def compute_epipolar_distances(normalized_coords_i1: np.ndarray,
normalized_coords_i2: np.ndarray,
i2Ei1: EssentialMatrix) -> Optional[np.ndarray]:
"""Compute symmetric point-line epipolar distances between normalized coordinates of correspondences.
Args:
normalized_coords_i1: normalized coordinates in image i1, of shape Nx2.
normalized_coords_i2: corr. normalized coordinates in image i2, of shape Nx2.
i2Ei1: essential matrix between two images
Returns:
Symmetric epipolar distances for each row of the input, of shape N.
"""
if (normalized_coords_i1 is None or normalized_coords_i1.size == 0
or normalized_coords_i2 is None or normalized_coords_i2.size == 0):
return None
# construct the essential matrix in the opposite directin
i2Ti1 = Pose3(i2Ei1.rotation(), i2Ei1.direction().point3())
i1Ti2 = i2Ti1.inverse()
i1Ei2 = EssentialMatrix(i1Ti2.rotation(), Unit3(i1Ti2.translation()))
# get lines in i2 and i1
epipolar_lines_i2 = feature_utils.convert_to_epipolar_lines(
normalized_coords_i1, i2Ei1)
epipolar_lines_i1 = feature_utils.convert_to_epipolar_lines(
normalized_coords_i2, i1Ei2)
# compute two distances and average them
return 0.5 * (feature_utils.compute_point_line_distances(
normalized_coords_i1, epipolar_lines_i1) +
feature_utils.compute_point_line_distances(
normalized_coords_i2, epipolar_lines_i2))
def test_convert_to_epipolar_lines_none_input(self):
"""Test conversion of None to epipolar lines using the essential matrix."""
points = None
E_matrix = EssentialMatrix(Rot3.RzRyRx(0, np.deg2rad(45), 0), Unit3(np.array([-5, 2, 0])))
F_matrix = E_matrix.matrix() # using identity intrinsics
computed = feature_utils.convert_to_epipolar_lines(points, F_matrix)
self.assertIsNone(computed)
def test_convert_to_epipolar_lines_valid_input(self):
"""Test conversion of valid 2D points to epipolar lines using the fundamental matrix, against manual
computation and with OpenCV's output."""
points = np.array([[10.0, -5.0], [3.5, 20.0],]) # 2d points in homogenous coordinates
E_matrix = EssentialMatrix(Rot3.RzRyRx(0, np.deg2rad(45), 0), Unit3(np.array([-5, 2, 0])))
F_matrix = E_matrix.matrix() # using identity intrinsics
expected_opencv = cv.computeCorrespondEpilines(points.reshape(-1, 1, 2), 1, F_matrix)
expected_opencv = np.squeeze(expected_opencv) # converting to 2D array as opencv adds a singleton dimension
computed = feature_utils.convert_to_epipolar_lines(points, F_matrix)
computed_normalized = computed / np.linalg.norm(computed[:, :2], axis=1, keepdims=True)
np.testing.assert_allclose(computed_normalized, expected_opencv)
def epipolar_inlier_correspondences(
keypoints_i1: Keypoints,
keypoints_i2: Keypoints,
intrinsics_i1: Cal3Bundler,
intrinsics_i2: Cal3Bundler,
i2Ti1: Pose3,
dist_threshold: float,
) -> Tuple[Optional[np.ndarray], Optional[np.ndarray]]:
"""Compute inlier correspondences using epipolar geometry and the ground truth relative pose.
Args:
keypoints_i1: keypoints in image i1.
keypoints_i2: corr. keypoints in image i2.
intrinsics_i1: intrinsics for i1.
intrinsics_i2: intrinsics for i2.
i2Ti1: relative pose
dist_threshold: max acceptable distance for a correct correspondence.
Returns:
is_inlier: (N, ) mask of inlier correspondences.
distance_squared: squared sampson distance between corresponding keypoints.
