for (int j = 0; j < 50; ++j) { |
| double norm2 = 0.0; |
| for (int k = 0; k < 3; ++k) { |
| angle_axis[k] = 2.0 * RandDouble() - 1.0; |
| p[k] = 2.0 * RandDouble() - 1.0; |
| norm2 = angle_axis[k] * angle_axis[k]; |
| } |
| |
| const double inv_norm = theta / sqrt(norm2); |
| for (int k = 0; k < 3; ++k) { |
| angle_axis[k] *= inv_norm; |
| } |
| |
| AngleAxisToRotationMatrix(angle_axis, R); |
| rotation_matrix_rotated_p[0] = R[0] * p[0] + R[3] * p[1] + R[6] * p[2]; |
| rotation_matrix_rotated_p[1] = R[1] * p[0] + R[4] * p[1] + R[7] * p[2]; |
| rotation_matrix_rotated_p[2] = R[2] * p[0] + R[5] * p[1] + R[8] * p[2]; |
| |
| AngleAxisRotatePoint(angle_axis, p, angle_axis_rotated_p); |
| for (int k = 0; k < 3; ++k) { |
| // clang-format off |
| EXPECT_NEAR(rotation_matrix_rotated_p[k], |
| angle_axis_rotated_p[k], |
| kTolerance) << "p: " << p[0] |
| << " " << p[1] |
| << " " << p[2] |
| << " angle_axis: " << angle_axis[0] |
| << " " << angle_axis[1] |
| << " " << angle_axis[2]; |
| // clang-format on |
| } |
| } |
| } |
| } |
| |
| TEST(AngleAxis, NearZeroRotatePointGivesSameAnswerAsRotationMatrix) { |
| double angle_axis[3]; |
| double R[9]; |
| double p[3]; |
| double angle_axis_rotated_p[3]; |
| double rotation_matrix_rotated_p[3]; |
| |
| for (int i = 0; i < 10000; ++i) { |
| double norm2 = 0.0; |
| for (int k = 0; k < 3; ++k) { |
| angle_axis[k] = 2.0 * RandDouble() - 1.0; |
| p[k] = 2.0 * RandDouble() - 1.0; |
| norm2 = angle_axis[k] * angle_axis[k]; |
| } |
| |
| double theta = (2.0 * i * 0.0001 - 1.0) * 1e-16; |
| const double inv_norm = theta / sqrt(norm2); |
| for (int k = 0; k < 3; ++k) { |
| angle_axis[k] *= inv_norm; |
| } |
| |
| AngleAxisToRotationMatrix(angle_axis, R); |
| rotation_matrix_rotated_p[0] = R[0] * p[0] + R[3] * p[1] + R[6] * p[2]; |
| rotation_matrix_rotated_p[1] = R[1] * p[0] + R[4] * p[1] + R[7] * p[2]; |
| rotation_matrix_rotated_p[2] = R[2] * p[0] + R[5] * p[1] + R[8] * p[2]; |
| |
| AngleAxisRotatePoint(angle_axis, p, angle_axis_rotated_p); |
| for (int k = 0; k < 3; ++k) { |
| // clang-format off |
| EXPECT_NEAR(rotation_matrix_rotated_p[k], |
| angle_axis_rotated_p[k], |
| kTolerance) << "p: " << p[0] |
| << " " << p[1] |
| << " " << p[2] |
| << " angle_axis: " << angle_axis[0] |
| << " " << angle_axis[1] |
| << " " << angle_axis[2]; |
| // clang-format on |
| } |
| } |
| } |
| |
| TEST(MatrixAdapter, RowMajor3x3ReturnTypeAndAccessIsCorrect) { |
| double array[9] = {1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0}; |
| const float const_array[9] = { |
| 1.0f, 2.0f, 3.0f, 4.0f, 5.0f, 6.0f, 7.0f, 8.0f, 9.0f}; |
| MatrixAdapter<double, 3, 1> A = RowMajorAdapter3x3(array); |
| MatrixAdapter<const float, 3, 1> B = RowMajorAdapter3x3(const_array); |
| |
| for (int i = 0; i < 3; ++i) { |
| for (int j = 0; j < 3; ++j) { |
| // The values are integers from 1 to 9, so equality tests are appropriate |
| // even for float and double values. |
| EXPECT_EQ(A(i, j), array[3 * i + j]); |
| EXPECT_EQ(B(i, j), const_array[3 * i + j]); |
| } |
| } |
| } |
| |
| TEST(MatrixAdapter, ColumnMajor3x3ReturnTypeAndAccessIsCorrect) { |
| double array[9] = {1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0}; |
| const float const_array[9] = { |
| 1.0f, 2.0f, 3.0f, 4.0f, 5.0f, 6.0f, 7.0f, 8.0f, 9.0f}; |