"""
i2Ei1 = EssentialMatrix(i2Ti1.rotation(), Unit3(i2Ti1.translation()))
i2Fi1 = verification_utils.essential_to_fundamental_matrix(
i2Ei1, intrinsics_i1, intrinsics_i2)
distance_squared = verification_utils.compute_epipolar_distances_sq_sampson(
keypoints_i1.coordinates, keypoints_i2.coordinates, i2Fi1)
is_inlier = distance_squared < dist_threshold**2 if distance_squared is not None else None
return is_inlier, distance_squared
def simulate_two_planes_scene(M: int, N: int) -> Tuple[Keypoints, Keypoints, EssentialMatrix]:
"""Generate a scene where 3D points are on two planes, and projects the points to the 2 cameras. There are M points
on plane 1, and N points on plane 2.
The two planes in this test are:
1. -10x -y -20z +150 = 0
2. 15x -2y -35z +200 = 0
Args:
M: number of points on 1st plane.
N: number of points on 2nd plane.
Returns:
keypoints for image i1, of length (M+N).
keypoints for image i2, of length (M+N).
Essential matrix i2Ei1.
"""
# range of 3D points
range_x_coordinate = (-5, 7)
range_y_coordinate = (-10, 10)
# define the plane equation
plane1_coeffs = (-10, -1, -20, 150)
plane2_coeffs = (15, -2, -35, 200)
# sample the points from planes
plane1_points = sample_points_on_plane(plane1_coeffs, range_x_coordinate, range_y_coordinate, M)
plane2_points = sample_points_on_plane(plane2_coeffs, range_x_coordinate, range_y_coordinate, N)
points_3d = np.vstack((plane1_points, plane2_points))
# define the camera poses and compute the essential matrix
wti1 = np.array([0.1, 0, -20])
wti2 = np.array([1, -2, -20.4])
wRi1 = Rot3.RzRyRx(np.pi / 20, 0, 0.0)
wRi2 = Rot3.RzRyRx(0.0, np.pi / 6, 0.0)
wTi1 = Pose3(wRi1, wti1)
wTi2 = Pose3(wRi2, wti2)
i2Ti1 = wTi2.between(wTi1)
i2Ei1 = EssentialMatrix(i2Ti1.rotation(), Unit3(i2Ti1.translation()))
# project 3D points to 2D image measurements
intrinsics = Cal3Bundler()
camera_i1 = PinholeCameraCal3Bundler(wTi1, intrinsics)
camera_i2 = PinholeCameraCal3Bundler(wTi2, intrinsics)
uv_im1 = []
uv_im2 = []
for point in points_3d:
uv_im1.append(camera_i1.project(point))
uv_im2.append(camera_i2.project(point))
uv_im1 = np.vstack(uv_im1)
uv_im2 = np.vstack(uv_im2)
# return the points as keypoints and the essential matrix
return Keypoints(coordinates=uv_im1), Keypoints(coordinates=uv_im2), i2Ei1
def test_convert_to_epipolar_lines_empty_input(self):
"""Test conversion of 0 2D points to epipolar lines using the essential matrix."""
points = np.array([]) # 2d points in homogenous coordinates
f_matrix = EssentialMatrix(Rot3.RzRyRx(0, np.deg2rad(45), 0), Unit3(np.array([-5, 2, 0]))).matrix()
computed = feature_utils.convert_to_epipolar_lines(points, f_matrix)
self.assertIsNone(computed)
def test_values(self):
values = Values()
E = EssentialMatrix(Rot3(), Unit3())
tol = 1e-9
values.insert(0, Point2(0, 0))
values.insert(1, Point3(0, 0, 0))
values.insert(2, Rot2())
values.insert(3, Pose2())
values.insert(4, Rot3())
values.insert(5, Pose3())
values.insert(6, Cal3_S2())
values.insert(7, Cal3DS2())
values.insert(8, Cal3Bundler())
values.insert(9, E)
values.insert(10, imuBias_ConstantBias())
# Special cases for Vectors and Matrices
# Note that gtsam's Eigen Vectors and Matrices requires double-precision
# floating point numbers in column-major (Fortran style) storage order,
# whereas by default, numpy.array is in row-major order and the type is
# in whatever the number input type is, e.g. np.array([1,2,3])
# will have 'int' type.
#
# The wrapper will automatically fix the type and storage order for you,
# but for performance reasons, it's recommended to specify the correct
# type and storage order.
# for vectors, the order is not important, but dtype still is
vec = np.array([1., 2., 3.])
values.insert(11, vec)
mat = np.array([[1., 2.], [3., 4.]], order='F')
values.insert(12, mat)
# Test with dtype int and the default order='C'
# This still works as the wrapper converts to the correct type and order for you
# but is nornally not recommended!
mat2 = np.array([[1, 2, ], [3, 5]])
values.insert(13, mat2)
self.assertTrue(values.atPoint2(0).equals(Point2(0,0), tol))
self.assertTrue(values.atPoint3(1).equals(Point3(0,0,0), tol))
self.assertTrue(values.atRot2(2).equals(Rot2(), tol))
self.assertTrue(values.atPose2(3).equals(Pose2(), tol))
self.assertTrue(values.atRot3(4).equals(Rot3(), tol))
self.assertTrue(values.atPose3(5).equals(Pose3(), tol))
self.assertTrue(values.atCal3_S2(6).equals(Cal3_S2(), tol))
self.assertTrue(values.atCal3DS2(7).equals(Cal3DS2(), tol))
self.assertTrue(values.atCal3Bundler(8).equals(Cal3Bundler(), tol))
self.assertTrue(values.atEssentialMatrix(9).equals(E, tol))
self.assertTrue(values.atimuBias_ConstantBias(
10).equals(imuBias_ConstantBias(), tol))
# special cases for Vector and Matrix:
actualVector = values.atVector(11)
self.assertTrue(np.allclose(vec, actualVector, tol))
actualMatrix = values.atMatrix(12)
self.assertTrue(np.allclose(mat, actualMatrix, tol))
actualMatrix2 = values.atMatrix(13)
self.assertTrue(np.allclose(mat2, actualMatrix2, tol))
def test_bundle_adjust(self):
"""Tests the bundle adjustment for relative pose on a simulated scene."""
two_view_estimator = TwoViewEstimator(matcher=None,
verifier=None,
inlier_support_processor=None,
bundle_adjust_2view=True,
eval_threshold_px=4)
# Extract example poses.
i1Ri2 = EXAMPLE_DATA.get_camera(1).pose().rotation()
i1ti2 = EXAMPLE_DATA.get_camera(1).pose().translation()
i2Ti1 = Pose3(i1Ri2, i1ti2).inverse()
i2Ei1 = EssentialMatrix(i2Ti1.rotation(), Unit3(i2Ti1.translation()))
# Perform bundle adjustment.
i2Ri1_optimized, i2Ui1_optimized, corr_idxs = two_view_estimator.bundle_adjust(
keypoints_i1=self.keypoints_i1,
keypoints_i2=self.keypoints_i2,
verified_corr_idxs=self.corr_idxs,
camera_intrinsics_i1=Cal3Bundler(),
camera_intrinsics_i2=Cal3Bundler(),
i2Ri1_initial=i2Ei1.rotation(),
i2Ui1_initial=i2Ei1.direction(),
i2Ti1_prior=None,
)
# Assert
rotation_angular_error = comp_utils.compute_relative_rotation_angle(
i2Ri1_optimized, i2Ei1.rotation())
translation_angular_error = comp_utils.compute_relative_unit_translation_angle(
i2Ui1_optimized, i2Ei1.direction())
self.assertLessEqual(rotation_angular_error, 1)
self.assertLessEqual(translation_angular_error, 1)
np.testing.assert_allclose(corr_idxs, self.corr_idxs)
def test_compute_epipolar_distances(self):
"""Test for epipolar distance computation using 2 sets of points and the essential matrix."""