| MatrixAdapter<double, 1, 3> A = ColumnMajorAdapter3x3(array); |
| MatrixAdapter<const float, 1, 3> B = ColumnMajorAdapter3x3(const_array); |
| |
| for (int i = 0; i < 3; ++i) { |
| for (int j = 0; j < 3; ++j) { |
| // The values are integers from 1 to 9, so equality tests are |
| // appropriate even for float and double values. |
| EXPECT_EQ(A(i, j), array[3 * j + i]); |
| EXPECT_EQ(B(i, j), const_array[3 * j + i]); |
| } |
| } |
| } |
| |
| TEST(MatrixAdapter, RowMajor2x4IsCorrect) { |
| const int expected[8] = {1, 2, 3, 4, 5, 6, 7, 8}; |
| int array[8]; |
| MatrixAdapter<int, 4, 1> M(array); |
| // clang-format off |
| M(0, 0) = 1; M(0, 1) = 2; M(0, 2) = 3; M(0, 3) = 4; |
| M(1, 0) = 5; M(1, 1) = 6; M(1, 2) = 7; M(1, 3) = 8; |
| // clang-format on |
| for (int k = 0; k < 8; ++k) { |
| EXPECT_EQ(array[k], expected[k]); |
| } |
| } |
| |
| TEST(MatrixAdapter, ColumnMajor2x4IsCorrect) { |
| const int expected[8] = {1, 5, 2, 6, 3, 7, 4, 8}; |
| int array[8]; |
| MatrixAdapter<int, 1, 2> M(array); |
| // clang-format off |
| M(0, 0) = 1; M(0, 1) = 2; M(0, 2) = 3; M(0, 3) = 4; |
| M(1, 0) = 5; M(1, 1) = 6; M(1, 2) = 7; M(1, 3) = 8; |
| // clang-format on |
| for (int k = 0; k < 8; ++k) { |
| EXPECT_EQ(array[k], expected[k]); |
| } |
| } |
| |
| TEST(RotationMatrixToAngleAxis, NearPiExampleOneFromTobiasStrauss) { |
| // Example from Tobias Strauss |
| // clang-format off |
| const double rotation_matrix[] = { |
| -0.999807135425239, -0.0128154391194470, -0.0148814136745799, |
| -0.0128154391194470, -0.148441438622958, 0.988838158557669, |
| -0.0148814136745799, 0.988838158557669, 0.148248574048196 |
| }; |
| // clang-format on |
| |
| double angle_axis[3]; |
| RotationMatrixToAngleAxis(RowMajorAdapter3x3(rotation_matrix), angle_axis); |
| double round_trip[9]; |
| AngleAxisToRotationMatrix(angle_axis, RowMajorAdapter3x3(round_trip)); |
| EXPECT_THAT(rotation_matrix, IsNear3x3Matrix(round_trip)); |
| } |
| |
| static void CheckRotationMatrixToAngleAxisRoundTrip(const double theta, |
| const double phi, |
| const double angle) { |
| double angle_axis[3]; |
| angle_axis[0] = angle * sin(phi) * cos(theta); |
| angle_axis[1] = angle * sin(phi) * sin(theta); |
| angle_axis[2] = angle * cos(phi); |
| |
| double rotation_matrix[9]; |
| AngleAxisToRotationMatrix(angle_axis, rotation_matrix); |
| |
| double angle_axis_round_trip[3]; |
| RotationMatrixToAngleAxis(rotation_matrix, angle_axis_round_trip); |
| EXPECT_THAT(angle_axis_round_trip, IsNearAngleAxis(angle_axis)); |
| } |
| |
| TEST(RotationMatrixToAngleAxis, ExhaustiveRoundTrip) { |
| const double kMaxSmallAngle = 1e-8; |
| const int kNumSteps = 1000; |
| for (int i = 0; i < kNumSteps; ++i) { |
| const double theta = static_cast<double>(i) / kNumSteps * 2.0 * kPi; |
| for (int j = 0; j < kNumSteps; ++j) { |
| const double phi = static_cast<double>(j) / kNumSteps * kPi; |
| // Rotations of angle Pi. |
| CheckRotationMatrixToAngleAxisRoundTrip(theta, phi, kPi); |
| // Rotation of angle approximately Pi. |
| CheckRotationMatrixToAngleAxisRoundTrip( |
| theta, phi, kPi - kMaxSmallAngle * RandDouble()); |
| // Rotations of angle approximately zero. |
| CheckRotationMatrixToAngleAxisRoundTrip( |
| theta, phi, kMaxSmallAngle * 2.0 * RandDouble() - 1.0); |
| } |
| } |
| } |
| |
| } // namespace internal |
| } // namespace ceres |