normalized_pts_i1 = np.array([[1.0, 3.5], [-2.0, 2.0]])
normalized_pts_i2 = np.array([[2.0, -1.0], [1.0, 0.0]])
i2Ri1 = Rot3.RzRyRx(0, np.deg2rad(30), np.deg2rad(10))
i2ti1 = np.array([-0.5, 2.0, 0])
i2Ei1 = EssentialMatrix(i2Ri1, Unit3(i2ti1))
expected = np.array([1.637, 1.850])
computed = verification_utils.compute_epipolar_distances(
normalized_pts_i1, normalized_pts_i2, i2Ei1)
np.testing.assert_allclose(computed, expected, rtol=1e-3)
def essential_to_fundamental_matrix(
i2Ei1: EssentialMatrix, camera_intrinsics_i1: Cal3Bundler,
camera_intrinsics_i2: Cal3Bundler) -> np.ndarray:
"""Converts the essential matrix to fundamental matrix using camera intrinsics.
Args:
i2Ei1: essential matrix which maps points in image #i1 to lines in image #i2.
camera_intrinsics_i1: intrinsics for image #i1.
camera_intrinsics_i2: intrinsics for image #i2.
Returns:
Fundamental matrix i2Fi1 as numpy array of shape (3x3).
"""
return np.linalg.inv(
camera_intrinsics_i2.K().T) @ i2Ei1.matrix() @ np.linalg.inv(
camera_intrinsics_i1.K())
def recover_relative_pose_from_essential_matrix(
i2Ei1: Optional[np.ndarray],
verified_coordinates_i1: np.ndarray,
verified_coordinates_i2: np.ndarray,
camera_intrinsics_i1: Cal3Bundler,
camera_intrinsics_i2: Cal3Bundler,
) -> Tuple[Optional[Rot3], Optional[Unit3]]:
"""Recovers the relative rotation and translation direction from essential matrix and verified correspondences
using opencv's API.
Args:
i2Ei1: essential matrix as a numpy array, of shape 3x3.
verified_coordinates_i1: coordinates of verified correspondences in image i1, of shape Nx2.
verified_coordinates_i2: coordinates of verified correspondences in image i2, of shape Nx2.
camera_intrinsics_i1: intrinsics for image i1.
camera_intrinsics_i2: intrinsics for image i2.
Returns:
relative rotation i2Ri1, or None if the input essential matrix is None.
relative translation direction i2Ui1, or None if the input essential matrix is None.
"""
if i2Ei1 is None:
return None, None
# obtain points in normalized coordinates using intrinsics.
normalized_coordinates_i1 = feature_utils.normalize_coordinates(
verified_coordinates_i1, camera_intrinsics_i1)
normalized_coordinates_i2 = feature_utils.normalize_coordinates(
verified_coordinates_i2, camera_intrinsics_i2)
# use opencv to recover pose
_, i2Ri1, i2ti1, _ = cv.recoverPose(i2Ei1, normalized_coordinates_i1,
normalized_coordinates_i2)
i2Ri1 = Rot3(i2Ri1)
i2Ui1 = Unit3(i2ti1.squeeze())
i2Ei1_reconstructed = EssentialMatrix(i2Ri1, i2Ui1).matrix()
# normalizing the two essential matrices
i2Ei1_normalized = i2Ei1 / np.linalg.norm(i2Ei1, axis=None)
i2Ei1_reconstructed_normalized = i2Ei1_reconstructed / np.linalg.norm(
i2Ei1_reconstructed, axis=None)
if not np.allclose(i2Ei1_normalized, i2Ei1_reconstructed_normalized):
logger.warning(
"Recovered R, t cannot create the input Essential Matrix")
return i2Ri1, i2Ui1
def convert_to_epipolar_lines(normalized_coordinates_i1: np.ndarray, i2Ei1: EssentialMatrix) -> Optional[np.array]:
"""Convert coordinates to epipolar lines in image i2.
The epipolar line in image i2 is given by i2Ei1 @ x_i1. A point x_i2 is on this line if x_i2^T @ i2Ei1 @ x_i1 = 0.
Args:
normalized_coordinates_i1: normalized coordinates in i1, of shape Nx2.
i2Ei1: essential matrix.
Returns:
Corr. epipolar lines in i2, of shape Nx3.
"""
if normalized_coordinates_i1 is None or normalized_coordinates_i1.size == 0:
return None
epipolar_lines = convert_to_homogenous_coordinates(normalized_coordinates_i1) @ i2Ei1.matrix().T
return epipolar_lines
def test_convert_to_epipolar_lines_valid_input(self):
"""Test conversion of valid 2D points to epipolar lines using the essential matrix."""
points = np.array([
[10.0, -5.0],
[3.5, 20.0],
]) # 2d points in homogenous coordinates
essential_mat = EssentialMatrix(Rot3.RzRyRx(0, np.deg2rad(45), 0),
Unit3(np.array([-5, 2, 0])))
computed = feature_utils.convert_to_epipolar_lines(
points, essential_mat)
expected = np.array([
[-2.36351579, -5.90878948, 1.75364193],
[-0.65653216, -1.64133041, -19.75129171],
])
np.testing.assert_allclose(computed, expected)
def count_correct_correspondences(
keypoints_i1: Keypoints,
keypoints_i2: Keypoints,
intrinsics_i1: Cal3Bundler,
intrinsics_i2: Cal3Bundler,
i2Ti1: Pose3,
epipolar_dist_threshold: float,
) -> int:
"""Checks the correspondences for epipolar distances and counts ones which are below the threshold.
Args:
keypoints_i1: keypoints in image i1.
keypoints_i2: corr. keypoints in image i2.
intrinsics_i1: intrinsics for i1.
intrinsics_i2: intrinsics for i2.
i2Ti1: relative pose
epipolar_dist_threshold: max acceptable distance for a correct correspondence.
Raises:
ValueError: when the number of keypoints do not match.
Returns:
Number of correspondences which are correct.
"""
# TODO: add unit test, with mocking.
if len(keypoints_i1) != len(keypoints_i2):
raise ValueError("Keypoints must have same counts")
if len(keypoints_i1) == 0:
return 0
normalized_coords_i1 = feature_utils.normalize_coordinates(
keypoints_i1.coordinates, intrinsics_i1)
normalized_coords_i2 = feature_utils.normalize_coordinates(
keypoints_i2.coordinates, intrinsics_i2)
i2Ei1 = EssentialMatrix(i2Ti1.rotation(), Unit3(i2Ti1.translation()))
epipolar_distances = verification_utils.compute_epipolar_distances(
normalized_coords_i1, normalized_coords_i2, i2Ei1)
return np.count_nonzero(epipolar_distances < epipolar_dist_threshold)
def test_VisualISAMExample(self):
# Create a factor
poseKey1 = symbol('x', 1)
poseKey2 = symbol('x', 2)
trueRotation = Rot3.RzRyRx(0.15, 0.15, -0.20)
trueTranslation = Point3(+0.5, -1.0, +1.0)
trueDirection = Unit3(trueTranslation)
E = EssentialMatrix(trueRotation, trueDirection)
model = gtsam.noiseModel.Isotropic.Sigma(5, 0.25)
factor = EssentialMatrixConstraint(poseKey1, poseKey2, E, model)
# Create a linearization point at the zero-error point
pose1 = Pose3(Rot3.RzRyRx(0.00, -0.15, 0.30), Point3(-4.0, 7.0, -10.0))
pose2 = Pose3(
Rot3.RzRyRx(0.179693265735950, 0.002945368776519,
0.102274823253840),
Point3(-3.37493895, 6.14660244, -8.93650986))
expected = np.zeros((5, 1))
actual = factor.evaluateError(pose1, pose2)
self.gtsamAssertEquals(actual, expected, 1e-8)
def generate_random_essential_matrix() -> EssentialMatrix:
rotation_angles = np.random.uniform(low=0.0, high=2 * np.pi, size=(3, ))
R = Rot3.RzRyRx(rotation_angles[0], rotation_angles[1], rotation_angles[2])
t = np.random.uniform(low=-1.0, high=1.0, size=(3, ))
return EssentialMatrix(R, Unit3(t))
